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sLGliHckR8	Some papers in the bibliography are cited with the arXiv version while there exists a peer-reviewed version. Is this because the arXiv version contains additional information not present in the peer-reviewed version, or is there another reason?	Drug Discovery with Dynamic Goal-Aware Fragments Anonymous authors Paper under double-blind review Abstract Fragment-based drug discovery is an effective strategy for discovering drug candidates in the vast chemical space, and has been widely employed in molecular generative models. However, many existing fragment extraction methods in such models do not take the target chemical properties into account or rely on heuristic rules. Additionally, the existing fragment-based generative models cannot update the fragment vocabulary with goal-aware fragments newly discovered during the generation. To this end, we propose a molecular generative framework for drug discovery, named Goal-aware fragment Extraction, Assembly, and Modification (GEAM). GEAM consists of three modules, each responsible for goal-aware fragment extraction, fragment assembly, and fragment modification. The fragment extraction module identifies important fragments that contribute to the desired target properties with the information bottleneck principle, thereby constructing an effective goal-aware fragment vocabulary. Moreover, GEAM can explore beyond the initial vocabulary with the fragment modification module, and the exploration is further enhanced through the dynamic goal-aware vocabulary update. We experimentally demonstrate that GEAM effectively discovers drug candidates through the generative cycle of the three modules in various drug discovery tasks. 1 Introduction The problem of drug discovery aims to find molecules with desired properties within the vast chemical space. Fragment-based drug discovery (FBDD) has been considered as an effective strategy in the recent decades as a means of exploring the chemical space and has led to the discovery of many potent compounds against various targets (Li, 2020). Inspired by the effectiveness of FBDD, many molecular generative models have also adopted it as a strategy to narrow down the search space and simplify the generation process, resulting in meaningful success (Jin et al., 2018; 2020a;b; Xie et al., 2020; Maziarz et al., 2022; Kong et al., 2022; Geng et al., 2023). In FBDD, the first step, fragment library construction, directly impacts the final generation results (Shi & von Itzstein, 2019) as the constructed fragments are used in the entire generation process. However, existing fragment extraction or motif mining methods suffer from two limitations: they 1) do not take the target chemical properties of drug discovery problems into account and/or 2) rely on heuristic fragment selection rules. For example, it is a common strategy to randomly select fragments (Yang et al., 2021) or extract fragments based on frequency (Kong et al., 2022; Geng et al., 2023) without considering the target properties. Jin et al. (2020b) proposed to find molecular substructures that satisfy the given properties, but the extraction process is computationally very expensive and the substructures cannot be assembled together. To this end, we first propose a novel deep learning-based goal-aware fragment extraction method, namely, Fragment-wise Graph Information Bottleneck (FGIB, Figure 1(a)). There is a strong connection between molecular structures and their activity, which is referred to as structure-activity relationship (SAR) (Crum-Brown & Fraser, 1865; Bohacek et al., 1996). Inspired by SAR, FGIB utilizes the graph information bottleneck (GIB) theory to identify important subgraphs in the given molecular graphs for predicting the target chemical property. These identified subgraphs then serve as building blocks in the subsequent generation. As shown in Figure 1(b), the proposed usage of goal-aware fragments extracted by FGIB improves the optimization performance by a significant margin compared to existing FBDD methods. Figure 1: (a) The architecture of FGIB. Using the GIB theory, FGIB aims to identify the important subgraphs that contribute much to the target chemical property in the given molecular graphs. The trained FGIB is then used to extract fragments in a molecular dataset in the goal-aware manner. (b) Performance comparison of GEAM and other FBDD methods on the jak2 ligand generation task. To effectively utilize the extracted fragments in molecular generation, we next construct a generative model consisting of a fragment assembly module and a fragment modification module. In this work, we employ soft-actor critic (SAC) for the assembly module and a genetic algorithm (GA) for the modification module. Through the interplay of the two modules, the generative model can both exploit the extracted goal-aware fragments and explore beyond the initial fragment vocabulary. Moreover, to further enhance molecular novelty and diversity, we propose to extract new fragments on-the-fly during the generation using FGIB and dynamically update the fragment vocabulary. Taken as a whole, the fragment extraction module, the fragment assembly module, and the fragment modification module in the form of FGIB, SAC, and GA, respectively, collectively constitute the generative framework which we refer to as Goal-aware fragment Extraction, Assembly, and Modification (GEAM). As illustrated in Figure 2, GEAM generates molecules through the iterative process that sequentially runs each module as follows: 1) After FGIB constructs an initial goal-aware fragment vocabulary, SAC assembles these fragments and generates a new molecule. 2) GEAM keeps track of the top generated molecules as the initial population of GA, and GA generates an offspring molecule from the population. 3) As a consequence of the crossover and mutation procedures, the offspring molecule contains new subgraphs that cannot be constructed from the current fragment vocabulary, and FGIB extracts the meaningful subgraphs from the offspring molecule and update the vocabulary. Through the collaboration of the three modules where FGIB provides goal-aware fragments to SAC, SAC provides high-quality population to GA, and GA provides novel fragments to FGIB, GEAM effectively explores the chemical space to discover novel drug candidates. We experimentally validate the proposed GEAM on various molecular optimization tasks that simulate real-world drug discovery scenarios. The experimental results show that GEAM significantly outperforms existing state-of-the-art methods, demonstrating its effectiveness in addressing real-world drug discovery problems. We summarize our contributions as follows: • We propose FGIB, a novel goal-aware fragment extraction method that applies the GIB theory to construct a fragment vocabulary for target chemical properties. • We propose to leverage SAC and GA jointly as a generative model to effectively utilize the extracted fragments while enabling exploration beyond the vocabulary. • We propose GEAM, a generative framework that combines FGIB, SAC, and GA to dynamically update the fragment vocabulary by extracting goal-aware fragments on-the-fly to further improve diversity and novelty. • We experimentally demonstrate that GEAM is highly effective in discovering drug candidates, outperforming existing molecular optimization methods. 2 RELATED WORK Fragment extraction Fragment extraction methods fragmentize the given molecules into molecular substructures, i.e., fragments, for subsequent generation. Yang et al. (2021) chose to randomly select fragments after breaking bonds in the given molecules with a predefined rule. Xie et al. (2020) and Maziarz et al. (2022) proposed to obtain fragments by breaking some of the bonds with a pre-defined rule (e.g., acyclic sing bonds), then select the most frequent fragments. Kong et al. (2022) and Geng et al. (2023) utilized merge-and-update rules to find the frequent fragments in the given molecules. All of these methods do not consider the target properties. On the other hand, Jin et al. (2020b) proposed to find molecular substructures that satisfy the given properties, but the approach requires an expensive oracle call to examine each building block candidate in a brute-force manner, and the substructures are not actually fragments in that they are already full molecules that have chemical properties and are not assembled together. Consequently, the found substructures are large in size and often few in number, resulting in low novelty and diversity of the generated molecules. Fragment-based molecule generation Fragment-based molecular generative models denote the models that use the extracted fragments as building blocks and learn to assemble the blocks into molecules. Xie et al. (2020) proposed to use MCMC sampling when assemble or delete the fragments. Yang et al. (2021) proposed to use a reinforcement learning (RL) model and view fragment addition as actions. Maziarz et al. (2022), Kong et al. (2022) and Geng et al. (2023) proposed to use a VAE to assemble the fragments. The model of Jin et al. (2020b) learns to complete the obtained molecular substructures into final molecules by adding molecular branches. Subgraph recognition Given a graph, subgraph recognition aims to find a compressed subgraph that contains salient information to predict the property of the graph. Graph information bottleneck (GIB) (Wu et al., 2020) approached this problem by considering the subgraph as a bottleneck random variable and applying the information bottleneck theory. Yu et al. (2022) proposed to utilize Gaussian noise injection into node representations to confine the information and recognize important subgraphs, while Miao et al. (2022) proposed to consider the subgraph attention process as the information bottleneck. Lee et al. (2023a) applied the GIB principle to molecular relational learning tasks. In practice, it is common for these methods to recognize disconnected substructures rather than connected fragments. Subgraph recognition by GIB has been only employed in classification and regression tasks, and this is the first work that applies GIB to fragment extraction. 3 METHOD We now introduce our Goal-aware fragment Extraction, Assembly, and Modification (GEAM) framework which aims to generate molecules that satisfy the target properties with goal-aware fragments. We first describe the goal-aware fragment extraction method in Section 3.1. Then we describe the fragment assembly method in Section 3.2. Finally, we describe the fragment modification method, the dynamic vocabulary update, and the resulting GEAM in Section 3.3. 3.1 GOAL-AWARE FRAGMENT EXTRACTION Assume that we are given a set of $N$ molecular graphs $G_i$ with its corresponding properties $Y_i \in [0, 1]$, denoted as $\mathcal{D} = \{(G_i, Y_i)\}_{i=1}^N$. Each graph $G_i = (X_i, A_i)$ consists of $n$ nodes with a node feature matrix $X_i \in \mathbb{R}^{n \times d}$ and an adjacency matrix $A_i \in \mathbb{R}^{n \times n}$. Let $\mathcal{V}$ be a set of all nodes from the graphs $ G = \{G_i\}_{i=1}^N $ and let $ E $ be a set of all edges from $ G $. Our goal is to extract goal-aware fragments from $ G $ such that we can assemble these fragments to synthesize graphs with desired properties. In order to achieve this goal, we propose Fragment-wise Graph Information Bottleneck (FGIB), a model that learns to identify salient fragments of $ G_i $ for predicting the target property $ Y_i $. Concretely, we first decompose a set of the graphs $ G $ into $ M $ candidate fragments, denoted as $ F $ with BRICS (Degen et al., 2008), a popular method that fragmentizes molecules into retrosynthetically interesting substructures. Each fragment $ F = (V, E) \in F $ is comprised of vertices $ V \subset V $ and edges $ E \subset E $. Then each graph $ G $ can be represented as $ m $ fragments, $ \{F_j = (V_j, E_j)\}_{j=1}^m $, with $ F_j \in F $. Inspired by graph information bottleneck (Wu et al., 2020), FGIB identifies a subgraph $ G_{\text{sub}} $ that is maximally informative for predicting the target property $ Y $ while maximally compressing the original graph $ G $: \[ \min_{G_{\text{sub}}} -I(G_{\text{sub}}, Y) + \beta I(G_{\text{sub}}, G), \] where $ \beta > 0 $ and $ I(X, Y) $ denotes the mutual information between the random variables $ X $ and $ Y $. FGIB first calculates the node embeddings $ \{h_i\}_{i=1}^n $ from the graph $ G $ with an MPNN (Gilmer et al., 2017) and use average pooling to obtain the fragment embedding $ e_j $ of the fragment $ F_j $ as follows: \[ [h_1 \cdots h_n]^\top = \text{MPNN}(X, A), \quad e_j = \text{AvgPool}(\{h_i : v_i \in V_j\}) \in \mathbb{R}^d, \] where $ v_i $ denotes the node whose corresponding node embedding is $ h_i $. Using an MLP with a sigmoid activation function, we obtain $ w_j \in [0, 1] $, the importance of the fragment $ F_j $ for predicting the target property $ Y $, as $ w_j = \text{MLP}(e_j) $. We denote $ \theta $ as the parameters of the MPNN and the MLP. Following Yu et al. (2022), we inject a noise to the fragment embedding $ e_j $ according to $ w_j $ to control the information flow from $ G $ as follows: \[ \tilde{e}_j = w_j e_j + (1 - w_j) \hat{\mu}_j + \epsilon, \quad w_j = \text{MLP}(e_j), \quad \epsilon \sim \mathcal{N}(0, (1 - w_j) \hat{\Sigma}), \] where $ \hat{\mu}_j \in \mathbb{R}^d $ and $ \hat{\Sigma} \in \mathbb{R}^{d \times d} $ denote an empirical mean vector and a diagonal covariance matrix estimated from $ \{e_j\}_{j=1}^m $, respectively. Intuitively, the more a fragment is considered to be irrelevant for predicting the target property (i.e., small weight $ w_j $), the more the transmission of the fragment information is blocked. Let $ Z = \text{vec}([\tilde{e}_1 \cdots \tilde{e}_m]) $ be the embedding of the perturbed fragments, which is a Gaussian-distributed random variable, i.e., $ p_\theta(Z\|G) = \mathcal{N}(\mu_\theta(G), \Sigma_\theta(G)) $. Here vec denotes a vectorization of a matrix, and $ \mu_\theta(G) $ and $ \Sigma_\theta(G) $ denote the mean and the covariance induced by the MPNN and the MLP with the noise $ \epsilon $, respectively. Assuming that there is no information loss in the fragments after encoding them, our objective function in Eq. (1) becomes optimization the parameters $ \theta $ such that we can still predict the property $ Y $ from the perturbed fragment embedding $ Z $ while minimizing the mutual information between $ G $ and $ Z $ as follows: \[ \min_{\theta} -I(Z, Y; \theta) + \beta I(Z, G; \theta) \] Following Alemi et al. (2017), we can derive the upper bound of $ L_{\text{IB}}(\theta) $ with variational inference: \[ L(\theta, \phi) := \frac{1}{N} \sum_{i=1}^N \left( -\log q_\phi(Y_i\|Z_i) + \beta D_{\text{KL}}(p_\theta(Z\|G_i) \\| u(Z)) \right), \] where $ q_\phi $ is a property predictor that takes the perturbed fragment embedding $ Z $ as an input, $ u(Z) $ is a variational distribution that approximates the marginal $ p_\theta(Z) $, and $ Z_i $ is drawn from $ p_\theta(Z\|G_i) = \mathcal{N}(\mu_\theta(G_i), \Sigma_\theta(G_i)) $ for $ i \in \{1, \ldots, N\} $. We optimize $ \theta $ and $ \phi $ to minimize the objective function $ L(\theta, \phi) $. Note that the variational distribution $ u(\cdot) $ is chosen to be Gaussian with respect to $ Z $, enabling analytic computation of the KL divergence. A detail proof is included in Appendix B. After training FGIB, we score each fragment $ F_j = (V_j, E_j) \in F $ with FGIB as follows: \[ \text{score}(F_j) = \frac{1}{\|S(F_j)\|} \sum_{(G,Y) \in S(F_j)} \frac{w_j(G, F_j)}{\sqrt{\|V_j\|}} \cdot Y \in [0, 1], \] where $ S(F_j) = \{(G,Y) \in D : F_j \text{ is a subgraph of } G\} $ and $ w_j(G, F_j) $ is an importance of the fragment $ F_j $ in the graph $ G $, computed as Eq. (3). Intuitively, the score quantifies the extent to which a fragment contributes to achieving a high target property. Specifically, the term $ w_j(G, F_j)/\sqrt{\|V_j\|} $ measures how much a fragment contributes to its whole molecule in terms of the target property, while the term $Y$ measures the property of the molecule. As the number of nodes of the fragment becomes larger, FGIB is more likely to consider it important when predicting the property. In order to normalize the effect of the fragment size, we include $\sqrt{\|V_j\|}$ in the first term. Based on the scores of all fragments, we choose the top-$K$ fragments as the goal-aware vocabulary $S \subset F$ for the subsequent generation of molecular graphs with desired properties. ### 3.2 Fragment Assembly The next step is to generate molecules with the extracted goal-aware fragment vocabulary. For generation, we introduce the fragment assembly module, which is a soft-actor critic (SAC) model that learns to assemble the fragments to generate molecules with desired properties. We formulate fragment assembly as an RL problem, following Yang et al. (2021). Given a partially generated molecule $g_t$ which becomes a state $s_t$ at time step $t$, a policy network adds a fragment to $g_t$ by sequentially selecting three actions: 1) the attachment site of $g_t$ to use in forming a new bond, 2) the fragment $F \in S$ to be attached to $g_t$, and 3) the attachment site of $F$ to use in forming a new bond. Following Yang et al. (2021), we encode the nodes of the graph $g_t$ with a GCN (Kipf & Welling, 2017) as $\mathbf{H} = \text{GCN}(g_t)$ and obtain the graph embedding with sum pooling as $\mathbf{h}_{g_t} = \text{SumPool}(\mathbf{H})$. Given $\mathbf{H}$ and $\mathbf{h}_{g_t}$, we parameterize the policy network $\pi$ with three sub-policy networks to sequentially choose actions conditioned on previous ones: $$p_{\pi_1}(\cdot \| s_t) = \pi_1(Z_1), \quad Z_1 = [z_{1,1} \cdots z_{1,m_1}]^\top = f_1(\mathbf{h}_{g_t}, \mathbf{H}_{\text{att}})$$ $$p_{\pi_2}(\cdot \| a_1, s_t) = \pi_2(Z_2), \quad Z_2 = [z_{2,1} \cdots z_{2,n_2}]^\top = f_2(z_{1,a_1}, \text{ECFP}(S))$$ $$p_{\pi_3}(\cdot \| a_1, a_2, s_t) = \pi_3(Z_3), \quad Z_3 = [z_{3,1} \cdots z_{3,n_3}]^\top = f_3(\text{SumPool}(\text{GCN}(F_{a_2})), \mathbf{H}_{\text{att}, F_{a_2}}),$$ where $\mathbf{H}_{\text{att}}$ denotes the node embeddings of the attachment sites. We employ multiplicative interactions (Jayakumar et al., 2020) for $f_1$, $f_2$ and $f_3$ to fuse two inputs from heterogeneous spaces. The first policy network $\pi_1$ outputs categorical distribution over attachment sites of the current graph $g_t$ conditioned on $\mathbf{h}_{g_t}$ and $\mathbf{H}_{\text{att}}$, and chooses the attachment site with $a_1 \sim p_{\pi_1}(\cdot \| s_t)$. The second policy network $\pi_2$ selects the fragment $F_{a_2} \in S$ with $a_2 \sim p_{\pi_2}(\cdot \| a_1, s_t)$, conditioned on the embedding of the previously chosen attachment site $z_{1,a_1}$ and the ECFPs of all the fragments $\text{ECFP}(S)$. Then we encode the node embeddings of the fragment $F_{a_2}$ with the same GCN as $\mathbf{H}_{F_{a_2}} = \text{GCN}(F_{a_2})$, and get the fragment embedding $\mathbf{h}_{F_{a_2}} = \text{SumPool}(\mathbf{H}_{F_{a_2}})$. The policy network $\pi_3$ chooses the attachment site of the fragment $F_{a_2}$ with $a_3 \sim p_{\pi_3}(\cdot \| a_1, a_2, s_t)$, conditioned on the fragment embedding $\mathbf{h}_{F_{a_2}}$ and the attachment site embeddings of the fragment $\mathbf{H}_{\text{att}, F_{a_2}}$. Finally, we attach the fragment $F_{a_2}$ to the current graph $g_t$ with the chosen attachment sites $a_1$ and $a_3$, resulting in a new graph $g_{t+1}$. With $T$ steps of sampling actions $(a_1, a_2, a_3)$ using the policy network, we generate a new molecule $g_T = G$, call the oracle to evaluate the molecule $G$ and calculate the reward $r_T$. With the SAC objective (Haarnoja et al., 2018), we train the policy network $\pi$ as follows: $$\pi^* = \arg\max_\pi \sum_t \mathbb{E}_{(s_t, a_t) \sim \rho_\pi}[r(s_t, a_t) + \alpha H(\pi(\cdot \| s_t))],$$ where $r(s_t, a_t)$ is a reward function$^1$, $H(\pi(\cdot \| s_t))$ is entropy of action probabilities given $s_t$ with a temperature parameter $\alpha > 0$, and $\rho_\pi(s_t, a_t)$ is a state-action marginal of the trajectory distribution induced by the policy $\pi(a_t \| s_t) = p_{\pi_3}(a_{3,t} \| a_{2,t}, a_{1,t}, s_t) \cdot p_{\pi_2}(a_{2,t} \| a_{1,t}, s_t) \cdot p_{\pi_1}(a_{1,t} \| s_t)$ with $a_t = (a_{1,t}, a_{2,t}, a_{3,t})$. In order to sample discrete actions differentiable for backpropagation, we use Gumbel-Softmax (Jang et al., 2017; Maddison et al., 2017) to optimize Eq. (10). ### 3.3 Fragment Modification and Dynamic Vocabulary Update With the fragment assembly module only, we are unable to generate molecules consisting of fragments not included in the predefined vocabulary, which hinders generation of diverse molecules and precludes exploration beyond the vocabulary. In order to overcome this problem, we introduce the fragment modification module, which utilizes a genetic algorithm (GA) to generate molecules that contain novel fragments. --- $^1$We set the intermediate rewards to 0.05, so that only final molecules are evaluated by the oracle. Specifically, we employ a graph-based genetic algorithm (GA) (Jensen, 2019). At the first round of the GA, we initialize the population with the top-$P$ molecules generated by the fragment assembly module. The GA then selects parent molecules from the population and generates offspring molecules by performing crossover and mutation. As a consequence of the crossover and mutation operations, the generated offspring molecules contain novel fragments not in the initial vocabulary. In the subsequent rounds, we choose the top-$P$ molecules generated so far by both SAC and GA to construct the GA population of the next round. We iteratively run the fragment assembly module described in Section 3.2 and the fragment modification in turn, and this generative scheme is referred to as GEAM-static. To further enhance molecular diversity and novelty, we propose incorporating the fragment extraction module into this generative cycle. Concretely, in each cycle after the fragment assembly and the fragment modification modules generate molecules, FGIB extracts novel goal-aware fragments $S'$ from the offspring molecules as described in Section 3.1. Then the vocabulary is dynamically updated as $S \cup S'$. When the size of the vocabulary becomes larger than the maximum size $L$, we choose the top-$L$ fragments as the vocabulary based on the scores in Eq. (6). The fragment assembly module assembles fragments of the updated vocabulary in the next iteration, and we refer to this generative framework as GEAM. The single generation cycle of GEAM is described in Algorithm 1 in Section A. 4 EXPERIMENTS We demonstrate the efficacy of our proposed GEAM in two sets of multi-objective molecular optimization tasks that simulate real-world drug discovery problems. We first conduct the experiment to generate novel molecules that have high binding affinity, drug-likeness, and synthesizability in Section 4.1. We then experiment on the practical molecular optimization (PMO) benchmark in Section 4.2. We further conduct extensive ablation studies and qualitative analysis in Section 4.3. 4.1 OPTIMIZATION OF BINDING AFFINITY UNDER QED, SA AND NOVELTY CONSTRAINTS Experimental setup Following Lee et al. (2023b), we validate GEAM in the five docking score (DS) optimization tasks under the quantitative estimate of drug-likeness (QED) (Bickerton et al., 2012), synthetic accessibility (SA) (Ertl & Schuffenhauer, 2009), and novelty constraints. In these tasks, the goal is to generate novel, drug-like, and synthesizable molecules that have a high absolute value of the docking score. Following Lee et al. (2023b), we set the property $Y$ as follows: $$Y(G) = \tilde{D}S(G) \times \tilde{Q}ED(G) \times \tilde{S}A(G) \in [0, 1],$$ where $\tilde{D}S$ and $\tilde{S}A$ are the normalized DS and the normalized SA, respectively (Eq. (16)). We use ZINC250k (Irwin et al., 2012) to train FGIB to predict $Y$ and extract initial fragments. Optimization performance is evaluated with 3,000 generated molecules using the following metrics. Novel hit ratio (%) measures the fraction of unique and novel hits among the generated molecules. Here, novel molecules is defined as the molecules that have the maximum Tanimoto similarity less than 0.4 with the molecules in the training set, and hit is the molecules that satisfy the following criteria: DS < (the median DS of known active molecules), QED > 0.5 and SA < 5. Novel top 5% DS (kcal/mol) measures the average DS of the top 5% unique, novel hits. parp1, fa7, 5ht1b, braf and jak2 are used as the protein targets the docking scores are calculated for. In addition, we evaluate the fraction of novel molecules, novelty (%), and the extent of chemical space covered, #Circles (Xie et al., 2023) of the generated hits. The details are provided in Section C.1 and Section C.2. Baselines REINVENT (Olivecrona et al., 2017) is a SMILES-based RL model with a pretrained prior. Graph GA (Jensen, 2019) is a GA-based model that utilizes predefined crossover and mutation rules. MORLD (Jeon & Kim, 2020) is an RL model that uses the MoIDQN algorithm (Zhou et al., 2019). HierVAE (Jin et al., 2020a) is a VAE-based model that uses the hierarchical motif representation of molecules. RationaleRL (Jin et al., 2020b) is an RL model that first identifies subgraphs that are likely responsible for the target properties (i.e., rationale) and then extends those to complete molecules. FREED (Yang et al., 2021) is an RL model that assembles the fragments obtained using CREM (Polishchuk, 2020). PS-VAE (Kong et al., 2022) is a VAE-based model that uses the mined principal subgraphs as the building blocks. MOOD (Lee et al., 2023b) is a diffusion model that incorporates an out-of-distribution (OOD) control to enhance novelty. The details are provided in Section C.2, and the results of additional baselines are included in Table 7 and Table 8. Table 1: Novel hit ratio (%) results. The results are the means and the standard deviations of 3 runs. The results for the baselines except for RationaleRL and PS-VAE are taken from Lee et al. (2023b). The best results are highlighted in bold. \| Method \| parp1 \| fa7 \| 5ht1b \| braf \| jak2 \| \|-------------------------\|-------------\|-------------\|------------\|-------------\|-------------\| \| REINVENT (Olivecrona et al., 2017) \| 0.480 (± 0.344) \| 0.213 (± 0.081) \| 2.453 (± 0.561) \| 0.127 (± 0.088) \| 0.613 (± 0.167) \| \| Graph GA (Jensen, 2019) \| 4.811 (± 1.661) \| 0.422 (± 0.193) \| 7.011 (± 2.732) \| 3.767 (± 1.498) \| 5.311 (± 1.667) \| \| MORLD (Jeon & Kim, 2020) \| 0.047 (± 0.050) \| 0.007 (± 0.013) \| 0.880 (± 0.735) \| 0.047 (± 0.040) \| 0.227 (± 0.118) \| \| HierVAE (Jin et al., 2020a) \| 0.553 (± 0.214) \| 0.007 (± 0.013) \| 0.507 (± 0.278) \| 0.207 (± 0.230) \| 0.227 (± 0.127) \| \| RationaleRL (Jin et al., 2020b) \| 4.226 (± 0.450) \| 0.900 (± 0.096) \| 2.926 (± 0.397) \| 0.101 (± 0.088) \| 2.970 (± 0.196) \| \| FREED (Yang et al., 2021) \| 4.627 (± 0.332) \| 1.332 (± 0.131) \| 16.767 (± 0.397) \| 2.940 (± 0.359) \| 5.800 (± 0.295) \| \| PS-VAE (Kong et al., 2022) \| 1.644 (± 0.389) \| 0.478 (± 0.140) \| 12.622 (± 1.437) \| 0.367 (± 0.047) \| 4.178 (± 0.933) \| \| MOOD (Lee et al., 2023b) \| 7.017 (± 0.428) \| 0.733 (± 0.141) \| 18.673 (± 0.423) \| 5.240 (± 0.285) \| 9.200 (± 0.524) \| \| GEAM-static (ours) \| 39.667 (± 4.493) \| 16.989 (± 1.999) \| 38.433 (± 2.103) \| 27.422 (± 0.941) \| 42.050 (± 1.855) \| \| GEAM (ours) \| 40.567 (± 0.825) \| 20.711 (± 1.873) \| 38.489 (± 0.350) \| 27.900 (± 1.822) \| 42.950 (± 1.117) \| Table 2: Novel top 5% docking score (kcal/mol) results. The results are the means and the standard deviations of 3 runs. The results for the baselines except for RationaleRL and PS-VAE are taken from Lee et al. (2023b). The best results are highlighted in bold. \| Method \| parp1 \| fa7 \| 5ht1b \| braf \| jak2 \| \|-------------------------\|-------------\|-------------\|------------\|-------------\|-------------\| \| REINVENT (Olivecrona et al., 2017) \| -8.702 (± 0.523) \| -7.205 (± 0.264) \| -8.770 (± 0.316) \| -8.392 (± 0.400) \| -8.165 (± 0.277) \| \| Graph GA (Jensen, 2019) \| -10.949 (± 0.532) \| -7.365 (± 0.326) \| -10.422 (± 0.670) \| -10.789 (± 0.341) \| -10.167 (± 0.576) \| \| MORLD (Jeon & Kim, 2020) \| -7.532 (± 0.260) \| -6.263 (± 0.165) \| -7.869 (± 0.650) \| -8.040 (± 0.337) \| -7.816 (± 0.133) \| \| HierVAE (Jin et al., 2020a) \| -9.487 (± 0.278) \| -6.812 (± 0.274) \| -8.081 (± 0.352) \| -8.978 (± 0.525) \| -8.285 (± 0.370) \| \| RationaleRL (Jin et al., 2020b) \| -10.061 (± 0.080) \| -5.186 (± 0.048) \| -9.005 (± 0.155) \| No hit found \| -9.005 (± 0.076) \| \| FREED (Yang et al., 2021) \| -10.579 (± 0.100) \| -8.378 (± 0.101) \| -10.417 (± 0.100) \| -9.562 (± 0.800) \| -9.735 (± 0.222) \| \| PS-VAE (Kong et al., 2022) \| -9.978 (± 0.091) \| -8.028 (± 0.050) \| -9.887 (± 0.115) \| -9.637 (± 0.049) \| -9.464 (± 0.129) \| \| MOOD (Lee et al., 2023b) \| -10.865 (± 0.113) \| -8.160 (± 0.071) \| -11.145 (± 0.042) \| -11.063 (± 0.034) \| -10.147 (± 0.060) \| \| GEAM-static (ours) \| -12.810 (± 0.124) \| -9.682 (± 0.026) \| -12.369 (± 0.084) \| -12.336 (± 0.157) \| -11.812 (± 0.055) \| \| GEAM (ours) \| -12.891 (± 0.158) \| -9.890 (± 0.116) \| -12.374 (± 0.036) \| -12.342 (± 0.065) \| -11.816 (± 0.067) \| Table 3: Novelty (%) results. The results are the means and the standard deviations of 3 runs. The results for the baselines except for RationaleRL and PS-VAE are taken from Lee et al. (2023b). The best results are highlighted in bold. \| Method \| parp1 \| fa7 \| 5ht1b \| braf \| jak2 \| \|-------------------------\|-------------\|-------------\|------------\|-------------\|-------------\| \| REINVENT (Olivecrona et al., 2017) \| 9.894 (± 2.178) \| 10.731 (± 1.516) \| 11.605 (± 3.688) \| 8.715 (± 2.712) \| 11.456 (± 1.793) \| \| MORLD (Jeon & Kim, 2020) \| 98.433 (± 1.189) \| 97.967 (± 1.764) \| 98.787 (± 0.743) \| 96.993 (± 2.787) \| 97.720 (± 0.995) \| \| HierVAE (Jin et al., 2020a) \| 60.453 (± 17.165) \| 24.853 (± 15.416) \| 48.107 (± 1.988) \| 59.747 (± 16.403) \| 85.200 (± 14.262) \| \| RationaleRL (Jin et al., 2020b) \| 9.300 (± 0.354) \| 9.802 (± 0.166) \| 7.133 (± 0.141) \| 0.000 (± 0.000) \| 7.389 (± 0.220) \| \| FREED (Yang et al., 2021) \| 74.640 (± 2.953) \| 78.787 (± 2.132) \| 75.027 (± 5.194) \| 73.653 (± 4.312) \| 75.907 (± 5.916) \| \| PS-VAE (Kong et al., 2022) \| 60.822 (± 2.251) \| 56.611 (± 1.892) \| 57.956 (± 2.181) \| 57.744 (± 2.710) \| 58.689 (± 2.307) \| \| MOOD (Lee et al., 2023b) \| 84.180 (± 2.123) \| 83.180 (± 1.516) \| 84.615 (± 0.822) \| 87.413 (± 0.830) \| 85.273 (± 1.455) \| \| GEAM-static (ours) \| 84.344 (± 5.290) \| 86.144 (± 6.807) \| 79.389 (± 3.903) \| 87.122 (± 2.163) \| 86.633 (± 1.817) \| \| GEAM (ours) \| 88.611 (± 3.107) \| 89.378 (± 2.619) \| 84.222 (± 2.968) \| 90.322 (± 3.467) \| 89.222 (± 1.824) \| Table 4: #Circles of generated hit molecules. The #Circles threshold is set to 0.75. The results are the means and the standard deviations of 3 runs. The results for the baselines except for RationaleRL and PS-VAE are taken from Lee et al. (2023b). The best results are highlighted in bold. \| Method \| parp1 \| fa7 \| 5ht1b \| braf \| jak2 \| \|-------------------------\|-------------\|-------------\|------------\|-------------\|-------------\| \| REINVENT (Olivecrona et al., 2017) \| 44.2 (± 15.5) \| 23.2 (± 6.6) \| 138.8 (± 19.4) \| 18.0 (± 2.1) \| 59.6 (± 8.1) \| \| MORLD (Jeon & Kim, 2020) \| 1.4 (± 1.5) \| 0.2 (± 0.4) \| 22.2 (± 16.1) \| 1.4 (± 1.2) \| 6.6 (± 3.7) \| \| HierVAE (Jin et al., 2020a) \| 4.8 (± 1.6) \| 0.8 (± 0.7) \| 5.8 (± 1.0) \| 3.6 (± 1.4) \| 4.8 (± 0.7) \| \| RationaleRL (Jin et al., 2020b) \| 6.1 (± 1.2) \| 2.0 (± 0.0) \| 31.9 (± 6.3) \| 3.0 (± 0.0) \| 19.9 (± 7.1) \| \| FREED (Yang et al., 2021) \| 34.8 (± 4.9) \| 21.2 (± 4.0) \| 88.2 (± 9.0) \| 34.0 (± 8.2) \| 59.3 (± 8.2) \| \| PS-VAE (Kong et al., 2022) \| 38.0 (± 5.4) \| 18.0 (± 1.9) \| 180.7 (± 11.6) \| 16.0 (± 0.8) \| 83.7 (± 11.9) \| \| MOOD (Lee et al., 2023b) \| 86.4 (± 11.2) \| 19.2 (± 4.0) \| 144.4 (± 15.1) \| 50.8 (± 3.8) \| 81.8 (± 5.7) \| \| GEAM-static (ours) \| 114.0 (± 2.9) \| 60.7 (± 4.2) \| 134.7 (± 8.5) \| 70.0 (± 2.2) \| 99.3 (± 1.7) \| \| GEAM (ours) \| 123.0 (± 7.8) \| 79.0 (± 9.2) \| 144.3 (± 8.6) \| 84.7 (± 8.6) \| 118.3 (± 0.9) \| Results The results are shown in Table 1 and Table 2. GEAM and GEAM-static significantly outperform all the baselines in all the tasks, demonstrating that the proposed goal-aware extraction method and the proposed combination of SAC and GA are highly effective in discovering novel, drug-like, and synthesizable drug candidates that have high binding affinity. GEAM shows comparable or better performance than GEAM-static, and as shown in Table 3 and Table 4, the usage of the dynamic vocabulary update enhances novelty and diversity without degrading optimization. Table 5: PMO MPO AUC top-100 results. The results are the means of 3 runs. The results for the baselines are taken from Gao et al. (2022). The best results are highlighted in bold. \| Method \| Benchmark \| Average \| \|-----------------\|-----------\|---------\| \| REINVENT \| \| \| \| (Olivercrona et al., 2017) \| 0.608 / 0.752 / 0.806 / 0.511 / 0.719 / 0.006 / 0.325 / 0.532 \| \| Graph GA \| \| \| \| (Jensen, 2019) \| 0.622 / 0.731 / 0.799 / 0.503 / 0.670 / 0.330 / 0.305 / 0.566 \| \| STONED \| \| \| \| (Nigam et al., 2021) \| 0.593 / 0.777 / 0.799 / 0.472 / 0.738 / 0.351 / 0.307 / 0.577 \| \| GEAM-static \| \| \| \| (ours) \| 0.602 / 0.796 / 0.828 / 0.501 / 0.703 / 0.346 / 0.397 / 0.596 \| \| GEAM \| \| \| \| (ours) \| 0.626 / 0.799 / 0.831 / 0.514 / 0.714 / 0.417 / 0.402 / 0.615 \| Table 6: PMO MPO novelty (%) / #Circles results. The #Circles threshold is set to 0.75. The results are the means of 3 runs. The best results are highlighted in bold. \| Method \| Benchmark \| Average \| \|-----------------\|-----------\|---------\| \| REINVENT \| \| \| \| (Olivercrona et al., 2017) \| 17.0 / 303.7 / 13.4 / 343.3 / 25.0 / 452.3 / 33.1 / 318.3 / 15.6 / 253.3 / 15.7 / 398.3 / 7.6 / 275.3 \| \| Graph GA \| \| \| \| (Jensen, 2019) \| 61.1 / 258.7 / 76.2 / 333.3 / 64.1 / 270.3 / 44.4 / 278.7 / 78.2 / 364.7 / 88.0 / 306.3 / 41.3 / 272.7 \| \| STONED \| \| \| \| (Nigam et al., 2021) \| 82.7 / 303.7 / 91.6 / 330.3 / 88.1 / 301.3 / 65.8 / 301.0 / 92.4 / 316.7 / 89.5 / 326.3 / 63.1 / 280.3 \| \| GEAM-static \| \| \| \| (ours) \| 83.1 / 412.0 / 97.6 / 397.7 / 94.5 / 315.3 / 93.2 / 318.0 / 68.9 / 256.7 / 73.7 / 233.0 / 76.2 / 267.0 \| \| GEAM \| \| \| \| (ours) \| 84.2 / 424.0 / 98.0 / 502.0 / 97.0 / 435.0 / 95.3 / 377.3 / 82.7 / 295.3 / 86.9 / 257.0 / 81.7 / 336.0 \| There is a general trend that the more powerful the molecular optimization model, the less likely it is to generate diverse molecules (Gao et al., 2022), but GEAM effectively overcomes this trade-off by discovering novel and high-quality goal-aware fragments on-the-fly. Note that the high novelty values of MORLD are trivial due to its poor optimization performance and very low diversity. In the same vein, the high diversity values of RationaleRL on the target proteins 5ht1b and jak2 are not meaningful due to its poor optimization performance and novelty. 4.2 Optimization of Multi-property Objectives in PMO Benchmark Experimental setup We validate GEAM in the seven multi-property objective (MPO) optimization tasks in the practical molecular optimization (PMO) benchmark (Gao et al., 2022), which are the tasks in the Guacamol benchmark (Brown et al., 2019) that additionally take the number of oracle calls into account for realistic drug discovery. The details are provided in Section C.1 and C.3. Baselines We use the top three models reported by Gao et al. (2022) as our baselines. In addition to REINVENT (Olivercrona et al., 2017) and Graph GA (Jensen, 2019), STONED (Nigam et al., 2021) is a GA-based model that manipulates SELFIES strings. Results As shown in Table 5, GEAM outperform the baselines in most of the tasks, demonstrating its applicability to various drug discovery problems. Note that GEAM distinctly improves the performance of GEAM-static in some tasks. Furthermore, as shown in Table 6, GEAM shows higher novelty and diversity than other methods. Especially, GEAM generates more novel and diverse molecules than GEAM-static, again verifying the dynamic vocabulary update of GEAM effectively improves novelty and diversity without degrading optimization performance. 4.3 Ablation Studies and Qualitative Analysis Effect of the goal-aware fragment extraction To examine the effect of the proposed goal-aware fragment extraction method with FGIB, in Figure 3(a), we compare FREED with FREED (FGIB), which is a variant of FREED that uses the fragment vocabulary extracted by FGIB as described in Section 3.1. FREED (FGIB) outperforms FREED by a large margin, indicating the proposed goal-aware fragment extraction method with FGIB largely boosts the optimization performance. We also compare GEAM against GEAM with different fragment vocabularies in Figure 3(b). GEAM (FREED), GEAM (MiCaM), GEAM (BRICS) are the GEAM variants that use the FREED vocabulary, the MiCaM (Geng et al., 2023) vocabulary, the random BRICS (Degen et al., 2008) vocabulary, respectively. GEAM (property) is GEAM which only uses the property instead of Eq. (6) when scoring fragments, i.e., $ \text{score}(F_j) = \frac{1}{\|S(F_j)\|} \sum_{(G,Y) \in S(F_j)} Y $. GEAM significantly outperforms all the variants, verifying the importance of our goal-aware fragment vocabulary. Notably, GEAM (property) uses the topmost fragments in terms of the target property, but performs worse than GEAM because it does not use FGIB to find important subgraphs that contribute to the property. Figure 3: (a-c) Ablation studies on FGIB, SAC and GA on the ligand generation task with the target protein jak2 and (d) the PLIP image showing hydrophobic interactions between an example molecule and jak2. Figure 4: The generation progress of GEAM and GEAM-static on the ligand generation task against jak2. Effect of the fragment assembly and modification To examine the effect of the proposed combinatorial use of the assembly and the modification modules, we compare GEAM with GEAM-w/o A and GEAM-w/o M in Figure 3(c). GEAM-w/o A does not use the assembly module and constructs its population as the top-$P$ molecules from ZINC250k, while GEAM-w/o M does not use the modification module. GEAM-random A uses random fragment assembly instead of SAC. We can observe GEAM-w/o A significantly underperforms as the fragment modification module alone cannot take the advantage of the goal-aware fragments, and GEAM-random A largely improves over GEAM-w/o A. GEAM outperforms all the ablated variants, demonstrating that jointly leveraging the fragment assembly module and the fragment modification module is crucial to the performance. Effect of the dynamic vocabulary update To thoroughly examine the effect of the proposed dynamic update of the fragment vocabulary, we compare the generation progress of GEAM with that of GEAM-static in Figure 4. GEAM-static-1000 is GEAM-static with the vocabulary size $K = 1,000$. When the initial vocabulary size $K = 300$ and the maximum vocabulary size $L = 1,000$, the vocabulary size of GEAM increases during generation from 300 to 1,000 as GEAM dynamically collects fragments on-the-fly while the vocabulary sizes of GEAM-static and GEAM-static-1000 are fixed to 300 and 1,000, respectively. As expected, GEAM-static-1000 shows the worst optimization performance since its vocabulary consists of top-1,000 fragments instead of top-300 from the same training molecules, and shows the highest diversity as it utilizes more fragments than GEAM and GEAM-static throughout the generation process. GEAM shows the best optimization performance and novelty thanks to its vocabulary update strategy that constantly incorporates novel fragments outside the training molecules, as well as improved diversity compared to GEAM-static. Qualitative analysis We qualitatively analyze the extracted goal-aware fragments. In Figure 3(d), we present an example of the binding interactions of a molecule and the target protein jak2 using the protein-ligand interaction profiler (PLIP) (Adasme et al., 2021). Additionally, we show the fragments of the molecule and $w$ of the fragments calculated by FGIB. We observe that the important fragments identified by FGIB with high $w$ (red and blue) indeed play crucial role for interacting with the target protein, while the fragments with low $w$ (gray) are not involved in the interactions. This analysis validates the efficacy of the proposed goal-aware fragment extraction method using FGIB and suggests the application of FGIB as a means to improve the explainability of drug discovery. 5 CONCLUSION In this paper, we proposed GEAM, a fragment-based molecular generative framework for drug discovery. GEAM consists of three modules, FGIB, SAC, and GA, responsible for goal-aware fragment extraction, fragment assembly, and fragment modification, respectively. In the generative cycle of the three modules, FGIB provides goal-aware fragments to SAC, SAC provides high-quality population to GA, and GA provides novel fragments to FGIB, enabling GEAM to achieve superior optimization performance with high molecular novelty and diversity on a variety of drug discovery tasks. These results highlight its strong applicability to real-world drug discovery. Ethics statement Given the effectiveness of GEAM to real-world drug discovery tasks, GEAM has the possibility to be used maliciously to generate harmful or toxic molecules. This can be prevented by setting the target properties to comprehensively consider toxicity and other side effects. Reproducibility statement The code to reproduce the experimental results of this paper is available at https://anonymous.4open.science/r/GEAM-45EF. Experimental details regarding the experiments of Section 4.1 are provided in Section C.1 and Section C.2. Experimental details regarding the experiments of Section 4.2 are provided in Section C.1 and Section C.3. REFERENCES Melissa F Adasme, Katja L Linnemann, Sarah Naomi Bolz, Florian Kaiser, Sebastian Salentin, V Joachim Haupt, and Michael Schroeder. Plip 2021: Expanding the scope of the protein–ligand interaction profiler to dna and rna. Nucleic acids research, 49(W1):W530–W534, 2021. Sungsoo Ahn, Junsu Kim, Hankook Lee, and Jinwoo Shin. Guiding deep molecular optimization with genetic exploration. Advances in neural information processing systems, 33:12008–12021, 2020. Alexander A. Alemi, Ian Fischer, Joshua V. Dillon, and Kevin Murphy. Deep variational information bottleneck. In International Conference on Learning Representations, 2017. Amr Alhossary, Stephanus Daniel Handoko, Yuguang Mu, and Chee-Keong Kwoh. Fast, accurate, and reliable molecular docking with quickvina 2. Bioinformatics, 31(13):2214–2216, 2015. 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0k85noSawb	- Besides, in general, it is widely believed that intermediate layers produce relatively low level features compared with the final output. Why do you think that regularization contributes to learn complex and high-level features?	Variance-Covariance Regularization Improves Representation Learning Anonymous authors Paper under double-blind review Abstract Transfer learning plays a key role in advancing machine learning models, yet conventional supervised pretraining often undermines feature transferability by prioritizing features that minimize the pretraining loss. Recent progress in self-supervised learning (SSL) has introduced regularization techniques that bolster feature transferability. In this work, we adapt an SSL regularization technique from the VICReg method to supervised learning contexts, introducing Variance-Covariance Regularization (VCReg). This adaptation encourages the network to learn a high-variance, low-covariance representation, promoting the learning of more diverse features. We outline best practices for implementing this regularization framework into various neural network architectures and present an optimized strategy for regularizing intermediate representations. Through extensive empirical evaluation, we demonstrate that our method significantly enhances transfer learning, achieving excellent performance across numerous tasks and datasets. VCReg also improves performance in scenarios like long-tail learning, and hierarchical classification. Additionally, we conduct analyses to suggest that its effectiveness may stem from its success in addressing challenges like gradient starvation and neural collapse. In summary, VCReg offers a universally applicable regularization framework that significantly advances the state of transfer learning, highlights the connection between gradient starvation, neural collapse, and feature transferability, and potentially opens a new avenue for regularization in this domain. 1 Introduction Transfer learning enables models to apply knowledge from one domain to enhance performance in another, particularly when data are scarce or costly to obtain [Pan & Yang, 2010; Weiss et al., 2016; Zhuang et al., 2020; Bonmassari et al., 2021]. One of the key challenges arises during the supervised pretraining phase. In this phase, models often lack detailed information about the downstream tasks to which they will be applied. Nevertheless, they must aim to capture a broad spectrum of features beneficial across various applications [Bengio, 2012; Caruana, 1997; Yosinski et al., 2014]. Without proper regularization techniques, these supervised pretrained models tend to overly focus on features that minimize supervised loss, resulting in limited generalization capabilities and issues such as gradient starvation and neural collapse [Zhang et al., 2016; Neyshabur et al., 2017; Zhang et al., 2021; Pezeshki et al., 2021; Papyan et al., 2020; Shwartz-Ziv, 2022]. To tackle these challenges we adapt the regularization techniques of the self-supervised VICReg method [Bardes et al., 2021] for the supervised learning paradigm. Our method, termed Variance-Covariance Regularization (VCReg), aims to encourage the learning of representations with high variance and low covariance, thus avoiding the overemphasis on features that merely minimize supervised loss. Crucially, our detailed studies reveal that the effectiveness of VCReg strongly depends on how well it is integrated into different neural network designs. Instead of simply applying VCReg to the final representation of the network, we explore the most effective ways to incorporate it throughout the intermediate representations of the network. The structure of the paper is as follows: We begin with an introduction of our method, including an outline of a fast implementation strategy designed to minimize computational overhead. Following this, we present a series of experiments aimed at validating the method’s efficacy across a wide range of tasks, datasets, and neural network architectures. Subsequently, we conduct analyses on the learned representations to demonstrate VCReg’s effectiveness in mitigating common issues in transfer learning, such as neural collapse and gradient starvation. This finding suggests a promising avenue for future research in transfer learning: focusing on resolving issues like gradient starvation and neural collapse, particularly in the context of transfer learning, has the potential to significantly improve performance. Our paper makes the following contributions: 1. We introduce a robust strategy for applying VCReg to neural networks, including integrating it into the intermediate layers. 2. We propose a computationally efficient implementation of VCReg. This implementation is optimized to ensure minimal impact from additional computational overhead, allowing for seamless integration into existing workflows while maintaining high training speed and resource efficiency. 3. Through extensive experiments on benchmark datasets, we demonstrate that using VCReg yields notable improvements in transfer learning performance across various network architectures, including ResNet (He et al., 2016), ConvNeXt (Liu et al., 2022), and ViT (Dosovitskiy et al., 2020). Moreover, with preliminary results, we also find that VCReg could improve performance in scenarios like long-tail learning, and hierarchical classification. 4. We investigate the learned representation of VCReg, revealing its effectiveness in combating challenges such as gradient starvation (Pezeshki et al., 2021), neural collapse (Papyan et al., 2020), and information compression (Shwartz-Ziv, 2022). 2 RELATED WORK 2.1 Variance-Invariance-Covariance Regularization (VICReg) VICReg (Bardes et al., 2021) is a novel SSL method. VICReg encourages learned representations to be invariant to data augmentation. However, by optimizing only the invariant criterion, the network will learn to generate a constant representation for all inputs. This means the representations will be invariant not only to data augmentation, but also to the input itself. VICReg primarily regularizes the network by using a combination of variance loss and covariance loss. The variance loss encourages high variance in the learned representations, thereby promoting the learning of a wide range of features. The covariance loss, on the other hand, aims to minimize redundancy in the learned features by reducing the overlap in information captured by different dimensions of the representation. This dual-objective optimization framework has been found to be effective in promoting diverse feature learning (Shwartz-Ziv et al., 2022). In this work, we borrow the feature collapse prevention mechanism from VICReg and propose the variance-covariance regularization method for supervised network training to improve transfer learning performance. To calculate the loss function of VICReg with a batch of data $ \{x_1, \ldots, x_n\} $, we first need to have a pair of inputs $(x'_i, x''_i)$ such that $x'_i$ and $x''_i$ are two augmented versions of the original input $x_i$. With the neural network $f_\theta(\cdot)$ and the final representations $z'_i = f_\theta(x'_i)$ and $z''_i = f_\theta(x''_i)$, the VICReg minimizes the following loss (we defer the detailed formulation of the variance and covariance loss terms to the subsequent section where we introduce our methods): \[ \ell_{\text{VICReg}}(z'_1, \ldots, z'_n, z''_1, \ldots, z''_n) = \alpha \ell_{\text{var}}(z'_1, \ldots, z'_n) + \alpha \ell_{\text{var}}(z''_1, \ldots, z''_n) \\ + \beta \ell_{\text{cov}}(z'_1, \ldots, z'_n) + \beta \ell_{\text{cov}}(z''_1, \ldots, z''_n) \\ + \sum_{i=1}^{n} \ell_{\text{inv}}(z'_i, z''_i) \] Notice that the only loss term that requires two augmented images is the invariance loss. We usually avoid using two augmented images for each training step in supervised learning. This is because it would approximately double the total computation, as we would need to perform two forward passes at each step. Furthermore, as discussed in some previous works (Shwartz-Ziv, 2022) the invariance term is not the essential factor that helps diversify the features. Therefore, in our adaptation to the supervised regime, we omit the invariance term from the regularization. 2.2 Representation Whitening and Feature Diversity Regularizers Representation whitening is a technique for processing inputs before they enter a network layer. It transforms the input so that its components are uncorrelated with unit variance (Kessy et al., 2018). This transformation achieves enhanced model optimization and generalization. It uses a whitening matrix derived from the data’s covariance matrix and results in an identity covariance matrix, thereby aiding gradient flow during training and acting as a lightweight regularizer to reduce overfitting and encourage robust data representations (LeCun et al., 2002). In addition to whitening as a processing step, additional regularization terms can be introduced to enforce decorrelation in the representations. Various prior works have explored these feature diversity regularization techniques to enhance neural network training (Cogswell et al., 2015; Ayinde et al., 2019; Laakom et al., 2023). These methods encourage diverse features in the representation by adding a regularization term. Recent methods like WLD-Reg (Laakom et al., 2023) and De-Cov (Cogswell et al., 2015) also employ covariance-matrix-based regularization to promote feature diversity, similar to our approach. However, the studies cited above primarily concentrate on the benefits of optimization and generalization for the source task, frequently overlooking their implications for transfer learning. VCReg sets itself apart by explicitly targeting enhancements in transfer learning performance. Our results indicate that such regularization techniques yield only modest performance improvements in in-domain evaluations. This may be attributed to the fact that modern optimizers and regularizers have already significantly alleviated challenges related to in-domain optimization and generalization. Therefore, the most impactful domain for these types of regularization appears to be transfer learning. 2.3 Gradient Starvation and Neural Collapse Gradient starvation and neural collapse are two recently recognized phenomena that can significantly affect the quality of learned representations and the network’s generalization ability (Pezeshki et al., 2021; Papyan et al., 2020; Ben-Shaul et al., 2023). Gradient starvation occurs when certain parameters in a deep learning model receive very little gradient during the training process, thereby leading to slower or non-existent learning for these parameters (Pezeshki et al., 2021). Neural collapse, on the other hand, is a phenomenon observed during the late stages of training where the internal representations of the network tend to collapse towards each other, resulting in a loss of feature diversity (Papyan et al., 2020). Both phenomena are particularly relevant in the context of transfer learning, where models are initially trained on a source task before being fine-tuned for a target task. Our work, through the use of VCReg, seeks to mitigate these issues, offering a pathway to more effective transfer learning. 3 Variance-Covariance Regularization 3.1 Vanilla VCReg: An Introduction to the Basic Formulation Consider a labeled dataset comprising $N$ samples, denoted as $\{(x_1, y_1), \ldots, (x_N, y_N)\}$ and a neural network $f_\theta(\cdot)$, which takes these inputs $x_i$ and produces final predictions $\hat{y}_i = f_\theta(x_i)$. In standard supervised learning, the loss is defined as $L_{\text{sup}} = \frac{1}{N} \sum_{i=1}^{N} \ell_{\text{sup}}(\hat{y}_i, y_i)$. The core objective of Vanilla VCReg is to ensure that the $D$-dimensional input representation $h_i$ to the last layer of the network exhibit both high variance and low covariance. To achieve this, we employ variance and covariance losses, same as mentioned in equation (1) $$\ell_{\text{vcreg}}(h_1, \ldots, h_N) = \alpha \ell_{\text{var}}(h_1, \ldots, h_N) + \beta \ell_{\text{cov}}(h_1, \ldots, h_N)$$ (2) The variance and covariance loss functions are defined as: \[ \ell_{\text{var}} = \frac{1}{D} \sum_{i=1}^{D} \max(0, 1 - \sqrt{C_{ii}}) \] \[ \ell_{\text{cov}} = \frac{1}{D(D-1)} \sum_{i \neq j} C_{ij}^2 \] where $ C = \frac{1}{N-1} \sum_{i=1}^{N} (h_i - \bar{h})(h_i - \bar{h})^T $ denotes the covariance matrix, and $ \bar{h} $ represents the mean vector, given by $ \bar{h} = \frac{1}{N} \sum_{i=1}^{N} h_i $. Intuitively speaking, the covariance matrix captures the interdependencies among the dimensions of the feature vectors $ z_i $. Maximizing $ \ell_{\text{var}} $ encourages each feature dimension to contain unique, non-redundant information, while minimizing $ \ell_{\text{cov}} $ aims to reduce the correlation between different dimensions, thus promoting feature independence. The overall training loss then becomes: \[ L_{\text{vanilla}} = \alpha \ell_{\text{var}}(z_1, \ldots, z_N) + \beta \ell_{\text{cov}}(z_1, \ldots, z_N) + \frac{1}{N} \sum_{i=1}^{N} \ell_{\text{sup}}(\hat{y}_i, y_i) \] Here, $ \alpha $ and $ \beta $ serve as hyperparameters to control the strength of each regularization term. ### 3.2 Extending VCReg to Intermediate Representations While regularizing the final layer in a neural network offers certain benefits, extending this approach to intermediate layers via VCReg provides additional advantages. (For empirical evidence supporting this claim, please refer to Appendix A.) Regularizing intermediate layers enables the model to capture more complex, higher-level abstractions. This strategy minimizes internal covariate shifts across layers, which in turn improves both the stability of training and the model’s generalization capabilities. Furthermore, it fosters the development of feature hierarchies and enriches the latent space, leading to enhanced model interpretability and improved transfer learning performance. To implement this extension, VCReg is applied at $ M $ strategically chosen layers throughout the neural network. For each intermediate layer $ j $, we denote the feature representation for an input $ x_i $ as $ h_i^{(j)} \in \mathbb{R}^{D_j} $. This culminates in a composite loss function, expressed as follows: \[ L_{\text{VCReg}} = \sum_{j=1}^{M} \left[ \alpha \ell_{\text{var}}(h_1^{(j)}, \ldots, h_N^{(j)}) + \beta \ell_{\text{cov}}(h_1^{(j)}, \ldots, h_N^{(j)}) \right] + \frac{1}{N} \sum_{i=1}^{N} \ell_{\text{sup}}(\hat{y}_i, y_i) \] Spatial Dimensions However, applying VCReg to intermediate layers of real-world neural networks presents challenges due to the spatial dimensions in these intermediate representations. Naively reshaping these representations into long vectors would lead to unmanageably large covariance matrices, thereby increasing computational costs and risking numerical instability. To address this issue, we adapt VCReg to accommodate networks with spatial dimensions. Each vector at a different spatial location is treated as an individual sample when calculating the covariance matrix. Both the variance loss and the covariance loss are then calculated based on this modified covariance matrix. In terms of practical implementation, a VCReg is usually applied subsequent to each block within the neural network architecture, often succeeding residual connections. This placement allows for seamless incorporation into current network architecture and training paradigms. Addressing Outliers with Smooth L1 Loss After treating spatial locations as independent samples for covariance computation, the resulting samples are no longer statistically independent. This can lead to outliers in the covariance matrix and unstable gradient updates. To address this, we introduce a smooth L1 penalty into the covariance loss term. Specifically, we replace the traditional squared covariance values $ C_{ij} $ in $ \ell_{\text{cov}} $ with a smooth L1 function: \[ \text{SmoothL1}(x) = \begin{cases} x^2, & \text{if } \|x\| \leq \delta \\ 2\delta\|x\| - \delta^2, & \text{otherwise} \end{cases} \] By implementing this modification, we ensure that the loss function increases in a more controlled manner with respect to large covariance values. Empirically, this minimizes the impact of outliers, thereby enhancing the stability of the training process. 3.3 FAST IMPLEMENTATION To optimize implementation speed, we take advantage of the fact that VCReg only affects the loss function and not the forward pass. This allows us to focus on directly modifying the backward function for improvements. Specifically, we sidestep the usual process of calculating the VCReg loss and subsequent backpropagation. Instead, we directly adjust the computed gradients, which is feasible since the VCReg loss calculation relies solely on the current representation. Further details of this speed-optimized technique are outlined in Appendix B. We quantify the computational overhead by measuring the average time required for one NVIDIA A100 GPU to execute both the forward and backward passes on the entire network for a batch size of 128 using the ImageNet dataset. These results are summarized in Table 1. For the sake of comparison, we also include the latencies associated with adding Batch Normalization (BN) layers, revealing that our optimized VCReg implementation exhibits similar latencies to BN layers. Table 1: Average Time Required for One Forward and Backward Pass with Various Layers Inserted \| Network \| Number of Inserted Layers \| Identity \| VCReg (Naive) \| VCReg (Fast) \| BN \| \|-----------------\|---------------------------\|----------\|---------------\|--------------\|--------\| \| ViT-Base-32 \| 12 \| 0.223s \| 1.427s \| 0.245s \| 0.247s \| \| ConvNeXt-T \| 18 \| 0.442s \| 2.951s \| 0.471s \| 0.468s \| 4 EXPERIMENTS In this section, we initially outline the experimental framework and findings to highlight the effectiveness of our proposed regularization approach, VCReg, within the realm of transfer learning that utilizes supervised pretraining. Subsequent to that discussion, we extend our experiments beyond the scope of supervised pretraining to suggest that VCReg could be applicable across various learning paradigms. For guidelines on reproducing these experiments, please consult Appendix C. 4.1 TRANSFER LEARNING WITH SUPERVISED PRETRAINING In this section, we adhere to evaluation protocols established by seminal works such as (Chen et al., 2020; Kornblith et al., 2021; Misra & Maaten, 2020) for our transfer learning experiments. Initially, we pretrained models using three different architectures: ResNet-50 (He et al., 2016), ConvNeXt-Tiny (Liu et al., 2022), and ViT-Base-32 (Dosovitskiy et al., 2020), on the full ImageNet dataset. We followed the standard PyTorch recipes (Paszke et al., 2019) for all networks and did not modify any hyperparameters other than those related to VCReg to ensure a fair baseline comparison. Subsequently, we performed a linear probing evaluation across a variety of datasets to evaluate the transfer learning performance. For ResNet-50, we included two other feature diversity regularizer methods, namely DeCov (Cogswell et al., 2015) and WLD-Reg (Laakom et al., 2023), for comparison. We conducted experiments solely with ResNet-50 because it is the principal architecture used in the WLD-Reg paper. To ensure a fair comparison, we sourced hyperparameters from Laakom et al. (2023) for both DeCov and WLD-Reg. The results presented in Table 2 depict significant improvements in transfer learning performance across all downstream datasets when VCReg is applied to the three architectures used. There is strong evidence to suggest that VCReg can help boost overall transfer learning performance, and it is effective for both ConvNet and Transformer architectures. 4.2 BEYOND TRANSFER LEARNING WITH SUPERVISED LEARNING In this section, we explore the versatility of the VCReg regularization method by extending its application beyond transfer learning with supervised pretraining. We focus on three specialized Table 2: Transfer Learning Performance with ImageNet Supervised Pretraining The table shows performance metrics for different architectures. Each model is pretrained on the full ImageNet dataset and then tested on different downstream datasets using linear probing. Application of VCReg consistently improves performance and beats other feature diversity regularizer. \| Architecture \| ImageNet \| iNat18 \| Places \| Food \| Cars \| Aircraft \| Pets \| Flowers \| DTD \| \|-----------------------\|----------\|--------\|--------\|------\|------\|----------\|------\|---------\|-----\| \| ResNet-50 \| 76.1% \| 42.8% \| 50.6% \| 69.1%\| 43.6%\| 54.8% \| 91.9%\| 77.1% \| 68.7%\| \| ResNet-50 (DeCov) \| 75.9% \| 43.1% \| 50.4% \| 69.0%\| 45.7%\| 55.5% \| 90.6%\| 79.2% \| 69.1%\| \| ResNet-50 (WLD-Reg) \| 76.5% \| 43.9% \| 51.2% \| 70.2%\| 43.9%\| 58.7% \| 91.4%\| 80.7% \| 69.0%\| \| ResNet-50 (VCReg) \| 76.3% \| 45.3% \| 51.2% \| 71.7%\| 54.1%\| 70.5% \| 92.1%\| 88.0% \| 70.8%\| \| ConvNeXt-T \| 82.5% \| 51.6% \| 53.8% \| 78.4%\| 62.9%\| 74.7% \| 93.9%\| 91.3% \| 72.9%\| \| ConvNeXt-T (VCReg) \| 82.4% \| 52.3% \| 54.7% \| 79.6%\| 64.2%\| 76.3% \| 94.1%\| 92.7% \| 73.3%\| \| ViT-Base-32 \| 75.9% \| 39.1% \| 47.9% \| 70.6%\| 51.2%\| 63.8% \| 90.3%\| 84.6% \| 66.1%\| \| ViT-Base-32 (VCReg) \| 76.3% \| 40.6% \| 48.1% \| 70.9%\| 52.0%\| 65.8% \| 91.0%\| 86.6% \| 66.5%\| learning scenarios: 1) class imbalance via long-tail learning, 2) synergizing with self-supervised learning frameworks, and 3) hierarchical classification problems. The objective is to assess the adaptability of VCReg across various data distributions and learning paradigms, thereby evaluating its broader utility in machine learning applications. Class Imbalance with Long-Tail Learning Class imbalance is a pervasive issue in many real-world datasets and poses a considerable challenge to standard neural network training algorithms. We conducted experiments to assess how well VCReg addresses this issue through long-tail learning. We evaluated VCReg using the CIFAR10-LT and CIFAR100-LT [Krizhevsky et al., 2009] datasets, both engineered to have an imbalance ratio of 100. These experiments were conducted using a ResNet-32 backbone architecture. The per-class sample sizes ranged from 5,000 to 50 for CIFAR10-LT and from 500 to 5 for CIFAR100-LT. Table 3: Performance Comparison on Class-Imbalanced Datasets Using VCReg This table shows the accuracy of standard ResNet-32 with and without VCReg when trained on class-imbalanced CIFAR10-LT and CIFAR100-LT datasets. The VCReg-enhanced models show improved performance, demonstrating the method’s effectiveness in addressing class imbalance. \| Training Methods \| CIFAR10-LT \| CIFAR100-LT \| \|------------------\|------------\|-------------\| \| ResNet-32 \| 69.6% \| 37.4% \| \| ResNet-32 (VCReg)\| 71.2% \| 40.4% \| Table 3 shows that models augmented with VCReg consistently outperformed the standard ResNet-32 models on imbalanced datasets. These results are noteworthy because they demonstrate that VCReg effectively enhances the model’s ability to discriminate between classes in imbalanced settings. This establishes VCReg as a valuable tool for real-world applications where class imbalance is often a concern. Enhancing Self-Supervised Learning with VCReg Our subsequent investigation focuses on examining the synergy between VCReg and existing self-supervised learning paradigms. We employed a ResNet-50 architecture, training it for 100 epochs under four different configurations: using either SimCLR loss or VICReg loss, coupled with the ImageNet dataset. For evaluation, we conducted linear probing tests on multiple downstream task datasets, following the protocols prescribed by [Misra & Maaten, 2020; Zbontar et al., 2021]. As indicated in Table 4, integrating VCReg into self-supervised learning paradigms such as SimCLR and VICReg resulted in consistent performance improvements for transfer learning. Specifically, the linear probing accuracies were enhanced across nearly all the evaluated datasets. These gains underscore the broad applicability and versatility of VCReg, demonstrating its potential to enhance various machine learning methodologies. Investigating Hierarchical Classification Capabilities To evaluate the efficacy of the learned representations across multiple levels of class granularity, we conducted experiments on the CIFAR100 dataset as well as five distinct subsets of ImageNet [Engstrom et al., 2019]. In each dataset, every data sample is tagged with both superclass and subclass labels, denoted as $(x_i, y_i^{\text{sup}}, y_i^{\text{sub}})$. Note Table 4: Impact of VCReg on Self-Supervised Learning Methods: This table presents a comparative analysis of ResNet-50 models pretrained with SimCLR and VICReg losses on ImageNet, both with and without the VCReg applied. The models are evaluated using linear probing on various downstream task datasets. The VCReg models consistently outperform the non-VCReg models, showcasing the method’s broad utility in transfer learning for self-supervised learning scenarios. \| Pretraining Methods \| ImageNet \| iNat18 \| Places \| Food \| Cars \| Aircraft \| Pets \| Flowers \| DTD \| \|---------------------\|----------\|--------\|--------\|------\|------\|----------\|------\|---------\|-----\| \| SimCLR \| 67.2% \| 37.2% \| 52.1% \| 66.4%\| 35.7%\| 62.3% \| 76.3%\| 82.6% \| 68.1%\| \| SimCLR (VCReg) \| 67.1% \| 41.3% \| 52.3% \| 67.7%\| 40.6%\| 61.9% \| 76.6%\| 83.6% \| 69.0%\| \| VICReg \| 65.2% \| 41.7% \| 48.2% \| 61.0%\| 27.3%\| 51.2% \| 79.1%\| 74.3% \| 65.4%\| \| VICReg (VCReg) \| 66.3% \| 41.4% \| 49.6% \| 61.6%\| 29.3%\| 54.2% \| 79.7%\| 74.5% \| 66.5%\| that while samples sharing the same subclass label also share the same superclass label, the reverse does not necessarily hold true. Initially, the model was trained using only the superclass labels, i.e., the $(x_i, y_i^{sup})$ pairs. Subsequently, linear probing was employed with the subclass labels $(x_i, y_i^{sub})$ to assess the quality of features abstracted at the superclass level. Table 5: Impact of VCReg on Hierarchical Classification in ConvNeXt Models: This table summarizes the classification accuracies obtained with ConvNeXt models, both with and without the VCReg regularization, across multiple datasets featuring hierarchical class structures. The models were initially trained using superclass labels and subsequently probed using subclass labels. VCReg consistently boosts performance in subclass classification tasks. \| Subsets of ImageNet \| CIFAR100 \| living_9 \| mixed_10 \| mixed_13 \| geirhos_16 \| big_12 \| \|---------------------\|----------\|----------\|----------\|----------\|------------\|--------\| \| Superclass Count \| 20 \| 9 \| 10 \| 13 \| 16 \| 12 \| \| Subclass Count \| 100 \| 72 \| 60 \| 78 \| 32 \| 240 \| \| ConvNeXt \| 60.7% \| 53.4% \| 60.3% \| 61.1% \| 60.5% \| 51.8% \| \| ConvNeXt (VCReg) \| 72.9% \| 62.2% \| 67.7% \| 66.0% \| 70.1% \| 61.5% \| Table 5 presents key performance metrics, highlighting the substantive improvements VCReg brings to subclass classification. The improvements are consistent across all datasets, with the CIFAR100 dataset showing the most significant gain—an increase in accuracy from 60.7% to 72.9%. These results underscore VCReg’s capability to assist neural networks in generating feature representations that are not only discriminative at the superclass level but are also well-suited for subclass distinctions. This attribute is particularly advantageous in real-world applications where class categorizations often exist within a hierarchical framework. 5 Exploring the Benefits of VCReg This section aims to thoroughly unpack the multi-faceted benefits of VCReg in the context of supervised neural network training. Specifically, we discuss its capability to address challenges such as gradient starvation (Pezeshki et al., 2021), neural collapse (Papyan et al., 2020), and the preservation of information richness during model training (Shwartz-Ziv, 2022). 5.1 Mitigating Gradient Starvation In line with the original study on gradient starvation (Pezeshki et al., 2021), we observe that most traditional regularization techniques fall short of capturing the vital features for the ‘two-moon’ dataset experiment. To assess the effectiveness of VCReg, we replicated this setting with a three-layer network and applied our method during training. Our visualized results in Figure 1 make it apparent that VCReg has a marked advantage over traditional regularization techniques, particularly in the aspects of separation margins. Thus, it is reasonable to conclude that VCReg can help mitigate gradient starvation. These results are significant for multiple reasons. Firstly, the encouraging outcomes with the ‘two-moon’ synthetic dataset set the stage for investigating VCReg’s applicability in more complex, high- dimensional tasks, thus cementing its status as a potent tool in contemporary machine learning. Second, VCReg’s capability to mitigate gradient starvation indicates that neural networks trained using this method excel at learning complex, non-linear mappings—an essential trait for tasks that require a sophisticated understanding of data distributions. Lastly, VCReg surpasses traditional regularization techniques by generating a feature space that is both discriminative and rich in information. This highlights its potential to boost the generalizability of neural networks, which is crucial in real-world scenarios where models need to be both robust and flexible. 5.2 Preventing Neural Collapse and Information Compression To deepen our understanding of VCReg and its training dynamics, we closely examine its learned representations. A recent study (Papyan et al., 2020) observed a peculiar trend in deep networks trained for classification tasks: The top-layer feature embeddings of training samples from the same class tend to cluster around their respective class means, which are as distant from each other as possible. However, this phenomenon could potentially result in a loss of diversity among the learned features (Papyan et al., 2020), thus curtailing the network’s capacity to grasp the complexity of the data and leading to suboptimal performance (Li et al., 2018) for transfer learning. Our investigation is based on two key metrics: Class-Distance Normalized Variance (CDNV) For a feature map $ f : \mathbb{R}^d \rightarrow \mathbb{R}^p $ and two unlabeled sets of samples $ S_1, S_2 \subset \mathbb{R}^d $, the CDNV is defined as \[ V_f(S_1, S_2) = \frac{\text{Var}_f(S_1) + \text{Var}_f(S_2)}{2\\|\mu_f(S_1) - \mu_f(S_2)\\|^2}, \] where $ \mu_f(S) $ and $ \text{Var}_f(S) $ signify the mean and variance of the set $ \{f(x) \mid x \in S\} $. This metric measures the degree of clustering of the features extracted from $ S_1 $ and $ S_2 $, in relation to the distance between their respective features. A value approaching zero indicates perfect clustering. Nearest Class-Center Classifier (NCC) This classifier is defined as \[ \hat{h}(x) = \arg\min_{c \in [C]} \\|f(x) - \mu_f(S_c)\\| \] According to this measure, during training, collapsed feature embeddings in the penultimate layer become separable, and the classifier converges to the ‘nearest class-center classifier’. Preventing Information Compression We next address the prevention of information compression during the learning process. Although effective compression often yields superior representations, overly aggressive compression might cause the loss of crucial information about the target task (Shwartz-Ziv et al., 2018; Shwartz-Ziv & Alemi, 2020; Shwartz-Ziv & LeCun, 2023). To investigate this, we use the mutual information neural estimation (MINE) (Belghazi et al., 2018), a method specifically designed to estimate the mutual information between the input and its corresponding embedded representation. This metric effectively gauges the complexity level of the representation, essentially indicating how much information (in terms of number of bits) it encodes. Table 6: VCReg learns richer representation and prevents neural collapse and information compression Metrics include Class-Distance Normalized Variance (CDNV), Nearest Class-Center Classifier (NCC), and Mutual Information (MI). Higher values in each metric for the VCReg model indicate reduced neural collapse and richer feature representations. \| Network \| CDNV \| NCC \| MI \| \|------------------\|------\|-----\|----\| \| ConvNeXt \| 0.28 \| 0.99\| 2.8\| \| ConvNeXt (VCReg) \| 0.56 \| 0.81\| 4.6\| We evaluate the learned representations of two ConvNeXt models (Liu et al., 2022), which are trained on ImageNet with supervised loss. One model was trained with VCReg, while the other was trained without VCReg. As demonstrated in Table 6, both types of collapse, measured by CDNV and NCC, and the mutual information estimation reveal that VCReg representations have significantly more diverse features (lower neural collapse) and contain more information compared to regular training. This suggests that not only does VCReg achieve superior results, but also its underlying representation contains more information. In summary, the VCReg method mitigates the neural collapse phenomenon and prevents excessive information compression, two crucial factors that often limit the effectiveness of deep learning models in transfer learning tasks. Our findings highlight the potential of VCReg as a valuable addition to the deep learning toolbox, significantly increasing the generalizability of learned representations. 6 CONCLUSION In this work, we addressed prevalent challenges in supervised pretraining for transfer learning by introducing Variance-Covariance Regularization (VCReg). Building on the regularization technique of the self-supervised VICReg method, VCReg is designed to cultivate robust and generalizable features. Unlike conventional methods that attach regularization only to the final layer, we strategically incorporate VCReg across intermediate layers to optimize its efficacy. Our key contributions are threefold: 1. 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mvGa1ikBD3	Eqn. (5): Is x^{t+1} computed from a^{t+1} or is it an independent variable? I am guessing that the network produces a^{t+1}, which is then used to compute x^{t+1}, and both a^{t+1} and x^{t+1} are fed into this loss function.	GRAPH NEURAL NETWORKS WITH DIRECTIONAL ENCODINGS FOR ANISOTROPIC ELASTICITY Anonymous authors Paper under double-blind review ABSTRACT Simulating the behavior of nonlinear and anisotropic materials is a central problem with applications across engineering, computer graphics, robotics, and beyond. While conventional mesh-based simulations provide accurate and reliable predictions, their computational overhead typically prevents their use in interactive applications. Graph neural networks (GNN) have recently emerged as a compelling alternative to conventional simulations for time-critical applications. However, existing GNN-based methods cannot distinguish between deformations in different directions and are thus limited to isotropic materials. To address this limitation, we propose a novel and easy-to-implement GNN architecture based on directional encodings of edge features. By preserving directional information during message passing, our method has access to the full state of deformation and can thus model anisotropic materials. We demonstrate through a set of qualitative and quantitative evaluations that our approach outperforms existing mesh-based GNN approaches for modeling anisotropic materials. 1 INTRODUCTION From plant leaves to animal muscle and from woven textiles to fiber-reinforced composites—many natural and engineered materials are strongly anisotropic, i.e., their stress response varies significantly depending on the direction of deformation. Simulating such anisotropic materials properties is crucial for many applications in science and engineering [1]. Conventional simulation methods typically rely on mesh-based finite element discretizations for numerical solutions of the underlying partial differential equations. While these methods can capture intricate material behavior with high accuracy, they come at a substantial computational cost. Striking a balance between accuracy and efficiency, learning-based methods have emerged as a promising alternative to conventional simulations. Arguably the closest analogy to mesh-based simulation is a mesh-based deep neural representation. Indeed, existing works built on mesh-based graph neural networks (MGNN) have shown promising results [2] [3]. While existing MGNN methods have focused on isotropic materials so far, accounting for anisotropy might seem a straightforward extension. Unfortunately, the message passing architectures of current MGNNs rely on spatial averaging of edge features, which discards all directional information on deformation. As we show in our analysis, discarding directional information means that existing MGNNs are unable to model anisotropic materials. In this work, we present a novel feature encoding scheme designed to preserve directional information during message passing. We decompose edge features into components along three material-space basis vectors and aggregate these components separately during message passing. In this way, feature averaging takes into account the material-space orientation of the edges, leading to significantly improved preservation of anisotropic information. This improvement requires minimal changes to standard mesh-based graph neural networks, thus allowing for easy integration into existing frameworks. We validate our approach on a set of qualitative and quantitative examples and demonstrate that our approach outperforms the state-of-the-art method for capturing material anisotropy. Figure 1: Anisotropic Elasticity. We apply our approach to model the nonlinear deformation of an elastic cantilever beam under gravitational load. The beam is made from an isotropic base material augmented with reinforcing fibers (see insets). On the left, fibers are oriented in parallel to the direction of gravity, which leads to only minor stiffening compared to the base material. On the right, fibers run along the axis of the beam, leading to significantly reduced deflection for this load case. Rest and deformed states are shown in orange and blue, respectively. 2 RELATED WORK Simulation of Deformable Objects Simulating deformable objects plays a pivotal role across various disciplines, including mechanical engineering, computer graphics, and robotics. Among existing approaches, which include particle-based [4, 5, 6], grid-based [7, 8, 9, 10], and hybrid methods [11, 12], mesh-based representations are arguably the most prevalent choice [13, 14, 15, 16]. The computer graphics community has made great strides in efficient, robust, and accurate mesh-based simulation of deformable bodies [17, 18, 19, 20, 21]. Although dimension reduction techniques exist [22, 23, 24, 25], the associated computational costs for native scale simulation are often too significant for real-time applications or rapid design explorations. Our approach falls into the same category of using mesh representation for the input geometry, however, we use mesh-based graph neural networks to reduce online computation time significantly. Simulation of Anisotropic Materials Realistic simulating of many phenomena in nature must take into account their inherent material anisotropy, e.g., muscle deformation [26], plant biomechanics [27], material fracture [28], etc. Within the scope of this work, we focus on simulating deformable objects within the hyperelastic regime. Within this realm of research, many forms of anisotropic energies have been extensively studied, for instance, transverse isotropic elasticity [29, 30, 31], orthotropic elasticity [32], and generalized anisotropic elasticity [33, 34]. We focus on transverse isotropic elastic material where a base isotropic material is augmented with freely oriented fibers to achieve directional-dependent properties. This allows for easy integration into existing isotropic formulations. While anisotropic material properties have been extensively studied for mesh-based simulation, representing directional-dependent behavior with neural representation remains unexplored. We identify a key limitation factor for existing mesh-based neural representations and propose a simple yet effective strategy for better capturing material anisotropy. Neural Representation Deep neural representations hold substantial promise as alternatives for modeling complex physical systems while significantly reducing computational requirements when compared to conventional approaches [35, 36, 37]. One stream of research relies on ground-truth simulation data for learning surrogate models, e.g., for fluid dynamics [38], character animation [39], and modeling nonlinear material properties [40, 41]. With the advancement of physics-informed learning [42], another line of research leverages physical laws directly as loss functions to enable self-supervised learning [43, 44, 45, 46]. In this manner, neural networks learn not only from existing data but also from the inherent physics governing the system. We also opt for an unsupervised training strategy where the variational formulation of the physics laws directly as loss functions. However, to the best of our knowledge, our work is the first to explore material anisotropy for neural representations of deformable solids with graph neural networks. Mesh-based Graph Neural Networks Recent advancements in graph-based neural network architectures (47, 48, 49) offer a new paradigm for soft-body simulations (2, 50, 51). Specifically, mesh graph networks have emerged as a promising alternative to conventional finite element methods for simulating, for instance, fluid (52, 53), solid (54, 55, 2), cloth (3), etc. Unlike grid-based methods (56, 57, 35), their unstructured nature allows for easy generation of simulation domains and resolutions. Most related to our approach is MeshGraphNet (2), where an encoder-processor-decoder network architecture is leveraged to predict accelerations per time step. While their approach is able to capture a range of phenomena governed by physics PDEs, challenges remain for material anisotropy. We propose a novel and easy-to-implement edge feature decomposition operation to encode directional information during training. As we demonstrate in the result section, this modification significantly improves the performance of learning anisotropic material properties. ### 3 METHOD In this section, we describe the machinery required for training GNNs with directional encodings. Our approach builds upon an encoder-processor-decoder network architecture with a novel edge feature decomposition scheme aimed at capturing material anisotropy (Sec. 3.1). We adopt a self-supervised training paradigm and use the variational formulation for implicit Euler as the loss function (3.2). We provide sampling, training, and implementation details in Sec. 3.3. #### 3.1 MODEL ARCHITECTURE ![Pipeline Diagram](image) Figure 2: Pipeline. Our method takes the current states of a deformable object and its boundary conditions as input and predicts end-of-time-step accelerations using a graph neural network. These accelerations are then used to obtain the deformed state for the next time step (first row). We leverage an encoder-processor-decoder architecture and propose a novel edge decomposition operation to encode directional information during message passing (second row). We define the simulation mesh as a graph $G = (V, E)$ with nodes $V$ and edges $E$. Each node is associated with a coordinate vector $x$ and additional physical parameters such as mass, external forces, and Dirichlet boundary conditions. We refer to these parameters as vertex features $v$. Likewise, we use $e$ to denote edge features, which include relative vertex positions and fiber orientations. Our neural representation builds on the encode-process-decode architecture (48), where two distinct multilayer perceptrons (MLPs) are used as encoders to extract vertex and edge features. The encoded features are then processed with a set of MLPs during a fixed number $L$ of message passing steps. In each step, all edge and vertex features are processed using the same MLPs, but each step has its own vertex and edge MLP. Finally, a decoder MLP is used to transform vertex features to end-of-time-step accelerations. The predicted accelerations are used to update vertex positions. See Figure 2 for an overview. Encoding and Decoding Our encoder and decoder MLPs largely follow MeshGraphNets (2). The input vertex and edge features are transformed into latent feature vectors through encoder MLPs $f_v$ and $f_e$, $$\tilde{v} = f_v(v), \quad \tilde{e} = f_e(e),$$ where $\tilde{v}$ and $\tilde{e}$ denote updated feature vectors. The vertex decoder $f_{v \rightarrow a}$ maps vertex features to end-of-time-step accelerations for a given vertex, $$a = f_{v \rightarrow a}(v),$$ which are then used to compute end-of-step positions. Direction-aware Message Passing Our key contribution lies in the message passing step where we leverage directional encodings to better preserve information on anisotropic states of deformation. Specifically, we update per vertex and edge feature as $$\tilde{e} = e + f_{e \rightarrow v}(e, v_0, v_1),$$ $$\tilde{v} = v + f_{v \rightarrow e}(v, \sum_{e_j \in N_i} \omega_{x,j} e_j, \sum_{e_j \in N_i} \omega_{y,j} e_j, \sum_{e_j \in N_i} \omega_{z,j} e_j),$$ where $v_0$ and $v_1$ are the vertex features for the two endpoints of a given edge, $e_j$ loops over the features for all edges incident to vertex $i$, and $f_{v \rightarrow e}$ and $f_{e \rightarrow v}$ are the edge and vertex processor MLPs respectively. We use $+$ to denote residual connections (58). It is important to note that MeshGraphNet (2) aggregates edge features directly to update vertex features. This operation, however, does not distinguish deformations in different directions. To understand this problem, consider a mesh edge that is oriented along the $x$-axis in material space. Since the edge stores relative position between its endpoints, it cannot sense deformation along the $y$- and $z$-directions, which leave relative positions along the $x$-axis unchanged. Nevertheless, the feature aggregation scheme used in MeshGraphNets does not consider this dependence of sensing capacity on edge orientation, which ultimately limits its ability to capture directional deformation and model material anisotropy. By contrast, our novel encoding scheme projects mesh edges onto an orthonormal material-space basis such as to measure their capacity to sense deformation along different coordinate axes. The resulting coordinates are then used to decompose the original edge feature into three weighted components that are averaged individually. Using this directional encoding, an edge that aligns well with a given direction of deformation is given more authority to determine the averaged feature than an edge that is almost orthogonal to that direction. As a result, our method is able to preserve directional deformation during message passing and can thus better model anisotropic materials. We note that the edge weights $\omega_{x,j}, \omega_{y,j},$ and $\omega_{z,j}$ are computed from the rest state edge vectors and remain constant during training. Concretely, the weights for a given edge $E_j$ are computed as $$\omega_{x,j} = \frac{E_j}{\|\|E_j\|\|} \cdot E_x, \quad \omega_{y,j} = \frac{E_j}{\|\|E_j\|\|} \cdot E_y, \quad \omega_{z,j} = \frac{E_j}{\|\|E_j\|\|} \cdot E_z,$$ where $E_x, E_y,$ and $E_z$ are unit-length basis vectors. We further note that this modification requires minimal changes to standard mesh-based graph neural network architectures, allowing for easy integration of our approach into an existing framework. As we demonstrate in the result section, our directional feature encoding scheme leads to significantly improved performance for learning material anisotropy. ### 3.2 Physics-based Loss Function Spatial Discretization We resort to tetrahedral finite elements with linear basis functions to model the nonlinear dynamics of deformable solids. Our network operates on the edges and nodes of the simulation mesh and performs message passing on the corresponding graph. Adhering to standard finite element practice, our loss functions by summation of per-element potentials. Loss Function To allow for efficient self-supervised learning, we formulate our loss function to directly penalize the violation of the dynamic equilibrium conditions. To enable robust time stepping for larger step sizes, we use backward Euler integration, i.e., a first-order accurate implicit time stepping scheme (59). Instead of directly solving the resulting system of nonlinear equations, we follow the variational formulation of Martin et al. (17) and convert the root finding problem into an energy minimization problem. We use the corresponding incremental potential as our physics-based loss function during training. Defining end-of-time-step positions and accelerations as $ x^{t+1} $ and $ a^{t+1} $, our total loss function reads \[ L_{\text{total}}(a^{t+1}, x^{t+1}) = L_{\text{elastic}}(x^{t+1}) + L_{\text{external}}(x^{t+1}) + L_{\text{kinetic}}(a^{t+1}). \] Our $ L_{\text{elastic}} $ term captures the elastic energies for both isotropic and anisotropic deformation. We focus on transversely isotropic materials, where anisotropic fibers are embedded in an isotropic base material. Such materials are widely used for physics-based modeling of, e.g., fiber-reinforced composites, and biological tissue. We adopt the widely used Saint Venant–Kirchhoff model (60) for the isotropic base material and augment it with an anisotropic term that models the effect of embedded fibers with a given orientation. The elastic energy for a given tetrahedron element is defined as \[ L_{\text{elastic}}(x^{t+1}) = \bar{v} \left( \frac{\lambda}{2} (\text{tr}(E))^2 + \mu \text{tr}(E^2) + \kappa (\mathbf{d}^\top F^\top F \mathbf{d} - 1)^2 \right), \] where $ E = \frac{1}{2}(F^\top F - I) $ is the nonlinear Green strain, $ F $ is the deformation gradient, and $ \mathbf{d} $ is the fiber direction. Furthermore, $ \bar{v} $ is the undeformed volume of an element, and $ \lambda $ and $ \mu $ are Lamé parameters for defining the material properties. Finally, $ \kappa $ is the Young’s modulus for fiber stiffness. The kinetic energy term is defined as \[ L_{\text{kinetic}}(a^{t+1}) = \frac{1}{2} (\Delta v^{t+1})^\top (\Delta v^{t+1} \odot m_v), \] where $ \Delta t $ is the simulation time step size, $ \Delta v^{t+1} = \Delta r^{t+1} $ are velocity increments and $ \odot $ denotes element-wise vector-vector multiplication between the velocity increments and masses for all vertices within an element. We further define the external energy corresponding to the work done by external loads as \[ L_{\text{external}}(x^{t+1}) = f_{\text{ext}} \cdot x^{t+1}, \] where $ f_{\text{ext}} $ is a vector containing all external forces. 3.3 Training and Implementation Details Sample Generation We generate our training samples using combinations of simple geometries, e.g. rectangular and cylindrical beams (36 in total) with different mesh topologies and resolutions. The training mesh resolution is between 60 to 120 elements. We uniformly sample the force direction and magnitude $(0 - 10kN/m^3)$ applied to each mesh element. For non-uniform loading scenarios, we add additional forces to each element with a probability of 5%. Magnitude and direction are randomly sampled for each element $(0 - 15kN/m^3)$. Finally, we include traction samples with a probability of 10% with fixed direction in $+z$ axis and amplitudes between $0 - 100kN/m^3$. For material anisotropy, we uniformly sample fiber orientations and magnitudes between $0 - 10E$ where $E$ is the Young’s Modulus of the base material which is fixed to be $100kPa$. To increase stability for long-time inference rollouts, we find it crucial to sample not only undeformed states with random forces but also deformed states with non-zero kinetic and elastic energies. We apply the above parameter sampling procedure to these pre-deformed samples as well. Training Our framework is implemented in C++ using LibTorch. We use the Adam (61) optimizer with a learning rate of $5 \times 10^{-5}$ and a weight decay rate of $10^{-4}$ per one hundred iterations. Each training sample is unique and randomly sampled. The batch size is set to 1 and we train a total of 672,000 epochs. All of our MLPs have two hidden layers with 128 neurons per layer and SiLU activation functions (62). Layer normalization is applied to all layers except the final decoder MLP. All input features except the one-hot encoded anchored vertices are normalized. The encoder MLPs produce output features of size 128, while the vertex decoder yields features of size 3. Following MeshGraphNets, we perform 15 message-passing steps. Vertex input features consist of a one-hot-ended vector containing Dirichlet boundary conditions, vertex velocities, vertex mass and vertex external forces. Edge input features consist of two vectors containing the edge direction of both undeformed configuration and current deformation. Both vectors are normalized and their norm is added as a separate feature. Additionally, all edges contain another vector with fiber direction and magnitude. During training, we introduce perturbations to both nodal velocity and positions using zero-mean noise. The variance for velocities is stochastically sampled from the range of $0 - 5 \times 10^{-2} \text{m/s}$, while the variance for position noise falls within the interval of $0 - 10^{-3} \text{m}$. This perturbation process, akin to MeshGraphNets, plays a pivotal role in ensuring the stability of the neural network for long rollouts. The network is trained on a workstation with an AMD Ryzen 7 5800X CPU and an NVIDIA GeForce RTX 3080Ti GPU. Training takes around 5 days, whereas inference takes 9ms for a mesh of 100 elements. The Lamé parameters are computed from Young’s modulus (100kPa) and Poisson’s ratio (0.48) of a soft rubber-like material. When performing time stepping, we use a step size $\Delta t$ of 0.02s. We will release our code upon acceptance. 4 RESULTS In this section, we compare our results to the state-of-the-art mesh graph neural network, MeshGraphNets [2] on a set of qualitative and quantitative experiments. Since MeshGraphNets are trained in a supervised fashion, for fair comparisons, we implemented an unsupervised version using their network architectures with only modifications to the loss function to accommodate self-supervised learning. We demonstrate that our approach outperforms this baseline in terms of convergence speed, the ability to capture material anisotropy, and volume preservation for nearly incompressible materials. We further use a standard finite element solver to generate ground truth data for reference. Convergence We begin by comparing our approach with MeshGraphNets for different numbers of test rollouts (Figure 3). We generate 15 random configurations, i.e., different mesh topologies, force magnitudes, and directions, as test sets for all approaches and compute the difference in energy with respect to the ground truth value obtained from our reference simulation. After each training iteration, we evaluate all networks on the same test sets for different numbers of rollout steps in order to gauge their stability for sequences of different lengths. In particular, longer rollouts are useful to test whether predictions are converging toward static equilibrium. As can be seen from Figure 3, our approach consistently improves on MeshGraphNets, showing substantially faster convergence in all cases. It can also be noted that our method converges to equilibrium states with lower total energy. ![Figure 3: Network convergence. We compare the convergence behavior for our approach with MeshGraphNets on a test set for different rollout lengths. As can be seen in these figures, our approach converges to lower energy states much faster while remaining stable for longer horizons.](image) We attribute the significant discrepancy of MeshGraphNets to its limited ability to capture the anisotropic fibers. To verify this hypothesis, we visualize the energy difference to ground truth data for the fiber term and the sum of all terms separately. In this example (Figure 4), a beam with fibers along its long axis is loaded along the fiber direction. As can be seen from the plot shown to the left, the error in the fiber term for MeshGraphNets dominates the overall energy profile, leading to 10 times larger error compared to our method. Figure 4: Fiber and total energy error. A beam under uni-axial tension with fibers aligned with the direction of loading (right). We report the fiber and total energy error compared to simulation references (left). Due to the limited capability of capturing material anisotropy, the error from the fiber term dominates the overall error leading to significant deviation from ground truth data. Our approach, on the other hand, demonstrates 10 times higher accuracy. Anisotropic Elasticity To quantify the difference in terms of capturing anisotropic elasticity, we compare our approach with MeshGraphNet on a set of uniaxial loading test cases with fiber reinforcements in different magnitudes and directions (see Figure 5). When fiber reinforcements are collinear with the loading direction, they introduce strong resistant forces upon tensioning. Consequently, larger stress magnitudes for a given strain rate. As can be seen in the slopes of the curves in Figure 5(a,b), our model successfully captures this highly anisotropic behavior for different fiber stiffness whereas MeshGraphNets leads to poor matching behavior. Note that for strong fibers, the predictions from MeshGraphNets deviate already for small strain. When fibers are aligned orthogonal to the loading directions, they have minimal effects on the directional stress magnitude. This behavior is again captured by our model (Figure 5(c)). (a) Strong fibers ($\kappa = 5E$) in parallel direction (b) Weak fibers ($\kappa = E/5$) in parallel direction (c) Fibers ($\kappa = E/5$) in orthogonal direction Figure 5: Strain-stress curves. We compare our approach with MeshGraphNets on a set of uniaxial loading cases with different fiber orientations and magnitudes. We use $E$ to denote Young’s modulus of the base material. The predictions from our approach track the ground truth solution consistently better than MeshGraphNets and do not suffer from instabilities for larger strain rates. Volume Preservation In addition to capturing explicit material anisotropy, direction encodings also facilitate learning volumetric effects pertaining to the Poisson ratio, i.e. when a tension load is applied in one direction, causing the orthogonal directions to contract in order to preserve the material volume. In this experiment, we compare our approach and MeshGraphNets to the reference simulation on volume preservation of a beam under a constant tensile force. We report the maximum relative percentage error over all elements in Figure 6. As can be seen from this plot, MeshGraphNets leads to volume change up to 60% whereas our approach exhibits almost zero volume changes. Tip Displacements Complementing previous examples where tension modes are examined, we now shift to bending modes for more analysis. In this example, we quantitatively validate our approach by comparing the tip displacement error for a cantilever beam to its reference simulation. Figure 6: Volume preservation error. We plot the maximum relative percentage error for all elements in a deformed beam under tension. While our directional feature encoding leads to almost zero volume change compared to the simulation baseline, MeshGraphNets permits volume changes up to 60%. We consider two extreme testing scenarios for fiber orientations, one that is aligned with gravity in its rest shape (more deformation) and one that is orthogonal to it (less deformation). They are referred to as parallel and orthogonal in Table 1. We test all approaches with two beam topologies, namely rectangular (row 1-6) and cylindrical beams (row 7-12). For this set of experiments, we use the same stiffness for both the base material and the fibers. As reported in Table 1, our approach consistently outperforms the baseline method in terms of accuracy across all tested scenarios and beam topologies. \| Fiber Orientation \| Beam Topology \| Method \| Tip Displacement Error (m) \| \|-------------------\|---------------\|----------------\|----------------------------\| \| parallel \| rectangular \| MeshGraphNets \| 0.0399 \| \| parallel \| rectangular \| ours \| 0.0119 \| \| orthogonal \| rectangular \| MeshGraphNets \| 0.0902 \| \| orthogonal \| rectangular \| ours \| 0.0510 \| \| parallel \| cylindrical \| MeshGraphNets \| 0.1111 \| \| parallel \| cylindrical \| ours \| 0.0776 \| \| orthogonal \| cylindrical \| MeshGraphNets \| 0.1400 \| \| orthogonal \| cylindrical \| ours \| 0.0977 \| Table 1: Tip displacement comparisons. We consider two types of beam structures under gravitational force with one end of the beams fixed and leaving the other end free. Reinforcement fibers are set to be either parallel or orthogonal to gravity. As can be seen from the tip displacement errors reported, our approach demonstrates significantly higher accuracy compared to MeshGraphNets. Imbalanced Forces In this experiment, we consider the physically imbalanced force in the configuration generated by MeshGraphNets and our approach. The gradient of our loss function w.r.t. nodal positions amounts to the force equilibrium condition governed by Newton’s second law of motion, which should vanish at stable configurations. We therefore refer to the nonzero gradients as imbalanced forces. In Table 2, we report the imbalanced force magnitude from network predictions for a cantilever beam under static force equilibrium configurations. Same as in the previous example, we use the same stiffness for both the base material and fiber reinforcements. We apply a force density with its direction with gravity with two magnitudes (1000 N/m³ and 5000 N/m³). The fiber directions are varied from 45 to 90 degrees with 90 being orthogonal to the force direction. As can be seen from the statistics for average and maximum nodal imbalance forces, our approach reduces the mean error by 80% on average and the maximum error up to 90%. Generalization Finally, we demonstrate that our network generalizes to unseen geometries with different fiber layouts (Figure 7). In the first example, we add fibers to a T-shaped deformable object to resist bending load whereas in the second one, the fibers resist compression force for a Y-shape geometry. The applied forces and fiber orientations are shown in the insets. \| Fiber Direction \| Force Density $(N/m^3)$ \| Method \| Imbalanced Force $(N)$ Max/Mean \| \|-----------------\|---------------------------\|----------------\|-----------------------------------\| \| 45° \| 5000 \| MeshGraphNets \| 77.84 / 16.71 \| \| 45° \| 5000 \| ours \| 14.01 / 3.747 \| \| 45° \| 1000 \| MeshGraphNets \| 46.40 / 12.20 \| \| 45° \| 1000 \| ours \| 4.321 / 1.446 \| \| 60° \| 5000 \| MeshGraphNets \| 76.36 / 16.83 \| \| 60° \| 5000 \| ours \| 18.92 / 4.205 \| \| 60° \| 1000 \| MeshGraphNets \| 42.98 / 12.07 \| \| 60° \| 1000 \| ours \| 4.760 / 1.608 \| \| 90° \| 5000 \| MeshGraphNets \| 67.89 / 16.25 \| \| 90° \| 5000 \| ours \| 18.78 / 4.447 \| \| 90° \| 1000 \| MeshGraphNets \| 44.69 / 11.86 \| \| 90° \| 1000 \| ours \| 4.105 / 1.841 \| Table 2: Physically imbalanced force. We compare the physically imbalanced force in the predictions from MeshGraphNets and our approach for different fiber orientations and force densities. Our approach significantly reduces both the average and the peak error. Figure 7: Network generalization. We apply our approach to geometries significantly different from our training set. As can be seen from these two examples, the embedded reinforcement fibers play a crucial role in determining the deformed configurations. This material anisotropy is faithfully captured by our approach. The rest and deformed states are shown in orange and blue respectively. 5 CONCLUSION We have presented a novel mesh-based graph neural network architecture for learning the elastodynamics of anisotropic elastic materials. Whereas state-of-the-art approaches are limited to isotropic materials, we propose a novel and easy-to-implement edge feature decomposition scheme that preserves directional information during message passing and thus allows for the modeling of material anisotropies. We demonstrate on a set of qualitative and quantitative examples that our approach outperforms the state-of-the-art method by significant margins. Although we focus on nonlinear elasticity in this work, we believe that our feature decomposition scheme can benefit other applications of graph neural networks that involve direction-dependent behavior. 5.1 LIMITATION AND FUTURE WORK While our approach generalizes well to unseen meshes with similar resolution, we would like to leverage hierarchical representations [51, 50] to apply our approach across a wider range of mesh resolutions. 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hRos9WldRK	The reviewer notes that within L2B, the weights assigned to the clean-label term seem to be the same as the weights for both the pseudo sample and label. Have the authors thought about using different weights for samples and labels? Adopting such a strategy might appear more rational.	L2B: Learning to Bootstrap Robust Models for Combating Label Noise Anonymous authors Paper under double-blind review Abstract Deep neural networks have shown great success in representation learning. However, when learning with noisy labels (LNL), they can easily overfit and fail to generalize to new data. To address this challenge, in this paper, we propose a novel machine learning method called Learning to Bootstrap (L2B) that leverages a joint reweighting mechanism to train models using their own predictions to bootstrap themselves without being adversely affected by erroneous pseudo-labels. Unlike conventional approaches, L2B dynamically adjusts the importance weight between real observed labels and pseudo-labels, as well as between different samples, to determine the appropriate weighting. Additionally, L2B conducts implicit relabeling concurrently, leading to significant improvements without incurring additional costs. L2B offers several benefits over the baseline methods. It yields more robust models that are less susceptible to the impact of noisy labels by guiding the bootstrapping procedure more effectively. It better exploits the valuable information contained in corrupted instances by adapting the weights of both instances and labels. Furthermore, L2B is compatible with existing noisy label learning methods and delivers competitive results on several benchmark datasets, including CIFAR-10, CIFAR-100, ISIC2019, and Clothing 1M datasets. Extensive experiments demonstrate that our method effectively mitigates the challenges of noisy labels, often necessitating few to no validation samples, and be well generalized to other tasks such as image segmentation. This not only positions it as a robust complement to existing LNL techniques but also underscores its practical applicability. The code and models are available at https://anonymous.4open.science/r/L2B-6006. 1 Introduction In computer vision, deep learning has made significant strides, especially when provided with extensive, high-quality datasets. However, the persistent issue of label noise in real-world datasets, which stem from factors such as inter-observer variability, human annotation errors, and adversarial rival, can significantly undermine performance (Nettleton et al., 2010). As the size of datasets for deep learning continues to grow, the impact of label noise may become more significant. Understanding and addressing label noise is crucial for improving the accuracy and reliability of deep learning models (Liu et al., 2020; Wang et al., 2020; Zheng et al., 2021; Yao et al., 2021; Zhu et al., 2021; Wu et al., 2021b; Zhou et al., 2021). Existing methods for learning with noisy labels (LNL) primarily focus on loss correction to counter noise effects. A common strategy is estimating the noise corruption matrix to adjust the loss function (Patrini et al., 2017; Goldberger & Ben-Reuven, 2017). However, correctly estimating the noise corruption matrix is usually challenging and often involves assumptions about the noise generation process (Xia et al., 2019; Liu & Tao, 2015; Hendrycks et al., 2018). Recent studies predominantly aim to identify and train on clean samples within noisy datasets (Jiang et al., 2018; Han et al., 2018; Yu et al., 2019), often considering low-loss samples as clean (Arpit et al., 2017). Rather than discarding noisy examples, meta-learning approaches have been to assign adaptive weights to each sample (Ren et al., 2018; Shu et al., 2019), with noisier samples given lower weights. However, this approach may compromise performance in high-noise scenarios by neglecting or underweighting portions of the training data. To fully exploit the corrupted samples, a popular direction is to leverage the network predictions (i.e., pseudo-labels [Lee et al., 2013]) to recalibrate the labels [Reed et al., 2015; Tanaka et al., 2018; Song et al., 2019; Yi & Wu, 2019; Arazo et al., 2019]. One representative work is the bootstrapping loss [Reed et al., 2015], which weights pseudo-labels in computing the training targets to counter the adverse effects of noisy samples. However, the weight for pseudo-labels is often static, potentially leading to overfitting and poor label correction [Arazo et al., 2019]. To tackle this challenge, Arazo et al. [2019] further designed a dynamic bootstrapping method, modulating the weight between actual and pseudo-labels by fitting a mixture model. In contrast to prior works that individually reweight labels or instances, our paper introduces a novel approach to concurrently adjust both, elegantly unified under a meta-learning framework. We term our method as Learning to Bootstrap (L2B), as our goal is to enable the network to self-boost its capabilities by harnessing its own predictions in combating label noise. Specifically, during each training iteration, L2B dynamically re-balances the importance between the true and pseudo labels as well as the per-sample weights, all of which are determined by the validation performance on a separated meta (clean) set in a meta-network. This differs from previous bootstrapping loss methods [Reed et al., 2015; Arazo et al., 2019; Zhang et al., 2020] that explicitly reassign labels using a weighted combination of pseudo and true labels. Importantly, unlike conventional reweighting mechanisms, L2B does not constrain these weights to sum to one. Furthermore, we empirically show that meta-learning algorithms’ need for a clean validation set can be removed by dynamically creating an online meta set from the training data using a Gaussian mixture model [Permuter et al., 2006]. This not only enhances our method’s practicality but also facilitates its integration with current LNL techniques like DivideMix [Li et al., 2020], UniCon [Karim et al., 2022], and C2D [Zheltonozhskii et al., 2022]. Consequently, L2B attains superior results without relying on a validation set. In addition, we theoretically prove that our formulation, which reweights different loss terms, can be reduced to the original bootstrapping loss and therefore conducts an implicit relabeling instead. Through a meta-learning process, L2B achieves significant improvements (e.g., +8.9% improvement on CIFAR-100 with 50% noise) compared with the instance reweighting baseline with almost no extra cost. Our comprehensive tests across both natural and medical image datasets such as CIFAR-10, CIFAR-100, Clothing 1M, and ISIC2019, covering various types of label noise and recognition tasks, highlight L2B’s superiority over contemporary label correction and meta-learning techniques. 2 RELATED WORKS Explicit relabeling. Existing works propose to directly identify noisy samples and relabel them through estimating the noise transition matrix [Xia et al., 2019; Yao et al., 2020; Goldberger & Ben-Reuven, 2017; Patrini et al., 2017] or modeling noise by graph models or neural networks [Xiao et al., 2015; Vahdat, 2017; Veit et al., 2017; Lee et al., 2018; Patrini et al., 2017; Hendrycks et al., 2018] which estimate the label corruption matrix to directly correct the loss function. However, these methods usually require assumptions about noise modeling. For instance, Hendrycks et al. [2018] assume that the noisy label is only dependent on the true label and independent of the data. Another line of approaches proposes to leverage the network prediction (pseudo-labels) for explicit relabeling. Tanaka et al. [2018]; Yi & Wu [2019] relabel the samples by directly using pseudo-labels in an iterative manner. Han et al. [2019] use generated prototypes as pseudo-labels to be more noise tolerant. Instead of directly using the pseudo-labels as supervision, Reed et al. [2015] propose to generate new training targets by a convex combination of the true and pseudo labels, furthered by Ortego et al. [2021a] for classification refinement. However, using a uniform weight for all samples, as in Reed et al. [2015], can exacerbate the influence of noisy data, impeding effective label correction. Semi-supervised LNL techniques like Li et al. [2020]; Zhang et al. [2020] segment training data into labeled “clean samples” and unlabeled noisy sets, subsequently relabeled using pseudo-labels. To bolster the reliability of these pseudo-labels, unsupervised contrastive learning approaches are employed [Li et al., 2021; Ghosh & Lan, 2021; Zheltonozhskii et al., 2022; Karim et al., 2022]. Instance reweighting. To counteract the adverse effects of corrupted examples, various strategies focus on reweighting or selecting training instances to minimize the influence of noisy sam- Based on the observation that deep neural networks tend to learn simple patterns first before fitting label noise (Arpit et al., 2017), many methods treat samples with small loss as clean ones (Jiang et al., 2018; Shen & Sanghavi, 2019; Han et al., 2018; Yu et al., 2019; Wei et al., 2020). Among those methods, Co-teaching (Han et al., 2018) and Co-teaching+ (Yu et al., 2019) train two networks to help select samples to train the other. Rather than directly selecting clean examples for training, meta-learning techniques (Ren et al., 2018; Shu et al., 2019; Xu et al., 2021) adjust instance weights, and curriculum learning (Jiang et al., 2018) sequences them by noise levels. Such strategies enhance robustness in medical imaging (Xue et al., 2019; Mirikhraj et al., 2019), but overlooking training subsets can affect performance in high-noise scenarios. Meta-learning. Meta-learning based methods (Ren et al., 2018; Shu et al., 2019; Xu et al., 2021; Li et al., 2019; Wu et al., 2021a; Zheng et al., 2021; Zhang et al., 2020) aim to optimize model weights and hyper-parameters through a meta-process leveraging a small clean validation set. Among them, Ren et al. (2018); Shu et al. (2019); Xu et al. (2021) employ instance reweighting, adjusting example weights and network parameters through bi-level optimization to determine the contribution of each training sample. Wu et al. (2021a); Zheng et al. (2021); Zhang et al. (2020) approach label correction as a distinct meta-process. Different from the aforementioned approaches which separately handle instance reweighting and label reweighting, we introduce a novel learning objective that concurrently meta-learns per-sample loss weights while implicitly relabeling the training data. ### 3 METHODOLOGY #### 3.1 PRELIMINARY Given a set of $N$ training samples, i.e., $\mathcal{D}_{tra} = \{(x_i, y_i) \| i = 1, ..., N\}$, where $x_i \in \mathbb{R}^{W \times H}$ denotes the $i$-th image and $y_i$ is the observed noisy label. In this work, we also assume that there is a small unbiased and clean validation set $\mathcal{D}_{val} = \{(x_i^v, y_i^v) \| i = 1, ..., M\}$ and $M \ll N$, where the superscript $v$ denotes the validation set. Let $\mathcal{F}(x, \theta)$ denote the neural network model parameterized by $\theta$. Given an input-target pair $(x, y)$, we consider the loss function of $\mathcal{L}(\mathcal{F}(x, \theta), y)$ (e.g., cross-entropy loss) to minimize during the training process. Our goal, in this paper, is to properly utilize the small validation set $\mathcal{D}_{val}$ to guide the model training on $\mathcal{D}_{tra}$, for reducing the negative effects brought by the noisy annotation. To establish a more robust training procedure, Reed et al. (2015) proposed the bootstrapping loss to enable the learner to “disagree” with the original training label, and effectively re-label the data during the training. Specifically, the training targets will be generated using a convex combination of training labels and predictions of the current model (i.e., pseudo-labels (Lee et al., 2013)), for purifying the training labels. Therefore, for a $L$-class classification problem, the loss function for optimizing $\theta$ can be derived as follows: $$y_i^{\text{pseudo}} = \arg\max_{l=1,...,L} P(x_i, \theta),$$ $$\theta^* = \arg\min_{\theta} \sum_{i=1}^{N} \mathcal{L}(\mathcal{F}(x_i, \theta), \beta y_i^{\text{real}} + (1 - \beta) y_i^{\text{pseudo}}),$$ where $\beta$ is used for balancing the weight between the real labels and the pseudo-labels. $P(x_i, \theta)$ is the model output. $y_i^{\text{real}}$ and $y_i^{\text{pseudo}}$ denote the observed label and the pseudo-label respectively. However, in this method, $\beta$ is manually selected and fixed for all training samples, which does not prevent fitting the noisy ones and can even lead to low-quality label correction (Arazo et al., 2019). Moreover, we observe that this method is quite sensitive to the selection of the hyper-parameter $\beta$. For instance, as shown in Figure 1(a), even a similar $\beta$ selection (i.e., $\beta = 0.6$ vs. $\beta = 0.8$) behaves differently under disparate noise levels, making the selection of $\beta$ even more intractable. Another limitation lies in that Eq. 2 treats all examples as equally important during training, which could easily cause overfitting for biased training data. Figure 1: (a) The original bootstrapping loss [Reed et al., 2015] is sensitive to the reweighting hyper-parameter $\beta$. Under different noise levels, the optimal $\beta$ is different ($NF$ stands for noise fraction). (b) Schematic description of our Learning to Bootstrap (i.e., L2B) method. The reweighting hyper-parameters are learned in a meta-process. 3.2 Learning to Bootstrap through Meta-Learning To address these above challenges, in this paper, we aim to learn to bootstrap the model by conducting a joint label reweighting and instance reweighting. To achieve this, we propose to generate meta-learned weights for guiding our main learning objective: $$\theta^(\alpha, \beta) = \arg\min_{\theta} \sum_{i=1}^{N} \alpha_i \mathcal{L}(F(x_i, \theta), y_i^{\text{real}}) + \beta_i \mathcal{L}(F(x_i, \theta), y_i^{\text{pseudo}}),$$ (3) with $\{\alpha_i, \beta_i\}_{i=1}^{N}$ being the balance weights. Here we note that this new learning objective can be regarded as a general form of the original bootstrapping loss, as Eq. 3 can be reduced to Eq. 2 when $\alpha_i + \beta_i = 1$ given that $\mathcal{L}(\cdot)$ is the cross-entropy loss (see details in Appendix B.1). By relaxing this constraint such that $\alpha, \beta \geq 0$, we can see that the optimization of Eq. 3 not only allows the main learner to explore the optimal combination between the two loss terms but also concurrently adjust the contribution of different training samples. In addition, compared with Eq. 2, the optimization of Eq. 3 does not rely on explicitly generating new training targets (i.e., $\beta y_i^{\text{real}} + (1 - \beta)y_i^{\text{pseudo}}$), but rather conducts implicit relabeling during training by reweighting different loss terms. We note that the key to L2B is that the sum of $\alpha_i$ and $\beta_i$ need not be 1, which results in +8.9% improvement on CIFAR-100 with 50% noise (Section 3.5). Note that this form is also similar to self-distillation in [Li et al., 2017]. But different from [Li et al., 2017] where the weights are determined by heuristics, our weights $\alpha, \beta$ are meta-learned based on its performance on the validation set $D_{\text{val}}$, that is $$\alpha^, \beta^* = \arg\min_{\alpha, \beta \geq 0} \frac{1}{M} \sum_{i=1}^{M} \mathcal{L}(F(x_i^, \theta^(\alpha, \beta)), y_i^).$$ (4) It is necessary to constrain $\alpha_i, \beta_i \geq 0$ for all $i$ to avoid potential unstable training [Ren et al., 2018]. Both the meta learner (i.e., Eq. 4) and the main learner (i.e., Eq. 5) are optimized concurrently, which allows the model to maximize the performance on the clean validation set $D_{\text{val}}$ by adjusting the importance weights of the observed and the pseudo-labels in a differentiable manner. Online Approximation. For each step $t$ at training, a mini-batch of training examples $\{(x_i, y_i), 1 \leq i \leq n\}$ with $n \ll N$ is sampled to estimate a temporary adjustment to the parameters based on the descent direction of the loss function. For simplicity, let $f_i(\theta)$ denote $\mathcal{L}(F(x_i, \theta), y_i^{\text{real}})$ and $g_i(\theta)$ denote $\mathcal{L}(F(x_i, \theta), y_i^{\text{pseudo}})$ in the following sections. Given any $\alpha, \beta$, we use $$\hat{\theta}_{t+1} = \theta_t - \lambda \nabla \left( \sum_{i=1}^{n} \alpha_i f_i(\theta) + \beta_i g_i(\theta) \right) \bigg\|_{\theta=\theta_t}$$ (5) to approach the solution of Eq. 2. Here $\lambda$ is the step size. We then estimate the corresponding optimal $\alpha, \beta$ as $$\alpha^_t, \beta^_t = \arg\min_{\alpha, \beta \geq 0} \frac{1}{M} \sum_{i=1}^{M} f_i^v(\hat{\theta}_{t+1}).$$ (6) Algorithm 1 Learning to Bootstrap Require: $\theta_0$, $D_{tra}$, $D_{val}$, $n$, $m$, $L$ Ensure: $\theta_T$ 1: for $t = 0 \ldots T - 1$ do 2: $\{x_i, y_i\} \leftarrow \text{SampleMiniBatch}(D_{tra}, n)$ 3: $\{x_i^v, y_i^v\} \leftarrow \text{SampleMiniBatch}(D_{val}, m)$ 4: For the $i$-th sample of $D_{tra}$, compute $y_i^{\text{pseudo}} = \arg\max_{l=1,\ldots,L} P(x_i, \theta_t)$ 5: Learnable weights $\alpha, \beta$ 6: Compute training loss $l_f \leftarrow \sum_{i=1}^{n} \alpha_i f_i(\theta_t) + \beta_i g_i(\theta_t)$ 7: $\hat{\theta}_{t+1} \leftarrow \theta_t - \lambda \nabla l_f \big\|_{\theta=\theta_t}$ 8: Compute validation loss $l_g \leftarrow \frac{1}{m} \sum_{i=1}^{m} f_i^v(\hat{\theta}_{t+1})$ 9: $(\alpha_t, \beta_t) \leftarrow -\eta \nabla l_g \big\|_{\alpha=0, \beta=0}$ 10: $\tilde{\alpha}_{t,i} \leftarrow \max(\alpha_{t,i}, 0), \tilde{\beta}_{t,i} \leftarrow \max(\beta_{t,i}, 0)$ 11: $\tilde{\alpha}_{t,i} \leftarrow \frac{\tilde{\alpha}_{t,i}}{\sum_{i=1}^{n} \tilde{\alpha}_{t,i} + \tilde{\beta}_{t,i}}, \tilde{\beta}_{t,i} \leftarrow \frac{\tilde{\beta}_{t,i}}{\sum_{i=1}^{n} \tilde{\alpha}_{t,i} + \tilde{\beta}_{t,i}}$ 12: Apply learned weights $\alpha, \beta$ to reweight the training loss as $\hat{l}_f \leftarrow \sum_{i=1}^{n} \tilde{\alpha}_{t,i} f_i(\theta_t) + \tilde{\beta}_{t,i} g_i(\theta_t)$ 13: $\theta_{t+1} \leftarrow \theta_t - \lambda \nabla \hat{l}_f \big\|_{\theta=\theta_t}$ 14: end for However, directly solving for Eq. 6 at every training step requires too much computation cost. To reduce the computational complexity, we apply one step gradient descent of $\alpha_t, \beta_t$ on a mini-batch of validation set $\{(x_i^v, y_i^v), 1 \leq i \leq m\}$ with $m \leq M$ as an approximation. Specifically, $$ (\alpha_{t,i}, \beta_{t,i}) = -\eta \nabla \left( \sum_{i=1}^{m} f_i^v(\hat{\theta}_{t+1}) \right) \bigg\|_{\alpha_t=0, \beta_t=0}, $$ where $\eta$ is the step size for updating $\alpha, \beta$. To ensure that the weights are non-negative, we apply the following rectified function: $$ \tilde{\alpha}_{t,i} = \max(\alpha_{t,i}, 0), \tilde{\beta}_{t,i} = \max(\beta_{t,i}, 0). $$ To stabilize the training process, we also normalize the weights in a single training batch so that they sum up to one: $$ \tilde{\alpha}_{t,i} = \frac{\tilde{\alpha}_{t,i}}{\sum_{i=1}^{n} \tilde{\alpha}_{t,i} + \tilde{\beta}_{t,i}}, \tilde{\beta}_{t,i} = \frac{\tilde{\beta}_{t,i}}{\sum_{i=1}^{n} \tilde{\alpha}_{t,i} + \tilde{\beta}_{t,i}}. $$ Finally, we estimate $\theta_{t+1}$ based on the updated $\alpha_t, \beta_t$ so that $\theta_{t+1}$ can consider the meta information included in $\alpha_t, \beta_t$: $$ \theta_{t+1} = \theta_t - \lambda \nabla \left( \sum_{i=1}^{n} \tilde{\alpha}_{t,i} f_i(\theta) + \tilde{\beta}_{t,i} g_i(\theta) \right) \bigg\|_{\theta=\theta_t}. $$ See Appendix B.2 for detailed calculation of the gradient in Eq. 10. A schematic description of our Learning to Bootstrap algorithm is illustrated in Figure 1(b) and the overall optimization procedure can be found in Algorithm 1. 3.3 Convergence Analysis In proposing Eq. 3, we show that with the first-order approximation of $\alpha, \beta$ in Eq. 7 and some mild assumptions, our method guarantees to convergence to a local minimum point of the validation loss, which yields the best combination of $\alpha, \beta$. Details of the proof are provided in Appendix B.3. Theorem 1. Suppose that the training loss function $f, g$ have $\sigma$-bounded gradients and the validation loss $f^v$ is Lipschitz smooth with constant $L$. With a small enough learning rate $\lambda$, the validation loss monotonically decreases for any training batch $B$, namely, $$ G(\theta_{t+1}) \leq G(\theta_t), $$ where $\theta_{t+1}$ is obtained using Eq. (10) and $G$ is the validation loss $$G(\theta) = \frac{1}{M} \sum_{i=1}^{M} f_i^v(\theta),$$ (12) Furthermore, Eq. (11) holds for all possible training batches only when the gradient of validation loss function becomes 0 at some step $t$, namely, $G(\theta_{t+1}) = G(\theta_t) \forall B \iff \nabla G(\theta_t) = 0$ ### 3.4 DATASETS CIFAR-10 & CIFAR-100.* Both CIFAR-10 and CIFAR-100 contain 50K training images and 10K test images of size $32 \times 32$. Following previous works (Tanaka et al., 2018; Kim et al., 2019; Li et al., 2020), we experimented with both symmetric and asymmetric label noise. In our method, we used 1,000 clean images in the validation set $D_{val}$ following Jiang et al. (2018); Ren et al. (2018); Shu et al. (2019); Hendrycks et al. (2018); Zheng et al. (2021). ISIC2019. Following Xue et al. (2019), we also evaluated our algorithm on a medical image dataset, i.e., skin lesion classification data, under different symmetric noise levels. Our experiments were conducted on the 25,331 dermoscopic images of the 2019 ISIC Challenge[^1], where we used 20400 images as the training set $D_{tra}$, 640 images as the validation set $D_{val}$, and tested on 4291 images. Clothing 1M. We evaluate on real-world noisy dataset, Clothing 1M (Xiao et al., 2015), which has 1 million training images collected from online shopping websites with labels generated from surrounding texts. In addition, the Clothing 1M also provides an official validation set of 14,313 images and a test set of 10,526 images. Implementation details can be found in Appendix A.1. ### 3.5 PERFORMANCE COMPARISONS Efficacy of L2B. We compare our method with different baselines: 1) Cross-Entropy (the standard training), 2) Bootstrap (Reed et al., 2015), which modifies the training loss by generating new training targets, and 3) L2RW (Ren et al., 2018), which reweights different instances through meta-learning under different levels of symmetric labels noise ranging from 20% ~ 50%. To ensure a fair comparison, we report the best epoch for all comparison approaches. All results are summarized in Table 1. Compared with the naive bootstrap method and the baseline meta-learning-based instance reweighting method L2RW, the performance improvement is substantial, especially under larger noise fraction, which suggests that using meta-learning to automatically bootstrap the model is more beneficial for LNL. For example, on CIFAR-100, the accuracy improvement of our proposed L2B reaches 7.6% and 8.9% under 40% and 50% noise fraction, respectively. We also show a set qualitative examples to illustrate how L2B adjust the weights to rectify the influence from the noisy labels in Figure 4. Comparison with the state-of-the-arts. We compare our method with SOTA methods on CIFAR 10 and CIFAR 100 in Table 2. We demonstrate our L2B is compatible with existing LNL methods. When integrated with existing LNL methods like DivideMix (Li et al., 2020), UniCon (Karim et al., 2022), C2D (Zheltonozhskii et al., 2022), L2B consistently enhances performance across varying noise ratios on both datasets. Notably, L2B-C2D surpasses all competing methods in various settings, achieving 94.4% and 60.7% accuracy under the noise ratio of 90% for CIFAR-10 and CIFAR-100. We also test our model with 40% asymmetric noise and summarize the testing accuracy in Table 3. Among all compared methods, we re-implement L2RW under the same setting and report the performance of all other competitors from previous papers including Kim et al. (2019, 2021); Li et al. (2020). Compared with previous meta-learning-based methods (e.g., Chen et al. (2019), Zhang & Yao (2020)), and other methods (e.g., Ren et al. (2018), Wu et al. (2021a), Shu et al. (2019)), our L2B achieves superior results. [^1]: https://challenge2019.isic-archive.com/data.html Table 1: Comparison in test accuracy (%) with the baseline methods on CIFAR-10/100 datasets with symmetric noise. \| Dataset \| CIFAR-10 \| CIFAR-100 \| ISIC \| \|---------\|----------\|-----------\|------\| \| Method/Noise ratio \| 20% \| 30% \| 40% \| 50% \| 20% \| 30% \| 40% \| 50% \| 20% \| 30% \| 40% \| 50% \| \| Cross-Entropy (CE) \| 86.9 \| 84.9 \| 83.3 \| 81.3 \| 59.6 \| 52.2 \| 49.2 \| 44.4 \| 79.4 \| 77.5 \| 75.3 \| 73.7 \| \| Bootstrap (Reed et al., 2015) \| 85.2 \| 84.8 \| 82.9 \| 79.2 \| 61.8 \| 54.2 \| 50.2 \| 45.8 \| 80.8 \| 77.7 \| 75.7 \| 74.8 \| \| L2RW (Ren et al., 2018) \| 90.6 \| 89.0 \| 86.6 \| 85.3 \| 67.8 \| 63.8 \| 59.7 \| 55.6 \| 80.1 \| 77.7 \| 76.3 \| 74.1 \| \| L2B (Ours) \| 92.2 \| 90.7 \| 89.9 \| 88.5 \| 71.8 \| 69.5 \| 67.3 \| 64.5 \| 81.1 \| 80.2 \| 78.6 \| 76.8 \| Table 2: Comparison in test accuracy (%) with state-of-the-art methods on CIFAR-10/100 datasets with symmetric noise. \| Dataset \| CIFAR-10 \| CIFAR-100 \| \|---------\|----------\|-----------\| \| Method/Noise ratio \| 20% \| 50% \| 80% \| 90% \| 20% \| 50% \| 80% \| 90% \| \| Co-teaching+ (Yu et al., 2019) \| 89.5 \| 85.7 \| 67.4 \| 47.9 \| 65.6 \| 51.8 \| 27.9 \| 13.7 \| \| Mixup (Zhang et al., 2018) \| 95.6 \| 87.1 \| 71.6 \| 52.2 \| 67.8 \| 57.3 \| 30.8 \| 14.6 \| \| PENCIL (Yi & Wu, 2019) \| 92.4 \| 89.1 \| 77.5 \| 58.9 \| 69.4 \| 57.5 \| 31.1 \| 15.3 \| \| Meta-Learning (Li et al., 2019) \| 92.9 \| 89.3 \| 77.4 \| 58.7 \| 68.5 \| 59.2 \| 42.4 \| 19.5 \| \| M-correction (Arazo et al., 2019) \| 94.0 \| 92.0 \| 86.8 \| 69.1 \| 73.9 \| 66.1 \| 48.2 \| 24.3 \| \| AugDesc (Nishi et al., 2021) \| 96.3 \| 95.4 \| 93.8 \| 91.9 \| 79.5 \| 77.2 \| 66.4 \| 41.2 \| \| GCE (Ghosh & Lan, 2021) \| 90.0 \| 89.3 \| 73.9 \| 36.5 \| 68.1 \| 53.3 \| 22.1 \| 8.9 \| \| Sel-CL+ (Li et al., 2022) \| 95.5 \| 93.9 \| 89.2 \| 81.9 \| 76.5 \| 72.4 \| 59.6 \| 48.8 \| \| MLC (Zheng et al., 2021) \| 92.6 \| 88.1 \| 77.4 \| 67.9 \| 66.8 \| 52.7 \| 21.8 \| 15.0 \| \| MSLC (Wu et al., 2021a) \| 93.4 \| 89.9 \| 69.8 \| 56.1 \| 72.5 \| 65.4 \| 24.3 \| 16.7 \| \| MOIT+ (Ortego et al., 2021b) \| 94.1 \| 91.8 \| 81.1 \| 74.7 \| 75.9 \| 70.6 \| 47.6 \| 41.8 \| \| DivideMix (Li et al., 2020) \| 96.1 \| 94.6 \| 93.2 \| 76.0 \| 77.3 \| 74.6 \| 60.2 \| 31.5 \| \| L2B-DivideMix \| 96.1 \| 95.4 \| 94.0 \| 91.3 \| 77.9 \| 75.9 \| 62.2 \| 35.8 \| \| UniCon (Karim et al., 2022) \| 96.0 \| 95.6 \| 93.9 \| 90.8 \| 78.9 \| 77.6 \| 63.9 \| 44.8 \| \| L2B-UniCon \| 96.5 \| 95.8 \| 94.7 \| 92.8 \| 78.8 \| 77.3 \| 67.6 \| 49.6 \| \| C2D (Zheltonozhskii et al., 2022) \| 96.3 \| 95.2 \| 94.4 \| 93.5 \| 78.7 \| 76.4 \| 67.8 \| 58.7 \| \| L2B-C2D \| 96.7 \| 95.6 \| 94.8 \| 94.4 \| 80.1 \| 78.1 \| 69.6 \| 60.7 \| Table 3: Comparison with 40% asymmetric noise in test accuracy on the CIFAR-10 dataset. \| Method \| Acc \| \|--------\|-----\| \| Cross-Entropy \| 85.0 \| \| F-correction (Patrini et al., 2017) \| 87.2 \| \| M-correction (Arazo et al., 2019) \| 87.4 \| \| Chen et al. (Chen et al., 2019) \| 88.6 \| \| P-correction (Yi & Wu, 2019) \| 88.5 \| \| REED (Zhang & Yao, 2020) \| 92.3 \| \| Tanaka et al. (Tanaka et al., 2018) \| 88.9 \| \| NLNL (Kim et al., 2019) \| 89.9 \| \| JNPL (Kim et al., 2021) \| 90.7 \| \| DivideMix (Li et al., 2020) \| 93.4 \| \| MLNT (Li et al., 2019) \| 89.2 \| \| L2RW (Ren et al., 2018) \| 89.2 \| \| MW-Net (Shu et al., 2019) \| 89.7 \| \| MSLC (Wu et al., 2021a) \| 91.6 \| \| Meta-Learning (Li et al., 2019) \| 88.6 \| \| Distilling (Zhang et al., 2020) \| 90.2 \| \| L2B-Naive (Ours) \| 91.8 \| \| L2B-C2D (Ours) \| 94.0 \| Table 4: Comparison with state-of-the-art methods in test accuracy (%) on Clothing 1M. \| Method \| Acc \| \|--------\|-----\| \| CrossEntropy \| 69.2 \| \| M-correction (Arazo et al., 2019) \| 71.0 \| \| PENCIL (Yi & Wu, 2019) \| 73.5 \| \| DivideMix (Li et al., 2020) \| 74.8 \| \| Nested (Chen et al., 2021) \| 74.9 \| \| AugDesc (Nishi et al., 2021) \| 75.1 \| \| RRL (Li et al., 2021) \| 74.9 \| \| GCE (Ghosh & Lan, 2021) \| 73.3 \| \| C2D (Zheltonozhskii et al., 2022) \| 74.3 \| \| MCNN (Li et al., 2019) \| 73.5 \| \| MLC (Zheng et al., 2021) \| 75.8 \| \| MSLC (Wu et al., 2021a) \| 74.0 \| \| Meta-Cleaner (Zhang et al., 2019) \| 72.5 \| \| Meta-Weight (Shu et al., 2019) \| 73.7 \| \| FaMUS (Xu et al., 2021) \| 74.4 \| \| MSLG (Algan & Ulusoy, 2021) \| 76.0 \| \| L2B-Naive (Ours) \| 77.5 ± 0.2 \| Generalization to real-world noisy labels. We test L2B on Clothing 1M (Xiao et al., 2015), a large-scale dataset with real-world noisy labels. The results of all competitors are reported from published papers. As shown in Table 4, our L2B-Naive attains an average performance of 77.5% accuracy from 3 independent runs with different random seeds, outperforming all competing methods. Figure 2: Comparison among different normalization functions (i.e., Eq. 9 Sigmoid function and Softmax function). Testing accuracy curve: (a) with different normalization functions under 40% symmetric noise label on the ISIC dataset. (b) with different normalization under 40% symmetric label noise on CIFAR-100. Generalization to image segmentation L2B can be easily generalized for segmentation tasks. Specifically, the learnable weights $\alpha$ and $\beta$ are replaced with pixel-wise weight maps corresponding to noisy labels and pseudo labels (model predictions). L2B dynamically assigns these weight maps, adjusting for both noisy and pseudo labels to optimize the bootstrapping process via a meta-process. To assess L2B’s performance in segmentation, we employed the PROMISE12 dataset [Litjens et al. (2014)] which contains 50 3D transversal T2-weighted MR images. Specifically, 40/10 cases were used for training/evaluation. 3 out of the 40 training cases are chosen randomly as the meta set. As presented in Table 5, we compare our method with 1) UNet++ [Zhou et al. (2018)], 2) UNet++ meta, which trains exclusively on the meta data, 3) NL reweighting [Mirikharaji et al. (2019)], which only reweights the noisy labels, 4) Mix-up [Zhang et al. (2017)], a regularization based method. L2B outperforms others in all evaluation metrics of Dice, Jaccard Index (JI), Hausdorff Distance (HD) and Average Surface Distance (ASD). More detailed analysis could be found in Appendix A.4 3.6 Ablation Study On the importance of $\alpha$, $\beta$. To understand why our proposed new learning objective can outperform previous meta-learning-based instance reweighting methods, we conduct the following analysis to understand the importance of hyper-parameter $\alpha$ and $\beta$ in our method. Specifically, we set $\alpha = 0$ and $\beta = 0$ respectively to investigate the importance of each loss term in Eq. equation 3. In addition, we also show how the restriction of $\alpha_i + \beta_i = 1$ (Eq. equation 2) would deteriorate our model performance as follows. • $\alpha = 0$. As shown in Table 6, in this case, the performance even decreases compared with the baseline approach. This is due to that when only pseudo-labels are included in the loss computation, the error which occurs in the initial pseudo-label will be reinforced by the network during the following iterations. • $\beta = 0$. From Eq. equation 3, we can see that setting $\beta$ as 0 is essentially equivalent to the baseline meta-learning-based instance reweighting method L2RW [Ren et al. (2018)]. In this case, the performance is largely improved compared to the baseline, but still inferior to our method, which jointly optimizes $\alpha$ and $\beta$. • $\alpha + \beta = 1$. We also investigate whether the restriction of $\alpha + \beta = 1$ is required for obtaining optimal weights during the meta-update, as in [Zhang et al. (2020)]. As shown in Table 6, L2B ($\alpha, \beta \geq 0$) consistently achieves superior results than L2B ($\alpha + \beta = 1$) under different noise levels on CIFAR-100. The reason may be the latter is only reweighting different loss terms, whereas the former not only explores the optimal combination between the two loss terms but also jointly adjusts the contribution of different training samples. Parameter normalization We note that the normalization of $\alpha$ and $\beta$ is one key component for accelerating the training process. However, we observe that different normalization methods of $\alpha$ and $\beta$ behave quite differently for different datasets. To further investigate this, we apply the following normalization functions to each $\alpha_i$ and $\beta_i$ on ISIC2019, CIFAR-100, and Clothing 1M: 1) Eq. 9 as in [Ren et al. (2018)], 2) Sigmoid function, $$\alpha_{t,i} = \frac{1}{1 + e^{-\alpha_{r,i}}}, \quad \beta_{t,i} = \frac{1}{1 + e^{-\beta_{r,i}}}.$$ (13) and 3) Softmax function, \[ \alpha_{t,i} = \frac{e^{\alpha_{t,i}/\tau}}{\sum_{i=1}^{n} e^{\alpha_{t,i}/\tau} + e^{\beta_{t,i}/\tau}}, \quad \beta_{t,i} = \frac{e^{\beta_{t,i}/\tau}}{\sum_{i=1}^{n} e^{\alpha_{t,i}/\tau} + e^{\beta_{t,i}/\tau}}, \] where $ t $ stands for the training iteration and $ \tau $ denotes the temperature parameter for scaling the weight distribution. $ \tau $ is set as 10.0 when using the Softmax function for normalization. The comparison among these three different normalization methods is summarized in Figure 2 on ISIC2019 and CIFAR-100 datasets with 40% symmetric noise. We can see that while Eq. 9 achieves the best result on CIFAR-100, it yields large training instability on the ISIC2019 dataset. Changing the normalization function to Sigmoid and Softmax can make the training procedure much more stable on the ISIC2019 dataset. ### Table 5: Performance comparison under noisy-supervision on PROMISE12. \| Method \| Dice (%)† \| JI (%)† \| HD (voxel)‡ \| ASD (voxel)‡ \| \|-------------------------\|-----------\|---------\|-------------\|--------------\| \| UNet+ [Zhou et al., 2018] \| 73.74 \| 58.90 \| 11.63 \| 3.70 \| \| UNet++ meta \| 73.04 \| 58.11 \| 17.06 \| 5.50 \| \| NL reweighting [Mousavi et al., 2019] \| 76.64 \| 62.62 \| 8.33 \| 2.75 \| \| Mix-up [Zhang et al., 2017] \| 69.18 \| 63.78 \| 13.25 \| 4.56 \| \| L2B (Ours) \| 80.83 \| 68.10 \| 6.68 \| 2.10 \| ### Table 6: Ablation of $ \alpha, \beta $. \| Method \| 20% \| 40% \| \|-----------------\|-----\|-----\| \| baseline (CE) \| 59.6\| 49.2\| \| $ \alpha = 0 $ \| 55.7\| 47.1\| \| $ \beta = 0 $ \| 63.2\| 57.5\| \| $ \alpha + \beta = 1 $ \| 64.8\| 59.1\| \| $ \alpha, \beta \geq 0 $ \| 71.8\| 67.3\| ### Table 7: Ablation on size of validation data on CIFAR-10 and CIFAR-100 datasets. \| Validation Size \| CIFAR-10 \| CIFAR-100 \| \|-----------------\|----------\|-----------\| \| \| 20% \| 50% \| 80% \| 90% \| 20% \| 50% \| 80% \| 90% \| \| baseline \| 96.1 \| 94.6 \| 93.2 \| 76.0 \| 77.3 \| 74.6 \| 60.2 \| 31.5 \| \| L2B-DivideMix \| 0 \| 96.3 \| 95.3 \| 93.5 \| 82.6 \| 77.6 \| 75.3 \| 60.8 \| 31.0 \| \| \| 500 \| 96.1 \| 95.3 \| 93.8 \| 91.1 \| 78.2 \| 75.3 \| 62.5 \| 34.0 \| \| \| 1000 \| 96.1 \| 95.4 \| 94.0 \| 91.3 \| 77.9 \| 75.9 \| 62.2 \| 35.8 \| \| L2B-UniCon \| baseline \| 96.0 \| 95.6 \| 93.9 \| 90.8 \| 78.9 \| 77.6 \| 63.9 \| 44.8 \| \| \| 0 \| 96.4 \| 95.6 \| 94.2 \| 92.5 \| 78.7 \| 77.4 \| 68.0 \| 48.6 \| \| \| 500 \| 96.3 \| 95.6 \| 94.5 \| 92.7 \| 78.5 \| 77.5 \| 67.8 \| 51.1 \| \| \| 1000 \| 96.5 \| 95.8 \| 94.7 \| 92.8 \| 78.8 \| 77.3 \| 67.6 \| 49.6 \| \| L2B-C2D \| baseline \| 96.4 \| 95.3 \| 94.4 \| 93.5 \| 78.7 \| 76.4 \| 67.8 \| 58.7 \| \| \| 0 \| 96.4 \| 95.6 \| 94.9 \| 93.7 \| 79.1 \| 77.8 \| 68.5 \| 60.3 \| \| \| 500 \| 96.6 \| 95.5 \| 94.9 \| 94.0 \| 79.5 \| 77.9 \| 69.0 \| 60.8 \| \| \| 1000 \| 96.7 \| 95.6 \| 94.8 \| 94.4 \| 80.1 \| 78.1 \| 69.6 \| 60.7 \| The number of clean validation samples In Table 7, our L2B method is shown to require few to no validation samples for LNL problems, highlighting its practicality. L2B consistently boosts baseline methods such as DivideMix, UniCon, and C2D. Specifically, L2B-DivideMix has showcased its efficacy, particularly at high noise levels. Specifically, in a scenario with 90% noise on CIFAR-10, our approach outstripped the baseline by 8.7%, achieving an accuracy of 82.6% compared to 76.0%, and this was achieved without the need for clean validation samples. The advantage of L2B-DivideMix becomes even more pronounced when we incorporate a minimal amount of clean labels. With just 500 clean labels (equivalent to 2% of the training data), our performance lead over the baseline extends to a remarkable 15.1%. However, as we double the clean samples to 1000, the incremental benefit tapers off, yielding a mere 0.2% boost. This behavior underscores the efficiency of L2B-DivideMix, demonstrating that it can deliver impressive results with minimal or even no clean validation data, making it a highly adaptable and practical solution for real-world applications. ## 4 Conclusion Our paper presents Learning to Bootstrap (L2B), a new technique using joint reweighting for model training. L2B dynamically balances weights between actual labels, pseudo-labels, and different samples, mitigating the challenges of erroneous pseudo-labels. Notably, L2B operates effectively without a clean validation set and can be well generalized to other tasks, highlighting its practicality in real-world settings. Extensive experiments on CIFAR-10, CIFAR-100, ISIC2019, and Clothing 1M datasets demonstrate the superiority and robustness compared to other existing methods under various settings. REFERENCES Görkem Algan and Ilkay Ulusoy. Meta soft label generation for noisy labels. 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hqUznsPMLn	In Section 3.2 you introduce a notion of interestingness, saying that functions R will map all uninteresting samples to the same point. This R notation is then never used, and the notion of interestingness is not further explored. What is meant by interesting, and how do the two representation functions (cosine distance and semantic descriptors) send uninteresting examples to the same point?	ACES: GENERATING DIVERSE PROGRAMMING PUZZLES WITH AUTOTELIC LANGUAGE MODELS AND SEMANTIC DESCRIPTORS Anonymous authors Paper under double-blind review ABSTRACT Finding and selecting new and interesting problems to solve is at the heart of curiosity, science and innovation. We here study automated problem generation in the context of the open-ended space of python programming puzzles. Existing generative models often aim at modeling a reference distribution without any explicit diversity optimization. Other methods explicitly optimizing for diversity do so either in limited hand-coded representation spaces or in uninterpretable learned embedding spaces that may not align with human perceptions of interesting variations. With ACES (Autotelic Code Exploration via Semantic descriptors), we introduce a new autotelic generation method that leverages semantic descriptors produced by a large language model (LLM) to directly optimize for interesting diversity, as well as few-shot-based generation. Each puzzle is labeled along 10 dimensions, each capturing a programming skill required to solve it. ACES generates and pursues novel and feasible goals to explore that abstract semantic space, slowly discovering a diversity of solvable programming puzzles in any given run. Across a set of experiments, we show that ACES discovers a richer diversity of puzzles than existing diversity-maximizing algorithms as measured across a range of diversity metrics. We further study whether and in which conditions this diversity can translate into the successful training of puzzle solving models. 1 INTRODUCTION Finding and selecting new and interesting problems to solve is at the heart of curiosity, science and innovation (Chu & Schulz, 2020; Schmidhuber, 2013; Herrmann et al., 2022). We propose to leverage machine learning, a set of tools usually targeted at solving problems, to automate the generation of an interesting diversity of solvable problems. Automated problem generation has a wide range of applications such as education (generating problems for students to solve), data augmentation (generating problems and solutions for AI model training), or automated scientific discoveries (e.g. discovering new scientific problems and their solutions). In this work, we focus on the generation of a diversity of Python programming puzzles, an open-ended space to explore that contains problems ranging from trivial string manipulations to open mathematical puzzles (Schuster et al., 2021). Importantly, puzzle-solution pairs produced by the search can be checked for correctness using a Python interpreter, providing a notion of ground truth that natural language problems lack (e.g. creative writing). The automated generation of diverse programming puzzles could benefit computer science education and be used as a data generation process for the training of large language models (LLMs). Pretraining on code indeed seems to be a major factor in LLMs’ reasoning abilities (Madaan et al., 2022; Liang et al., 2022; Fu et al., 2022). Standard generative models do not explicitly optimize for diversity but are instead trained to fit the distribution of a reference dataset (e.g. Goodfellow et al., 2014; Brown et al., 2020; Chen et al., 2020; Ho et al., 2020). Measuring and optimizing for diversity requires the definition of a behavioral characterization (BC) of the generated artefacts on which to evaluate the measure. Early diversity-producing methods often used hand-coded low-dimensional representation functions, which focused and restricted the diversity search along features one could easily compute. More recent methods leverage pretrained embedding functions allowing them to work with higher-dimensional data. (e.g., image, text, programs) at the expense of interpretability and control over the axes of variations (Nair et al., 2018; Laversanne-Finot et al., 2018; Cully, 2019; Etcheverry et al., 2020). We propose to leverage semantic descriptors: a hand-defined list of abstract features evaluated by LLMs. Semantic descriptors allow us to work with high-dimensional inputs (here programs) while focusing the diversity search along interpretable semantic features of interest. Specifically, we represent any puzzle by the set of programming skills required to solve it among 10 possible skills (e.g., graphs, dynamic programming, recursion). Evaluating descriptors with LLMs allows us to define more abstract features that better capture our intuitive perception of axes of variation, descriptors that would have been hard or even impossible to code by hand. The compositionality of language and linguistic categories further allow us to easily define sets of orthogonal conceptual categories that can be almost arbitrarily combined (Colas et al., 2020). This work introduces a new diversity-producing algorithm called ACES for Autotelic Code Exploration with Semantic descriptors. ACES leverages an LLM for puzzle generation, solution generation and novelty evaluation. It slowly grows an archive of discovered puzzle-solution pairs, where each cell of the archive contains puzzles that share a given semantic representation—a 10D binary vector obtained by the semantic descriptors. At each new cycle of the algorithm, ACES targets a cell randomly in the archive (semantic goal) and generates a candidate puzzle and solution by prompting the LLM-based puzzle generator with the target semantic representation and a set of examples from the archive. The generated puzzle-solution pair is then evaluated for validity using a Python interpreter and, if valid, gets encoded by the puzzle labeler into a corresponding semantic representation used to store the newly discovered puzzle in the right archive cell, see Figure 1. Our experiments study the evolution of several diversity metrics over time and compares ACES with state-of-the-art baselines (Lehman et al., 2022; Haluptzok et al., 2023). To summarize, our contributions in this paper are the following: • We define the notion of semantic descriptors to leverage LLMs for the encoding of high-dimensional textual data into hard-to-compute, abstract and interpretable features of interest. • We introduce a set of such semantic descriptors to characterize the diversity of programming puzzles based on classical programming ontologies. • We propose Autotelic Code Exploration with Semantic descriptors (ACES), a new diversity-producing method building on these semantic descriptors that leverages the few-shot learning abilities of LLMs to generate an interesting diversity of programming puzzles; • We evaluate the ability of ACES and its baselines to achieve various kinds of diversities and provide a comprehensive analysis of the interactions between diversity and finetuned performance on a held out test set. 2 RELATED WORK Diversity-producing algorithms were originally proposed within the field of evolutionary computing. Beginning with novelty search (Lehman & Stanley, 2011b,a), this line of research expanded with the invention of quality-diversity algorithms (QD: Mouret & Clune, 2015a; Cully & Demiris, 2018a), a set of methods striving to evolve a diverse population of locally-performant individuals via the undirected mutation of existing solutions. A parallel line of research introduced goal-directed exploration processes, also called autotelic learning, where exploring agents learn to represent and sample their own goal as a way to direct the diversity search (Baranes & Oudeyer, 2013; Forestier et al., 2022; Colas et al., 2022). Although autotelic methods were first developed to model the open-ended development of children in skill learning robots Moulin-Frier et al. (2014); Oudeyer & Smith (2016), they have also proved effective in the automatic exploration of complex systems, either simulated (Reinke et al., 2019; Etcheverry et al., 2020) or physical (Grizou et al., 2020). In all these methods, one must define a BC space to characterize novelty. The earliest works used predefined low-dimensional descriptors to represent generated artefacts (Lehman & Stanley, 2011b; Baranes & Oudeyer, 2013; Mouret & Clune, 2015b), which constrains the search along a handful of features one can code a descriptor for. More recent works have relied on higher-dimensional learned or pretrained embedding functions (Nair et al., 2018; Laversanne-Finot et al., 2018; Reinke et al., 2020), and even hierarchies of such spaces, each representing different perceptual features of the generated artefacts (Cully & Demiris, 2018b; Etcheverry et al., 2020). Diversity-search algorithms sometimes need to be adapted to work with such high-dimensional spaces whose discretization leads to an exponential number of cells (Vassiliades et al., 2017). But the main issue is that they are hardly interpretable and might not always align with the dimensions of variation humans find meaningful. With ACES, we propose an autotelic diversity-producing algorithm that constrains the search along a set of abstract, interpretable and hard-to-compute features of interest evaluated by LLMs. This work follows a recent trend on leveraging feedback or generations from LLMs to improve older learning architectures. The original inspirations for this paper come from the QD literature, where LLMs are now used to suggest mutations and crossover in diversity-producing evolutionary algorithms (Lehman et al., 2022; Bradley et al., 2023b; Meyerson et al., 2023). Several approaches also rely on LLMs to replace human feedback (AI feedback): e.g. to finetune other LLM models (Bai et al., 2022; Lee et al., 2023), to characterize generated poems (Bradley et al., 2023a), to revise the policy of autotelic agents (Wang et al., 2023a), to suggest goals for them (Colas et al., 2023; Du et al., 2023) or measure their interestingness (Zhang et al., 2023). With ACES, we use AI feedback to compute abstract and interpretable representations of programming puzzles so as to optimize for diversity in that space. In a similar way, QDAIF (parallel work) uses 2D LLM-generated characterisations in the context of poem generation, a space where there is not such a clear notion of feasibility and solvability (Bradley et al., 2023a). 3 METHODS We first present the programming puzzles (Section 3.1) and discuss measures of interesting diversity (Section 3.2). Then, we introduce ACES, our new method for diversity generation (Section 3.3) and define relevant baselines (Section 3.4). We will open-source the implementation of the algorithms and the datasets of generated puzzles and solutions with the camera-ready version. 3.1 PROGRAMMING PUZZLES AND THE P3 DATASET. The Python Programming Puzzles dataset (P3) contains 1715 puzartefactszle-solution pairs where each puzzle is defined by a short test program $ f $ designed to verify the validity of solution programs $ g $ such that valid solutions satisfy $ f(g()) == True $ when run in an interpreter, see example in Figure 2 (Schuster et al., 2021). P3 puzzles span problems of various difficulties that involve different programming skills: e.g. string manipulation, classic (e.g. Tower of Hanoi) and more complex programming problems (e.g. involving dynamic programming or factoring), or even open problems in computer science or mathematics. The P3 dataset is split into training and testing datasets ($ N = 636 $ and 1079 respectively). Traditionally, a solver model is trained on puzzle-solution pairs from the train set and evaluated on the test set. Both datasets are pre-filtered to examples shorter than 1024 tokens to accommodate for restricted context windows. def f(ls: List[str]): """Divide the decimal representation of 8^88 up into strings of length eight.""" return "".join(ls)==str(888) and all(len(s)==8 for s in ls) def g(): return [str(888)[i:i+8] for i in range(0,80,8)] assert f(g()) == True Figure 2: Example of a simple programming puzzle and its solution from the P3 dataset (Schuster et al., 2021). A solution function $ g $ must return a valid solution such that $ f(g()) == True $. ### 3.2 Measuring Interestingness and Diversity Diversity search aims to generate collections of artefacts that are both diverse and interesting, two subjective measures that strongly depend on the observer’s point of view. Let’s define them. #### Defining Interesting Representation Spaces One can generate interesting diversity by generating it in a representation space where everything is interesting, meaning that all uninteresting samples collapse to small regions. This requires the careful definition of a representation function $ R $ mapping each artefact $ p $ (here puzzle) to a numerical representation $ z_p = R(p) $ and a metric $ m $ to compute distances between them. What should these functions be in the context of our programming puzzles? First, we use cosine distance computed in three different continuous embedding spaces — a standard approach for representing programs and text in general (Reimers & Gurevych, 2019). Here, we use `salesforce/codellp-110m-embedding` (Wang et al., 2023c), `wizardlm/wizardcoder-1b-v1.0` and `wizardlm/wizardcoder-3b-v1.0` (Luo et al., 2023) embedding models from the HuggingFace’s Hub (Wolf et al., 2020) to obtain 256D, 2048D, and 2816D continuous embedding representation vectors. Second, we propose to represent programming puzzles using semantic descriptors — a set of hand-defined features selected from a standard computer science textbook to capture interesting differences in the programming puzzles (Cormen, 2009). We define 10 of these: Sorting and Searching, Counting and Combinatorics, Tree and Graph problems, Mathematical Foundations, Bit Manipulation, String Manipulation, Geometry and Grid Problems, Recursion and Dynamic Programming, Stacks and Queues, Optimization Algorithms, see definitions in Appendix Section A.2. A puzzle is then represented as a 10D binary semantic vector $ z_p $, where each value $ z_i^p $ evaluates whether the puzzle requires skill $ s_i $ (1) or not (0) to be solved. It is unclear how we could write a piece of code to label puzzles along these dimensions. Instead, we ask ChatGPT to assign these labels (version `gpt-3.5-turbo-0613`). In the example of Figure 2, ChatGPT assigns labels for Sorting and Searching, Counting and Combinatorics as well as String Manipulation (encoding 1100010000, see other examples in Appendix Section A.2). We use Hamming distance in that semantic space. The produced diversity will necessarily be shaped and constrained by the subjective choice of the representation function $ R $. A good representation function conflates uninteresting objects in small areas of the representation space (e.g. all puzzles that require no skills are mapped to the 0000000000 cell of our semantic space) and “spreads out” interesting objects. In such a space, most uninteresting puzzles look the same (low inter-puzzle distance) while interesting puzzles look different (high inter-puzzle distance) and optimizing for diversity leads to the generation of diverse interesting puzzles. The experimenter controls specifies semantic descriptors of interests and thus controls the resulting diversity produced. Optimizing for diversity in pretrained embedding spaces is also a subjective choice (different embedding representation will lead to different diversities), but it is made implicitly: the experimenter does not really know what they are signing for. This semantic representation function is a contribution of this paper. We hypothesize that it is more aligned with human perception of programming puzzles than continuous embedding functions. Our proposed algorithms are designed to maximize this form of diversity (see Section 3.3). We hypothesize that this will achieve not only higher diversity scores in this semantic representation space, but also higher scores in the continuous embedding representation spaces. Measuring diversity. We measure the diversity of sets of generated puzzles in different ways. We use counts of discovered puzzles and cells: 1) the number of discovered cells (filled with at least 1 puzzle), 2) the number of cells discovered beyond the ones covered by the train set, 3) the number of valid puzzles that were generated, 4) the number of valid puzzles generated beyond the cells covered by the train set. We also track measures of density or entropy: 5) the average pairwise distance between embedding representations, 6) the entropy of the distribution of semantic representations. An utilitarian take on measuring interesting diversity. Interesting puzzles must be solvable, which is why we filter out invalid puzzle-solution pairs. One could also be interested in training a puzzle solver to achieve high-performance on a specific problem distribution. In this case, we would perceive a collection of generated puzzle-solution pairs as more interesting than others if the solver finetuned on this set outperforms the same solver finetuned on the other sets when tested on the target distribution (e.g. P3’s test set). Section 4.4 will look at correlations between various metrics and the final performance of a LLaMA model (openlm-research/open_llama_3b_v2 on HF’s hub, Geng & Liu, 2023) after finetuning for two epochs on the generated set of puzzle-solutions. In line with previous research (Chen et al., 2021), we will report the Pass@k performance metric on the testing set of P3 for $ k \in [1..10] $: the percentages of puzzles for which at least one valid solution is generated within $ k $ attempts. In addition to the diversity metrics listed above, we will look at various metrics measuring how well the generated set of puzzle-solution pairs covers the testing distribution. 3.3 AUTOTELIC GENERATION OF INTERESTING DIVERSE PUZZLES This section introduces ACES, a new diversity-producing algorithm that generates an interesting diversity of programming puzzles by optimizing for the novelty of each new generation in the semantic space described above, see Figure 1. ACES grows an archive of diverse puzzle-solution pairs. It repeatedly: samples a semantic goal from the archive, generates a new puzzle-solution pair conditioned on that goal and, if the pair is valid, labels and adds it to the archive. We use the ChatGPT LLM (gpt-3.5-turbo-0613, Schulman et al.). Algorithm 1, Figure 1 and Appendix Section A.2 respectively present the pseudo-code, illustration and prompts of ACES. Algorithm 1: Pseudo-code of ACES Initialize an archive $ A $ (with labeled puzzle-solution pairs from the P3 train set) \[ \text{for } i = 1 \text{ to } N \text{ do} \] Sample a goal: $ z_g \sim \text{Uniform}(A) $ (uniform sample of a semantic goal) Sample examples: $ e \sim E(A, z_g) $ (nearest neighbor sampling with Hamming distance) Generate puzzle and solution: $ (p, s) \sim \text{LLM}(\text{prompt}_{\text{gen}}(g, e)) $ (see Appendix A.2) Test puzzle-solution pair: $ \text{pass} = p(s()) == \text{True} $ (using the interpreter) if pass then Label the puzzle $ z_p \sim \text{LLM}(\text{prompt}_{\text{lab}}; p) $ (see Appendix A.2) Add $ (p, s, z_p) $ to the archive $ A $ in cell $ c_{z_p} $ Sampling a goal and relevant examples. ACES selects a semantic goal by sampling uniformly among the set of $ 2^{10} $ possible semantic representations. We then greedily select three closest examples from the archive using the Hamming distance in the semantic space: two from the generated puzzle-solution pairs and one from P3’s train set to always keep a well-formatted puzzle example. Puzzle generator. The puzzle generator is implemented by an LLM. Conditioned on the semantic goal and the three examples, we ask the LLM to generate a puzzle-solution pair that would be labeled with the target semantic vector. For each cycle of the algorithm and to leverage the parallelization of LLM calls, we repeat the process of sampling a goal, examples, puzzles and solutions 10 times before considering the addition of these candidates to the archive. Using a Python interpreter, we filter the set of valid puzzle-solution pairs and send them to the puzzle labeler. Puzzle labeler. The puzzle labeler computes the semantic representation vector of each valid puzzle-solution pair. The prompt details the task to the LLM and presents the complete list of skills, then asks it to compute its semantic representation (see Section 3.2). The puzzle-solution pair is finally added to its corresponding cell in the archive. Note that, although we aim for a particular target semantic representation, whether we achieve the goal or not is not that important. What is important is that the generated puzzle is valid and falls into a new cell. This is the driving principle behind the hindsight learning (Andrychowicz et al., 2017). What’s new? In addition of the semantic descriptors (whose novelty is discussed in Section 3.2), ACES is the first algorithm to leverage a goal-directed LLM for the generation of diverse artefacts via in-context learning. The LLM is here used as the generation engine and is steered towards generating novel and interesting puzzles via its goal-directedness and example selection (in-context learning). The algorithm ELM already used an LLM to suggest mutations of existing artefacts (Lehman et al., 2022). But like other quality-diversity algorithms, it is not goal-directed: it samples a previously-generated artefact from its archive, mutates it with the LLM in the hope of generating a new artefact that would fill a new cell. 3.4 Baselines Static generative model (Static Gen). This baseline was proposed in Haluptzok et al. (2023): it repeatedly prompts the LLM to generate a new puzzle-solution pair given three examples uniformly sampled from P3’s train set. Ablation of goal-directedness (ELM semantic). Instead of sampling a goal and asking the puzzle generator to reach it, we uniformly sample two puzzle-solution pairs from the archive that serve as examples. We then sample a cell in the archive and a puzzle from that cell, and ask the language model to output a mutation of this puzzle. The resulting algorithm is not autotelic anymore but becomes a variant of the QD algorithm Map-Elites (Mouret & Clune, 2015b). In fact, this implementation is a variant of the ELM algorithm (Lehman et al., 2022) where the explored representation space is our proposed semantic space. Ablation of goal-directedness and semantic representations (ELM). We can further ablate ACES by removing the use of the semantic representation space. Instead, this baseline uses the continuous embedding space described in Section 3.2 (Salesforce/codet5p-110m-embedding, Wang et al., 2023c). This ablation is a variant of ELM (Lehman et al., 2022) where the explored representation space is a pretrained embedding space. To define a limited number of cells in this high-dimensional space, we use the method proposed in CVT-Map-Elites, a variant of MapElites that uses centroidal Voronoi tessellations (CVTs) to partition the space into a tractable number of well-distributed cells (Vassiliades et al., 2017). The partition is conducted in two steps. We first sample with replacement 40k puzzles from P3’s train set and perturb their embeddings with a Gaussian noise $N(\mu = 0, \sigma^2 = 1.2)$ before normalizing them to unit-length. Then, we use the K-means algorithm (Steinhaus, 1957; MacQueen, 1967) to identify 1024 centroids and obtain the same number of cells as ACES in the archive. Once this archive is created, we simply run the ELM algorithm. ELM and ELM-semantic share their mutation operator but differ in the way the archive is maintained (CVT archive on continuous embedding features for ELM, archive of semantic cells for ELM-semantic). Generating an interesting diversity of textual artefacts. Although the current paper focuses puzzle generation, ACES can in principle be used to generate an interesting diversity of any type of textual artefacts. For each new type of textual artefacts we want to generate a diversity of, we need to provide a set of features of interest the LLM will be able to evaluate on each new generation. Natural language provides a rich and flexible descriptive space. Compared to traditional representation functions that rely on hand-engineered features, the abstract nature of language allows us to describe the semantic qualities of textual artefacts in an open-ended way. 4 Results 4.1 Is LLM labeling faithful? Our proposition of leveraging LLMs to semantically characterize generated programming puzzles is only meaningful to the extent that the LLM faithfully labels the puzzles. To make sure this is the case on the distribution of puzzles we end up generating, we compare the LLM-generated labels to hand-defined labels on a set of 80 puzzles sampled from the generated set of the seed of ACES. which has the highest label diversity. Details of how the puzzles were sampled for the computation of the confusion matrix can be found in Appendix Section A.3. Figure 3 reports the confusion matrix and the number of puzzles containing the ground truth label for each row. The puzzle labeler demonstrates high true negative rates on most dimensions but sometimes struggles to detect present skills (low true positive rate), e.g., for the Stacks and Queues and the Geometry and Grid dimensions. Note that annotating puzzle with semantic labels is also hard for humans and that the classification does not need to be perfect to drive diversity (see Section 4.2). However, this poses a challenge when using the same labeler for evaluation purposes. We thus choose to report diversity metrics in three different embedding spaces that were not used for training. ![Confusion Matrix](image) Figure 3: Faithfulness of semantic labeling. Confusion matrices for the multi-label classification task performed by the puzzle labeler. For each semantic descriptor, we report the confusion matrix where rows indicate the ground truth presence (1) or absence (0) of the skill while the column indicates its detection (1) or non-detection (0). We thus read from top left to bottom right: true negative, false positive, false negative, true positive rates (sample size in parenthesis). ### 4.2 Measuring Diversity Diversity in semantic space. Figures 4a to 4e report the evolution of various diversity measures as a function of the number of puzzle-solution pairs generated by the puzzle generator. Semantic algorithms (ACES and ELM semantic) better explore the semantic representation space: they discover more cells beyond the cells covered by P3’s train set (4b), more cells in general (4a) and generate more puzzles beyond the cells covered by the train set (4d) and their cell distributions have higher entropy (4e). ELM algorithms generate more valid puzzles in general, but the non-semantic ELM mostly generates puzzles falling in cells covered by the train set (4c vs 4d). Our goal-directed ACES generates puzzles whose cell distribution has higher entropy than other baselines (4e). Algorithms that optimize for diversity in the semantic space were expected to achieve higher diversity in that space, but does it translate to higher diversity in other representation spaces? Diversity in embedding spaces. Figure 5a to 5c report a diversity metric (Friedman & Dieng, 2022) computed over three different embedding spaces (see Section 3.2). ELM demonstrates relatively high diversity in the embedding space it uses for optimization (5a) but lower diversity on other embedding spaces (5b, 5c). ACES achieves the highest diversity in the two WizardCoder embedding spaces (5b, 5c) while ELM semantic reaches the highest diversity in the Code5P embedding space (5a). These results demonstrate that optimizing for diversity in our custom semantic space also leads to higher diversity in other representation spaces. ### 4.3 Qualitative Inspection of Samples We here describe the most remarkable trends that we have observed by manually inspecting the data (see Appendix A.4). One tendency of the generation process across all experiments is to shift the definition of what a puzzle is. In the original formulation, the problem $ f $ implements a test that verifies a certain number of conditions are met, and $ g $ implements an algorithm that produces a value that satisfies the conditions. What the generation processes do in many cases is shift the algorithmic load from $ g $ to $ f $, in which case $ g $ only provides arguments for $ f $. We hypothesise this originally Figure 4: Diversity of generated puzzles in semantic space. We report the evolution of several diversity metrics computed in the semantic space as a function of the number of puzzle-solution pairs generated by the puzzle generator. Semantic algorithms (ACES and ELM semantic) achieve higher diversity in the semantic space. Figure 5: Diversity of generated puzzles in embedding spaces. We report the evolution of the pairwise distance between puzzle-solution pair embeddings as a function of the number of generated puzzle-solution pairs, for three different embedding representation spaces (average across seeds). comes from the docstring description of puzzles in the seed examples, which are included in $\varepsilon$ but describe the behavior of $g$. Another important difference is what the puzzles look like: puzzles generated by the Static Gen baseline tend to be short and more math-heavy, while samples generated by ACES tend to be longer and more creative (see Appendix A.4 again for examples). Appendix Section A.5 provides additional visualizations: 2D projections of the generated puzzles embeddings using UMAP (Appendix Figures 11 to 14), depictions of the evolution of the archives as a function of time (Appendix Figure 10), as well as histograms for the distribution of skills and number of skills across generated puzzles for each of the algorithms (Appendix Figures 7). The UMAP projections, in particular, give a good sense of the difference in distribution between methods. 4.4 Looking for performance predictors Our contributions lead to larger diversities of interesting programming puzzles. Could this translate into higher performance on an arbitrary target distribution we might be interested in? We consider P3’s testing set of puzzles to be our target distribution and measure the Pass@k performance of LLaMA 3B models finetuned on the generated sets of puzzle-solution pairs. Figure 6 shows Pass@k metrics for various values of $k$. Static Gen performs higher despite having the lowest diversity of all algorithms on all considered metrics. Other algorithms spend more time exploring the full-extent of the space of programming puzzles (see Figures 4 and 5) and thus less time focusing on generating puzzles close to the training distribution. Generating more puzzle-solution pairs and a higher diversity of them does not lead to higher performance in our setup. Note that this can be perfectly fine. Our algorithms did not optimize for final performance on an arbitrary target distribution (here P3’s test set) but optimize for pure diversity. This said, we might be interested in figuring out how to constrain diversity-search to maximize performance on a target test distribution down the line. We thus test for correlations between the Pass@10 performance and both diversity metrics and metrics assessing the distance between generated puzzle-solution pairs from P3’s test set (target distribution coverage). Over the test metrics we only found: an anti-correlation with the number of valid puzzles generated, an anti-correlation with the number of cells discovered and an anti-correlation with the average pairwise distance between generated puzzles in the Code5P embedding space, all mostly driven by Static Gen’s low numbers and high performance. The FID score in embedding space (Heusel et al., 2017), number of puzzles in cells covered by the test set, number of test cells discovered or the average distance between test puzzles and their nearest neighbors in the generated sets computing in embedding space all measure how much the generated puzzle-solution pairs cover the test distribution of puzzles but none of them were found to be significantly correlated with the downstream performance metric. 5 DISCUSSION This paper introduced ACES, an autotelic generative algorithm that optimizes for diversity in a custom semantic description space adapted for the generation of programming puzzles. ACES produces stronger diversity in semantic and several embedding spaces, which results in the generation of interesting and creative puzzles as detected by manual inspection (Section 4.3). Our last set of experiments uncovered a counter-intuitive result calling for more research: generating more puzzles of higher diversity does not translate into the higher downstream performance of an LLM finetuned on the generated data as measured over a target distribution of problems (here P3’s test set). Surprisingly, we could not find any significant correlation between downstream performance and metrics measuring how much the generated set covers the target distribution. These results might be explained by more superficial drifts between the generated data and the test data: e.g. by the shift of the computational burden from the solution function to the problem function exposed in Section 4.3. This raises questions for future research: what are causal predictors of final downstream performance? Can we design the goal sampler of our autotelic diversity search to both search for maximal diversity while exploring the space of puzzle that will lead to good downstream performance? Can we characterize the trade-off between diversity and downstream performance? Future work could also improve various aspects of ACES. One could replace the default uniform goal sampling with more sophisticated autotelic sampling methods (e.g. using novelty or learning progress intrinsic motivations, Colas et al., 2022) or improve on the selection of in-context examples to help the puzzle generator. ACES currently explores a fixed set of semantic features and is thus somewhat constrained to combinatorial forms of creativity (Boden, 1998). Moving forward, we could give it the ability to come up with new semantic features or prune others as the exploratory process unfolds, opening the door to more exploratory and transformational forms of creativity or even historical creativity if this system were to interact with human cultural evolution processes, as defined in Boden’s work on human and artificial creativity (Boden, 1996; 1998). Pure diversity-search is useful beyond its data augmentation applications. On the first end, it could be used to generate diverse and interesting problems to educate the new generations of programmers. It could also be combined with autotelic solving systems where it would optimize for both interesting diversity and quality. High-quality problems could for instance be the ones that are intermediately difficult for the learner (Florensa et al., 2018), or those that maximize its learning progress (Oudeyer & Kaplan, 2007). 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DiG14qg4ok	Why the reported experimental results of the compared methods (such as ORTHOG-SUBSPACE Chaudhry et al. (2020)) are different from their papers? It seems the network backbone, datasets, and settings are the same.	LOW-COHERENCE SUBSPACE PROJECTION: ENHANCE THE LEARNING CAPACITY OF ORTHOGONAL PROJECTION METHODS ON LONG TASK SEQUENCES Anonymous authors Paper under double-blind review ABSTRACT Gradient Orthogonal Projection (GOP) is an efficient strategy in continual learning to mitigate catastrophic forgetting. Despite its success so far, GOP-based methods often suffer from the learning capacity degradation problem with an increasing number of tasks. To address this problem, we propose a novel and plug-and-play method to learn new tasks in low-coherence subspaces rather than orthogonal subspaces. Specifically, we construct a unified cost function with the DNN parameters lying on the Oblique manifold. A corresponding gradient descent algorithm is developed to jointly minimize the cost function that involves both inter-task and intra-task coherence. We then provide a theoretical analysis to show the advantages of the proposed in the stability and plasticity. Experimental results show that the proposed method has prominent advantages in maintaining the learning capacity, when the number of tasks increases, especially on a large number of tasks, compared with baselines. 1 INTRODUCTION Although Deep Neural Networks (DNNs) have achieved promising performance in many tasks, their applications are limited for continual learning, suffering from catastrophic forgetting [French (1999)]. When tasks are to be learned sequentially, catastrophic forgetting refers to the phenomenon of new knowledge interfering with old knowledge. Research in continual learning, also known as incremental learning [Aljundi et al. (2018a); Chaudhry et al. (2018a); Chen & Liu (2018); Aljundi et al. (2017)], and sequential learning [Aljundi et al. (2018b); McCloskey & Cohen (1989)], aims to find effective algorithms that enable DNNs to simultaneously achieve plasticity and stability, i.e., to achieve both high learning capacity and high memory capacity. Various methods have been proposed to avoid or mitigate the catastrophic forgetting [De Lange et al. (2019)], either by replaying training samples [Rolnick et al. (2019); Ayub & Wagner (2020); Saha et al. (2021)], or reducing mutual interference of model parameters, features or model architectures between different tasks [Zenke et al. (2017); Mallya & Lazebnik (2018); Wang et al. (2021)]. Among these methods, Gradient Orthogonal Projection (GOP) [Chaudhry et al. (2020); Zeng et al. (2019); Farajtabar et al. (2020); Li et al. (2021)] is an efficient continual learning strategy that advocates projecting gradients with the orthogonal projector to prevent the knowledge interference between tasks. GOP-based methods have achieved encouraging results in mitigating catastrophic forgetting. However, from Fig. 1, we observe that these methods suffer from the learning capacity degradation problem. Namely, their learning capacity is gradually degraded as the number of tasks increases and eventually becomes unlearnable. Specifically, when learning multiple tasks, e.g., more than 30 tasks in Fig. 1, their performance on new tasks dramatically decreases. These results suggest that the GOP-based methods focus on the stability and somewhat ignore the plasticity. Ignoring the plasticity may limit the task-learning capacity of models, i.e., The performance of the model on a new task when learning multiple tasks consecutively. To address this issue, we propose a novel projection-based method, called Low-coherence Subspace Projection (LcSP), which learns new tasks in low-coherence subspaces rather than orthogonal subspaces. Specifically, LcSP utilizes low-coherence projectors at each layer to project both features and gradients into subspaces with low coherence. To achieve this, we construct a unified cost function to find projectors and develop a gradient descent algorithm on the Oblique manifold to jointly minimize inter-task coherence and intra-task coherence among the projectors. Minimizing the inter-task coherence can reduce the mutual interference between tasks while minimizing the intra-task coherence can enhance the model’s expressive power. Restricting projectors on the Oblique manifold can avoid the scale ambiguity [Aharon et al., (2006); Wei et al., (2017)], i.e., preventing the parameters of the projector from being extremely large or extremely small. Moreover, the algorithm we propose for constructing low-coherence projectors is a plug-and-play module. By reusing this module, LcSP can be easily extended to most GOP methods. For example, based on LcSP, we provide the algorithm pseudo-code for continual learning with GPM [Saha et al., (2021)], which can be used in both task-incremental and class-incremental settings, in the Appendix A.2. 2 RELATED WORK In this section, we briefly review some existing works of continual learning and the GOP-based methods. Replay-based Strategy. The basic idea of replay-based approaches is to use limited memory to store small amounts of data (e.g., raw samples) from previous tasks, called episodic memory, and to replay them when training a new task. Some of the existing works focused on selecting a subset of raw samples from the previous tasks [Rolnick et al., (2019); Isele & Cosgun, (2018); Chaudhry et al., (2019); Zhang et al., (2020)]. In contrast, others concentrated on training a generative model to synthesize new data that can substitute for the old data [Shin et al., (2017); Van de Ven & Tolias, (2018); Lavda et al., (2018); Ramapuram et al., (2020)]. Regularization-based Strategy. This strategy prevents catastrophic forgetting by introducing a regularization term in the loss function to penalize the changes in the network parameters. Existing works can be divided into data-focused and prior-focused methods [De Lange et al., (2021)]. The Data-focused methods take the previous model as the teacher and the current model as the student, transferring the knowledge from the teacher model to the student model through knowledge distillation. Typical methods include Lwf [Li & Hoiem, (2017), LFL Jung et al., (2016), EBLL Rannen et al., (2017), DMC Zhang et al., (2020) and GD-WILD Lee et al., (2019)]. The prior-focused methods estimate a distribution over the model parameters, assigning an importance score to each parameter and penalizing the changes in significant parameters during learning. Relevant works include SI Zenke et al., (2017), EWC Kirkpatrick et al., (2017), RWalk Chaudhry et al., (2018a), AGS-CL Jung et al., (2020) and IMM Lee et al., (2017). Parameter Isolation-based Strategy. This strategy considers dynamically modifying the network architecture by pruning, parameter mask, or expansion to greatly or even completely reduce catastrophic forgetting. Existing works can be roughly divided into two categories. One is dedicated to isolating separate sub-networks for each task from a large network through pruning techniques and parameter masks, including PackNet Mallya & Lazebnik, (2018), PathNet Fernando et al., (2017), HAT Serra et al., (2018) and Piggyback Mallya et al., (2018). Another class of methods dynamically expands the network architecture, increasing the number of neurons or sub-network branches, to break the limits of expressive capacity [Rusu et al., (2016); Aljundi et al., (2017); Xu & Zhu, (2018); Rosenfeld & Tsotsos, (2018)]. However, as the number of tasks growing, this approach also complicates the network architecture and increases the computation and memory consumption. Gradient Orthogonal Projection-based Strategy. Methods based on GOP strategies, which reduce catastrophic forgetting by projecting gradient or features with orthogonal projectors, have been shown to be effective in continual learning with encouraging results [Farajtabar et al., (2020); Zeng et al., (2019); Saha et al., (2021); Wang et al., (2021); Chaudhry et al., (2020)]. According to the different ways of finding the projector, we can further divide the existing works into Context Orthogonal Projection (COP) and Subspace Orthogonal Projection (SOP). Methods based on COP, such as OWM Zeng et al., (2019), Adam-NSCL Wang et al., (2021), and GPM Saha et al., (2021), always rely on the context of previous tasks to build projectors. In contrast to COP, SOP-based methods such as ORTHOG-SUBSPACE Chaudhry et al., (2020) use hand-crafted, task-specific orthogonal projectors and yield competitive results. A related work to ours is TRGP Lin et al., (2022), which leverages the parameters of the most relevant old tasks for the new task to enhance forward knowledge propagation. The task-correlation is computed by the norm of gradient projection onto the input subspace of each task. Unlike TRGP, LcSP does not depend on the Single Value Decomposition (SVD) algorithm to obtain the projector. Instead, LcSP derives the projector by minimizing the task coherence on the Oblique Manifold, where task coherence is the measure of alignment between projectors. Our experiments show that LcSP surpasses TRGP on such as Split CIFAR100 and miniImageNet benchmarks. Moreover, LcSP has low computational overhead and is faster than TRGP. 3 Continual Learning Setup In continual learning, the learner needs to learn multiple tasks sequentially. Let us assume that there are $T$ tasks, denoted by $\mathcal{T}_t$ for $t = 1, \ldots, T$ with its training data $\mathcal{D}_t = \{(x_i, y_i, \tau_t)\}_{i=1}^{N_t}$. Here, the data $(x_i, y_i) \in \mathcal{X} \times \mathcal{Y}$ is assumed to be drawn from some independently and identically distributed random variables, and $\tau_t \in \mathcal{T}$ denotes the task identifier. In the TIL setting, the data $\mathcal{D}_t$ can be accessed if and only if task $\mathcal{T}_t$ arrives. When episodic memory is adopted, a limited number of data samples drawn from old tasks can be stored in the replay buffer $\mathcal{M}$ so that $\mathcal{D}_t \cup \mathcal{M}$ can be used for training when task $\mathcal{T}_t$ arrives. Assuming that a network $f$ parameterized with $\Phi = \{\theta, \varphi\}$ consists of two parts, where $\theta$ denotes the parameters of the backbone network and $\varphi$ denotes the parameters of the classifier. Let $f(x; \theta) : \mathcal{X} \times \mathcal{T} \rightarrow \mathcal{H}$ denote the backbone network parameterized with $\theta = \{W_l\}_{l=1}^L$, which encodes the data samples $x$ into feature vector. Let $f(x; \varphi) : \mathcal{H} \rightarrow \mathcal{Y}$ denote the classifier parameterized with $\varphi = w$ which returns the classification result of the feature vector obtained by $f(x; \theta)$. The goal of TIL is to learn $T$ tasks sequentially with the network $f$ and finally achieve the optimal loss on all tasks. Evaluation Metrics Once the training on all tasks is finished, we evaluate the performance of algorithm by calculating the average accuracy $\mathcal{A}$ and forgetting $\mathcal{F}$ [Chaudhry et al., 2020] of the network on the $T$ tasks $\{\mathcal{T}_1, ..., \mathcal{T}_T\}$. Suppose all tasks come sequentially, let $\text{Acc}_{i,j}$ denote the test accuracy of the network on task $\mathcal{T}_j$ after learning task $\mathcal{T}_i$, where $i \leq j$. The average accuracy is defined as $$\mathcal{A} = \frac{1}{T} \sum_{i=1}^{T} \text{Acc}_{i,T},$$ and the forgetting is defined as $$\mathcal{F} = \frac{1}{T-1} \sum_{i=1}^{T-1} \max_{j \in \{i, ..., T-1\}} (\text{Acc}_{i,j} - \text{Acc}_{i,T}).$$ 4 Continual Learning in Low-coherence Subspaces In this section, we describe the details of the LcSP algorithm based on hierarchical projection. In addition, LcSP can also be extended to more GOP methods. In the Appendix A.2, we provide the LcSP algorithm based on GPM, which can be used in class-incremental setting. In the following, we begin by introducing how to find task-specific, low-coherence projectors for LcSP on the Oblique manifold. We then describe how to use it in a specific DNN architecture to project features and gradients. Finally, we analyze the factors that enable LcSP to maintain plasticity and stability. 4.1 Preliminary Since our proposed algorithm involves knowledge related to optimization on oblique manifold, we first introduce the related mathematical definitions and concepts here to help readers better understand the context. Optimization on the Oblique manifold, i.e., the solution lies on the Oblique manifold, is a well-established area of research [Absil et al., 2009; Absil & Gallivan, 2006; Selvan et al., 2012]. Here, we briefly summarize the main steps of the optimization process. We refer readers who are interested in the relevant content to [Absil et al., 2009] for more details. Formally, the Oblique manifold $\mathcal{OM}(n,p)$ is defined as $$\mathcal{OM}(n,p) \triangleq \{ X \in \mathbb{R}^{n \times p} : \text{diag}(X^\top X) = I_p \},$$ representing the set of all $n \times p$ matrices with normalized columns. $\mathcal{OM}$ can also be considered as an embedded Riemannian manifold of $\mathbb{R}^{n \times p}$, endowed with the canonical inner product $$\langle X_1, X_2 \rangle = \text{trace}(X_1^\top X_2),$$ where $\text{trace}(\cdot)$ represents the sum of the diagonal elements of the given matrix. For a given point $X$ on $\mathcal{OM}$, the tangent space at $X$, denoted by $T_X \mathcal{OM}$, is defined as $$T_X \mathcal{OM}(n,p) = \{ U \in \mathbb{R}^{n \times p} : \text{diag}(X^\top U) = 0 \}.$$ Further, the tangent space projector $P_X$ at $X$ which projects $H \in \mathbb{R}^{n \times p}$ into $T_X \mathcal{OM}$, is represented as $$P_X(H) = H - X \text{ddiag}(X^\top H),$$ where $\text{ddiag}$ sets all off-diagonal entries of a matrix to zero. When optimizing on $\mathcal{OM}$, the $k$th iteration $X_k$ must move along a descent curve on $\mathcal{OM}$ for the cost function, such that the next iteration $X_{k+1}$ will be fixed on the manifold. This is achieved by the retraction $$R_{X_k}(U) = \text{normalize}(X_k + U),$$ where $\text{normalize}$ scales each column of the input matrix to have unit length. Finally, with this knowledge, we can extend the gradient descent algorithm to solve any unconstrained optimization problems on $\mathcal{OM}$, which can be summarized as $$U = P_{X_k}(\nabla_{X_k} J),$$ $$X_{k+1} = R_{X_k}(-\alpha U),$$ where $J$ denotes the cost function and $\nabla_{X_k} J$ denotes the Euclidean gradient at the $k$th iteration and $\alpha$ is the step size. ### 4.2 Constructing Low-coherence Projectors on Oblique Manifold In the following, we first introduce the concept of coherence metric. The coherence metric is usually used in compressed sensing and sparse signal recovery to describe the correlation of the columns of a measurement matrix [Candes et al., 2011; Candes & Romberg, 2007]. Formally, the coherence of a matrix $M$ is defined as $$\mu(M, N) = \begin{cases} \max_{j < k} \frac{\|\langle M_j, M_k \rangle\|}{\\|M_j\\|_2 \\|M_k\\|_2}, & M = N; \\ \max_{i,j} \frac{\|\langle M_i, N_j \rangle\|}{\\|M_i\\|_2 \\|N_j\\|_2}, & M \neq N, \end{cases}$$ where $M_j$ and $M_k$ denote the column vectors of matrix $M$. Without causing confusion, we use $\mu(M)$ to denote $\mu(M, M)$. To measure the coherence between different projectors, we introduce the Babel function [Li & Lin, 2018], measuring the maximum total coherence between a fixed atom and a collection of other atoms in a dictionary, which can be described as follows. $$B(P, M) = \max_{i \in M} \sum_{j \in P} \frac{\|\langle M_i, P_j \rangle\|}{\\|M_i\\| \\|P_j\\|}$$ where $M$ denotes a fixed atom and $P$ denotes target projector. With the concept of a coherence metric in mind, we then introduce the main optimization objective in finding projectors. Specifically, suppose that the DNN has learned the task $T_1, T_2, ..., T_{t-1}$ in the subspace $S_1, S_2, ..., S_{t-1}$, respectively, $P_1, P_2, ..., P_{t-1}$ denote the projectors of all previous tasks. When learning task $T_t$, we project features and gradients into a $d_t$-dimensional low-coherence subspace $S_t$ with projector $P_t$ so that the LcSP can prevent catastrophic forgetting. The projector $P_t$ can be found by optimizing $$\arg \min_M B(P_t, M),$$ subject to $$P_t \in \mathbb{R}^{m \times m}, \quad \text{rank}(P_t) = d_t.$$ Here $M = \{P_1, ..., P_{t-1}\}$ denotes the collection of projectors of previous tasks. Two considerations need to be taken in solving Eq. (11), i.e., considering the rank constraint and the column vector’s scale (L2 norm) constraint. Empirically, extremely large or small length (L2 norm) of the projected column vector can lead to unstable training, as shown in Appendix A.1. We constrain the length of the projected column vectors to be equal to 1 because when the projection matrix $ P_t $ satisfies the constraints and is orthogonal, the length of the gradient does not change after it has been projected, and thus does not affect the convergence rate. Therefore, we rephrase the rank and scale constrained problem as a problem on the Oblique manifold $ OM(m, d_t) $, by setting $ P_t = O_t O_t^\top, O_t \in \mathbb{R}^{m \times d_t} $, and normalizing the columns of $ O_t $, i.e., $ \text{diag}(O_t^\top O_t) = I_n $, where $ \text{diag}(\cdot) $ represents the diagonal matrix and $ I_n $ is the $ n \times n $ identity matrix. With these settings, the new cost function $ J(\cdot) $ and the optimization problem can be described as follows: \[ J(O_t, M) = \begin{cases} \lambda \cdot B(O_t O_t^\top, M) + \gamma \cdot \mu(O_t O_t^\top), & t > 1 \\ \mu(O_t O_t^\top), & t = 1 \end{cases} \] \[ O_t = \arg \min J(O_t, M), \quad \text{s.t.} \quad O_t \in OB(m, d_t). \] In the cost function $ J(O_t, M) $, we define an inter-task optimization objective $ B(O_t O_t^\top, M) $, which measures the coherence between the current task projector $ P_t $ and the previous task projectors $ P_i $ ($ i < t $). Following the intuition of the GOP method, we minimize $ B(O_t O_t^\top, M) $ to reduce the interference between tasks, thereby overcoming catastrophic forgetting. In contrast, we define an intra-task optimization objective $ \mu(O_t O_t^\top) $, which measures the coherence of the current task projector $ P_t $ itself. We observe that using a projector with a particularly low rank means selecting a small portion of parameters to learn new tasks and will severely limit the model’s plasticity, resulting in poor performance on new tasks. Therefore, we make $ O_t $ full-rank by minimizing $ \mu(O_t O_t^\top) $, in order to maintain the model’s learning ability for new tasks. For the first task, since there is no interference from other tasks, we only need to focus on $ \mu(O_t O_t^\top) $. When learning the $ t $-th task ($ t > 1 $), we consider both $ B(O_t O_t^\top, M) $ and $ \mu(O_t O_t^\top) $, and to cope with different scenarios more flexibly, we utilize parameters $ \gamma $ and $ \lambda $ to provide a trade-off between them. In the Appendix A.1 we provide relevant ablation experiments and numerical analysis and summarize our algorithm for finding $ O_t $ in $ OM $ for task $ T_t $. ### 4.3 Application of Low-coherence Projectors in DNNs With the LcSP at hand, the following introduces some technical details of applying LcSP in DNNs. When learning task $ T_t $, LcSP first constructs task-specific projector $ P_t^l $ for each layer before training, and freezes them during training. These projectors are used to project the features and gradients, ensuring that the DNN learns in the low-coherence subspace. Specifically, suppose that a network $ f $ with $ L $ linear layers admits a DNN architecture, let $ W_t^l, x_t^l, z_t^l, \sigma^l, $ and $ P_t^l $ denote the model parameters, the input features, the output features, the activation function, and the introduced low-coherence projector in layer $ l \in \{1, ..., L\} $, respectively. LcSP introduces $ P_t^l $ immediately after $ W_t^l $ such that the pre-activation features are projected into the subspace, i.e., \[ z_t^l = (x_t^l W_t^l) P_t^l, \] \[ x_t^{l+1} = \sigma(z_t^l). \] According to the chain rule of derivation, the gradients at $ W_t^l $ will also be multiplied with $ P_t^l $ in backpropagation, as follows \[ \frac{\partial L}{\partial(W_t^l)(i,:)} = \frac{\partial L}{\partial z_t^l} \frac{\partial z_t^l}{\partial(W_t^l)(i,:)} \] \[ = \frac{\partial L}{z_t^l} \prod_{k=l}^{L-1} \frac{\partial z_t^{k+1}}{\partial z_t^k} \cdot (x_t^l)_i \cdot P_t^l, \] where $ (W_t^l)(i,:) $ represents the $ i $th row of $ W_t^l $ and $ (x_t^l)_i $ is the $ i $th element of $ x_t^l $. In Convolutional Neural Networks (CNNs), the input and the output typically represent the image features and have more than two dimensions, e.g., input channel, output channel, height, and width. In this case, we reshape $ z_t^l \in \mathbb{R}^{c_{\text{out}} \times (c_{\text{in}} \cdot h \cdot w)} $ to $ z_t^l \in \mathbb{R}^{(c_{\text{in}} \cdot h \cdot w) \times c_{\text{out}}} $ and align the dimension of projector with the output channel so that $ P_t^l \in \mathbb{R}^{c_{\text{out}} \times c_{\text{out}}} $. After the projection, we recover the shape of $ z_t^l $ so that it can be used as input for the next layer. 4.4 Method Analysis In this section, we provide an analysis on the plasticity and the stability of LcSP. Stability Analysis. Let $\theta = \{W_l^t\}_{l=1}^L$ denote the parameter set of $f$; $\Delta \theta = \{\Delta W_1^t, \ldots, \Delta W_L^t\}$ denote set of variation values of parameters after learning task $T_t$; $P_t = \{P_l^t\}_{l=1}^L$ denote the projectors set obtained by LcSP; $x_{q,t}^l$ and $z_{q,t}^l$ denote the input and output when feeding the data of task $T_q$ ($q \leq t$) into the network $f$, which has been optimized in learning task $T_t$. Lemma 1. Assume that $f$ is fed the data of task $T_t$ ($q < t$), then $f$ can effectively overcome catastrophic forgetting if $$z_{q,q}^l \approx z_{q,t}^l, \quad \forall q \leq t$$ (15) holds for $l \in \{1, 2, ..., L\}$. Lemma 1 suggests that $f$ can overcome catastrophic forgetting if the output of $f$ to previous tasks is invariant. In the following, we prove that LcSP achieves approximate invariance to the output of previous tasks. Proof. Suppose $q = t - 1$. When $l = 1$, $x_{q,t}^l = x_{q,q}^l$. Then $$z_{q,t}^l = x_{q,t}^l(W_q^l + \Delta W_q^t)P_q^l$$ $$= x_{q,t}^lW_q^lP_q^l + x_{q,t}^l\Delta W_q^tP_q^l$$ $$= z_{q,q}^l + x_{q,t}^l\Delta W_q^tP_q^l.$$ (16) Let $g_l^t$ denote the gradient when training the network on task $T_t$. In backpropagation, $\Delta W_q^t = g_l^tP_l^t$. Then $x_{q,t}^l\Delta W_q^tP_q^l = x_{q,t}^l g_l^t P_l^t P_q^l$. If the inter-task coherence $\mu(P_l^t, P_q^l) \approx 0$, then $P_l^t P_q^l \approx 0$. Projectors satisfying this condition can be found by LcSP. We can prove that $z_{q,q}^l \approx z_{q,t}^l$ holds for all layers by repeating the above process. This proof can also be generalized to any previous task $T_q$. Plasticity Analysis. Let $\hat{g}_l^t = g_l^t P_l^t$ denote the projected gradient at $W_l^t$. $f$ can achieve optimal loss on task $T_t$ if $\langle g_l^t, \hat{g}_l^t \rangle > 0$ holds for each $l \in \{1, \ldots, L\}$, where $\langle \cdot, \cdot \rangle$ represents the inner product. Here, we prove that $\langle g_l^t, \hat{g}_l^t \rangle > 0$ holds for each $l \in \{1, \ldots, L\}$. Proof. Let $\hat{g}_l^t = g_l^t P_l^t$ denote the projected gradient, we have $$\langle g_l^t, \hat{g}_l^t \rangle = g_l^t \hat{g}_l^{t\top} = g_l^t O_l^t (O_l^t)^{\top} g_l^t$$ $$= \\|g_l^t O_l^t\\|^2 > 0.$$ (17) Note that $\\|g_l^t O_l^t\\|$ is always positive unless $g_l^t O_l^t$ is 0. This result is easy to generalize to each layer. 5 Experiments In this section, we evaluate our approach on several popular continual learning benchmarks and compare LcSP with previous state-of-the-art methods. The result of accuracy and forgetting demonstrate the effectiveness of our LcSP, especially when the number of tasks is large. 5.1 Benchmarks We evaluate the effectiveness of our algorithms in several widely used continuous learning datasets: Permuted MNIST, Rotated MNIST, Split CIFAR100, and Split miniImageNet. The Permuted MNIST dataset is derived from MNIST [LeCun (1998)] by randomly permuting the image pixels with different seeds for different tasks. The Rotated MNIST dataset is another variation of MNIST that rotates the images by a random angle between $[0, \pi]$ for each task. For both Permuted MNIST and Rotated MNIST, we generate 10 sequential tasks with 10 classes each. The Split CIFAR100 dataset is obtained by dividing CIFAR100 into 20 tasks, where each task contains five randomly selected classes (without replacement) from the total of 100 classes. The Split miniImageNet dataset, used in [Chaudhry et al. Table 1: The average accuracy and forgetting results of the proposed LcSP and baselines. \| Methods \| Permutated MNIST \| Rotated MNIST \| Split CIFAR100 \| Split miniImageNet \| \|---------\|------------------\|---------------\|----------------\|--------------------\| \| \| A(%) F \| A(%) F \| A(%) F \| A(%) F \| \| EWC \| 89.97 0.04 \| 92.68 0.03 \| 68.80 0.02 \| 52.01 0.12 \| \| A-GEM \| 83.56 0.14 \| 93.36 0.02 \| 63.98 0.15 \| 57.24 0.12 \| \| ER-Res \| 87.24 0.11 \| 94.16 0.01 \| 71.73 0.06 \| 58.94 0.07 \| \| HAT \| - \| - \| 72.06 0.00 \| 59.78 0.03 \| \| OWM \| 90.71 0.01 \| 93.35 0.01 \| 50.94 0.39 \| - \| \| GPM \| 93.91 0.03 \| 95.22 0.01 \| 72.48 0.00 \| 60.41 0.00 \| \| Adam-NSCL \| Wang et al. (2021) \| - \| 75.95 0.04 \| 63.27 0.06 \| \| TRGP \| 96.34 0.01 \| 96.79 0.01 \| 74.46 0.01 \| 61.78 0.01 \| \| ORTHOG-SUBSPACE \| Chaudhry et al. (2020) \| - \| 64.30 0.07 \| 51.40 0.10 \| \| LcSP (ours) \| 95.16 0.02 \| 96.12 0.01 \| 76.47 0.00 \| 67.90 0.00 \| Table 2: Total training time measured on a single GPU after learning all the tasks. The training time is normalized with respect to the value of GPM. We refer Saha et al. (2021) for a specific value. \| Methods \| Permutated MNIST \| Split CIFAR100 \| Split miniImageNet \| \|---------\|------------------\|----------------\|--------------------\| \| \| Training Time [s] \| \| \| \| EWC \| 2.63 \| 1.76 \| 1.22 \| \| A-GEM \| 1.82 \| 3.48 \| 2.19 \| \| ER-Res \| 1.06 \| 1.49 \| 0.82 \| \| HAT \| - \| 1.62 \| 0.90 \| \| OWM \| 6.77 \| 2.41 \| - \| \| Adam-NSCL \| Wang et al. (2021) \| - \| 1.20 \| 1.51 \| \| ORTHOG-SUBSPACE \| Chaudhry et al. (2020) \| 1.72 \| 1.90 \| 3.69 \| \| TRGP \| 3.03 \| 3.36 \| 4.39 \| \| GPM \| 1.00 \| 1.00 \| 1.00 \| \| LcSP (ours) \| 0.90 \| 0.71 \| 0.95 \| (2018b), is created by splitting 100 classes of miniImageNet into 20 sequential tasks with 5 classes each. In addition, we conducted fair comparison experiments using the same settings on the longer Permutated MNIST (containing 150 tasks) and Permutated CIFAR10 (containing 64 tasks). Compared to baselines, we verified that LcSP is better able to maintain the learning capability on long task sequences. 5.2 Baselines We compare the proposed method with several state-of-the-art approaches that consider sequential task learning in fixed network architecture. These approaches include GOP-based methods, such as Orthogonal Weight Modulation (OWM) Zeng et al. (2019), Adam-NSCL Wang et al. (2021), Gradient Projection Memory (GPM) Saha et al. (2021), Trust Region Gradient Projection (TRGP) Lin et al. (2022) and ORTHOG-SUBSPACE Chaudhry et al. (2020), regularization-based methods, such as HAT Serra et al. (2018) and Elastic Weight Consolidation (EWC) Kirkpatrick et al. (2017), and replay-based methods, such as Experience Replay with reservoir sampling (ER-Res) Chaudhry et al. (2019) and Averaged GEM (A-GEM) Chaudhry et al. (2018b). 5.3 Implementation Details For experiments on Permutated MNIST, we use a fully connected network with two hidden layers, each of which is with 256 neurons, by utilizing ReLU activations. Consistent with GPM, we use a 5-layer AlexNet Krizhevsky et al. (2012) for experiments on CIFAR100 and a standard ResNet18 for experiments on miniImageNet. For experiments on MNIST, all tasks share the same classifier. For experiments on CIFAR and miniImageNet, each task requires a task-specific classifier. For all experiments, LcSP does not use episodic memory to store data samples for data replay. For all methods, we use Stochastic Gradient Descent (SGD) uniformly. The learning rate is set to 0.01 for experiments on MNIST and 0.003 for experiments on CIFAR and ImageNet. Both $\lambda$ and $\gamma$ in Eq. (12) are set to 1. All experiments were run five times with five different random seeds. 5.4 Main Results Permutated MNIST and Rotated MNIST. In this experimental setup, a single-head classifier is employed for all tasks. HAT and Adam-NSCL are excluded from the comparison as require a Figure 1: (a) and (b) show the average accuracy and forgetting of the last 10 tasks on Permutated MNIST when learning 150 tasks. (c) and (d) show the average accuracy and forgetting of the last 5 tasks on Permutated CIFAR10 when learning 64 tasks. Figure 2: The accuracy of the last task on Permutated MNIST (left) and Permutated CIFAR10 (right), respectively. separate classifier for each task. Tab.3 presents that LcSP obtained competitive results on MNIST. LcSP outperformed other baselines while was slightly inferior to TRGP, with average accuracies of 95.16% and 96.12%, respectively. We found that LcSP outperformed other methods mainly due to its hierarchical projection mechanism, which effectively minimized task interference. Moreover, by keeping the projectors with low coherence, LcSP could utilize the network capacity more efficiently. However, LcSP also had some limitations. It did not apply projection to the classification layer, which resulted in more severe forgetting on this layer compared to TRGP. Although LcSP was exceeded by TRGP in average accuracy, it had a considerably shorter training time than TRGP, as shown in Tab.2, which made LcSP more efficient and practical in comparison. Split CIFAR100. In this experiment, we adopted the multi-head setup, which enabled us to compare with HAT and Adam-NSCL. As shown in Tab.3 LcSP outperformed all baselines, achieving an average accuracy of 76.47%, exceeding the baselines by 23.53% ~ 0.52%, and marginally surpassing Adam-NSCL. Moreover, our results showed that LcSP achieved zero forgetting. This was explained by two factors. First, by using the distinct classification heads for each task, LcSP avoided forgetting on the classification layer. Second, due to the ample network capacity to accommodate all tasks, LcSP did not need to compromise some stability for plasticity. Split miniImageNet. In this experiment, we assessed the effectiveness of our algorithm on a deeper network (standard ResNet18). Tab. 3 shows that our method achieved a remarkable improvement in average accuracy over the baseline methods, from 15.89% to 4.53%. This indicates that LcSP has good scalability on deep neural networks and can be applied to large datasets and more complex tasks. Comparisons of Learning 150 Tasks and 64 Tasks. To demonstrate the promising advantage of the proposed methods in learning a long sequence of tasks, the following experiments compare the results with 64 tasks and 150 tasks. Note that, in Fig. 1, LcSP (orthogonal) is a variant that uses orthogonal projectors, while LcSP (low-coherence) uses low-coherence projectors. Figs. 1(a) and 1(b) report the average accuracy and forgetting of the last 10 tasks, with learning 150 tasks on Permutated MNIST. Figs. 1(c) and 1(d) report the average accuracy and forgetting of the last 5 tasks with learning 64 tasks on Permutated CIFAR10. The average accuracy of all methods, except LcSP (low-coherence), dramatically declines or remains low as the number of tasks increases. Furthermore, it can be seen from Fig. 1(d) that all methods except ORTHOG-SUBSPACE have almost no forgetting. This result indicates that methods using orthogonal projectors gradually lose their learning capacity with the increasing number of tasks. The proposed method uses the low-coherence projector to relax the orthogonal constraint, effectively solving this problem. Efficiency Analysis. To evaluate the practicality of the LcSP, we measured the total training time of all algorithms on a single GPU, normalized by the time of GPM. As shown in Tab. 2, the proposed LcSP trains faster than all baselines on MNIST and Split miniImageNet, and slightly slower than HAT and ER-Res on Split miniImageNet. The main reasons why LcSP training is faster than other GOP-based baselines are as follows. Firstly, LcSP uses dimensionally aligned projectors to project features, which is faster than manual projection of gradients (e.g., Adam-NSCL, GPM, and TRGP) during backpropagation. Secondly, LcSP trains the network parameters directly in Euclidean space, whereas ORTHOG-SUBSPACE trains the network parameters on Stiefel manifolds. This makes LcSP faster than ORTHOG-SUBSPACE, especially on deep neural networks. Thirdly, in contrast to TRGP which uses the post-projection gradient length to calculate task correlation and trust region projections, LcSP only needs to calculate the coherence of the projectors to evaluate task coherence. Since the number of parameters of the projectors is much smaller than the number of parameters of the model, LcSP can be trained faster than TRGP. ### 5.5 Ablation Studies Learning Capacity Degradation in Gradient Orthogonal Projection. To further investigate the learning capacity degradation problem, we give the accuracy of baselines for the last task on Permutated MNIST and Permutated CIFAR10. As shown in Fig. 2, all baselines, except LcSP, suffer from this problem with different degrees and show some decrease in accuracy compared to the initial (66.16% ~ 24.63% on Permutated MNIST and 24.8% ~ 3.48% on Permutated CIFAR10). These results suggest that learning capacity degradation is the critical factor that results in degrading the performance of GOP-based methods in the case of a large number of tasks. ### 6 Conclusion This paper experimentally observes that GOP methods in continual learning suffer from the learning capacity degradation problem. Specifically, the performance of the GOP methods on new tasks gradually decreases as the number of tasks increases. This paper proposed a novel method, namely LcSP, to address this problem. Instead of learning in orthogonal subspace, LcSP projects features and gradients via low-coherence projectors to minimize inter-task and intra-task coherence. Extensive experiments show that our approach works well in alleviating forgetting and has a significant advantage in maintaining learning capacity, especially in learning long-sequence tasks. 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030cjlZm4a	It is shown that the use of the fairness regularizer works in order to minimize both FNR and FPR, but it is not discussed whether or not the constraint impacts the performances of the checklist, so there is no way to truly understand if its usage is really beneficial.	LEARNING PREDICTIVE CHECKLISTS WITH PROBABILISTIC LOGIC PROGRAMMING Anonymous authors Paper under double-blind review ABSTRACT Checklists have been widely recognized as effective tools for completing complex tasks in a systematic manner. Although originally intended for use in procedural tasks, their interpretability and ease of use have led to their adoption for predictive tasks as well, including in clinical settings. However, designing checklists can be challenging, often requiring expert knowledge and manual rule design based on available data. Recent work has attempted to address this issue by using machine learning to automatically generate predictive checklists from data, although these approaches have been limited to Boolean data. We propose a novel method for learning predictive checklists from diverse data modalities, such as images and time series, by combining the power of dedicated deep learning architectures with the interpretability and conciseness of checklists. Our approach relies on probabilistic logic programming, a learning paradigm that enables matching the discrete nature of checklist with continuous-valued data. We propose a regularization technique to tradeoff between the information captured in discrete concepts of continuous data and permit a tunable level of interpretability for the learned checklist concepts. We demonstrate that our method outperforms various explainable machine learning techniques on prediction tasks involving image sequences, medical time series, and clinical notes. 1 INTRODUCTION In recent years, machine learning models have gained popularity in the healthcare domain due to their impressive performance in various medical tasks, including diagnosis from medical images and early prediction of sepsis from clinical time series, among others (Davenport & Kalakota [2019], Esteva et al. [2019]). Despite the proliferation of these models in the literature, their wide adoption in real-world clinical practice remains challenging (Futoma et al. [2020], Ahmad et al. [2018], Ghassemi et al. [2020], De Brouwer et al. [2022]). Ensuring the level of robustness required for healthcare applications is difficult for deep learning models due to their inherent black box nature. Non-interpretable models make stress testing arduous and thus undermine the confidence required to deploy them in critical applications such as clinical practice. To address this issue, recent works have focused on developing novel architectures that are both human-interpretable and retain the high performance of black box models (Ahmad et al. [2018]). One such approach is learning medical checklists from available medical records. Due to their simplicity and ability to assist clinicians in complex situations, checklists have become increasingly popular in medical practice (Haynes et al. [2009]). However, the simplicity of using checklists typically contrasts with the complexity of their design. Creating a performant checklist requires domain experts who manually collect evidence about the particular clinical problem of interest (Hales et al. [2008]), and subsequently reach consensus on meaningful checklist rules (Hales et al. [2008]). As the number of available medical records grows, the manual collection of evidence becomes more tedious, bringing the need for partially automated design of medical checklists. Recent works have taken a step in that direction by learning predictive checklists from Boolean, categorical, or continuous tabular data (Zhang et al. [2021], Makhija et al. [2022]). Nevertheless, many available clinical data, such as images or time series, are nor categorical nor tabular by nature. They therefore fall outside the limits of applicability of previous approaches for learning checklists from data. This work aims at addressing this limitation. Prior work leverages integer programming to generate checklists, but the discrete (combinatorial) nature of solving integer programs makes it challenging to learn predictive checklists from images or time series data. Deep learning architectures rely on gradient-based optimization which differs in style and is difficult to reconcile with integer programming (Shvo et al., 2021). We instead propose to formulate predictive checklists within the framework of probabilistic logic programming. This enables us to extract binary concepts from high-dimensional modalities like images, time series, and text data according to a probabilistic checklist objective, while propagating derivatives throughout the entire neural network architecture. Unlike existing approaches, ProbChecklist doesn’t rely on fixed summary extractors such as mean or standard deviation of time series; instead, it learns concepts using neural networks (concept learners). Our architecture, ProbChecklist, operates by creating binary concepts from high-dimensional inputs, which are then used for evaluating the checklist. However, they are learnt with deep neural networks and are not necessarily interpretable. We therefore investigate two different strategies for providing predictive yet interpretable concepts. The first relies on using inherently interpretable concept extractors, which only focus on specific aspects of the input data (Johnson et al., 2022). The second adds regularization penalties to enforce interpretability in the neural network by design. Several regularization terms have been coined to ensure the concepts are unique, generalizable, and correspond to distinctive input features (Jeffares et al., 2023) (Zhang et al., 2018). Clinical practice is a highly stressful environment where complex decisions with far-reaching consequences have to be made quickly. In this context, the simplicity, robustness, and effectiveness of checklists can make a difference (Hales et al., 2007). Healthcare datasets contain sensitive patient information, including ethnicity and gender, which should not cause substantial differences in the treatment provided. Nevertheless, machine learning models trained on clinical data have been shown to exhibit unacceptable imbalance of performance for different population groups, resulting in biased predictions. When allocating scarce medical resources, fairness should be emphasized more than accuracy to avoid targeting minority subgroups (Fawzy et al., 2022). In an attempt to mitigate this problem, we study the impact of including a fairness regularization into our architecture and report significant reductions in the performance gap across sensitive populations. We validate our approach empirically on several classification tasks using various data modalities such as images and clinical time series. We show that ProbChecklist outperforms previous learnable predictive checklist approaches as well as several interpretable machine learning baselines. We showcase the capabilities of our method on two healthcare case studies, learning interpretable checklists to early predict the occurrence of sepsis and mortality prediction for intensive care patients. Contributions. • We propose the first framework to learn predictive checklists from arbitrary input data modalities. Our approach can learn checklists and extract meaningful concepts from time series and images, among others. In contrast with previous works that used (mixed-)integer programming, our approach formulates the predictive checklist learning within the framework of probabilistic logical programming. • We investigate the impact of different schemes for improving the interpretability of the concepts learnt as the basis of the checklist. We employ regularization techniques to encourage the concepts to be distinct, so they can span the entire input vector and be specialized, i.e. ignore the noise in the signal and learn sparse representations. We also investigate the impact of incorporating fairness constraints into our architecture. • We validate our learning framework on different data modalities such as images, text and clinical time series, displaying significantly improved performance compared to state-of-the-art checklist learning schemes. 2 RELATED WORKS A major motivation for our work is the ability to learn effective yet interpretable predictive models from data, as exemplified by the interpretable machine learning literature. Conceptually, our method builds upon the recent body of work on learning predictive checklists from data. The implementation of our solution is directly inspired by the literature on probabilistic logic programming. Interpretable machine learning. Motivated by the lack of robustness and trust of black box models, a significant effort has been dedicated to developing more human-interpretable machine learning models in the last years (Ahmad et al., 2018; Murdoch et al., 2019). Among them, one distinguishes between intrinsic (i.e. when the model is itself interpretable such as decision trees) and posthoc (i.e. when trained models are interpreted a posteriori) methods (Du et al., 2019). Checklists belong to the former category as they are an intuitive and easy to use decision support tool. Compared to decision trees, checklists are more concise (there is no branching structure) and can thus be potentially more effective in high stress environments (a more detailed argument is presented in Appendix D). Our approach also relies on building concepts from the input data. Because the concepts are learnt from data, they may themselves lack a clear interpretation. Both intrinsic and posthoc interpretability techniques can then be applied for the concept extraction pipeline (Jeffares et al., 2023). Concept Bottleneck Models (Koh et al., 2020) insert a concept layer before the last fully connected layer, assigning a human-understandable concepts to each neuron. However, a major limitation is that it requires expensive annotated data for predefined concepts. Rule-based learning. Boolean rule mining and decision rule set learning is a well-studied area that has garnered considerable attention spurred by the demand for interpretable models. Some examples of logic-based models include Disjunctive Normal Forms (OR of ANDs), Conjunctive Normal Forms (AND of ORs), chaining of rules in the form of IF-THEN-ELSE conditions in decision lists, and decision tables. Most approaches perform pre-mining of candidate rules and sample rules using integer programs (IP), simulated annealing, performing local search algorithm for optimizing simplicity and accuracy (Lakkaraju et al., 2016), and Bayesian framework for constructing a maximum a posteriori (MAP) solution (Wang et al., 2017). Checklist learning. Checklists, pivotal in clinical decision-making, are typically manually designed by expert clinicians (Haynes et al., 2009). Increasing medical records make manual evidence collection tedious, prompting the need for automated medical checklist design. Recent works have taken a step in that direction by learning predictive checklists from Boolean or categorical medical data (Zhang et al., 2021). Makhija et al. (2022) have extended this approach by allowing for continuous tabular data using mixed integer programming. Our work builds upon these recent advances but allows for complex input data modalities. What is more, in contrast to previous works, our method does not rely on integer programming and thus exhibits much faster computing times and is more amenable to the most recent deep learning stochastic optimization schemes. Probabilistic logical programming. Probabilistic logic reasoning combines logic and probability theory. It represents a refreshing framework from deep learning in the path towards artificial intelligence, focusing on high-level reasoning. Examples of areas relying on these premises include statistical artificial intelligence (Raedt et al., 2016; Koller et al., 2007) and probabilistic logic programming (De Raedt & Kimmig, 2015). More recently, researchers have proposed hybrid architectures, embedding both deep learning and logical reasoning components (Santoro et al., 2017; Rocktäschel & Riedel, 2017; Manhaeve et al., 2018). Probabilistic logic reasoning has been identified as important component for explainable or interpretable machine learning, due to its ability to incorporate knowledge graphs (Arrieta et al., 2020). Combination of deep learning and logic reasoning programming have been implemented in interpretable computer vision tasks, among others (Bennetot et al., 2019; Oldenhof et al., 2023). 3 BACKGROUND Problem Statement: We consider a supervised learning problem where we have access to $N$ input data points $x_i \in \mathcal{X}$ and corresponding binary labels $y_i \in \{0, 1\}$. Each input data point consists of a collection of $K$ data modalities: $x_i = \{x_{i1}, x_{i2}, \ldots, x_{iK}\}$. Each data modality can either be continuous ($x_{ik}^c \in \mathbb{R}^{d_k}$) or binary ($x_{ik}^b \in \{0, 1\}^{d_k}$). Categorical data are assumed to be represented in expanded binary format. We set $d$ as the overall dimension of $x_i$. That is, $d = \sum_{k=1}^{K} d_k$. The $N$ input data points and labels are aggregated in a data structure $\mathbf{X}$ and a vector $\mathbf{y}$ respectively. Our objective is to learn an interpretable decision function $ f : \mathcal{X} \rightarrow \{0, 1\} $ from some domain $ \mathbb{F} $ that minimizes some error criterion $ d $ between the predicted and the true label. The optimal function $ f^* $ then is: \[ f^* = \arg\min_{f \in \mathcal{F}} \mathbb{E}_{x,y \sim D}[d(f(x), y)], \] where $ D $ stands for the observational data distribution. We limit the search space of decision functions $ \mathcal{F} $ to the set of predictive checklists, which are defined below. Predictive checklists: Generally, we define a predictive checklist as a linear classifier applying on a list of $ M $ binary concepts $ c_i \in \{0, 1\}^M $. A checklist will predict a data point, consisting of $ M $ concepts $ c_i = \{c_i^1, \ldots, c_i^M\} $, as positive if the number of concepts such that $ c_i^m = 1 $ is larger or equal to a threshold $ T $. That is, given a data point with concepts $ c_i $, the predicted label of a checklist with threshold $ T $ is expressed as: \[ \hat{y}_i = \begin{cases} 1 & \text{if } \sum_{m=1}^{M} c_i^m \geq T \\ 0 & \text{otherwise} \end{cases} \] The only parameter of a checklist is the threshold $ T $. Nevertheless, the complexity lies in the definition of the list of concepts that will be given as input to the checklist. This step can be defined as mapping $ \phi $ that produces the binary concepts from the input data: $ c_i = \psi(x_i) $. Existing approaches for learning checklists from data differ by their mapping $ \psi $. Zhang et al. (2021) assume that the input data is already binary. In this case, the mapping $ \psi_M $ is then a binary matrix $ \Psi \in \{0, 1\}^{M \times k} $ such that $ \Psi 1_k = 1_k $, where $ 1_k $ is a column vector of ones. One then computes $ c_i $ as $ c_i = \Psi_M x_i $. The element of $ \Psi_M $ as well as the number of concepts $ M $ (hence the dimension of the matrix) are learnable parameters. Previous approaches (Makhija et al., 2022) relax the binary input data assumption by allowing for the creation of binary concepts from continuous data through thresholding. Writing $ x_i^b $ and $ x_i^c $ for the binary and real parts of the input data respectively, the concept creation mechanism transforms the real data to binary with thresholding and then uses the same matrix $ \Psi_M $. We have \[ c_i = \Psi_M [x_i^b, \text{sign}(x_i^c - t_i)], \] where $[,\cdot]$ is the concatenation operator, $ t_i $ is a vector of thresholds, $ \text{sign}(\cdot) $ is an element-wise function that returns 1 is the element is positive and 0 otherwise. In this formulation one learns the number of concepts $ M $, the binary matrix $ \Phi_M $ as well as the thresholds values $ t_i $. Probabilistic Logic Programming: Probabilistic logical reasoning is a knowledge representation approach that involves the use of probabilities to encode uncertainty in knowledge. This is encoded in a probabilistic logical program (PLP) $ P $ connected by a set of $ N $ probabilistic facts $ U = \{U_1, \ldots, U_N\} $ and $ M $ logical rules $ F = \{f_1, \ldots, f_M\} $. PLP enables inference on knowledge graphs $ P $ by calculating the probability of a query. This query is executed by summing over the probabilities of different "worlds" $ w = u_1, \ldots, u_N $ (i.e., individual realizations of the set of probabilistic facts) that are compatible with the query $ q $. The probability of a query $ q $ in a program $ P $ can be inferred as \[ P_P(q) = \sum_w P(w) \cdot \mathbb{I}[F(w) \equiv q], \] where $ F(w) \equiv q $ indicates that the propagation of the realization $ w $ across the knowledge graph, according to the logical rules $ F $, leads to $ q $ being true. The motivation behind using PLP is to navigate the tradeoff between discrete checklists and learnable soft concepts. Incorporating a neural network into this framework enables the generation of probabilistic facts denoted as the neural predicate $ U^\theta $, where $ \theta $ represents the weights. These weights can be trained to minimize a loss that depends on the probability of a query $ q $: \[ \hat{\theta} = \arg\min_\theta L(P(q \mid \theta)). \] ### 4 ProbChecklist: Learning Fair and Interpretable Predictive Checklists #### 4.1 Architecture Overview Our method first applies concept extractors on each data modality. Each concept extractor outputs a list of concept probabilities for each data modality. These probabilities are then concatenated to form a vector of probabilistic concepts $ (p_i) $ for a given data sample. This vector is dispatched to a probabilistic logic module that implements a probabilistic checklist with query $ q := P(y_1 = \hat{y}_1) $. We can then compute the probability of the label of each data sample and backpropagate through the whole architecture. At inference time, the checklist inference engines discretize the probabilistic --- 1This corresponds effectively to every row of $ \Psi $ summing to 1. checklist to provide a complete predictive checklist. A graphical depiction of the overall architecture is given in Figure 2. 4.2 Data Modalities and Concepts Data modalities refer to the distinct sets of data that characterize specific facets of a given process. For instance, in the context of healthcare, a patient profile typically includes different clinical time series, FMRI and CT-scan images, as well as prescriptions and treatment details in text format. The division in data modalities is not rigid but reflects some underlying expert knowledge. Concepts are characteristic binary variables that are learnt separately for each modality. 4.3 Concept Extractor Instead of directly learning binary concepts, we extract soft concepts that we subsequently discretize. For each of the $K$ data modalities, we have a soft concept extractor $\psi_k : \mathbb{R}^{d_k} \rightarrow [0, 1]^{d'_k}$ that maps the input data to a vector of probabilities $p_i^k$, where $d'_k$ is the number of soft concepts to be extracted from data modality $k$. Concatenating the outputs of the $K$ concept extractors results in a vector of probabilities $p_i \in [0, 1]^{d'}$, with the $d'$ the total number of soft concepts. 4.4 Checklist Learning The checklist prediction formula of Equation (3) can be understood as logical rules in a probabilistic logical program. Together with the probabilities of each concepts, encoded in vector $p_i$ that represent $d'$ probabilistic facts, this represents a probabilistic logical program $P_\theta$. We refer to $\theta$ as the set of learnable parameters in the probabilistic logical program. We want to maximize the probability of the prediction being correct. That is, we want to maximize the probability of the query $q := \hat{y}_i = y_i$, $$\hat{\theta} = \arg\min_\theta - P_{P_\theta}(\hat{y}_i = y_i) = \arg\min_\theta - \sum_w P(w) \cdot I[F(w) \equiv (\hat{y}_i = y_i)]$$ (1) By interpreting the probabilities $p_i$ as the probability that the corresponding binary concepts are equal to 1 (i.e., $p_i[j] = P(c_i[j] = 1)$, where $[j]$ indexes the $j$-th component of the vector), we can write the probability of query $q$ as follows. Proposition 4.1. The probability of the query $\hat{y}_i = y_i$ in the predictive checklist is given by $$P_{P_\theta}(\hat{y}_i = 1) = 1 - P_{P_\theta}(\hat{y}_i = 0) = \sum_{d=T} \sum_{\sigma \in \Sigma_d} \prod_{j=1}^{d'} (p_i[j])^{\sigma(j)} (1 - p_i[j])^{1-\sigma(j)}$$ (2) where $\Sigma_d$ is the set of selection functions $\sigma : [d'] \rightarrow \{0, 1\}$ such that $\sum_{j=1}^{d'} \sigma(j) = d$. The detailed derivations are presented in Appendix A. We use the log-likelihood as the loss function, which leads our final loss: $L = y_i \log(P_{P_\theta}(\hat{y}_i = 1)) + (1 - y_i) \log(P_{P_\theta}(\hat{y}_i = 0))$. The parameters $\theta$, include multiple elements: the parameters of the different soft concept extractors ($\theta_k$), the number of concepts to be extracted for each data modality $d'_k$, the checklist threshold $T$. As the soft concept extractors are typically parameterized by neural networks, optimizing $L$ with respect to $\theta_k$ can be achieved via gradient based methods. $d'_k$ and $T$ are constrained to be integers and are thus treated as hyper-parameters in our experiments. 4.5 Checklist Inference ProbChecklist relies on soft concepts extraction for each data modality. Yet, at test time, a checklist operates on binary input data. We thus binarize the predicted soft concepts by setting $c_i[j] =$ \[ \mathbb{I}[P_i[j] > \tau] \]. The thresholding parameter $ \tau $ is an hyperparameter that can be tuned based on validation data. After training, we construct the final checklist by pruning the concepts that are never used in the training data (i.e., concepts $ j $ such that $ c_i[j] = 0, \forall i $, are pruned). This step offers users the flexibility to tune between sensitivity-specificity depending on the application. The optimal checklist can be obtained by varying $ \tau $ to optimize the desired metric on the validation data. ### 4.6 Interpretability of the Concept Extractors The checklist concepts are learnt using deep neural networks and are, therefore, not interpretable in general. To address this issue, we propose two mechanisms to improve the interpretability of the learnt concepts: focused concept learners and regularization terms that incorporate explainability in the structure of the concept learners. Focused models limit the range of features that can contribute to a concept. This can be done via manual specification of the models e.g., using different LSTMs for each time series (Johnson et al., 2022). Regularization terms such as TANGOS helps unveil each input signal’s contribution to the learnt concepts for a given sample, making the concepts interpretable. It ensures that the concepts are obtained from distinct and sparse subsets of the input vector, avoiding overlap. Sparsity is achieved by taking the L1-norm of the concept gradient attributions with respect to the input vector. To promote decorrelation of signal learned in each concept, the loss is augmented by incorporating the inner product of the gradient attributions for all pairs of concepts. This scheme compels the models to learn unique concepts. More details about TANGOS and its mathematical formulation can be found in the Appendix [F.1]. We additionally introduce a regularization term which propels the learnt concept probabilities towards either 0 or 1. This term helps in identifying characteristic concepts for each patient: $ L_{\text{prob-reg}} = \sum_j \sum_i p_i[j] $. ### 4.7 Fairness Regularization We encourage fairness of the learnt checklists by equalizing the error rate across subgroups of protected variables. This is achieved by penalizing significant differences in False Positive and False Negative Rates for sensitive subgroups (Pessach & Shmueli, 2022). For a binary classification problem, with protected attribute $ S $, predicted labels $ \hat{y} \in \{0, 1\} $, and actual label $ y \in \{0, 1\} $, we define separations as follows (Corbett-Davies & Goel, 2018): \[ \Delta FPR = \\| P(\hat{y}_i = 1 \| y = 0, S = s_i) - P(\hat{y}_i = 1 \| y = 0, S = s_j) \\|_1 \quad \forall s_i, s_j \in S \] \[ \Delta FNR = \\| P(\hat{y}_i = 0 \| y = 1, S = s_i) - P(\hat{y}_i = 0 \| y = 1, S = s_j) \\|_1 \quad \forall s_i, s_j \in S \] and combine these in a fairness regularizer $ L_{\text{Fair}} = \lambda (\Delta FPR + \Delta FNR) $. ### 5 Experiments We investigate the performance of ProbChecklist along multiple axes. We first compare the classification performance against a range of interpretable machine learning baselines. Second, we investigate the importance of several key hyperparameters of our method. Lastly, we demonstrate how we can tune the interpretability of the learnt concepts and how we can enforce fairness constraints into ProbChecklist. Complete details about the datasets, baselines used in our experiments, and hyperparameter tuning are available in Appendix [E.3]. #### Baselines. We compare our method against the following baselines. Mixed Integer Programming (MIP) (Makhija et al., 2022). This approach allows to learn predictive checklists from continuous inputs. For images or time series, we typically apply MIP on top of an embedding obtained from a pre-trained deep learning model. Integer Linear Program (ILP) (Zhang et al., 2021). ILP learns predictive checklists with Boolean inputs. We apply these to tabular data by categorizing the data using feature means as threshold. CNN/LSTM/BERT + Logistic Regression (LR). This consists in using a CNN, LSTM or pre-trained BERT on the input data and applying a logistic regression on the combination of the last layer’s embeddings of each modality. CNN/LSTM/BERT + Multilayer perceptron (MLP). This is similar to the previous approach but where we apply an MLP on the combination of the last layer’s embeddings of each modality. #### Datasets. A crucial strength of our method resides in its ability to learn predictive from high dimensional input data. We briefly describe the MNIST synthetic dataset created here and defer the descriptions of other datasets (PhysioNet sepsis tabular dataset, MIMIC mortality dataset, Medical Abstracts TC Corpus) to the Appendix [E.3]. Synthetic MNIST checklist. Due to the absence of real-world datasets with ground-truth checklists, we first validate our idea on a synthetic setup created using MNIST image sequences as input and a checklist defined on digit labels. Each sample consists of a sequence of $K = 4$ MNIST images (treating each image as a separate modality). We then assign a label to each sample according to the following ground-truth checklist. (i) Digit of Image 1 $\in \{0, 2, 4, 6, 8\}$, (ii) Image 2 $\in \{1, 3, 5, 7, 9\}$, (iii) Image 3 $\in \{4, 5, 6\}$, (iv) Image 4 $\in \{6, 7, 8, 9\}$. If at least 3 of the rules are satisfied, the label is 1, and 0 otherwise. ### 5.1 Checklist performance We evaluate the classification performance of the different models according to accuracy, precision, recall and specificity. For the checklist baselines, we also report the total number of concepts used ($M$) and the threshold for calling a positive sample ($T$). Results are presented in Table 1. Additional results and details about hyperparameter tuning are provided in the Appendix. \| Dataset \| Model \| Accuracy \| Precision \| Recall \| Specificity \| $d_k$ \| T \| M \| \|-------------------------\|------------------------\|----------------\|----------------\|---------------\|-----------------\|------\|-----\|------\| \| MNIST Checklist \| CNN + MLP \| 94.72 ± 4.32 \| 0.895 ± 0.1 \| 0.835 ± 0.13 \| 0.976 ± 0.02 \| 4 \| - \| - \| \| \| CNN + LR \| 95.04 ± 0.31 \| 0.914 ± 0.01 \| 0.836 ± 0.016 \| 0.99 ± 0.003\| 8 \| 13.5 ± 0.5 \| \| \| pretrained CNN + MIP \| 79.2 ± 0.4 \| \| \| \| \| \| \| \| ProbChecklist \| \| 96.608 ± 0.24 \| 0.917 ± 0.015 \| 0.929 ± 0.01 \| 0.978 ± 0.004 \| 4 \| 8.4 ± 1.2 \| 16 \| \| PhysioNet Tabular \| Logistic Regression \| 62.555 ± 1.66 \| 0.07 ± 0.043 \| 0.03 ± 0.0393 \| 0.9995 ± 0.0003\| 1 \| 3.2 ± 1.16 \| 9.6 ± 0.8 \| \| \| Unit Weighting \| 55.230 ± 3.80 \| 0.523 ± 0.093 \| 0.438 ± 0.297 \| 0.868 ± 0.251 \| 1 \| 2.8 ± 0.748 \| 4.4 ± 1.01 \| \| \| ILP mean thresholds \| 62.992 ± 0.82 \| 0.544 ± 0.087 \| 0.1196 ± 0.096\| 0.9326 ± 0.0623 \| 1 \| 3.6 ± 0.8 \| 8 ± 1.095 \| \| \| MIP Checklist \| 63.688 ± 2.437 \| 0.563 ± 0.050 \| 0.403 ± 0.082 \| 0.7918 ± 0.06 \| 1 \| 3.6 ± 0.8 \| 8 ± 1.095 \| \| ProbChecklist \| \| 62.579 ± 2.58 \| 0.61 ± 0.076 \| 0.345 ± 0.316 \| 0.815 ± 0.1855 \| 1 \| 3.6 ± 1.2 \| 10 \| \| MIMIC III \| Unit Weighting \| 73.681 ± 0.972 \| 0.469 ± 0.091 \| 0.223 ± 0.206 \| 0.889 ± 0.026 \| 1 \| 6.1 ± 0.830 \| 8.9 ± 0.627 \| \| \| ILP mean thresholds \| 75.492 ± 0.318 \| 0.545 ± 0.028 \| 0.142 ± 0.059 \| 0.959 ± 0.019 \| 1 \| 3.6 ± 0.894 \| 3.6 ± 0.894 \| \| \| MIP Checklist \| 74.988 ± 0.025 \| 0.232 ± 0.288 \| 0.014 ± 0.017 \| 0.997 ± 0.004\| 1 \| 4.5 ± 2.082 \| 4.5 ± 2.082 \| \| \| LSTM + MLP \| 66.58 ± 0.69 \| 0.446 ± 0.223 \| 0.107 ± 0.164 \| 0.962 ± 0.043 \| 1 \| - \| - \| \| \| LSTM + MLP (all features) \| 76.128 ± 0.737 \| 0.446 ± 0.223 \| 0.23 ± 0.132 \| 0.939 ± 0.036 \| 1 \| - \| - \| \| ProbChecklist \| \| 77.58 ± 0.481 \| 0.642 ± 0.075 \| 0.247 ± 0.032 \| 0.953 ± 0.019 \| 2 \| 9.6 \| 20 \| \| Medical Abstracts Corpus\| BERT + ILP \| 72.991 ± 0.06 \| 0.292 ± 0.29 \| 0.197 ± 0.26 \| 0.879 ± 0.17 \| 1 \| 1.2 ± 0.4 \| 1.2 ± 0.4 \| \| \| BERT + MIP \| 69.32 ± 8.1 \| 0.583 ± 0.14 \| 0.059 ± 0.008 \| 0.991 ± 0.009 \| 6 \| 2.5 ± 0.6 \| 4 ± 0.8 \| \| \| BERT + LR \| 80.193 ± 0.84 \| 0.798 ± 0.051 \| 0.138 ± 0.065 \| 0.998 ± 0.007\| 1 \| - \| - \| \| \| BERT + MLP \| 81.782 ± 0.31 \| 0.611 ± 0.040 \| 0.061 ± 0.010 \| 0.964 ± 0.010 \| 1 \| - \| - \| \| ProbChecklist \| \| 83.213 ± 0.23 \| 0.616 ± 0.006 \| 0.623 ± 0.01 \| 0.891 ± 0.003 \| 6 \| 3 \| 6 \| Table 1: Performance results for all the models and baselines on all the datasets. We report accuracy, precision, recall as well as conciseness of the learnt checklist. To facilitate visualization and comparison, we plot these results in Section I of the Appendix (Figure 10). MNIST Checklist. We used a simple three-layered CNN model as the concept learner for each image. In Table 1, we report the results of the baselines and ProbChecklist for $d_k = 4$ ($M = 16$) on the test samples. Our method outperforms all the baselines, in terms of accuracy and recall, indicating that it identifies the minority class better than these standard approaches. The MIP failed to find solutions for some folds of the dataset and didn’t generalise well on the test samples. Sepsis Prediction from Tabular Data. This setup is ideal for comparison with existing checklist method as they only operate on tabular dataset. In Figure 4, we visualize learnt by ProbChecklist in one of the experiments. We observe that ProbChecklist exhibits similar performance to checklist baselines. We want to emphasize that ProbChecklist provides a significantly broader applicability to multimodal datasets while maintaining comparable performance on tabular datasets, thus making it valuable. Neoplasm Detection from Clinical Abstracts. We use a pretrained BERT model (Alsentzer et al., 2019) with frozen weights as our concept learner. This was a BioBERT model pretrained on clinical notes of MIMIC-III dataset. Our checklist has a much better recall and accuracy than previous methods. Both checklist learning and deep learning methods give poor performance on the minority class. Mortality Prediction using Time Series Data. To learn representations from clinical timeseries, we initialize $K$ two-layered LSTMs. We highlight our key results in Table 1. For ProbChecklist, we report the checklist which attains the highest accuracy on validation data. We surpass existing methods in terms of accuracy and precision with a significant margin. We find that a checklist with better recall can be constructed by optimizing over F1-Score instead of accuracy. Sensitivity analysis: We investigate the evolution of performance of ProbChecklist with increasing number of learnt concepts $d_k$. On Figure 3a, we show the accuracy, precision, recall, and specificity... in function of the number of concepts per image on the MNIST dataset. We observe a significant improvement in performance when $d'_k$ increases from 1 to 2, which suggests that having learning one concept per image is inadequate to capture all the signal in the sample. It is also interesting to note that the performance reaches a saturation point after $d'_k = 3$. This suggests held-out loss can be used to tune the value of $d'_k$ to find the optimal number of concepts for a given data modality. (a) Performance of ProbChecklist with varying $d'_k$ on MNIST Checklist Dataset (b) We plot images and corresponding gradient attributions heat maps for seven inputs samples of the Image 2 modality of the MNIST dataset. We used a checklist with two learnable concepts per image. The intensity of red denotes the positive contribution of each pixel, whereas blue indicates the negative. If a concept predicted as true for an image, then we represent that with plus (+) sign, and with a negative sign (-) otherwise. Figure 3 5.2 Concepts Interpretation We investigate the concepts learnt from image and timeseries datasets with an interpretability regularization as in Section 4.6. To gain insight into what patterns of signals refer to each individual concept, we examine the gradient of each concept with respect to each dimension of the input signal. Intuitively, the interpretability regularization enforces the concepts to focus on a sparse set of features of the input data. MNIST Images: We analyze the gradient of our checklist on individual pixels of the input images. We use a checklist with two concepts per image. On Figure 3b, we show example images of the Image 2 of the MNIST dataset along with the gradient heat map for each learnt concept of the checklist. The ground truth concept for this image is Image 2 $\in \{1, 3, 5, 7, 9\}$. First, we see that the 7, 9, and 5 digits are indeed the only ones for which the predicted concepts of our checklist are positive. Second, we infer from the gradient heat maps that concepts 1 and 2 focus on the image’s upper half and centre region, respectively. Concept 1 is true for digits 5, 8, 9 and 7, indicating that it corresponds to a horizontal line or slight curvature in the upper half. Since Digits 0 and 2 have deeper curvature than the other images, and there is no activity in that region in the case of 4, concept 1 is false for them. Concept 2 is true for images with a vertical line, including digits 9, 4, 5 and 7. Therefore, concept 2 is false for the remaining digits (0, 2, 8). The checklist outcome matches the ground truth when both concepts are true for a given image. Complementary analyses on MNIST and MIMIC III are provided in Appendix F.2 and F.3. This analysis ensures interpretability at the individual sample level. As illustrated in the previous example, recognizing and comprehending these concepts at the dataset level relies on visual inspection. Medical Abstracts: Compared to images and time series, interpreting concepts learned from textual data is easier because its building blocks are tokens which are already human understandable. For the Neoplasm detection task, we adopt an alternative method by conducting a token frequency analysis across the entire dataset. This approach has yielded a more lucid checklist shown in Figure 1. We identified key tokens associated with positive and negative concepts (positive and negative tokens). Each concept is defined by the presence of positive words and the absence of negative words. 5.3 Fairness We evaluate the fairness of ProbChecklist on the MIMIC-III Mortality Prediction task and show that we can reduce the performance disparities between sensitive attributes by incorporating fairness regularization (FR) terms, as introduced in Section 4.7. We set the sensitive features as gender and ethnicity ∈ {Black, White, Others}. Our results are displayed on Tables 2 and 11. These disparities in performance across different sub-populations are significantly reduced after fairness regularization is used. To see the effectiveness of the regularizer, we report the percentage decrease in ΔFNR and ΔFPR observed with respect to the unregularized checklist predictions for all pairs of sensitive subgroups. Similar fairness constraints (FC) can also be added to the ILP mean-thresholds baseline (Jin et al., 2022). We include a separate constraint for each pair that restricts \|ΔFNRI\| and \|ΔFPRI\| to be less than ε = 0.05. It is important to note that our approach minimizes the summation of ΔFNR and ΔFPR across all pairs of sub-groups, but in the ILP we can specify a strict upper bound for each pair. Due to this, we might observe an increase in the gap for certain pairs in case of ProbChecklist, but adjusting the relative weights of these terms in the loss equation helps in achieving optimal performance. Although ProbChecklist had higher initial FNR/FPR values, the regularizer effectively reduces them to be comparable to those of ILP, particularly for the ethnicity pairs. 6 DISCUSSION Performance of ProbChecklist. Through these experiments, we aim to showcase that ProbChecklist surpasses existing checklist methods and achieves comparable performance to MLP (non-interpretable) methods. The switch to learnable concepts explains the improvement in accuracy over checklist methods. These concepts capture more signal than fixed summary/concept extractors used in prior works to create binarized tabular data. It’s important to note that a checklist, due to its binary weights, has a strictly lower capacity and is less expressive than deep learning but possesses a more practical and interpretable structure. Despite this, it exhibits similar performance to an MLP. Interpretability of checklist structure and learnt concepts. Although ProbChecklist employs a probabilistic objective for training the concept learners, the end classifier used for inference is, in fact, a discrete checklist. While this makes the classifier highly interpretable, it also shifts the focus of interpretability to the learnt concepts. We fully realize this trade-off and investigate existing techniques to maintain feature-space interpretability. For time series and images, we employ regularization terms (4.6) to enforce sparsity, avoid redundancy, and learn strongly discriminative features with high probability. We also use focused concept learners to avoid learning concepts that are functions of multiple modalities. Identifying patterns from the binarized concepts is primarily based on visual inspection and expert knowledge. We noticed it is easier to source and comprehend the key tokens contributing to each concept for text data. Lastly, we want to highlight that ProbChecklist is a flexible framework, and other interpretable models can be easily integrated as concept learners. Limitations. We have taken the first step towards learning checklists from complex modalities, whereas the existing methods are restricted to tabular data. Even though we have a mechanism to learn interpretable checklist classifiers using logical reasoning, more work is needed on the interpretability of the learnt concepts. Another drawback is the exponential memory complexity of the training. A fruitful future direction would be to study approximations to explore a smaller set of combinations of concepts. Detailed complexity analysis can be found in Appendix B. Societal Impact. As discussed initially in the paper, manually designed checklists are extensively used in hospitals for decision-making under complex situations and help automate certain aspects of the treatment. With more research on the interpretability of concepts, ProbChecklist can replace the existing manual procedure and reduce the burden on the healthcare system. Table 2: Improvement in fairness metrics across gender and ethnicity on MIMIC III for the mortality prediction task after adding fairness regularization. We report ΔFNR and ΔFPR for all pairs of subgroups of sensitive features and the percentage decrease (%) ↓ wrt unregularized checklist. REFERENCES Muhammad Aurangzeb Ahmad, Carly Eckert, and Ankur Teredesai. 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EKEcYL7gaf	Can you provide more details on the conducted user study? How many images were assessed by each human evaluator? What is reported, e.g., majority decision? Why are eight raters an appropriate and sufficient number of participants?	Predicated Diffusion: Predicate Logic-Based Attention Guidance for Text-to-Image Diffusion Models Anonymous authors Paper under double-blind review Abstract Diffusion models have achieved remarkable results in generating high-quality, diverse, and creative images. However, when it comes to text-based image generation, they often fail to capture the intended meaning presented in the text. For instance, a specified object may not be generated, an unnecessary object may be generated, and an adjective may alter objects it was not intended to modify. Moreover, we found that relationships indicating possession between objects are often overlooked. While users’ intentions in text are diverse, existing methods tend to specialize in only some aspects of these. In this paper, we propose Predicated Diffusion, a unified framework to express users’ intentions. We consider that the root of the above issues lies in the text encoder, which often focuses only on individual words and neglects the logical relationships between them. The proposed method does not solely rely on the text encoder, but instead, represents the intended meaning in the text as propositions using predicate logic and treats the pixels in the attention maps as the fuzzy predicates. This enables us to obtain a differentiable loss function that makes the image fulfill the proposition by minimizing it. When compared to several existing methods, we demonstrated that Predicated Diffusion can generate images that are more faithful to various text prompts, as verified by human evaluators and pretrained image-text models. 1 Introduction The recent advancements in deep learning have paved the way for the generation of images that are high-quality, diverse, and creative. This progress is primarily attributed to diffusion models (Ho et al., 2020; Song et al., 2021), which recursively update images to remove noise and to make them more realistic. Diffusion models are significantly more stable and scalable compared to previous methods, such as generative adversarial networks (Goodfellow et al., 2014; Radford et al., 2016) or autoregressive models (van den Oord et al., 2016; Kolesnikov & Lampert, 2017). Moreover, the field of text-based image generation is attracting considerable attention, with the goal being to generate images that are faithful to a text prompt given as input. Even in this area, the contributions of diffusion models are notable (Ramesh et al., 2021). We can benefit from commercial applications such as DALL-E2 (Ramesh et al., 2022) and Imagen (Saharia et al., 2022), as well as the state-of-the-art open-source model, Stable Diffusion (Rombach et al., 2022). These models are trained on large-scale and diverse text-image datasets, which allows them to respond to a variety of prompts and to generate images of objects with colors, shapes, and materials not found in the existing datasets. However, many previous studies have pointed out that these models often generate images that ignore the intended meanings of a given prompt, as exemplified in Fig. 1 (Feng et al., 2023; Chefer et al., 2023; Rasskin et al., 2023; Wang et al., 2023). When multiple objects are specified in a prompt, only some are generated, with the others disappearing (see the column missing objects in Fig. 1). Also, two specified objects are sometimes mixed together to form one object in the generated image (object mixture). Given an adjective in a prompt, it alters a different object than the one the adjective was originally intended to modify (attribute leakage). We have found a novel challenge that, when a prompt specifies an object being held by someone, the object is depicted as if discarded on the ground (possession failure). Although these challenges need to be addressed, retraining diffusion models on large-scale datasets is prohibitively expensive. Many studies have proposed methods... offering guidance for the image generation process of pre-trained diffusion models, ensuring that the images are updated to become more faithful to the prompt. However, these guidances vary widely, and a unified solution to address the diverse range of challenges has yet to be established. We hypothesize that the root cause of such challenges lies in the text encoder within diffusion models failing to correctly capture the logical statements presented in the given prompt. If we could represent such logical statements by using predicate logic and integrate it into the diffusion model, it might make the generated images more faithful to the statements. Motivated by this idea, we introduce Predicated Diffusion in this paper. Herein, we represent the relationships between the words in the prompt by propositions using predicate logic. By employing attention maps and fuzzy logic (Hájek [1998], Prokopowicz et al. [2017]), we measure the degree to which the image under generation fulfills the propositions, providing guidance for images to become more faithful to the prompt. See the conceptual diagram in Fig. 2. The contribution of this paper is threefold. Theoretical Justification and Generality: Most existing methods have been formulated based on deep insights, which makes it unclear how to combine them effectively or how to apply them in slightly different situations. In contrast, Predicated Diffusion can resolve a variety of challenges based on the same foundational theory, allowing us to deductively expand it to address challenges not summarized in Fig. 1. High Fidelity to Prompt: The images generated by the proposed Predicated Diffusion and comparison method were examined by human evaluators and pretrained image-text models (Radford et al. [2021], Li et al. [2022]). We confirmed that Predicated Diffusion generates images that are more faithful to the prompts and is more likely prevent the issues shown in Fig. 1. New Challenge and Solution: This paper introduces a new challenge, named possession failure, which occurs when the generated image fails to correctly depict a prompt indicating a subject in possession of an object. Thus, we broaden the horizons of the current research, which has mainly focused on the presence or absence of objects and attributes, to encompass actions. The fact that Predicated Diffusion can successfully address this new challenge is worthy of attention. ## 2 RELATED WORK Conditional Image Generation A diffusion model was proposed as a parameterized Markov chain (Sohl-Dickstein et al. [2015], Ho et al. [2020]). Taking a given image $x$ as the initial state $x_0$, the forward process $q(x_{t+1}\|x_t)$ adds noise to the state $x_t$ repeatedly. The model learns the reverse process $p(x_{t-1}\|x_t)$, reproducing the data distribution $p(x) = p(x_0)$. Intuitively speaking, it repeatedly denoises images to be more realistic. The reverse process resembles a discretized stochastic differential equation, akin to the Langevin dynamics, which ascends the gradient of the log-probability, $\nabla \log p(x)$ (Song et al., 2021). With a separate classifier $p(y\|x)$ for class label $y$, the diffusion model can reproduce the conditional probability $p(x\|y)$, by ascending the gradient of the conditional log-probability, $\nabla \log p(x\|y) = \nabla \log p(y\|x) + \nabla \log p(x)$. Although grounded in probability theory, what it practically offers is additional guidance $\nabla \log p(y\|x)$ for updating images, which is generalized as classifier guidance (Dhariwal & Nichol, 2021). The diffusion model can learn the conditional probability $p(x\|c)$ directly. The condition $c$ might be text, images, or other annotations (Ramesh et al., 2021; Rombach et al., 2022). The difference between conditional and unconditional updates serves as classifier-free guidance, which can adjust the fidelity of the generated image to condition $c$ (Ho & Salimans, 2021). Liu et al. (2022) proposed Composable Diffusion, inspired by energy-based models (Du et al., 2020). It generates an image conditioned on two concepts, $c_0$ and $c_1$, by summing their respective conditional updates. It negates or removes a concept $c_n$ from generated images by subtracting the update conditioned on $c_n$, termed as a negative prompt. Text-Based Image Generation by Cross-Attention Mechanism One of the leading models, Stable Diffusion, employs the cross-attention mechanism for conditioning (Vaswani et al., 2017). A convolutional neural network (CNN), U-Net (Ronneberger et al., 2015), transforms the image $x$ into an intermediate representation. For text conditions, a text encoder, CLIP (Radford et al., 2021), transforms text prompt $y$ into a sequence of intermediate representations, each linked to a word $c$ within the prompt $y$. Given these representations, the cross-attention mechanism creates an attention map $A_c$ for each word $c$. Using these maps as weights, U-Net then updates the image $x$. Technically, these processes target not the image $x$ but the latent variable $z$ extracted by a variational autoencoder (Kingma & Welling, 2014). Despite its sophistication, Stable Diffusion sometimes fails to capture the intended meaning of the text prompt, as discussed in the Introduction. The novelty of Stable Diffusion primarily lies in its structure, which is compatible with existing guidances such as Composable Diffusion and has inspired new guidances. Structure Diffusion feeds segmented text prompts to the text encoder to emphasize each clause (Feng et al., 2023). High pixel intensity in the attention map $A_c$ suggests the presence of a corresponding object or concept $c$ at that pixel. Hence, recent studies offer guidances based on the attention map, termed as attention guidance. Attend-and-Excite enhances the intensity of at least one pixel in the attention map $A_c$ to ensure the existence of the corresponding object $c$ (that is, address missing objects) (Chefer et al., 2023). SynGen equalizes the intensity distributions for related nouns and adjectives, while differentiating others, thus addressing attribute leakage (Rassin et al., 2023). While these methods are based on deep insights, they lack comprehensive theoretical justification and generality. Another branch of studies has proposed attention guidances using external annotations, such as bounding boxes (Xie et al., 2023; Ma et al., 2023; Mao & Wang, 2023) and segmentation masks (Park et al., 2023). While effective in intentionally controlling image layout, these methods sometimes limit the diversity of the generated images. Table 1: Propositions and Attention Map. \| Proposition \| Attention Map \| \|-------------\|---------------\| \| true \| 1 \| \| false \| 0 \| \| $P(x)$ \| $A_P[i]$ \| \| $\neg P(x)$ \| $1 - A_P[i]$ \| \| $P(x) \land Q(x)$ \| $A_P[i] \times A_Q[i]$ \| \| $P(x) \lor Q(x)$ \| $1 - (1 - A_P[i]) \times (1 - A_Q[i])$ \| \| $P(x) \rightarrow Q(x)$ \| $1 - A_P[i] \times (1 - A_Q[i])$ \| \| $\forall x. P(x)$ \| $\prod_i A_P[i]$ \| \| $\exists x. P(x)$ \| $1 - \prod_i (1 - A_P[i])$ \| Table 2: Statements that Predicated Diffusion Can Express. \| Statements \| Example Prompts \| Loss \| \|-----------------------------\|--------------------------\|------\| \| Existence \| There is a dog \| 1 \| \| Concurrent existence \| There are a dog and a cat\| 2 \| \| Adjective \| A black dog \| 3 \| \| One-to-one correspondence \| A black dog and a white cat \| 5 \| \| Possession \| A man holding a bag \| 6 \| \| Multi-color \| A green and grey bird \| A1 \| \| Negation \| without snow \| A2 \| 3 Method: Predicated Diffusion Predicate Logic First-order predicate logic is a formal language for expressing knowledge (Genesereth & Nilsson [1987]). Variables like $x$ and $y$ denote unspecified objects. Predicates like $P$ and $Q$ indicate properties or relationships between objects. Using variables and predicates, we can express logical statements that define object properties. For example, the proposition $P(x)$ represents the statement that "$x$ has property $P$." If the predicate $P$ indicates the property "being a dog," the proposition $P(x)$ represents the statement that "$x$ is a dog." The existential quantifier, denoted by $\exists$, declares the existence of objects satisfying a given property. Thus, the proposition $\exists x. P(x)$ asserts the existence of at least one object $x$ that satisfies the predicate $P$, representing that "There is a dog." Predicate Logic in Attention Map and Resulting Guidance Through predicate logic, we propose an attention-based guidance termed Predicated Diffusion. In a diffusion model for text-based image generation, a cross-attention mechanism creates attention maps. Each map is linked to a word in a text prompt and assigns weights to specific regions of the image. We denote the attention map linked to the word $P$ by $A_P$. The value $A_P[i] \in [0, 1]$ refers to the intensity of the $i$-th pixel in $A_P$. We treat the intensity $A_P[i]$ as a continuous form of a proposition $P(x)$. $A_P[i] = 1$ indicates that the proposition $P(x)$ holds, whereas $A_P[i] = 0$ implies that it does not. $1 - A_P[i]$ indicates the negation of the proposition, $\neg P(x)$. The correspondences are summarized in Table 1, which are inspired by the strong conjugation, strong negation, and material implication of the product fuzzy logic (Hájek [1998], Prokopowicz et al. [2017]). Given another proposition $Q(x)$, we consider the conjunction $Q(x) \land P(x)$ to correspond to the product $A_Q[i] \times A_P[i]$ in the attention maps. One can derive any logical operations using both negation and conjunction. For example, because the disjunction $Q(x) \lor P(x)$ is equivalent to $\neg(\neg Q(x) \land \neg P(x))$, it corresponds to $1 - (1 - A_Q[i]) \times (1 - A_P[i])$. The universal quantifier $\forall$ asserts that a predicate holds for all object. Thus, $\forall x. P(x) = \land_x P(x)$ corresponds to $\prod_i A_P[i]$. Using this, the existential proposition $\exists x. P(x)$ can be re-expressed as $\neg(\forall x. \neg P(x))$, corresponding to $1 - \prod_i (1 - A_P[i])$. For simplicity, we will treat italicized nouns and adjectives as predicates. Specifically, we will use $Dog(x)$ to represent "$x$ is a dog" rather than $P(x)$. A text prompt "There is a dog" is represented by the proposition $\exists x. Dog(x)$. Then, we expect that $1 - \prod_i (1 - A_{Dog}[i]) = 1$. To encourage this, we consider its negative logarithm, \[ L[\exists x. Dog(x)] = -\log(1 - \prod_i (1 - A_{Dog}[i])), \] and adopt it as the loss function, making the intensity of at least one pixel approach 1. This loss function is inspired by the negative log-likelihood for Bernoulli random variables. Moving forward, we will denote the loss function resulting from the proposition $R$ by $L[R]$. We provide an overview of prompts and their corresponding loss functions in Table 2. The reverse process of a diffusion model, $q(x_{t-1}\|x_t,c)$, is typically modeled as a Gaussian distribution $q(x_{t-1}\|x_t,c) = N(x_{t-1}\|\mu_\theta(x_t,t,c), \Sigma_\theta(x_t,t,c))$. The parameters for this are determined by neural networks $\mu_\theta$ and $\Sigma_\theta$ which consider the current image $x_t$, time $t$, and condition $c$. The attention map $A_P$ serves as a component of these neural networks. We take the gradient of the loss function with respect to the input image, $\nabla_x L[R]$, and subtract it from the mean of the reserve process as $q(x_{t-1}\|x_t,c) = N(x_{t-1}\|\mu_\theta(x_t,t,c) - \nabla_x L[R], \Sigma_\theta(x_t,t,c))$, which decreases the loss. function $ L[R] $ and guides the image toward fulfilling the proposition $ R $. This modification of the reverse process is referred to as guidance. In general, to encourage a proposition to hold, one can convert it to an equation of the attention map intensity, take its negative logarithm, use it as a loss function, and integrate it into the reverse process. A visual representation of this is found in Fig. 2. Concurrent Existence by Logical Conjunction In practice, when text prompts include multiple objects, one of the objects often disappears. Take, for instance, the prompt “There are a dog and a cat.” This can be decomposed into two statements: “There is a dog” and “There is a cat.” Given that a set of statements can be represented through the conjunction of propositions, the prompt can be represented by the proposition $ (\exists x. Dog(x)) \land (\exists x. Cat(x)) $. The corresponding loss function is \[ L[(\exists x. Dog(x)) \land (\exists x. Cat(x))] = L[\exists x. Dog(x)] + L[\exists x. Cat(x)]. \tag{2} \] Minimizing this loss function encourages the concurrent existence of both a dog and a cat. Adjective by Logical Implication We develop these ideas into logical implication. For a prompt such as “There is a black dog,” it can be decomposed into: “There is a dog” and “The dog is black.” The former statement has been previously discussed. The latter can be represented with the proposition $ \forall x. Dog(x) \rightarrow Black(x) = \forall x. \neg(Dog(x) \land \neg Black(x)) $. Thus, the loss function is \[ L[\forall x. Dog(x) \rightarrow Black(x)] = - \sum_i \log(1 - ADog[i] \times (1 - ABlack[i])). \tag{3} \] To ensure the both statements, we can sum the loss functions (1) and (3). One-to-One Correspondence As far as we have confirmed, the existing models rarely fail to generate an object with a specified color. These models might struggle when handling prompts with multiple adjectives and nouns; one of the specified objects may not be generated properly, one of the specified adjectives may be ignored, or an adjective may modify a wrong noun. The first two issues can be addressed using the loss functions (2) and (3). The last issue is often referred to as attribute leakage. For example, given the prompt “a black dog and a white cat,” leakage could lead to the generation of a white dog or a black cat. To prevent leakage, we must deduce statements implicitly suggested by the original prompt. From the prompt, we can deduce not only “The dog is black” but also “The black object is a dog.” The latter can be represented by the proposition $ \forall x. Black(x) \leftrightarrow Dog(x) $. When combined, these two statements can be represented as a bimplication: $ \forall x. Dog(x) \leftrightarrow Black(x) = (\forall x. Dog(x) \rightarrow Black(x)) \land (\forall x. Black(x) \rightarrow Dog(x)) $. This leads to the loss function \[ L[\forall x. Dog(x) \leftrightarrow Black(x)] = L[\forall x. Dog(x) \rightarrow Black(x)] + L[\forall x. Black(x) \rightarrow Dog(x)]. \tag{4} \] Furthermore, we can deduce a negative statement, “The dog is not white,” represented by $ \forall x. Dog(x) \rightarrow \neg White(x) $. Thus, the comprehensive loss function for the original statement is: \[ L_{one-to-one} = L[\forall x. Dog(x) \leftrightarrow Black(x)] + L[\forall x. Cat(x) \leftrightarrow White(x)] + \alpha L[\forall x. Dog(x) \rightarrow \neg White(x)] + \alpha L[\forall x. Cat(x) \rightarrow \neg Black(x)], \tag{5} \] where the hyperparameter $ \alpha \in [0, 1] $ adjusts the weight of the implicitly negative statements. To further ensure the existence of objects, the loss function (2) can also be applied. Possession by Logical Implication We introduce another type of implication. Consider the text prompt “a man holding a bag.” This implies that the bag forms part of the man. Such a relationship can be represented by the proposition $ Bag(x) \rightarrow Man(x) $, leading to the loss function \[ L[\forall x. Bag(x) \rightarrow Man(x)] = - \sum_i \log(1 - ABag[i] \times (1 - AMan[i])). \tag{6} \] Not limited to the word holding, other words indicating possession such as having, grasping, and wearing can be represented by the logical implication. Discussions and Potential Extensions Several studies have introduced loss functions or quality measures for machine learning methods by drawing inspiration from fuzzy logic (Hu et al., 2016; Diligenti et al., 2017; Mordido et al., 2021; Marra et al., 2023) (see also Giunchiglia et al., 2022 for a survey). In this context, Predicated Diffusion is the first method to establish the correspondence between the attention map and the predicates. The propositions and corresponding loss functions can be adapted to a variety of scenarios, including, but not limited to, the concurrent existence of more than two objects, a single object modified by multiple adjectives, the combination of one-to-one correspondence and possession, and the negation of existence, modifications, and possessions, as we will show in the following sections. Some previous studies extracted the structure of sentences using syntactic parsers or obtained relationships between words from additional data such as scene graphs (Feng et al., 2023). Such methods can be combined with Predicated Diffusion. The (weak) conjugation of Gödel fuzzy logic and the product fuzzy logic is achieved by the minimum operation (Hájek, 1998; Prokopowicz et al., 2017). If we employ this operation and define the loss function by taking the negative instead of the negative logarithm, the proposition asserting the concurrent existence, $(\exists x. Dog(x)) \land (\exists x. Cat(x))$, leads to the loss function $\max(1 - \max_i A_{Dog}[i], 1 - \max_j A_{Cat}[j])$. This is equivalent to the one used for Attend-and-Excite (Chefer et al., 2023). This comparison suggests that our approach considers Attend-and-Excite as Gödel fuzzy logic, replaces the underlying logic with the product fuzzy logic, and broadens the scope of target propositions. Similar to the loss function (5), SynGen equalizes the attention map intensities for related nouns and adjectives (Rassin et al., 2023). SynGen additionally differentiates those for all word pairs except for the adjective-noun pairs. In contrast, the loss function (5) differentiates those for only specific pairs which could trigger attribute leakage based on inferred propositions, thereby preventing the disruption of the harmony, as shown in the following section. 4 EXPERIMENTS AND RESULTS Experimental Setting We implemented Predicated Diffusion by adapting the official implementation of Attend-and-Excite (Chefer et al., 2023[^1]). The reverse process spans 50 steps; following Attend-and-Excite and SynGen, we applied the guidance of Predicated Diffusion only to the initial 25 steps. See Appendix [A.1] for more details. For comparative evaluation, we also prepared Composable Diffusion (Liu et al., 2022), Structure Diffusion (Feng et al., 2023), and SynGen (Rassin et al., 2023), in addition to Stable Diffusion and Attend-and-Excite. All models used the officially pretrained Stable Diffusion (Rombach et al., 2022[^2]) as backbones. We conducted four experiments for assessing each method’s performance. We provided each method with the same prompt and random seed, and then generated a set of images. Human evaluators were tasked with the visual assessment of these generated images. Instructions and evaluation criteria provided to the evaluators are detailed in Appendix [A.2]. (i) Concurrent Existence: We prepared 400 random prompts, each mentioning “[Object A] and [Object B]”, and generated 400 sets of images. The evaluators identified cases of “missing objects,” where one or both of the specified two objects were not generated. Some cases involved an “object mixture”, where, although the two objects were generated, their boundaries were unclear. The evaluators tallied the cases of “missing objects” based on two criteria: a lenient criterion where “object mixture” was not counted as “missing objects”, and a strict criterion where it was. For Predicated Diffusion, we used the loss function (2). (ii) One-to-One Correspondence: Similarly, we prepared 400 random prompts, each mentioning “[Adjective A] [Object A] and [Adjective B] [Object B]”. In addition to identifying missing objects, the evaluators identified the cases where adjectives incorrectly altered unrelated objects and tallied the number of such cases as “attribute leakage”. For Predicated Diffusion, we used the loss function (2) + (5) with $ \alpha = 0.3 $. (iii) Possession: We prepared 10 prompts, each mentioning “[Subject A] is [Verb C]-ing [Object B]”, [Verb C] can be “have,” “hold,” “wear,” or the like. We generated 20 images for each of these prompts. In addition to identifying missing objects, the evaluators identified the cases where [Verb C] was not executed appropriately, and tallied the number of such cases as “possession failure”. For Predicated Diffusion, we used the loss function (2) + (6). (iv) Complicated: To demonstrate the generality of Predicated Diffusion, we prepared diverse prompts, some of which were taken from the ABC-6K dataset (Feng et al., 2023). Images were generated after manually extracting propositions and their respective loss functions. While a summary of generated images is presented, numerical evaluations were not undertaken due to the diversity of the prompts. [^1]: https://github.com/yuval-alaluf/Attend-and-Excite (MIT license) [^2]: https://github.com/CompVis/stable-diffusion (CreativeML Open RAIL-M) Table 3: Results of Experiments (i) and (ii) \| Models \| Experiment (i) \| \| \| \| \| \| \| \|----------------------\|----------------\|----------\|----------\|----------\|----------\|----------\|----------\| \| \| Missing† \| Fidelity \| Similarity‡ \| Missing† \| Attribute Leakage \| Fidelity \| Similarity‡ \| \| \| Objects \| \| \| Objects \| \| \| \| \| Stable Diffusion \| 54.7 / 66.0 \| 11.0 \| 0.325 / 0.770 \| 64.8 / 73.5 \| 88.5 \| 6.0 \| 0.343 / 0.741 \| \| Composable Diffusion \| 44.5 / 82.3 \| 2.5 \| 0.318 / 0.740 \| 49.3 / 83.5 \| 88.5 \| 3.8 \| 0.347 / 0.725 \| \| Structure Diffusion \| 56.0 / 64.5 \| 12.0 \| 0.320 / 0.760 \| 64.3 / 69.5 \| 86.5 \| 5.8 \| 0.342 / 0.737 \| \| Attend-and-Excite \| 25.3 / 36.3 \| 29.5 \| 0.337 / 0.814 \| 28.0 / 35.8 \| 64.5 \| 19.3 \| 0.367 / 0.781 \| \| SynGen \| — \| — \| — \| 23.3 / 29.3 \| 40.3 \| 36.8 \| 0.365 / 0.792 \| \| Predicated Diffusion \| 18.5 / 28.5 \| 30.3 \| 0.340 / 0.837 \| 10.0 / 16.5 \| 33.0 \| 44.8 \| 0.375 / 0.808 \| † Using the lenient and strict criterions. ‡ Text-image similarity and text-text similarity. Figure 3: Example results of Experiment (i) for concurrent existence. See also Fig. A2 Figure 4: Example results of Experiment (ii) for one-to-one correspondence. See also Fig. A3 Table 4: Results of Experiment (iii) for possession \| Models \| Missing Objects† \| Possession Failure \| Fidelity \| \|----------------------\|------------------\|--------------------\|----------\| \| Stable Diffusion \| 31.5 / 36.0 \| 52.5 \| 33.5 \| \| Attend-and-Excite \| 7.5 / 17.0 \| 51.5 \| 27.5 \| \| Predicated Diffusion \| 4.0 / 7.0 \| 29.5 \| 52.0 \| † Using the lenient and strict criterions. Figure 5: Example results of Experiment (iii) for possession. See also Fig. A4 The first two experiments were inspired by previous works: Feng et al. (2023); Chefer et al. (2023); Rassin et al. (2023). In the first three experiments, the evaluators also assessed the fidelity of the generated images to the prompts. To measure fidelity automatically, we evaluated the similarity using the pretrained image-text encoder, CLIP, in two manners, following Chefer et al. (2023). We provided both the prompts and the generated images with CLIP to extract their embedding vectors, and then calculated their cosine similarity, referred to as text-image similarity. In addition, we fed the generated images into the pretrained image-captioning model, BLIP, to obtain captions (Li et al., 2022). Then, we provided both the prompts and the obtained captions with CLIP to extract their embedding vectors, and then calculated their cosine similarity, referred to as text-text similarity. The chosen objects, adjectives, and prompts can be found in Appendix A.3. Results for Concurrent Existence and One-to-One Correspondence Table 3 summarizes the results of the quantitative assessments from Experiments (i) and (ii). Values except for the similarity are expressed in percentages. Higher scores are desirable for the fidelity and similarity, while lower values are preferred for the remaining criteria. Predicated Diffusion notably outperforms other methods, as it achieved the best outcomes across all 11 criteria. Figures 3 and 4 show example images for visual evaluation, where images in each column are generated using the same random seed. See also Figs. A2 and A3 in Appendix. Stable Diffusion, Composable Diffusion, and Structure Diffusion often exhibit missing objects and attribute leakage. The absent of objects is particularly evident when prompts feature unusual object combinations like “a crown and a rabbit” and “a yellow car and a blue bird.” When the prompts specify visually similar objects, such as “a bird and a cat,” the two objects often get mixed together. While Attend-and-Excite effectively prevents the issue of missing objects, it struggles with attribute leakage in Experiment (ii) due to its lack of a dedicated mechanism to address this. While SynGen has achieved relatively good results, Predicated Diffusion outperforms it by further preventing missing objects and attribute leakage and producing images that are the most faithful to the prompt. Although this aspect was not explicitly part of the evaluation, SynGen often generates multiple instances of small objects, such as birds and balloons. Results for Possession Table 4 summarizes the results from Experiment (iii). Compared to the vanilla Stable Diffusion, Attend-and-Excite succeeds in preventing missing objects, but on the contrary, fails to prevent the possession failure and loses fidelity to the prompt. Figures 5 and A4 show visual samples of generated images. If [Subject A] is an animal, Attend-and-Excite succeeds more frequently than Stable Diffusion in depicting both objects but often depicts [Object B] as discarded on the ground or suspended in the air. If [Subject A] is a human, the vanilla Stable Diffusion often produces satisfactory results. Then, Attend-and-Excite, however, tends to deteriorate the overall image quality. With the possession relationship, [Subject A] and [Object B] often overlap. Attend-and-Excite makes both stand out competitively and potentially disrupts the overall harmony. In contrast, the loss function (6) is designed to encourage overlap, and hence Predicated Diffusion adeptly depicts subjects in possession of objects. See Section B in Appendix for an ablation study. Qualitative Analysis on Complicated Prompts Figures 6, A5, and A6 show example results from Experiment (iv), along with the propositions used for Predicated Diffusion. Both the vanilla Stable Diffusion and Structure Diffusion plagued by missing objects and attribute leakage. Experiments (i) and (ii) confirmed that SynGen often generates numerous small objects. Hence, when tasked with generating “A black bird with a red beak,” it produced multiple red objects. When generating “A white teddy bear with a green shirt and a smiling girl,” comparison methods other than Predicated Diffusion often mistakenly identified the girl, not the teddy bear, as the owner of the green shirt. In comparison to the vanilla Stable Diffusion, SynGen reduced the size of the teddy bear’s shirt because it differentiates between the intensity distributions on the attention maps of different objects. A similar tendency is evident in the third case in Fig. 6, where the adjective “green” often modifies wrong objects, and the green hair is not placed on the baby’s head. Predicated Diffusion performed well in these scenarios, which include the concurrent existence of more than two objects with specified colors and possession relationships simultaneously. See Section B in Appendix for further results and discussions, where we also examined the cases of multiple colors and negation. 5 CONCLUSION This paper proposed Predicated Diffusion, where the intended meanings in a text prompt are represented using propositions using predicate logic, offering guidance for text-based image generation by diffusion models. Experiments using Stable Diffusion as a backbone demonstrated that Predicated Diffusion generates images that are more faithful to the prompt compared to other existing methods and addresses challenges observed in diffusion models: missing objects, attribute leakage, and possession failure. Moreover, due to the generality of predicate logic, Predicated Diffusion has the ability to fulfill complicated prompts that include multiple objects, adjectives, and their relationships. Although predicates cannot represent all meanings present in natural languages, they can handle most scenarios for adjusting the layout of generated images. In future work, we plan to explore the automatic extraction of propositions from prompts by syntactic parsers and explore 2-ary predicates that assert the relationships, such as Above(x, y), which implies “x is above y.” REPRODUCIBILITY STATEMENT Details on experimental settings can be found in the first subsection of Section 4. For further information, refer to Section A in Appendix. The code and pretrained models on which our experiments rely are noted in the footnotes. We also provide the experiment code as supplementary material. REFERENCES Mohammadreza Armandpour, Ali Sadeghian, Huangjie Zheng, Amir Sadeghian, and Mingyuan Zhou. Re-imagine the Negative Prompt Algorithm: Transform 2D Diffusion into 3D, alleviate Janus problem and Beyond. arXiv, 2023. Hila Chefer, Yuval Alaluf, Yael Vinker, Lior Wolf, and Daniel Cohen-Or. Attend-and-Excite: Attention-Based Semantic Guidance for Text-to-Image Diffusion Models. In ACM SIGGRAPH, 2023. Prafulla Dhariwal and Alex Nichol. 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Diederik P. Kingma and Max Welling. Auto-Encoding Variational Bayes. In International Conference on Learning Representations (ICLR), 2014. Alexander Kolesnikov and Christoph H. Lampert. PixelCNN Models with Auxiliary Variables for Natural Image Modeling. In International Conference on Machine Learning (ICML), 2017. Junnan Li, Dongxu Li, Caiming Xiong, and Steven Hoi. BLIP: Bootstrapping Language-Image Pretraining for Unified Vision-Language Understanding and Generation. In International Conference on Machine Learning (ICML), pp. 12888–12900. PMLR, 2022.
atQqW27RMQ	I can understand that the method of generating proxies using a well-trained model as a guide contains many characteristics of the majority class. However, it seems that there is a lack of analysis on how the approach of training both the classification model and the generation model together addresses the issue of imbalanced characteristics. Ultimately, it appears that the authors' idea is correct, but I'm curious about why that is the case.	GENIU: A RESTRICTED DATA ACCESS UNLEARNING FOR IMBALANCED DATA Anonymous authors Paper under double-blind review ABSTRACT With the increasing emphasis on data privacy, the significance of machine unlearning has grown substantially. Class unlearning, which involves enabling a trained model to forget data belonging to a specific class learned before, is important as classification tasks account for the majority of today’s machine learning as a service (MLaaS). Retraining the model on the original data, excluding the data to be forgotten (also known as forgetting data), is a common approach to class unlearning. However, the availability of original data during the unlearning phase is not always guaranteed, leading to the exploration of class unlearning with restricted data access, which has attracted considerable attention. While current unlearning methods with restricted data access usually generate proxy sample via the trained neural network classifier, they typically focus on training and forgetting balanced data. However, the imbalanced original data can cause trouble for these proxies and unlearning, particularly when the forgetting data consists predominantly of the majority class. To address this issue, we propose the GENerative Imbalanced Unlearning (GENIU) framework. GENIU utilizes a Variational Autoencoder (VAE) to concurrently train a proxy generator alongside the original model. These generated proxies accurately represent each class and are leveraged in the unlearning phase, eliminating the reliance on the original training data. To further mitigate the performance degradation resulting from forgetting the majority class, we introduce an “in-batch tuning” strategy which works with the generated proxies. GENIU is the first practical framework for class unlearning in imbalanced data settings and restricted data access, ensuring the preservation of essential information for future unlearning. Experimental results confirm the superiority of GENIU over existing methods, establishing its effectiveness in empirical scenarios. 1 INTRODUCTION Given the rising concerns on data privacy, and legal protections (European Parliament & Council of the European Union [BUKATY] 2019) the practice of machine unlearning (Nguyen et al., 2020; Brophy & Lowd, 2021; Sekhari et al., 2021), which allows a model to forget specific data, has become increasingly important. In specific, class unlearning has been considered significant to many real-world applications and can effectively addresses many privacy and usability needs, as classification services play an important role (Li et al., 2019; Guzella & Caminhas, 2009; Lu & Weng, 2007) in machine learning as a service (MLaaS) (Ribeiro et al., 2015). For example, in facial recognition, each individual’s face is considered as a distinct class. Thus, when a model forgets a person’s face, it essentially unlearns the class associated with that face (Masi et al., 2018). Similarly, in online shopping, products from a specific brand can be considered to all belong to an individual class – the brand. When a long-term customer of this brand loses interest, it is essential for the online shopping system to forget the customer’s preference for this brand, i.e. unlearn the class quickly. Generally, the class unlearning refers to a process of modifying or updating a well-trained model by forgetting or disregarding specific classes that it has learned previously. The data for the classes we want to forget is termed ‘forgetting data’, while the data for the classes we retain is called ‘retaining data’. A straightforward unlearning method usually retrains a new model from scratch using the original data with the forgetting data excluded. Such exact unlearning (Bourtoule et al., 2021; Chen et al., 2022; Liu et al., 2021) is widely accepted but not efficient and requires the availability of full data which is challenging in real-world, i.e., SISA (Bourtoule et al., 2021), RecEraser (Chen et al., Approximate unlearning (Thudi et al., 2022; Graves et al., 2021) is usually more efficient as it focuses on updating parameters of the well-trained model to achieve class unlearning without the retraining of a new model, i.e., the Amnesiac (Graves et al., 2021) and Unrolling (Thudi et al., 2022). They all based on a strong assumption that the original data can be fully accessed during the unlearning phase. However, such assumption cannot hold in real-world applications due to considerations of storage efficiency and privacy. For example, in data sensitive applications, the original data will be deleted after the training for preserving data privacy. Also in some streaming service scenarios, data will not be saved for a long time due to the limited storage space. To combat the unavailability of the original data, generative based approximate unlearning methods such as zero-shot (Chundawat et al., 2023) and zero-glance (Tarun et al., 2021) unlearning have been proposed. Both of these approaches limit the retention of the original training data to some extent by employing a generative approach to create a limited set of proxies for each class. The generative method must be capable of producing proxies that faithfully capture the characteristics unique to each class. In the unlearning phase, such generative methods create class proxies to facilitate forgetting, and assume balanced data to ensure accurate class representation. However, in reality, there are a lot of scenarios when data are imbalanced (Spelmen & Porkodi, 2018; Rout et al., 2018). The presence of imbalanced data can significantly affect the performance of these generative methods by leading to biased representations and inadequate coverage of minority classes, resulting in suboptimal generation of proxies for those classes. The challenge posed by imbalanced data becomes even more pronounced when examining existing approximate class unlearning methods such as (Chundawat et al., 2023; Graves et al., 2021; Tarun et al., 2021). For the generative based methods (Chundawat et al., 2023; Tarun et al., 2021), as minority-class proxy samples might unintentionally carry characteristics of the majority class, the proxy samples may not accurately reflect the class characteristics sufficiently. This causes the model to use the unreliable proxies when unlearning, resulting in the inability to unlearning effectiveness. What is more, methods (Graves et al., 2021; Tarun et al., 2021) typically involve two steps: impairment, which erases the knowledge related to the forgetting data, and repair, which aims to restore performance on the retained data. If the majority class constitutes the forgetting data and is subjected to impairment, it results in the removal of a substantial portion of the model’s task-specific knowledge, making it difficult to fully recover the performance on the remaining data. To address the challenge of handling imbalanced data in class unlearning with limited data access, this study introduces a novel generative-based class unlearning approach. To tackle the issue of inaccurate proxy brought by imbalance, we present the innovative Generative Imbalanced Unlearning (GENIU) framework. Different with prior researches (Chundawat et al., 2023; Tarun et al., 2021), we leverage a generator structured with a Variational Autoencoder (VAE) (Kingma & Welling, 2014) which trained concurrently with the original model to produce reliable proxies for each class. Since the unlearning method cannot access data samples from original dataset, we employ carefully crafted noise samples, one for each class, as proxy generating prompts and will be stored for generating proxy with the trained generator in the unlearning phase. These noise samples are determined as designed class representations by the original model and rendered indistinguishable from human-generated data. This approach enhances privacy by thwarting attempts to recover features associated with the forgotten class. To further mitigate the adverse effects of unlearning the majority class on model performance, we introduce in-batch tuning. This technique simultaneously considers impairment and repair as a unified objective during the updating of the original model, contributing to a more effective and seamless unlearning process. Our contributions can be summarized as: 1) We are the first to explore the challenges presented by the application of data access restricted class unlearning methods within an imbalanced data setting. To the best of our knowledge, the proposed GENIU is also the first non-retrain-based unlearning framework for imbalanced data. 2) GENIU train the proxy generator and the original model at the same time, which ensures the generated proxy adequately represents its corresponding class by avoiding the minority class proxies unintentionally carrying the characteristics of the majority class. We also innovatively propose the in-batch tuning strategy during the unlearning phase to further mitigate the negative effect on the model performance as forgetting the majority class. 3) Through experimental results, we illustrate that existing unlearning methods, which restrict access to historical training data, struggle to perform well in an imbalanced data context. In contrast, GENIU shows superior performance over these baselines when tested on several widely used datasets, with high efficiency in both storage and time. 2 RELATED WORKS Machine unlearning. Machine unlearning (Cao & Yang, 2015; Baumhauer et al., 2022; Nguyen et al., 2022a) is a new machine learning paradigm which allows data owners to completely delete their data from a machine learning model and enable their “right to be forgotten”. Many existing unlearning works (Baumhauer et al., 2022; Brophy & Lowd, 2021; Cauwenberghs & Poggio, 2000; Chen et al., 2019; Mahadevan & Mathioudakis, 2021; Li et al., 2021) have found analytical optimization solutions by identifying the impact of data on model for traditional machine learning models, however, these unlearning methods are only suitable for machine learning methods with a convex problem nature. For deep neural networks in unlearning (Nguyen et al., 2022b), the non-convex nature of the problem and the stochasticity of the learning process have become the challenges which makes it hard to model the impact of data on the trained model and further eliminate such impact from model. A straightforward approach is to retrain a new model from scratch with a dataset that has no forgetting data. However, this retraining method is time-consuming, requires numerous data storage, and is infeasible when original training data is unavailable. To speed up the retraining process, SISA (Bourtoule et al., 2021) splits the complete dataset into several partitions and trains a model for each partition, thus it only needs to perform retraining on partitions that was containing unlearned data. Similar methods have been applied in recommender system (Chen et al., 2022) and federated learning (Liu et al., 2021) scenarios and this type of retrain-based method can be categorized as exact unlearning. Another type of method that requires no retraining of a new model from scratch is called approximate unlearning. The approximate unlearning can makes the parameters of the unlearned model closer to that of the retrained model by updating the original model for a few rounds. The Unrolling SGD (Thudi et al., 2022) and Amnesiac unlearning (Graves et al., 2021) record the changes of the parameter during the training of the data to be unlearned and recovers these changes during unlearning. However, all these methods require full access to the historical training data which cannot be satisfied in many real practices. Data restricted unlearning methods. Most training data are often deleted or archived post-training due to storage costs and privacy concerns. Storing large amounts of data is expensive and poses security risks, especially with sensitive information. Data breaches or unauthorized access can lead to legal, ethical, and reputational consequences. Therefore, in a wider range of real practices, the unlearning method has no access to full or even partial of the historical training data. The zero-glance and zero-shot unlearning settings take such restrictions into account. The former can only access the retaining data in unlearning phase, while the latter is more strict and requires no access to any original data. The solutions corresponding to them, UNSIR (Tarun et al., 2021) and GRT (Chundawat et al., 2023) respectively, adopt the idea of generating proxies for the training data to provide a basis for unlearning. Detailedly, they use the well-trained classification model to generate proxies for inaccessible data, then use these proxies to represent actual data and perform unlearning. Therefore, these proxies trained through the knowledge of the well-trained classifiers are critical for unlearning. However, they both assume that the data used to train the original model is balanced. Due to data imbalance, the knowledge of the classifier can be biased, which in turn affects the generated proxies. Imbalanced data poses significant challenges to these generative-based methods as they may produce proxy samples for minority classes that inadvertently carry majority class traits, leading to unreliable unlearning. Learning and unlearning from imbalanced data. An imbalanced dataset considers when there are some classes containing considerably more amount of samples (majority) than other classes (minorities). Learning from such an imbalanced dataset can make the predictions of minority classes inaccurate (Spelmen & Porkodi, 2018; Rout et al., 2018). An existing work (Koch & Soll) investigated the impact of imbalanced class setting on SISA (Bourtoule et al., 2021) unlearning method, when full original data is accessible during the unlearning phase they found that the imbalance in each data shard will lead corresponding retraining model unreliable. For example, in the case of imbalanced data, when the data is divided into various shards, some shards may be composed of the majority class or contain only a few samples of other classes. This will cause the model trained on this shard lacks or even has no data when retraining. This impact is more severe when access to training data is restricted, as less learning material is available for model retraining. 3 PRELIMINARIES AND PROBLEM FORMALISATION In this section, we will first introduce preliminary notations and terms, i.e., class unlearning and imbalanced unlearning, and then formalise the problem of this work at the end of this section. Class unlearning. Let $ D = \{(x_i, y_i)\}_{i=1}^n \in X \times Y $ be a dataset containing $ n $ data samples that belong to $ K $ classes. The $ i $-th pair of the data sample and its associated label can be denoted as $(x_i, y_i)$, where $ x_i \in X \subseteq \mathbb{R}^d $ and $ y_i \in Y = \{1, \ldots, K\} $. We denote $ D^k = \{(x_i, y_i)\|y_i = k\} $ as a subset of $ D $ that contains samples of the $ k $-th class. When a class unlearning request is issued, it requires the classifier to forget knowledge on the forgetting class $ Y_f $ and maintain knowledge learned on the retain class $ Y_r $, where $ Y_f, Y_r \subset Y, Y_f \cap Y_r = \emptyset $ and $ Y_r \cup Y_f = Y $. Then, we can further denote their corresponding dataset $ D_f = \{(x_i, y_i)\|y_i \in Y_f\} $ and $ D_r = \{(x_i, y_i)\|y_i \in Y_r\} $, where $ D_f \cup D_r = D $ and $ D_f \cap D_r = \emptyset $. A deep learning neural network $ f(x, \theta) $, which is parameterized by $ \theta $, can output a vector $ p \in [0, 1]^K $, where the $ j $th element of $ p $ represents the posterior probability of the $ j $th label given $ x $, i.e., $ p_j $ is interpreted as $ P(y = j\|x) $. In the context of unlearning, an original model $ f(\cdot, \theta_{or}) $ is trained with $ D $. A retrained model $ f(\cdot, \theta_{re}) $ is trained with $ D_r $. An unlearning method $ U $ is expected to make $ f(\cdot, \theta_{or}) $ forget the knowledge about $ D_f $ and output an unlearned model $ f(\cdot, \theta_{un}) $ which has the similar performance as a retrained model, i.e., $ f(\cdot, \theta_{un}) \approx f(\cdot, \theta_{re}) $. In retrain-based methods (Bourtoule et al., 2021; Chen et al., 2022; Liu et al., 2021), the unlearned model $ f(\cdot, \theta_{un}) $ is directly retrained with $ D_r $. However, as discussed above, they are computationally cost and infeasible when original training data is unavailable as retraining requires access to numerous training data to train a new model from scratch. Non-retrain methods (Thudi et al., 2022; Tarun et al., 2021; Chundawat et al., 2023), although more efficient, still assume that original data can be accessed when performing unlearning, i.e., \[ f(\cdot, \theta_{un}) = U(D, f(\cdot, \theta_{or})). \] (1) Imbalanced unlearning. In the imbalanced unlearning setting, we assume the complete dataset $ D $ is imbalanced and contains a set of majority class, i.e., $ Y_m $. Then, we have $ D_m = \{(x_i, y_i)\|y_i \in Y_m\} $ that contains data of a majority class. We also have $ D_l = \{(x_i, y_i)\|y_i \notin Y_m\} $ that contains data of a class other than majority class. To facilitate the control of the variables, without special instructions, we assume all minority class have similar number of data and the number is far less than that of the majority class data. Then we have \[ \|D^{k_1}\| \gg \|D^{k_2}\| \quad \forall k_1 \in Y_m, \forall k_2 \notin Y_m \text{ and } \|D^{k_3}\| \approx \|D^{k_4}\| \quad k_3 \neq k_4, k_3 \notin Y_m, k_4 \notin Y_m \] (2) The imbalance rate can be denoted as $ r = \|D^{k_1}\|/\|D^{k_2}\| $, where $ k_1 \in Y_m, k_2 \notin Y_m $. In this work, we assume that $ D_f $ contains one or more majority classes, that is $ D_m \subseteq D_f $, which also means the unlearning request asks the model to forget the majority class(es). Target problem: class unlearning with restricted data access and imbalanced data setting. Full access to $ D $ in the Eq[1] cannot be satisfied in many practical cases. Therefore, we follow the generative-based unlearning pipeline (Chundawat et al., 2023), which does not require the original training data and is applicable to a wider range of scenarios, using a set of generated proxy data $ D_p $ to provide approximate information about data features and make unlearning feasible. We need to design an unlearning method $ U $ that, upon receiving an unlearning request which requires the forgetting of a majority class, i.e., the $ k $-th class, is able to take the original model $ f(\cdot, \theta_{or}) $ as input and output an unlearned model $ f(\cdot, \theta_{reun}) $ without using any data in $ D $, such that $ f(\cdot, \theta_{un}) $ is able to perform similarly to a model $ f(\cdot, \theta_{re}) $ retrained on data without the $ k $-th class, i.e., the $ D_r $. \[ f(\cdot, \theta_{un}) = U(D_p, f(\cdot, \theta_{or})). \] (3) It is noteworthy that, unlike generative based unlearning, we aim at the situation where the $ f(\cdot, \theta_{or}) $ is learned from an imbalanced data distribution. It can be inferred from the Eq[3] that the proxy set $ D_p $ is critical for unlearning, and existing generative methods cannot generate $ D_p $ well enough in the situation of imbalanced data. 4 Our Method We show an overall view of GENIU in Figure 1. There are two main phases in GENIU i.e. the training phase and the unlearning phase. In the context of imbalanced data and no access to actual data samples, if the generator were trained by the well-trained $ f(\cdot, \theta_{or}) $ after the training phase, as existing works have done, the generated proxies cannot accurately represent the characteristic of its designed class, because most of the knowledge of $ f(\cdot, \theta_{or}) $ comes from the majority class and the generator learns some biased knowledge. Therefore, we need to record the correct feature when actual samples appear. We train and store the noise samples $ \{z_i\}_{i=1}^K $ (one for each class) and a generator $ g(\cdot, \phi) $ in the training phase to preserve valuable information about the features of the samples for proxy generating. In the unlearning phase, both $ z $'s and $ g(\cdot, \phi) $ will work together to generate reliable proxy samples, then a proposed in-batch tuning method will leverage these proxies to update the $ f(\cdot, \theta_{or}) $. This is a softer update method, other than the existing impair-repair update, that can eliminate the performance deduction on other knowledge when the model forgets most of the knowledge under the imbalanced unlearning problem. In the following subsections, we are going to detail these shown components one by one. Then, we provide the algorithms for both training and unlearning phase of the proposed GENIU. 4.1 Proxy Generator Under conditions of no access to original data, we need to generate proxies for original data to provide the information for unlearning. Considering the imbalanced data, existing proxy generating methods, which directly use the $ f(\cdot, \theta_{or}) $ as a guider and update a random noise sample with minimum error target, cannot get the proxies that can correctly express the characteristics of designed classes. Variational Autoencoders (VAE) (Kingma & Welling, 2014) is an impressive technology, in which the decoder can reconstruct a sample by giving a latent code and making the reconstructed sample $ x' $ (also named as proxy in this work) look like a data sample in the training set. However, the belonged class of $ x' $ depends on the given latent code. To generate data belonging to a particular class, the latent code needs to be specified. That is if we want to get a proxy $ x'_i $, where $ \{(x'_i, y'_i)\|y'_i = k\} $, an ideal way is taking a real sample $ x_i $ whose associated label $ y_i = k $ as input of the generator’s encoder and naturally get the appropriate code for the decoder. But it is infeasible when original data is unavailable. Therefore, we introduce a VAE structure as the proxy generator $ g(\cdot, \phi) $ (Figure 2) and feed a carefully designed noise $ z $ as a prompt for proxies’ generating. The generating processing can be formalized as $ x' = g(z, \phi) $, where $ z $ is the carefully designed noise which can be determined as a designed class by $ f(\cdot, \theta_{or}) $ and will be detailed in Section 4.2. It is difficult to train the $ g(\cdot, \phi) $ in the unlearning phase, because the knowledge of the $ g(\cdot, \phi) $ cannot be accurately obtained in the unlearning phase as there are no samples available that can accurately describe the class characteristics. Thus, we intend to train the generator in the training phase alongside the training of $ f(\cdot, \theta) $. Detailedly, given a set of noise $ D_z = \{(z_k, y_k)\|y_k = k\}_{k=1}^K $ which contains only $ K $ pairs of noise and label, and a set of selected samples from training dataset... Figure 2: The proxy generator $ g(\cdot, \phi) $ used in GENIU. $ D_s = \{(x_k, y_k) \| y_k = k\}_{k=1}^K $. The reconstruction loss $ L_{rec} $ can be defined as \[ L_{rec} = \frac{1}{K} \sum_{k=1}^{K} \\|g(z_k, \phi) - x_k\\|. \] (4) To make the learned Gaussian distribution more accurate, a distribution loss $ L_{dis} $ can be defined as \[ L_{dis} = \frac{1}{2K} \sum_{k=1}^{K} \sum_{j=1}^{l} (1 + \log((\sigma_j^k)^2 - (\mu_j^k)^2)) \] (5) where the $ \mu \in \mathbb{R}^l $ and $ \sigma \in \mathbb{R}^l $ are learnable gaussian distribution parameters for modeling the latent code, and the $ l $ is the dimension of the latent code. Finally, the overall objective of learning generator $ g(\cdot, \phi) $ is \[ \min_{\phi} L_{gen} = L_{rec} - \lambda L_{dis} \] (6) where $ \lambda $ is a hyperparameter that used to trade-off the impact of $ L_{rec} $ and $ L_{dis} $. Optimizing Eq. 6 could give the generator. The details on how to select $ x_k $'s will be introduced in Section 4.4. 4.2 TRAINING THE NOISE PROMPT. To avoid using a historical data sample as a guide to reconstruct a proxy samples of a specific class in the unlearning phase, we intend to train a noise $ z_k $ as the prior knowledge for constructing a proxy sample of the specific class $ k $. Specifically, the trained noise $ z_k $ should be correctly determined as the interesting class $ k $ by the classifier $ f(\cdot, \theta) $, that is $ y_k = f(z_k, \theta) $ and $ y_k = k $. To achieve this goal, we update a randomly initialized noise $ z_{init} $ by minimizing the classification error, which satisfies \[ z_{init} \in \mathbb{R}^d \sim \mathcal{N}(0, 1) \in \mathbb{R}^d. \] (7) The optimization objective of noise $ z_k $ is basically the original task objective. For the classification task, this objective should be \[ z_k = \min_z \text{CrossEntropy}(f(z, \theta), y_k), \quad y_k = k. \] (8) In this work, we use the Adam optimizer (Kingma & Ba, 2015) to update the randomly initialized noise $ z_{init} $ according to the objective equation [8]. It is worth noting that noise and classifier are updated independently, and the training of noise will not affect the training of the classifier. 4.3 IN-BATCH TUNING FOR UNLEARNING. To further mitigate the performance degradation of the model by forgetting the majority class in the imbalance unlearning, we make the mini-batch of each unlearning step containing proxies of each class. It is noteworthy that, in the unlearning phase, we need only one mini-batch which includes $ K $ proxies. A proxy $ x'_k $ is generated by $ g(\cdot, \phi) $ with a given trained noise $ z_k $. Therefore, the dataset used for unlearning is \[ D_u = \{(x'_k, y_k) \| y_k = k\}_{k=1}^K, \quad \text{where } x'_k = g(z_k, \phi). \] (9) In the process of model tuning, we hope that the proxies that need to be unlearned can make the model change in the direction of increasing error, and the proxies that need to be retained can make the model continue to change on the direction of reducing error. In consideration of this, we design the following loss \[ L_u = \sum_{(x'_k, y_k) \in D_u, y_k \in Y_r} L(f(x'_k, \theta), y_k) + \sum_{(x'_k, y_k) \in D_u, y_k \in Y_f} \frac{1}{L(f(x'_k, \theta), y_k)}; \] (10) where the used loss $ L(\cdot, \cdot) $ should be the same as the loss on which the original model is trained. 4.4 Supervision sample selection. Since we use a tuning style method to perform unlearning only with generated proxies $ x' $, if the $ x' $ can be correctly classified by $ f(\cdot, \theta) $ with high confidence, the tuning step would be small, since in this situation the $ x' $ is away from the decision boundary and results in a small value of classification loss. Therefore, we prefer the selected supervision samples $ x_k $ (Eq.4) near to the decision boundary. Specifically, we select an $ x $ with maximum logit entropy for each class. The logits entropy $ E(x) $ can be calculated as $ E(x) = -\sum_{k=1}^{K} p_k \cdot \log(p_k) $, where the $ p_k $ is the output probability of $ x $ belonging to the $ k $-th class. The higher the $ E(x) $, the closer each probability in $ p $ and also the higher the uncertainty of determining $ x $. Therefore, to supervise the training of $ g(\cdot, \phi) $, we need a set of supervision samples $ D_s $, whose items are selected as $ x_k = \max_{x_i \in D} E(x_i) $ and $ y_i = k $. 4.5 GENIU algorithm. The proposed GENIU is divided into training phase and unlearning phase. During the training phase (Appendix. A Algorithm. 1), the classifier $ f(\cdot, \theta) $ will be trained normally. In each epoch of $ f(\cdot, \theta) $ training, additional training on noise $ z $'s is performed. If the trained noise $ z $'s in an epoch can be correctly classified by $ f(\cdot, \theta) $, these noises will be used together with the selected sampled $ x $'s to train the generator $ g(\cdot, \phi) $, otherwise the training of the generator will be skipped in this epoch. In the unlearning phase (Appendix. A Algorithm. 2), only the trained noise $ z $'s and generator $ g(\cdot, \phi) $ will be used. The generator will reconstruct $ z $ into proxy $ x' $, and then the in-batch tuning will use these proxies to adjust $ f(\cdot, \theta_{or}) $ and finally output the unlearned model $ f(\cdot, \theta_{un}) $. 5 Experiments Datasets. We evaluate the effectiveness of the proposed GENIU on four benchmark datasets, i.e., Digits-MNIST (LeCun et al., 1998), Fashion-MNIST (Xiao et al., 2017), Kuzushiji-MNIST (Clanuwat et al., 2018) and CIFAR-10 (Krizhevsky et al., 2009). Detailedly, all these three MNIST style dataset contains 60,000 samples in their training set and 10,000 samples in their test set. Each sample of these MNIST style dataset is a 28 × 28 grayscale image associated with a label from ten classes. In the Digits-MNIST, the classes are handwritten digits from 0 to 9. In the Fashion-MNIST, the classes are ten different fashion items (i.e. T-shirts, shoes). In the Kuzushiji-MNIST, the classes are ten different Hiragana characters. CIFAR-10 contains 50,000 training samples and 10,000 test samples each of which is an RGB image in the shape of 32 × 32 and associated with one of ten semantic classes. To make the imbalanced dataset, we set the imbalance rate $ r = 0.1 $ in this work. Specifically, we keep the number of samples of majority class the same as the raw dataset and select 10% samples for each of the minority classes. Baselines. We conduct comparison experiments on two types of methods, one can access the original data that includes I-R (Graves et al., 2021) and Unrolling SGD (Thudi et al., 2022), the other cannot access the original data that includes GKT (Chundawat et al., 2023) and UNSIR (Tarun et al., 2021). In expectation, methods which can access training data should have better performance than methods cannot access training data. Specifically, 1) I-R (Graves et al., 2021). Amnesiac records the changes of the parameter during the training of the data to be unlearned and recovers these changes during unlearning. 2) Unrolling SGD (Thudi et al., 2022). In unlearning phase, it arranges forgetting data in the first batch and performs incremental training with both unlearned training data and retain training data. It records gradients when learning the first batch and adds recorded gradients on weights after the incremental training. 3) GKT (Chundawat et al., 2023), which is the SOTA zero-shot unlearning method. The GKT generates the error maximized noise to proxy $ D_f $ and generates error minimized noise to proxy $ D_r $. Then, it initializes a new network called the student and teaches the student with the original model. 4) UNSIR (Tarun et al., 2021), which is the SOTA zero-glance unlearning method. It generates the error maximized noises to proxy $ D_f $ and mixes these noises with a part of $ D_r $. Then, it performs impair-repair steps to tune the original model. Implementation details. For all experiments, we use the AllCNN (Springenberg et al., 2015) as the base classification model as it has been widely used for image data and been used by baselines. Following the baselines’ setting, the training batch size for all dataset is set as 256, and the learning rate and weight decay are set as 0.01 and $10^{-4}$, respectively. We also follow the default setting in the VAE (Kingma & Welling [2014]) and set the learning rate for training of noise $z$ and generator $g(\cdot, \phi)$ as 0.02 and 0.005, respectively, and $\lambda = 2.5 \times 10^{-4}$ (Eq.6). Then, 1) for all MNIST style datasets, in the training phase, we train the AllICNN for 20 epochs, and train the initialized noise $z$ as well as the generator $g(\cdot, \phi)$ for 100 steps in each epoch. In the unlearning phase, we conduct in-batch tuning for 100 rounds. 2) For CIFAR-10, in the training phase, we train the AllICNN for 40 epochs, and train the initialized noise $z$ for 100 steps and the generator $g(\cdot, \phi)$ for 200 steps in each epoch. In the unlearning phase, we conduct in-batch tuning for 45 rounds. For all dataset, in the unlearning phase, we set the learning rate for tuning $f(\cdot, \theta_{or})$ as $4 \times 10^{-4}$. In the generator, we set a CNN structure with increasing channels for the encoder, i.e. [32, 64, 128, 256], and the decoder is a CNN structure symmetrical to the encoder. The dimension of the latent code is 128 for MNIST style dataset and 256 for CIFAR-10. All other parameters of baseline methods follow their default settings. All the experiments are conducted with NVIDIA RTX A5000 GPU and the reported results are the average of five trials of experiments using different seeds. ### 5.1 Results and Analysis Table 1: Unlearning performance. The direction of the arrow indicates the desired direction of value change. The up arrow means higher is better, the down arrow means lower is better. \| Dataset \| Acc \| Original Model \| Retrain Model \| GKT \| UNSIR \| GENIU (ours) \| I-R \| Unrolling \| \|-----------\|-----\|----------------\|---------------\|-----\|-------\|--------------\|-----\|-----------\| \| D-MNIST \| $D_r \uparrow$ \| 0.9494 \| 0.9405 \| 0.4116 \| 0.3502 \| 0.9286 \| 0.9766 \| 0.8555 \| \| \| $D_f \downarrow$ \| 0.9913 \| 0.0 \| 0.0258 \| 0.0001 \| 0.0065 \| 0.0 \| 0.1466 \| \| F-MNIST \| $D_r \uparrow$ \| 0.8057 \| 0.16 \| 0.2595 \| 0.3002 \| 0.712 \| 0.8368 \| 0.7571 \| \| \| $D_f \downarrow$ \| 0.9681 \| 0.0 \| 0.0 \| 0.0016 \| 0.0002 \| 0.0 \| 0.0106 \| \| K-MNIST \| $D_r \uparrow$ \| 0.8172 \| 0.3641 \| 0.3537 \| 0.2566 \| 0.7012 \| 0.8788 \| 0.8073 \| \| \| $D_f \downarrow$ \| 0.9764 \| 0.0 \| 0.0029 \| 0.0 \| 0.0004 \| 0.0 \| 0.0550 \| \| CIFAR-10 \| $D_r \uparrow$ \| 0.5952 \| 0.6347 \| 0.273 \| 0.1778 \| 0.4948 \| 0.4838 \| 0.3971 \| \| \| $D_f \downarrow$ \| 0.9452 \| 0.0 \| 0.0 \| 0.0327 \| 0.0103 \| 0.0 \| 0.0136 \| Effectiveness. We conduct unlearning experiments with each class as forgetting class (majority class) on each dataset and report the mean accuracy performance in Table 1. From the performance of the original model on $D_r$ and $D_f$, it can be seen that the imbalanced dataset will cause a corresponding imbalance in the performance of the original model. The model will perform significantly better in the majority class than in other classes. Among all methods with limited access to original data, the proposed method GENIU performs best. GKT and UNSIR, relying on the original model for proxy generation, their generated proxies are affected by this imbalance, impacting unlearning quality. I-R and Unrolling, with full historical data access, generally outperformed GENIU, but GENIU showed better results on CIFAR-10. Detailed results from Fashion-MNIST (Appendix B, Table 6) demonstrate unlearning performance when each class is the majority. Further tests with multiple classes as majority for deletion (0-th and 1-st classes) also confirm these findings, as reported in Appendix C (Table 7). Why existing generative based unlearning methods failed with imbalanced data? To further prove that GENIU can obtain more reliable noise in the case of imbalance data, we try to observe the origin model’s perception on noises generated by different methods. Intuitively, the noise generated by leveraging the well-trained $f(\cdot, \theta_{or})$ will have the characteristics of the majority class, since the knowledge from majority class dominates the model. Therefore, the origin model’s perception of noise of other classes will be closer to that of the majority class. Specifically, the distribution of the model’s logits output of minority classes will be closer to that of the majority class. To verify this, we sample some training examples of the majority class and feed them to $f(\cdot, \theta_{or})$ to obtain reference perception $p_{ref}$, which is basically the output logits. Then, we feed the noise of other classes generated by different unlearning methods to $f(\cdot, \theta_{or})$ to obtain observation perception $p_{obs}$. We then try to fit the distribution of the reference perception with the distribution of these observation perceptions, which is a common use of KL divergence $D_{kl}(p_{obs}\|\|p_{ref})$, to observe the difference between the $p_{obs}$ and the $p_{ref}$ of different methods. According to the property of KL divergence, the greater the $D_{kl}(p_{obs}\|\|p_{ref})$, the more significant the difference between the model’s perception of generated noise and its perception of the majority class, that is, the better. Since UNISR only generates noise for the forgetting class and does not generate noise for other classes, here we only compare the GKT and GENIU methods. From Table 2, we can observe that when producing noise... for the four data sets, the $D_{kl}(p_{obs}\|\|p_{ref})$ of the noise generated by GENIU is greater than that of GKT. This shows that the origin model’s perception of the noise generated by GKT is closer to the majority class, and it carries more characteristics of the majority class than the noise generated by GENIU. We also reconstruct more specific proxies for GKT by using its generated noises and the trained VAE of GENIU. Since the noise generated by GKT carries more features of the majority class, these reconstructed proxies will make these features more specific. As can be seen from Table 3, it is more difficult for GKT to use such reconstructed proxies to eliminate the knowledge of the majority class. Some visualized samples are provided in Appendix D. Table 2: Comparing origin model’s perception of noise generated by different methods in $D_{kl}(p_{obs}\|\|p_{ref})$. \| Noise Generator \| D-MNIST \| F-MNIST \| K-MNIST \| CIFAR-10 \| \|-----------------\|---------\|---------\|---------\|----------\| \| GKT \| 11.7565 \| 11.4835 \| 12.6639 \| 12.2472 \| \| GENIU \| 12.2526 \| 11.8418 \| 13.2708 \| 12.9941 \| Table 3: Reconstruct proxy with noise generated by existing method. \| Method \| Acc_D-MNIST \| Acc_F-MNIST \| Acc_K-MNIST \| Acc_CIFAR-10 \| \|--------------\|-------------\|-------------\|-------------\|--------------\| \| GKT_vac \| 0.6115 \| 0.4854 \| 0.269 \| 0.1429 \| \| GENIU \| 0.9286 \| 0.7711 \| 0.7012 \| 0.4948 \| Table 4: Time cost in unlearning phase. \| Dataset \| Time cost \| GKT \| UNSIR \| GENIU \| I-R \| Unrolling \| \|-------------\|-----------\|-----\|-------\|-------\|-----\|-----------\| \| D-MNIST \| ms \| 39086 \| 1804 \| 326 \| 17005 \| 483 \| \| F-MNIST \| ms \| 39702 \| 1854 \| 327 \| 16848 \| 608 \| \| K-MNIST \| ms \| 37312 \| 1758 \| 330 \| 16254 \| 411 \| \| CIFAR-10 \| ms \| 33633 \| 2515 \| 159 \| 16601 \| 195 \| Unlearning efficiency. We compare the time consumption among various unlearning methods. Experiments were conducted under identical conditions, measuring the time in milliseconds from inputting the original model $f(\cdot, \theta_{or})$ to outputting the unlearned model $f(\cdot, \theta_{un})$. Results in Table 4 show that GENIU is more time-efficient in the unlearning phase, as it doesn’t require training a generation network and only uses a small number of proxies equal to the class count for adjustments. Regarding storage costs, retaining original data for a MNIST-like dataset needs 45MB, and CIFAR-10 needs 169MB. But storing a generator instead requires only 4.6MB for MNIST and 6.1MB for CIFAR-10. Ablation studies We also conduct ablation studies to assess the impact of different technologies on GENIU. It starts by evaluating two main techniques, as shown in Table 5. Further investigations focus on the type of supervision sample selection and the number of in-batch tuning rounds, with findings and analyses detailed in Appendices G.2 and G.3. The study examines how two technical components affect unlearning performance: 1) training a proxy generator alongside the original model, and 2) in-batch tuning during the unlearning phase. It compares the first with post-training generated proxies and the second with an impair-repair process, using identical learning rates and rounds. From the results which are reported in Table 5 when the proxy generated by the GENIU framework is applied, the impair-repair process will first forget the knowledge related to the majority class, however, this part of knowledge is most of the knowledge of the model about the classification task and makes the model hard to maintain the performance on retain classes in the subsequent repair stage. Additionally, when using post-training generated proxies, the imbalance in original training data causes these proxies to exhibit characteristics of the majority class, reducing the model’s ability to distinguish between classes to be retained after forgetting the majority class. 6 CONCLUSION In this work, we explore the challenges presented by the applications of restricting data access unlearning methods within an imbalanced data setting. The proposed framework, Generative Imbalanced Unlearning (GENIU) offers an effective solution to these challenges. GENIU requires neither training a new model from scratch nor access to any historical training data. The unique approach of training the proxy generator and the original model concurrently ensure the proxies accurately represent their corresponding classes. The in-batch tuning strategy that we introduce in the unlearning phase effectively mitigates the performance degradation as the model unlearns the majority class. 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auguNUCto5	I still have some doubts regarding the use of RNN-TCN for extracting global information. Both GCN and TGN employ similar information aggregation approaches, aggregating nodes up to n-hops away. Why is RNN-TCN considered to be more effective in representing global information?	Boosting Temporal Graph Learning From Global and Local Perspectives Anonymous authors Paper under double-blind review Abstract Extensive research has been dedicated to learning on temporal graphs due to its wide range of applications. Some works intuitively merge GNNs and RNNs to capture structural and temporal information, while recent works propose to aggregate information from neighbor nodes in local subgraphs based on message passing or random walk. These methods produce node embeddings from a global or local perspective and ignore the complementarity between them, thus facing limitations in capturing complex and entangled dynamic patterns when applied to diverse datasets or evaluated by more challenging evaluation protocols. To address the challenges, we propose the Global and Local Embedding Network (GLEN) for effective and efficient temporal graph representation learning. Specifically, GLEN dynamically generates embeddings for graph nodes by considering both global and local perspectives. Then, global and local embeddings are elegantly combined by a cross-perspective fusion module to extract high-order semantic relations in graphs. We evaluate GLEN on multiple real-world datasets and apply several negative sampling strategies. Sufficient experimental results demonstrate that GLEN outperforms other baselines in both link prediction and dynamic node classification tasks. 1 Introduction Graph representation learning (Hamilton et al., 2017b; Battaglia et al., 2018) has attracted tremendous research interest in both academic (Perozzi et al., 2014; Tang et al., 2015) and industrial (Wang et al., 2018; Rossi et al., 2019) communities owing to its powerful capabilities of mining and discovering abundant information in the non-Euclidean space (Asif et al., 2021; Wu et al., 2020). However, general methods consider the graphs to be static. Nothing is eternal except change itself. In the real world, most graph systems are usually dynamic and constantly change over time, making temporal graphs ubiquitous (Longa et al., 2023; Ma et al., 2020). In such temporal graphs, the topologies of networks evolve as nodes and edges appear or disappear across different timestamps, along with the attributes of nodes and edges changing dynamically (Zhu et al., 2022; Du et al., 2018). Learning on temporal graphs has received substantial research attention (Kazemi et al., 2020) since the ability to process dynamic networks can be useful for a wider range of scenarios like recommender systems (Wang et al., 2021a; Zhang et al., 2021), biology and medicine (Loo et al., 2023; Lim et al., 2019), traffic forecasting (Zhao et al., 2019), pandemic forecasting (Panagopoulos et al., 2021), etc. There has been a surge of solutions for temporal graph learning (Souza et al., 2022; Wang et al., 2021c; Rossi et al., 2020). Many works sophisticatedly combine graph neural networks (GNNs) (Kipf & Welling, 2016; Velickovic et al., 2017) and recurrent neural networks (RNNs) (Medsker & Jain, 1999) to obtain structural and temporal information. These GNN-RNN methods take the entire graph at each time step as the input of GNNs and dynamically update the weight parameters of GNNs (Pareja et al., 2020; Chen & Hao, 2023) or the node features (Liu et al., 2020; Chen et al., 2022) through RNNs. Whereas temporal graph networks (TGNs) (Souza et al., 2022) generate dynamic node representations through aggregating temporal subgraphs triggered by events. Such approaches utilize memory modules with message passing mechanism (MP-TGNs) (Xu et al., 2020; Rossi et al., 2020) or aggregate temporal walks (WA-TGNs) (Wang et al., 2021c; Bastas et al., 2019b). GNN-RNN methods model temporal graphs from a global perspective, but lack the information of micro variation. TGNs obtain the features of each node by aggregating information from a limited neighboring region without perceptions of global structural dependencies. The unidimen- Figure 1: In the temporal graph example (a), edges $e_{AB}$ and $e_{AC}$ occur at $t_1$, while $e_{CD}$ occurs at $t_2$. Different methods (b) and (c) produce node embeddings in different ways and have different properties. Node embeddings are used to capture the correlation between nodes, so as to make predictions (e.g., whether nodes $B$ and $D$ will interact at $t_3$). The functionality of the aforementioned methods could result in less accurate inferences (Lu et al., 2019). In this paper, we present that modeling temporal graphs from both global and local perspectives is advantageous (Jin et al., 2019) in the following aspects. The neighborhood information gathered in two ways has certain complementarities. Different methods retain or discard different neighborhood information. As shown in Figure 1, GNN-RNN methods (Pareja et al., 2020; Liu et al., 2020; Manessi et al., 2020) retain all events of each time step without filtering, so that all edges are fully utilized when generating node embeddings, but noisy or useless edges are also retained. As for TGNs methods, the neighborhood size is generally limited by a given constant via sampling operations (Rossi et al., 2020; Wang et al., 2021c; Zheng et al., 2021). Sampling of the temporal neighbors provides the opportunity to avoid noisy and irrelevant edges but may cause some interactions to be ignored or futilely reused when updating the states of nodes. The temporal information acquired from two perspectives complements each other. The diversity of graph topologies across different domains leads to the complexity of temporal properties. Due to the regularity and abruptness of events, the pattern of events can also vary across time. Therefore, modeling at different time granularities have to be taken into account. RNNs learn the evolution patterns between adjacent graph snapshots at a coarse level. In contrast, MP-TGNs and WA-TGNs encode timestamps simultaneously while aggregating neighborhood contextual information (Xu et al., 2020; Rossi et al., 2020; Wang et al., 2021c). These two types of approaches model the temporal relevance of event occurrence in different forms and with different granularities as indicated in Figure 1, thus the acquired temporal information can complement each other. The two types of methods retain different integrities of graph structures. Since the endogenous and exogenous factors driving the generative process of networks are frequently complex and variable, temporal graphs across diverse domains tend to exhibit a variety of properties (Zheng et al., 2021). For instance, social networks and international trade networks may have extremely different characteristics (e.g., varying sparsities and edge recurrence patterns) (Pourasfæi et al., 2022). GNN-RNN methods with the global perspective are more likely to consider the overall nature of a temporal graph since GNNs maintain the complete graph structure at different time steps. In contrast, fine-grained patterns in motifs (Paranjape et al., 2017; Liu et al., 2021) such as the triadic closure process (Zhou et al., 2018; Liu et al., 2022) are better reflected in the encoding of local subgraphs by TGNs. Based on the aforementioned insights, we propose GLEN[^1] (short for Global and Local Embedding Network) to learn representations for temporal graphs by considering both global and local perspectives. Our method fills the research gap in existing temporal graph methods that only focus on one perspective and highlights the importance of considering both perspectives. Unlike conventional global-view methods that model sequences using RNNs, we employ a temporal convolution network (TCN) for more efficient and stable training. From the local perspective, we devise a weighted sum algorithm based on time interval to distinguish the impact of events at different time. Since neither GNN-RNN methods, MP-TGNs, nor WA-TGNs can extract high-order features in graphs (Mao et al., 2023; Xu et al., 2018; Talati et al., 2021), simply fusing embeddings of two perspectives via summation or concatenation is empirically less than ideal. To tackle this issue, we devise a cross-perspective fusion module for GLEN to combine the node features embedded from global and local perspectives. The fusion module employs a devised attention mechanism to capture the semantic relevance between each two nodes’ global and local embeddings. We summarize our contributions as follows: [^1]: GLEN is available at [https://anonymous.4open.science/r/GLEN/](https://anonymous.4open.science/r/GLEN/) • New Finding. We innovatively present that modeling from both global and local perspectives is indispensable for temporal graph representation learning. To the best of our knowledge, we are the first in the subfield of temporal graph learning to propose a method that simultaneously models the graph structure from an entire global perspective and a local subgraph perspective, and fuses all node embeddings across views. • New Method. From the global perspective, we innovatively employ TCN instead of conventionally adopted RNNs for more stable and efficient training. From the local perspective, a new weighted sum algorithm based on time interval is devised to effectively aggregate neighborhood information. To better combine globally and locally acquired node embeddings, we introduce a cross-perspective fusion module based on a devised attention mechanism. • SOTA Performance. Extensive experimental results on diverse real-world datasets for several predictive tasks demonstrate the advantages of GLEN. Moreover, multiple negative edge sampling strategies are employed for link prediction, which are proved to reflect real-world considerations for temporal graphs. 2 RELATED WORKS Static graph methods. With a wide variety of applications, graph embedding has emerged as a focal point of increasing research interest (Zhou et al., 2020). Classical methods leverage matrix factorizations (Cao et al., 2015; Ou et al., 2016) or autoencoders (Pan et al., 2018; Hajiramezanali et al., 2019) to generate node embeddings. Random-walk-based methods such as DeepWalk (Perozzi et al., 2014), Node2Vec (Grover & Leskovec, 2016), LINE (Tang et al., 2015), and SDNE (Wang et al., 2016) employ a flexible and stochastic measure of node similarity and preserve the structural identity of nodes. Recent years have witnessed a burst of GNNs (graph neural networks) like GCN (Kipf & Welling, 2016), GAT (Veličković et al., 2017), and GraphSAGE (Hamilton et al., 2017a) that automatically learn to encode graph structure by aggregating neighboring features. Temporal graph methods. GNN-RNN-based temporal graph methods such as EvolveGCN (Pareja et al., 2020), CTGCN (Liu et al., 2020), and GCRN (Seo et al., 2018) learn constituent representations through GNNs in each snapshot and capture the temporal patterns across snapshots through RNNs. Message-passing temporal graph networks (MP-TGNs) such as JODIE (Kumar et al., 2019), TGAT (Xu et al., 2020), TGN (Rossi et al., 2020), APAN (Wang et al., 2021b), and TPGNN (Wang et al., 2022) aggregate local information through the message passing machinism. Walk-aggregating temporal graph networks (WA-TGNs) such as evolve2vec (Bastas et al., 2019a), STWalk (Pandire et al., 2018), and EVONRL (Heidari & Papagelis, 2020) rely on temporal walks unfolding as the evolution of the graph. CAWN (Wang et al., 2021c) proposes causal anonymous walks using relative node identities. There are also some methods (Souza et al., 2022; Makarov et al., 2021) that leverage the advantages of both MP-TGNs and WA-TGNs. 3 PRELIMINARIES A temporal graph contains a set of nodes: $ V = \{1, 2, \ldots, n\} $. To simplify the problem and be consistent with other works, we assume that the number of nodes in the graph remains constant (Wang et al., 2021b; Xu et al., 2020; Rossi et al., 2020). Throughout we use $ l $ ($ l \in \{0, 1, \ldots, L\} $) to denote the layer index of the network. Interaction events occur temporally between nodes, which is represented as an event stream $ E = \{e_{uv}(t)\} $ ordered by time. $ e_{uv}(t) $ denotes a featured interaction between node $ u $ and node $ v $ at timestamp $ t $ and is modeled as an edge in the graph. Each edge may disappear if it is not present in the dataset at some time. When two nodes interact at $ t $, they are each other’s temporal neighbor and multiple interactions can occur between each two nodes. We follow TGN (Rossi et al., 2020) to keep a memory module $ s_u(t) \in \mathbb{R}^d $ for each node $ u $, which is a $ d $-dimensional vector that summarizes the history information of $ u $ and is updated as events occur. According to the message passing mechanism, when an interaction event $ e_{uv}(t) $ between $ u $ and $ v $ occurs at $ t $, two messages are generated: \[ m_u(t) = \text{msg}(s_u(t^-), s_v(t^-), \phi(t - t_u), e_{uv}(t)), \] \[ m_v(t) = \text{msg}(s_v(t^-), s_u(t^-), \phi(t - t_v), e_{uv}(t)). \] (1) Here, $ s_u(t^-) $ denotes the memory of node $ u $ just before $ t $. $ t_u $ is the time of the last event involving $ u $. $ \phi(\cdot) $ is a generic time encoding method (Xu et al., 2020; Rossi et al., 2020) that maps the time interval into a $ d $-dimensional vector: \[ \phi(t) = [\cos(\omega_1 t), \sin(\omega_1 t), \ldots, \cos(\omega_d t), \sin(\omega_d t)], \] where $ \omega_i $ is learnable. msg is a message function such as concatenation or MLPs. Due to the batch processing in temporal graphs, all events involving node $ u $ in a batch need to be aggregated as: \[ \overline{m}_u(t) = \text{agg}(\overline{m}_u(t_i) \mid t_i \leq t), \] where agg is implemented by keeping only the most recent message for a given node $ u $, which is the same as TGN-attn (Rossi et al., 2020). Then, the memory of node $ u $ is updated as: \[ s_u(t) = \text{upd}(\overline{m}_u(t), s_u(t^-)), \] where upd indicates a recurrent neural network (Chung et al., 2014). For another node $ v $ involved in the event, its memory $ s_v(t) $ is updated in the same way. ### 4 Proposed Method #### 4.1 Overall Framework As shown in Figure 2, the framework of GLEN includes three major components: a GCN-TCN-based global embedding module, a local embedding module based on time interval weighting, and a cross-perspective fusion module. The global and local embedding modules respectively generate node embeddings from the global or local perspective. The cross-perspective fusion module is designed to effectively fuse the global and local node embeddings based on the attention mechanism, allowing the high-order information in a temporal graph (Liu et al., 2022) to be captured. ![Figure 2: The overall framework of the proposed Global and Local Embedding Network (GLEN).](image) #### 4.2 GCN-TCN-based Global Embedding Module The global embedding module of GLEN applies GCN (Kipf & Welling, 2016) to the graph composed of edges within a period (i.e., interaction events of a batch) to generate embeddings for graph nodes. The main reason for choosing GCN rather than other GNNs (e.g., GAT) is that GCN has higher computation efficiency. The obtained embedding matrices of several time steps are fed into TCN (Bai et al., 2018) to capture the temporal patterns of global graph evolution. Graph Convolutional Network (GCN). Let $ b $ denote the index of each batch and the events a batch occur at the same time step. The corresponding input of GCN includes the adjacency matrix $ A_b $ of the graph consisting of edges in the $b$-th batch and the matrix of node features $X_b \in \mathbb{R}^{n \times d}$. Since each row of $X_b$ represents the attributes of a corresponding node, we take the sum of memory $s_u$ and its temporal node features as the $d$-dimensional representation of node $u$ in $X_b$. $s_u$ indicates the updated node memory of $u$ after the events of the $b$-th batch, as temporal graph models assume that events of a single batch arrive simultaneously (Wang et al., 2021b). GCN consists of $L$ layers of graph convolution. At each time step $b$, the $l$-th GCN layer takes $A_b$ and the node embedding matrix $H_b^{(l)}$ as input. The node embedding matrix is updated to $H_b^{(l+1)}$ using the weight matrix $W_b^{(l)}$. In each GCN layer, $A_b$ is normalized to $\tilde{A}_b$ first, defined as (for brevity, here we omit the subscript $b$): $$\tilde{A} = A + I, \quad \tilde{D} = \text{diag}\left(\sum_v \tilde{A}_{uv}\right), \quad \tilde{A} = \tilde{D}^{-\frac{1}{2}} A \tilde{D}^{-\frac{1}{2}},$$ where $I$ is the identity matrix for adding self-loops and $\tilde{D}$ is the diagonal matrix used for propagating the features of each node’s neighbors. Then the process of a single graph convolutional layer in GCN is described as the mathematical formula below: $$H_b^{(0)} = X_b, \quad H_b^{(l+1)} = \sigma\left(\tilde{A}_b H_b^{(l)} W_b^{(l)}\right),$$ where $\sigma(\cdot)$ is the relu activation function. The output of GCN is denoted as $H_b^{(L)}$. Temporal Convolutional Network (TCN). RNNs (Medsker & Jain, 1999; Chung et al., 2014; Hochreiter & Schmidhuber, 1997) generally suffer from inefficiency and unstable training (Ribeiro et al., 2020; Bengio et al., 1994). To avoid the problems, we innovatively adopt TCN (Bai et al., 2018) to model the sequential effect across snapshots, since it allows for parallel computation and uses techniques such as residual connection (He et al., 2016) and weight normalization (Salimans & Kingma, 2016) to make training more stable. For the output of GCN, we consider the chronological embeddings $\{H_b^{(L)}[u], H_b^{(L)}[u], ..., H_b^{(L)}[u]\}$ of node $u$ as a temporal sequence with $d$ channels. To mitigate the so-called staleness problem, we set a time window constraint with a length of $\Gamma$ to limit the temporal range. Only the sequence elements of the $b$ and preceding $(\Gamma - 1)$ time steps are input into TCN. If the window size is 1, the current output of GCN is directly used as global node embeddings. For node $u$ and each channel $c \in \{1, 2, ..., d\}$, the input sequence of TCN is: $$X = \{x_0, x_1, ..., x_{\Gamma-1}\} = \{H_b^{(L)}[u][c], H_b^{(L)}[(b-\Gamma+1)][u][c], ..., H_b^{(L)}[(b-\Gamma+2)][u][c], ..., H_b^{(L)}[u][c]\}.$$ Then TCN applies the dilated convolution operation (Oord et al., 2016) on the sequence at each layer, and the formula is as follows: $$\hat{y}_i = (X * f)[i] = \sum_{j=0}^{k-1} f(j) \cdot x_{i-\delta j},$$ where $*$ is the convolution operator, $k$ is the size of the filter $f : \{0, 1, ..., k-1\} \rightarrow \mathbb{R}$ and $\delta$ is the dilation factor of each layer increasing exponentially with the depth of TCN (i.e., at the $l$-th TCN layer, $\delta = 2^l$). TCN predicts the corresponding sequence $\{\hat{y}_0, \hat{y}_1, ..., \hat{y}_{\Gamma-1}\} = \text{TCN}(\{x_0, x_1, ..., x_{\Gamma-1}\})$ and we take $\hat{y}_{\Gamma-1}$ as the output. The receptive field of one TCN layer is $(k - 1) \times \delta$, thereby increasing the kernel size or using a deep network for a larger dilation factor enables richer historical information to be captured. Both the numbers of input channels and output channels of TCN are set to $d$. For node $u$, the outputs of $d$ kernels are considered the global embedding $z_u^{\text{Global}}$ that evolves over time. For time step $b$, the global embeddings of $n_b$ nodes involved in the events of the corresponding $b$-th batch are denoted as: $$Z^{\text{Global}} \in \mathbb{R}^{n_b \times d} = \{z_1^{\text{Global}}, z_2^{\text{Global}}, ..., z_{n_b}^{\text{Global}}\}.$$ ### 4.3 Local Embedding Module Based on Time Interval Weighting This module generates the local embedding $z_u^{\text{Local}}$ that evolves over time for each node $u$ from the local perspective. There is a common pattern in temporal graphs: recent events tend to have more important potential information. Therefore, we devise a weighted sum algorithm based on time interval to effectively aggregate the information of temporal neighbors. To control the computation cost and ensure a fair comparison, we restrict the neighborhood size of each node like other works (Rossi et al., 2020; Wang et al., 2021c). We denote the neighbor set of $ u $ at $ t $ as $ N_u(t) $, which contains a certain number $ \|N\| $ of the most recent neighbors that interact with $ u $ before $ t $. If the timestamp $ t_{uv} $ of the event $ e_{uv} $ is far from the current time $ t $, the impact of $ e_{uv} $ and $ v $ on node $ u $ should be reduced. Thus, different temporal weights for neighbors are computed as: \[ w(v,u,t) = \frac{\exp(-(t - t_{uv}))}{\sum_{(v',t_{uv'}) \in N_u(t)} \exp(-(t - t_{uv'}))}. \] The temporal weight decreases as the time interval increases. We generate the corresponding representation vector for the neighbor $ v $ and event $ e_{uv} $ through a linear layer: \[ z_{uv}(l)(t) = \text{Tanh}\left(\text{Linear}_1(h_v^{(l-1)}(t)\\|e_{uv}(t)\\|\phi(t - t_{uv}))\right), \] where $ h_v^{(l-1)}(t) $ is the input of the $ l $-th network layer, and $ h_v^{(0)}(t) $ is the sum of $ s_v(t) $ and temporal node features. The activation function Tanh is used to provide nonlinear transformations and limit the values in a certain range to facilitate the subsequent summation. We then utilize the temporal weights to aggregate neighborhood information for node $ u $ through a weighted sum: \[ \tilde{h}_u(l)(t) = \sum_{(v,t_{uv}) \in N_u(t)} w(v,u,t) \cdot z_{uv}(l)(t). \] The node embedding of $ u $ is generated by combining its own representation with aggregated neighborhood information through a linear layer: \[ h_u(l)(t) = \text{Linear}_2(h_u^{(l-1)}(t)\\|\tilde{h}_u(l)(t)). \] After all events of the $ b $-th batch are processed, the output of the module is taken as the local embedding of node $ u $: $ z_u^{\text{Local}} = h_u^{(L)}(t) $ that evolves over time. Similar to Eq. 9, the local embedding matrix of the $ b $-th time step is denoted as: \[ Z^{\text{Local}} \in \mathbb{R}^{n_b \times d} = \{z_1^{\text{Local}}, z_2^{\text{Local}}, \ldots, z_{n_b}^{\text{Local}}\}. \] ### 4.4 Cross-Perspective Fusion Module ![Figure 3](image) Figure 3: The cross-perspective fusion module of GLEN calculates embeddings $ Z $ according to the relevance between $ z_u^{\text{Global}} $ and $ z_u^{\text{Local}} $ of each two nodes $ u $ and $ v $. For GLEN to combine global and local node embeddings, as illustrated in Figure 3, multi-head attention allows the model to jointly attend to crucial information in different representation subspaces. We use $ \eta $ to indicate the number of heads and $ i $ to denote the index of each head. In each single attention head, we forward global embeddings to a linear projection to obtain the 'query', and local embeddings to another two different linear projections to obtain the 'key' and 'value': \[ Q_i = Z^{\text{Global}} W_i^Q, \quad K_i = Z^{\text{Local}} W_i^K, \quad V_i = Z^{\text{Local}} W_i^V, \] where $ W_i^Q \in \mathbb{R}^{d \times d_k}, W_i^K \in \mathbb{R}^{d \times d_k}, W_i^V \in \mathbb{R}^{d \times d_v} $ are three transformation matrices and $ d_k = d_v = d/\eta $. The attentive output of each head is denoted as: \[ \tilde{Z}_i = \text{softmax}\left(\frac{Q_i K_i^T}{\sqrt{d_k}}\right)V_i. \] The attention coefficient of each two nodes $ u $ and $ v $ implies the correlation between $ z_u^{\text{Global}} $ and $ z_v^{\text{Local}} $ and increases with relevance. The outputs of all heads are concatenated as the output of the attention mechanism: \[ \tilde{Z} = \text{MultiHead}(Q, K, V) = \text{Concat}(\tilde{Z}_1, \tilde{Z}_2, ..., \tilde{Z}_\eta)W^O, \] where $ W^O \in \mathbb{R}^{nd_v \times d} $. Since GLEN chooses the linear projection of local embeddings as the 'value', $ \tilde{Z} $ actually gives hidden representations of the local embeddings. To further combine these latent representations with global node embeddings, we concatenate $ \tilde{z}_u \in \tilde{Z} $ with $ z_u^{\text{Global}} $ and input them to a feedforward neural network to further capture the nonlinear correlation between local and global embeddings of the same node: \[ z_u = \text{FFN}(\tilde{z}_u \|\| z_u^{\text{Global}}). \] The attention mechanism captures the correlation between each two nodes, allowing for the retention of high-order information in temporal graphs. The intent representations influenced by the affinity weight matrix enable the model to selectively focus on pairs of nodes with high relevance and ignore mostly unimportant information. To empirically prove that both global and local perspectives are important and enhance the interpretability of our fusion module, we additionally conduct a case study on the correlation of node embeddings in Appendix A. Experimental results reveal that both global and local views are essential since considering only one is not comprehensive. 5 EXPERIMENTS 5.1 DATASETS AND BASELINES We totally use seven public real-world temporal graph datasets to extensively validate the effectiveness of GLEN, including Wikipedia (Kumar et al., 2019), Reddit (Kumar et al., 2019), Enron (Shetty & Adibi, 2004), UCI (Panzarasa et al., 2009), UN Trade (MacDonald et al., 2015), MOOC (Kumar et al., 2019), and Flights (Schäfer et al., 2014). Descriptions and statistics of the datasets are reported in Appendix C.1. We choose eight state-of-the-art approaches for temporal graph representation learning as strong baselines to compare with, including DyRep (Trivedi et al., 2019), JODIE (Kumar et al., 2019), TGAT (Xu et al., 2020), TGN (Rossi et al., 2020), CAWN (Wang et al., 2021c), PINT (Souza et al., 2022), GraphMixer (Cong et al., 2023), and TIGER (Zhang et al., 2023). Introductions of baselines are available in Appendix C.2. For the settings of baselines, we use their recommended configurations. We use the same data processing and splitting procedures as TGAT (Xu et al., 2020) and TGN (Rossi et al., 2020). For fairness, we evaluate all the methods in the same environment and on the same Nvidia Tesla V100-SXM2 GPU to obtain experimental results. 5.2 IMPLEMENTATION DETAILS AND EVALUATION PROTOCOL We conduct experiments on two predictive tasks: link prediction (Zhang et al., 2020; Srinivasan & Ribeiro, 2019; Li & Zhou, 2011) and dynamic node classification (Aggarwal & Li, 2011; Xu et al., 2019). For all datasets, we split edges chronologically by 70%, 15%, and 15% for training, validation, and testing. We use the Adam optimizer and early stopping with a patience of 5 for training. For both link prediction and dynamic node classification, we use BCE loss. All the settings are consistent with those set by baselines (Xu et al., 2020; Rossi et al., 2020; Wang et al., 2021c). More implementation details of GLEN can be found in Appendix C.3. The pseudo-code of GLEN can be seen in Appendix C.4. In addition, random, historical, and inductive negative sampling strategies (Poursafaei et al., 2022) are applied for evaluation, which proves to reflect real-world considerations for temporal graphs. More details of the evaluation protocol are introduced in Appendix C.5. 5.3 QUANTITATIVE RESULTS Table 2 presents the results of inductive link prediction experiments, where NS is the abbreviation of negative sampling. Since PINT takes too long on the largest Flights dataset, we did not include PINT’s results on Flights in Table 2. In the link prediction task, GLEN obviously outperforms the baselines on all datasets under the evaluation of all negative sampling strategies. Baselines have a significant performance drop with both historical and inductive sampling strategies, as their one-sidedness makes it difficult to correctly predict the pattern of interactive appearance in temporal graphs. Whereas, GLEN can keep the performance relatively stable owing to the complementarity between information acquired globally and locally, and the effective cross-perspective fusion. The results of dynamic node classification are shown in Table 1 where GLEN also obtains the best results on all datasets. The results of transductive link prediction are reported in Appendix D.1 where GLEN also shows the state-of-the-art performance. Table 2: Average Precision (%) of link prediction under different negative sampling strategies in the inductive setting (over 5 runs). (First second) \| NS Strategy \| Methods \| Wikipedia \| Reddit \| Enron \| UCI \| UN Trade \| MOOC \| Flights \| \|-------------\|---------\|-----------\|--------\|-------\|------\|----------\|------\|--------\| \| Random \| DyRep \| 72.17±1.14 \| 55.13±1.04 \| 59.99±3.77 \| 60.55±1.29 \| 59.22±0.75 \| 64.21±0.59 \| 92.47±0.72 \| \| \| JODIE \| 97.97±0.00 \| 99.26±0.74 \| 81.68±0.10 \| 98.06±0.23 \| 57.96±6.18 \| 83.23±6.50 \| 94.85±0.64 \| \| \| TGAT \| 94.03±0.20 \| 96.62±0.15 \| 56.01±2.46 \| 74.39±5.33 \| 59.80±0.83 \| 71.21±0.41 \| 89.02±0.06 \| \| \| TGN \| 98.00±0.18 \| 94.09±1.07 \| 75.28±3.37 \| 83.04±2.37 \| 57.42±1.73 \| 81.51±3.31 \| 84.11±0.61 \| \| \| CAWN \| 89.46±0.38 \| 99.82±0.10 \| 92.05±1.77 \| 98.45±0.66 \| 91.64±0.26 \| 87.59±1.88 \| 98.67±0.14 \| \| \| PINT \| 98.30±0.08 \| 99.04±0.39 \| 92.05±2.23 \| 92.04±0.25 \| 91.04±0.60 \| 87.59±1.88 \| 98.67±0.14 \| \| \| GraphMixer \| 96.49±0.08 \| 95.22±0.03 \| 58.67±0.55 \| 90.79±0.32 \| 56.47±2.82 \| 80.95±0.65 \| 83.00±0.07 \| \| \| TIGER \| 98.30±0.02 \| 98.64±0.53 \| 83.40±1.13 \| 92.98±0.23 \| 55.29±0.11 \| 84.72±1.48 \| 91.84±0.86 \| \| \| GLEN \| 99.95±0.05 \| 99.85±0.28 \| 96.15±1.61 \| 99.11±0.30 \| 96.09±0.12 \| 96.48±4.02 \| 99.36±0.17 \| \| Historical \| DyRep \| 69.45±1.16 \| 52.40±1.68 \| 56.96±3.12 \| 52.67±0.87 \| 59.55±0.81 \| 60.93±0.58 \| 62.00±1.81 \| \| \| JODIE \| 60.46±0.32 \| 49.68±0.22 \| 51.26±0.67 \| 54.23±2.32 \| 58.07±2.47 \| 47.14±5.81 \| 60.41±2.39 \| \| \| TGAT \| 71.35±0.93 \| 63.25±0.78 \| 53.45±2.52 \| 61.62±0.49 \| 51.85±3.03 \| 59.60±0.59 \| 64.43±0.32 \| \| \| TGN \| 81.96±1.10 \| 61.29±1.30 \| 61.90±2.01 \| 72.31±1.54 \| 54.4±1.00 \| 63.70±2.02 \| 58.27±1.73 \| \| \| CAWN \| 80.14±0.52 \| 82.10±1.33 \| 58.58±4.36 \| 81.81±1.75 \| 87.00±0.75 \| 97.33±0.23 \| 51.84±0.13 \| \| \| PINT \| 64.97±1.12 \| 68.27±1.53 \| 78.66±0.68 \| 84.78±0.91 \| 58.50±0.23 \| 67.33±4.25 \| 65.23±0.40 \| \| \| GraphMixer \| 88.04±0.39 \| 64.48±0.36 \| 61.10±1.20 \| 80.29±0.31 \| 58.92±2.67 \| 74.07±0.73 \| 65.23±0.40 \| \| \| GLEN \| 96.25±0.27 \| 97.31±2.46 \| 97.28±0.53 \| 95.94±2.00 \| 95.78±2.32 \| 99.53±0.93 \| 76.96±0.54 \| \| Inductive \| DyRep \| 69.36±1.20 \| 52.48±1.11 \| 57.16±3.34 \| 52.68±0.90 \| 59.57±0.90 \| 60.92±0.62 \| 61.99±1.81 \| \| \| JODIE \| 40.58±0.18 \| 49.73±0.16 \| 51.46±0.42 \| 54.61±2.58 \| 57.88±2.56 \| 47.15±5.77 \| 59.47±1.36 \| \| \| TGAT \| 71.46±0.79 \| 63.29±0.64 \| 53.98±3.02 \| 62.66±0.84 \| 51.94±2.83 \| 59.65±0.68 \| 64.42±0.32 \| \| \| TGN \| 81.90±1.28 \| 62.15±1.46 \| 62.37±2.47 \| 72.25±1.55 \| 54.48±1.07 \| 63.61±2.00 \| 58.14±1.77 \| \| \| CAWN \| 68.70±1.48 \| 78.34±1.37 \| 62.22±6.60 \| 83.32±7.21 \| 89.83±1.64 \| 90.93±1.38 \| 53.84±0.04 \| \| \| PINT \| 64.86±7.09 \| 72.79±5.75 \| 78.59±0.73 \| 84.72±1.03 \| 54.39±1.88 \| 67.36±4.35 \| 63.13±0.15 \| \| \| GraphMixer \| 83.91±0.54 \| 63.96±0.26 \| 72.19±1.19 \| 80.33±0.31 \| 58.89±2.66 \| 74.08±0.73 \| 63.13±0.15 \| \| \| GLEN \| 96.13±0.29 \| 97.28±2.48 \| 97.38±0.46 \| 95.43±2.63 \| 95.76±2.32 \| 99.54±0.91 \| 77.23±0.61 \| Table 1: Average ROC AUC (%) of dynamic node classification (over 5 runs). (First second) \| Methods \| Wikipedia \| Reddit \| MOOC \| \|---------\|-----------\|--------\|------\| \| DyRep \| 80.79±1.86 \| 50.01±2.27 \| 66.08±0.24 \| \| JODIE \| 84.46±2.84 \| 61.57±4.34 \| 69.46±0.51 \| \| TGAT \| 85.98±1.45 \| 65.87±1.45 \| 54.05±0.20 \| \| TGN \| 87.33±0.30 \| 60.09±1.64 \| 64.09±0.68 \| \| GraphMixer \| 86.26±1.83 \| 63.24±1.91 \| 68.65±1.09 \| \| GLEN \| 85.55±0.30 \| 68.83±1.62 \| 70.99±0.05 \| \| \| 90.16±0.32 \| 70.21±0.27 \| 71.49±0.33 \| 5.4 Efficiency We further evaluate the ability to trade off the precision and efficiency of GLEN, which is illustrated in Figure 4. The AP (Average Precision) is computed with the random negative sampling strategy and the inductive setting in a percentage format. Methods closer to the upper left corner of the figure are more ideal. Note that the training time of PINT here does not include precomputing the positional features, otherwise its training time will be unbearably long. The efficiency of GLEN is comparable to the fastest baselines, and the performance is improved. The complexity analysis and more experimental results about efficiency are reported in Appendix B and Appendix D.2 respectively. Overall, GLEN strikes an impressive balance between inference precision and training speed, which can be attributed to the training efficiency of TCN. 5.5 Hyper-parameter Investigation We systematically analyze the effect of hyper-parameters related to GLEN, including the time window size $\Gamma$, number of sampled neighbors $\|\mathcal{N}\|$, number of layers in TCN, kernel size of TCN, number of heads $\eta$ in the multi-head attention mechanism, and the dropout ratio. Figure 5 illustrates the impact of various hyper-parameters on GLEN. We combine the hyper-parameters in pairs. The reason for jointly considering dropout and attention heads is that they both mainly affect the cross-perspective fusion module of GLEN. Both the number of layers in TCN and the kernel size of TCN affect the receptive field and the temporal convolution operations of TCN, so we consider them together. GLEN exhibits its robustness as the fluctuation of AP is small. The effect of $\Gamma$ and $\|V\|$ on GLEN is shown in Figure 6. An interesting insight is that GLEN tends to achieve the maximum AP with a small time window size, which means crucial recent information is sufficient for GLEN to capture the evolution patterns of temporal graphs. $\|V\|$ barely makes a difference to GLEN, while other local-view TGNs methods typically require a certain number (usually 10 or 20) of neighbor nodes to achieve their best performance (Rossi et al., 2020; Wang et al., 2021c). This indicates that global embeddings supplement local embeddings through GLEN’s fusion module. More experimental results on hyper-parameter investigation are reported in Appendix D.3. Figure 6: Performance of GLEN with different time window sizes and numbers of sampled neighbors with different negative sampling strategies. 5.6 Ablation Study We further analyze GLEN by performing an ablation study to manifest the contributions of different components of GLEN. More details of the ablation study are reported in Appendix C.6. We summarize the results of the ablation study on link prediction in Table 3. From the results, we can observe that removing any of GLEN’s components will cause performance degradation, indicating that the modules we designed are indispensable for temporal graph representation learning. The ablation study further proves the effectiveness of the cross-perspective fusion module and provides a certain degree of interpretability for the complementarity between global and local modeling of temporal graphs. Results of the ablation study on dynamic node classification are reported in Appendix D.4. Table 3: Average Precision (%) for ablation study of GLEN in inductive link prediction. \| Ablation \| Earon \| UCI \| UN Trade \| MOOC \| \|----------\|-------\|-----\|----------\|------\| \| \| Random \| Historical \| Inductive \| Random \| Historical \| Inductive \| Random \| Historical \| Inductive \| Random \| Historical \| Inductive \| \| w/o GCN \| 85.87 \| 73.74 \| 75.09 \| 96.28 \| 95.42 \| 95.42 \| 91.61 \| 91.72 \| 91.73 \| 87.26 \| 84.88 \| 84.88 \| \| w/o TCN \| 87.40 \| 70.09 \| 71.23 \| 95.36 \| 81.97 \| 81.97 \| 95.79 \| 95.33 \| 95.32 \| 94.87 \| 97.02 \| 97.01 \| \| w/o Global \| 90.58 \| 93.00 \| 93.07 \| 80.31 \| 70.95 \| 70.95 \| 90.45 \| 90.59 \| 90.70 \| 62.66 \| 56.29 \| 56.29 \| \| w/o Local \| 85.55 \| 89.75 \| 89.78 \| 81.16 \| 69.23 \| 69.23 \| 90.95 \| 91.07 \| 91.07 \| 77.42 \| 66.43 \| 66.43 \| \| GLEN \| 96.15 \| 97.28 \| 97.38 \| 98.47 \| 95.94 \| 95.43 \| 96.09 \| 95.78 \| 95.76 \| 96.48 \| 99.53 \| 99.54 \| 6 Conclusion In this paper, we proposed the Global and Local Embedding Network (GLEN), an adventurous method for temporal graph representation learning. Specifically, GLEN consists of three main components: the GCN-TCN-based global embedding module, the local embedding module based on time interval weighting, and the cross-perspective fusion module. The global embedding module models temporal graphs from a global perspective, while the local embedding module does so from a local perspective. Then, the fusion mechanism combines global and local embeddings based on a novel attention mechanism. By taking both global and local perspectives, GLEN outperforms all the baselines in extensive experiments. REFERENCES Charu C Aggarwal and Nan Li. On node classification in dynamic content-based networks. In Proceedings of the 2011 SIAM international conference on data mining, pp. 355–366. SIAM, 2011. Nurul A Asif, Yeahia Sarker, Ripon K Chakrabortty, Michael J Ryan, Md Hafiz Ahamed, Dip K Saha, Faisal R Badal, Sajal K Das, Md Firoz Ali, Sumaya I Moyeen, et al. Graph neural network: A comprehensive review on non-euclidean space. 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J2TZgj3Tac	How do the proposed algorithms compare to other state-of-the-art methods, including non-PSRO methods such as population-based training and league training (e.g., AlphaStar), in terms of exploitability reduction and convergence speed?	TOWARD OPTIMAL POLICY POPULATION GROWTH IN TWO-PLAYER ZERO-SUM GAMES Stephen McAleer∗1, JB Lanier2, Kevin A. Wang2, Pierre Baldi2, Tuomas Sandholm1 and Roy Fox2 1Department of Computer Science, Carnegie Mellon University 2Department of Computer Science, University of California, Irvine ∗Corresponding author: smcaleer@cs.cmu.edu ABSTRACT In competitive two-agent environments, deep reinforcement learning (RL) methods like Policy Space Response Oracles (PSRO) often increase exploitability between iterations, which is problematic when training in large games. To address this issue, we introduce anytime double oracle (ADO), an algorithm that ensures exploitability does not increase between iterations, and its approximate extensive-form version, anytime PSRO (APSRO). ADO converges to a Nash equilibrium while iteratively reducing exploitability. However, convergence in these algorithms may require adding all of a game’s deterministic policies. To improve this, we propose Self-Play PSRO (SP-PSRO), which incorporates an approximately optimal stochastic policy into the population in each iteration. APSRO and SP-PSRO demonstrate lower exploitability and near-monotonic exploitability reduction in games like Leduc poker and Liar’s Dice. Empirically, SP-PSRO often converges much faster than APSRO and PSRO, requiring only a few iterations in many games. 1 INTRODUCTION In competitive two-agent environments, also known as zero-sum games, deep reinforcement learning (RL) methods based on the double oracle (DO) algorithm (McMahan et al., 2003), such as Policy Space Response Oracles (PSRO) (Lanctot et al., 2017), are some of the most promising methods for finding approximate Nash equilibria in large games. One reason is that such methods are simple to use with existing RL methods and naturally provide a measure of approximate exploitability. The exploitability of a policy is defined as the performance against a worst-case opponent, and it is optimal at zero when the policy is a Nash equilibrium. A second reason is that these methods effectively prune the game tree by only considering mixtures over policies that are already trained to be effective best responses. Finally, they can be used in games with large or continuous action spaces because they do not require full game-tree traversals. Methods based on PSRO such as AlphaStar (Vinyals et al., 2019) and Pipeline PSRO (McAleer et al., 2020) have achieved state-of-the-art performance on Starcraft and Barrage Stratego, respectively. PSRO-based methods iteratively add RL best-response policies to a population. The best response for each player trains against a restricted distribution over the opponent’s existing population of policies. To find this restricted distribution, a Nash equilibrium (a pair of mutually best-responding policies) is computed in a restricted single-step game where each action corresponds to choosing a policy from the population. As PSRO iterations progress, an optimal distribution over these population policies will approximate a Nash equilibrium in the full game. In practice, however, PSRO is terminated early in large games. This can be a problem because the PSRO restricted distribution over the population policies is not guaranteed to decrease in exploitability every iteration. As a result, if PSRO is terminated early, the final restricted distribution could potentially be arbitrarily more exploitable than the initial one. In this paper, we first propose a new double oracle variant, anytime double oracle (ADO) that, in each iteration, finds the least-exploitable restricted distribution over the population policies of each player. This algorithm is called anytime, in the sense that it can be stopped in any iteration and return a solution that is not worse than the previous iteration. We then present an approximate extensive-form RL version called anytime PSRO (APSRO). Anytime double oracle (ADO) can be viewed as a modification of the range of skill (ROS) algorithm (Zinkevich et al., 2007) that finds a restricted Nash equilibrium over two restricted games, one per player. Each player’s restricted game is defined such that their strategies are restricted to be within their population, but the opponent is unrestricted. For each player, ADO adds to the opponent’s population a best response to the player’s NE restricted distribution. ADO is guaranteed not to increase exploitability from one iteration to the next, while also being guaranteed to converge to a Nash equilibrium in a number of iterations at most equal to the number of pure strategies in the game. Anytime policy-space response oracles (APSRO) updates the restricted distribution using a no-regret algorithm trained against a single approximate best response from the opponent. This opponent approximate best response is itself being continually trained via reinforcement learning against the restricted distribution. We find empirically that APSRO tends not to increase exploitability as much as PSRO and can greatly outperform PSRO in some domains. However, because common implementations of PSRO add pure-strategy (i.e. deterministic) best responses in each iteration, PSRO may still need to add many policies to the population before they can support a Nash equilibrium. In fact, in certain games, all pure strategies will be added before finding a Nash equilibrium. This is because many games require mixing over a large number of pure strategies to arrive at a Nash equilibrium. Furthermore, before termination, the restricted distribution over population policies can be arbitrarily exploitable, even if it decreases monotonically until then. In addition to introducing APSRO, we also build on APSRO by adding to the population in each iteration a stochastic policy that is trained via an off-policy procedure. A key insight is that mixed strategies (i.e. stochastic policies) can lower the exploitability of a population more than pure strategies. To see this, note that a Nash equilibrium strategy is an optimal strategy to add because the least-exploitable distribution over the resulting population will also be a Nash equilibrium strategy. If all Nash equilibria are mixed, as is often the case, then no pure strategy can be added to the population that reduces exploitability as much as the mixed strategy Nash equilibrium. Although finding the optimal strategy to add is as hard as solving the original game, we find that adding a rough approximation to the optimal strategy can offer striking empirical benefits in quickly reducing the exploitability of the restricted distribution. We present Self-Play PSRO (SP-PSRO), which, similarly to APSRO, learns a restricted distribution over the population via no regret against the opponent’s best response. Additionally, SP-PSRO trains off-policy a new strategy against the opponent’s best response. At the end of each iteration, SP-PSRO adds two strategies to the population: (1) the time-average of this new strategy and (2) the best response to the opponent’s restricted distribution. Section 3 clarifies this algorithm using formal notation. In large games like dark chess and Starcraft, where PSRO may never converge, the early performance holds paramount importance. Our approach with SP-PSRO is tailored to this reality, ensuring robust performance from the outset. Recognizing that the completion of the full training procedure in such extensive games is a rare occurrence, the anytime property of our proposed method takes on a critical role, delivering viable strategies at any stage of the iterative process. By training the new strategy off-policy, SP-PSRO requires the same amount of experience in each iteration as APSRO and PSRO. Experiments on normal-form games and extensive-form games such as Liar’s Dice, Battleship, and Leduc Poker suggest that SP-PSRO can learn policies that are dramatically less exploitable than APSRO and PSRO. Our empirical results demonstrate SP-PSRO’s superior performance in reducing exploitability before convergence across various games, a testament to its practical effectiveness. While APSRO serves as a foundational concept in our research, the leap to SP-PSRO marks a significant advancement, particularly in terms of reducing exploitability before PSRO has neared convergence. To summarize, our contributions are as follows: • We introduce a version of double oracle that does not increase in exploitability, called anytime double oracle (ADO) and its extensive-form approximation, anytime PSRO (APSRO). • We present an enhancement to APSRO, termed Self-Play PSRO (SP-PSRO). In each iteration, without requiring extra environment steps, it incorporates an additional mixed strategy aimed at reducing our population’s exploitability. 2 BACKGROUND We consider extensive-form games with perfect recall (Hansen et al., 2004). An extensive-form game progresses through a sequence of player actions and has a world state $ w \in W $ at each step. In an $ N $-player game, $ A = A_1 \times \cdots \times A_N $ is the space of joint actions for the players. $ A_i(w) \subseteq A_i $ denotes the set of legal actions for player $ i \in N = \{1, \ldots, N\} $ at world state $ w $ and $ a = (a_1, \ldots, a_N) \in A $ denotes a joint action. At each world state, after the players choose a joint action, a transition function $ T(w, a) \in \Delta^W $ determines the probability distribution of the next world state $ w' $. Upon transition from world state $ w $ to $ w' $ via joint action $ a $, player $ i $ makes an observation $ o_i = O_i(w, a, w') $. In each world state $ w $, player $ i $ receives a reward $ R_i(w) $. The game ends when the players reach a terminal world state. In this paper, we consider games that are guaranteed to end in a finite number of actions. A history is a sequence of actions and world states, denoted $ h = (w^0, a^0, w^1, a^1, \ldots, w^t) $, where $ w^0 $ is the known initial world state of the game. $ R_i(h) $ and $ A_i(h) $ are, respectively, the reward and set of legal actions for player $ i $ in the last world state of a history $ h $. An information set for player $ i $, denoted by $ s_i $, is a sequence of that player’s observations and actions up until that time $ s_i(h) = (a^0_i, o^1_i, a^1_i, \ldots, o^t_i) $. Define the set of all information sets for player $ i $ to be $ I_i $. The set of histories that correspond to an information set $ s_i $ is denoted $ H(s_i) = \{h : s_i(h) = s_i\} $, and it is assumed that they all share the same set of legal actions $ A_i(s_i(h)) = A_i(h) $. A player’s strategy $ \pi_i $ is a function mapping from an information set to a probability distribution over actions. A strategy profile $ \pi $ is a tuple $ (\pi_1, \ldots, \pi_N) $. All players other than $ i $ are denoted $ -i $, and their strategies are jointly denoted $ \pi_{-i} $. A strategy for a history $ h $ is denoted $ \pi_i(h) = \pi_i(s_i(h)) $ and $ \pi(h) $ is the corresponding strategy profile. When a strategy $ \pi_i $ is learned through RL, we refer to the learned strategy as a policy. The expected value (EV) $ v_i^\pi(h) $ for player $ i $ is the expected sum of future rewards for player $ i $ in history $ h $, when all players play strategy profile $ \pi $. The EV for an information set $ s_i $ is denoted $ v_i^\pi(s_i) $ and the EV for the entire game is denoted $ v_i(\pi) $. A two-player zero-sum game has $ v_1(\pi) + v_2(\pi) = 0 $ for all strategy profiles $ \pi $. The EV for an action in an information set is denoted $ v_i^\pi(s_i, a_i) $. A Nash equilibrium (NE) is a strategy profile such that, if all players played their NE strategy, no player could achieve higher EV by deviating from it. Formally, $ \pi^* $ is a NE if $ v_i(\pi^) = \max_{\pi_i} v_i(\pi_i, \pi^_{-i}) $ for each player $ i $. The exploitability $ e(\pi) $ of a strategy profile $ \pi $ is defined as $ e(\pi) = \sum_{i \in N} \max_{\pi'_i} v_i(\pi'_i, \pi_{-i}) $. A best response (BR) strategy $ BR_i(\pi_{-i}) $ for player $ i $ to a strategy $ \pi_{-i} $ is a strategy that maximally exploits $ \pi_{-i} $: $ BR_i(\pi_{-i}) = \arg \max_{\pi'_i} v_i(\pi'_i, \pi_{-i}) $. An $ \epsilon $-best response ($ \epsilon $-BR) strategy $ BR_i^\epsilon(\pi_{-i}) $ for player $ i $ to a strategy $ \pi_{-i} $ is a strategy that is at most $ \epsilon $ worse for player $ i $ than the best response: $ v_i(BR_i(\pi_{-i}), \pi_{-i}) \geq v_i(BR_i^\epsilon(\pi_{-i}), \pi_{-i}) - \epsilon $. An $ \epsilon $-Nash equilibrium ($ \epsilon $-NE) is a strategy profile $ \pi $ in which, for each player $ i $, $ \pi_i $ is an $ \epsilon $-BR to $ \pi_{-i} $. A normal-form game is a simultaneous-move single-step extensive-form game. An extensive-form game induces a normal-form game in which the legal actions for player $ i $ are its deterministic strategies $ X_{s_i \in I_i} A_i(s_i) $. These deterministic strategies are called pure strategies of the normal-form game. A mixed strategy is a distribution over a player’s pure strategies. 3 ANYTIME DOUBLE ORACLE ALGORITHM (ADO) Double oracle (DO) (described in [B, J]) is guaranteed to converge because in the worst case, it will expand all pure strategies, at which point it terminates at a Nash equilibrium (NE). Unfortunately, before convergence, there is no guarantee on the exploitability of the restricted-game NE. In fact, DO can increase exploitability arbitrarily from one iteration to the next. To see this, consider the game in Figure 1. If both players start with a population consisting only of the first strategy (top row and left column), then the best response for each player is the second strategy, giving that player value 1, for a total exploitability of 2. In the next iteration (Figure 1), when both the first and second strategies are in the population for both players, the restricted-game NE of DO will give probability 1 to the second strategy for each player. This restricted NE has exploitability of 4. In Appendix D, we show empirically that DO does indeed increase exploitability arbitrarily before terminating in this class of games. PSRO inherits this property. Figure 1: Top: In DO, a single restricted game is created and solved in gray. Since this restricted game does not consider strategies outside of the population, it can lead to exploitable restricted distributions. In this example, the DO restricted distribution $\pi$ places all mass on the second strategy, resulting in total exploitability of 4. Bottom: Conversely, ADO creates two restricted games where the opponent is unrestricted, player 1’s restricted game is shown in green and player 2’s restricted game is shown in red. Solving these modified restricted games results in the least-exploitable restricted distributions. In this example, the restricted distribution $\pi$ for ADO puts $\frac{2}{3}$ mass on the first strategy and $\frac{1}{3}$ mass on the second strategy, resulting in the optimal exploitability for this restricted game of $\frac{4}{3}$. Algorithm 1 Anytime Double Oracle (ADO) Result: Nash Equilibrium Input: Initial population $\Pi^0$ repeat {for $t = 0, 1, \ldots$} for $i \in \{1, 2\}$ do $\pi^r_i \leftarrow$ NE in restricted game $G^i$ (eq. (1)) for $i \in \{1, 2\}$ do Find a novel best response $\beta_i \leftarrow \mathbb{BR}_i(\pi^r_{-i})$ $\Pi^{t+1}_i = \Pi^t_i \cup \{\beta_i\}$ until No novel best response exists for either player Return: $\pi^r$ In this paper, we introduce anytime double oracle (ADO) (Algorithm 1), which is guaranteed not to increase exploitability from one iteration to the next. ADO primarily serves as a foundational element for the development of our subsequent algorithm, APSRO. This foundational role is crucial as it lays the groundwork for APSRO’s convergence guarantees. Like DO, ADO maintains a population $\Pi^t_i$ for player $i$ in iteration $t$, and in each iteration computes a Nash equilibrium on a restricted game and adds to each population a best response to the restricted NE. However, unlike DO, ADO creates a different restricted game for each player. The restricted game $G^i$ for player $i$ is created by restricting that player to only play strategies included in their population $\Pi_i$, while the opponent can play any strategy in the full game. The game value of $G^i$ for player $i$ is $$\max_{\pi_i \in \Pi_i} \min_{\pi_{-i}} v_i(\pi_i, \pi_{-i}).$$ (1) The restricted game $G^i$ for player $i$ is then solved for both players to get a NE for each restricted game. We refer to player $i$’s NE strategy in their restricted game as their restricted NE $\pi^r_i$. The restricted NE for player $i$ is the least exploitable mixed strategy supported by player $i$’s population. Note that in large games this restricted game will be prohibitively large to solve and will require approximation with APSRO, introduced later in this paper. Next, a best response $\beta_i = \text{BR}(\pi_{-i})$ is computed for each player $i$ against the restricted-NE mixed strategy of the restricted opponent, and is added to the player’s population. If there are multiple best responses, a novel best response $\beta_i \not\in \Pi_i$ is chosen that is not currently in that player’s population. ADO is guaranteed to terminate because there are finitely many pure strategies in the original game. When ADO terminates, the restricted NE is a NE in the original game (Proposition 2). Unlike DO, the exploitability of the restricted NE does not increase from iteration to iteration (Proposition 1). Proposition 1. The exploitability of ADO is monotonically non-increasing. Proof. All proofs are contained in Appendix H. To illustrate this property of ADO, consider the algorithm dynamics on the DO bad case given in Figure 1. Similar to DO, ADO adds the second strategy to the population in the first iteration. Now, however, instead of taking the second strategy with probability 1 as DO does, ADO solves the restricted game where one player is restricted to the first two strategies and the other is unrestricted and can play any of the three strategies. The Nash equilibrium of this game for the restricted player is to play the first strategy with probability $\frac{4}{7}$ and the second strategy with probability $\frac{3}{7}$. This strategy results in a total exploitability of $\frac{4}{7}$, compared with the DO exploitability of 4 and the initial ADO exploitability of 2. In addition to this property of never-increasing exploitability, ADO is guaranteed to converge to a Nash equilibrium, as shown below. Proposition 2. When ADO terminates, the restricted NE of both players is a Nash equilibrium in the full game. ### 4 ANYTIME PSRO ALGORITHM (APSRO) In this section we introduce a scalable extensive-form version of ADO, which we coin anytime PSRO (APSRO) (Algorithm 2). Rather than computing the exact NE for each player’s ADO restricted game $G^i$, APSRO approximates this solution by simultaneously optimizing each player’s restricted distribution $\pi^r_i$ via a regret minimization algorithm against a continuously trained RL best response $\beta_{-i}$. In this work, we update $\pi^r_i$ via the exponential-weight algorithm (Exp3) (Auer et al., 2002) or the Multiplicative Weights Update (MWU) algorithm (Cesa-Bianchi & Lugosi, 2006; Freund & Schapire, 1999). Algorithm 2 Anytime PSRO Result: $\epsilon$-Nash Equilibrium Input: Initial population $\Pi^0$ while Not Terminated $\{t = 0, 1, \ldots\}$ do Initialize $\pi^r_i$ to uniform over $\Pi^t_i$ for $i \in \{1, 2\}$ Initialize policies $\beta_i$ for $i \in \{1, 2\}$ for $i \in \{1, 2\}$ do for $n$ inner iterations do for $m$ iterations do Update policy $\beta_{-i}$ toward $\text{BR}_{-i}(\pi^r_i)$ (e.g. via Q-learning) Update $\pi^r_i$ via regret minimization vs. $\beta_{-i}$ (e.g. via Exp3 or MWU) $\Pi^{t+1}_i = \Pi^t_i \cup \{\beta_i\}$ for $i \in \{1, 2\}$ Return: $\pi^r$ Instead of recomputing an exact best response between regret minimization updates, APSRO maintains an approximate best response RL policy $\beta_{-i}$ for each player and updates it for a small number $m$ of steps in each inner-loop iteration. We allow $\beta_{-i}$ to be an approximate best response, and we set the hyperparameter $m$ to a smaller value than may be necessary to fully converge to $\text{BR}_{-i}(\pi^r_i)$. In practice, this trades off the theoretical guarantees of exact best responses with a considerable computational speedup. We include details about the no-regret procedure in Appendix C. The updates to the best response can be made through a variety of algorithms. In this paper we show experiments with updates via tabular Q-learning as well as experiments via the deep reinforcement learning algorithm DDQN (Van Hasselt et al., 2016). Importantly, compared to PSRO, APSRO uses the same amount of episodes and environment interactions. The only difference is that APSRO changes the restricted distribution dynamically during training via a no-regret procedure. 4.1 APSRO THEORY In this section we present theory for APSRO where we assume the best response is exact in every inner iteration. We show that under this assumption APSRO converges to an approximate Nash equilibrium and never increases exploitability by much. The following proposition shows that APSRO with exact best responses approximately finds the least-exploitable restricted distribution. Proposition 3. Assume $ \beta_{-i} = \text{BR}_{-i}(\pi^_j) $ in every inner iteration of APSRO. Then APSRO with a regret minimizing algorithm that has regret $ R_j $ at inner iteration $ j $ will output a policy $ \pi^n $ such that $ e(\pi^n) \leq \frac{R_n}{n} $. By this proposition we know that APSRO with exact best responses will approximately find the least-exploitable restricted distribution for each player in each outer iteration. Since the population grows in every iteration, the least-exploitable distribution of a later iteration is never more exploitable than the least-exploitable distribution of an earlier iteration, simply because exploitability is later minimized over a superset of population mixtures. The following proposition formalizes this intuition. Proposition 4.* Assume APSRO with exact inner-loop best responses runs sufficiently many inner-loop updates in each iteration such that the exploitability in each restricted game is at most $ \epsilon $. Then the exploitability of APSRO will never increase by more than $ 2\epsilon $ from one iteration to the next. 5 SELF-PLAY PSRO Although ADO and APSRO mitigate increases in exploitability from one iteration to the next by adding to each player’s population the pure-strategy best response $ \beta_i $ to the opponent’s restricted distribution $ \pi_{-i} $, they are not guaranteed to decrease exploitability. $ \beta_i $ may not be the myopically optimal pure strategy whose addition to $ \Pi_i $ decreases exploitability the most. Moreover, adding mixed strategies can generally reduce exploitability faster than adding pure strategies. For example, consider the generalized Rock–Paper–Scissors game shown in Figure 2. In this game, the NE mixes equally over all pure strategies. As a result, any DO method that only adds pure strategies, such as common implementations of PSRO, will have to enumerate all pure strategies in the game before supporting the NE. Ideally, we would like to add a mixed strategy that decreases exploitability the most. A single-iteration objective would then be able to find the strategy such that after it is added to the population and the least-exploitable distribution is computed over this new population, the exploitability of the resulting distribution is the lowest. In this example game, a mixed strategy that mixes over the pure strategies equally is optimal and will lower exploitability more than any pure strategy. In general, the Nash equilibrium of the original game would be the optimal mixed strategy to add to the population, however finding a Nash equilibrium of the original game is very expensive and is our main goal in the first place. By trying to add a rough approximation of a Nash equilibrium of the original game to our population, we can still expect to improve our population exploitability a great deal. The closer this new mixed strategy is to being a Nash equilibrium of the original game, the more we would expect it to lower the resulting exploitability of the population. Motivated by this, we propose Self-Play PSRO, a PSRO method that learns and adds to the population an additional new mixed strategy each iteration. This new strategy is learned by best-responding to the opponent best response via off-policy reinforcement learning in a self play fashion and calculating a mixed-strategy time-average of it. While this self play process won’t necessarily produce a Nash Figure 3: SP-PSRO. In this diagram we show how SP-PSRO works within an iteration from the perspective of the column player. The fixed population is shown in blue and the new strategy is shown in green. Every inner iteration, three things happen. (1) The opponent best response updates toward a best response to the current distribution over both the fixed population and the new strategy. (2) The new strategy updates toward a best response against the opponent best response. (3) The restricted distribution updates via no regret against the opponent best response. In the final iteration, the time-average of the new strategy and the player’s best response to the opponent’s restricted distribution (which is trained in a symmetric manner) are added to the population, and the cycle starts again. Algorithm 3 Self-Play PSRO Result: Approximate Nash Equilibrium Input: Initial population $\Pi^0$ while Not Terminated $\{t = 0, 1, \ldots\}$ do for $i \in \{1, 2\}$ do Initialize new strategy $\nu_i$ arbitrarily Initialize $\pi^_i$ to uniform over $\Pi^t_i \cup \{\nu_i\}$ for $n$ iterations do for $m$ iterations do Update policy $\beta_{-i}$ toward $\text{BR}_{-i}(\pi^_i)$ (e.g. via Q-Learning) Update new strategy $\nu_i$ toward $\text{BR}_i(\beta_{-i})$ (e.g. via Q-Learning) Update $\pi^_i$ via regret minimization vs. $\beta_{-i}$ (e.g. via Exp3 or MWU) $\Pi^{t+1}_i = \Pi^t_i \cup \{\beta_i, \bar{\nu}_i\}$ for $i \in \{1, 2\}$ Return: $\pi^_r$ SP-PSRO works by maintaining a restricted distribution $\pi^_i$ over a population. Unlike PSRO, where $\pi^_i$ is the NE of the restricted game, SP-PSRO trains $\pi^_i$ in the same way as in APSRO, via regret minimization. In addition, at the beginning of each iteration, a new strategy $\nu_i$ is initialized and added to the population. During an iteration, three training processes unfold concurrently. First, as in APSRO, the opponent’s best response $\beta_i$ takes multiple update steps toward a best response to the current restricted distribution $\text{BR}_{-i}(\pi^_i)$. Second, the new strategy $\nu_i$ is updated toward a best response to the opponent best response $\text{BR}_i(\beta_{-i})$. Third, the restricted distribution $\pi^_i$ is trained via regret minimization; this includes updating the probability of the new population strategy $\nu_i$, even as $\nu_i$ is also trained. This procedure can be thought of a form of self-play, in which the new strategy is updating against the opponent best response, while the opponent best response is updating against the restricted distribution, which also contains the new strategy. When the iteration is finished, the time-average $\bar{\nu}_i$ of $\nu_i$ is added to the population. We include further details on SP-PSRO in Appendix L.3 Averaging over the updates of $\nu_i$ can be accomplished by checkpointing the policy over time and uniformly sampling checkpoints, or by training a neural network to distill a buffer of experience generated by $\nu_i$ as it trains. Since the new strategy is trained via off-policy reinforcement learning, SP-PSRO uses the same amount of environment experience as APSRO, but does require more compute to train the new network. Additionally, since it still adds best responses $\beta$, similar to APSRO and PSRO, it will also converge to an optimal population that supports a NE. 6 EXPERIMENTS 6.1 NORMAL FORM EXPERIMENTS In this section we describe experiments on normal form games. To emulate the process of a strategy $\pi$ learning a best response to another policy $\pi'$, in every inner loop iteration $t$ we update $\pi$ by the following learning rule: $\pi_{t+1} = (1 - \lambda)\pi_t + \lambda \times BR_\pi(\pi')$. We show three normal form games. The first, described in Figure 4a, is a large generalized Rock–Paper–Scissors game. The second is a Hex restricted game (Perez-Nieves et al., 2021). The third game is the final restricted game of the AlphaStar population (Vinyals et al., 2019). More normal form games are included in Appendix E. As shown in Figure 4, SP-PSRO vastly outperforms both PSRO and APSRO. Note that APSRO and SP-PSRO only reach an $\epsilon$-NE because they use a finite number of regret minimization updates to determine the restricted distribution, while PSRO is able to exactly compute a NE. We have included further details in the Appendix. 6.2 TABULAR EXPERIMENTS We evaluated SP-PSRO with tabular methods in a variety of games. We applied tabular SP-PSRO to the domains of Leduc Poker (9,457 states), a tiny version of Battleship (1,573 states), and 4x-Repeated Rock Paper Scissors (9,841 states). The experiments used game implementations and tools from the OpenSpiel library (Lanctot et al., 2019). In extensive form tabular experiments, the new population strategy $\nu_i$ and the best response $\beta_{-i}$ are represented by tabular Q-learning agents. When training the Q-learning agent for $\beta_{-i}$, experience from the same episodes is also used to train the agent for $\nu_i$ in an off-policy manner. The tabular Q-learning agents are $\epsilon$-greedy, and we use a constant value of $\epsilon$ for both agents. Because experience for $\beta_{-i}$ and $\nu_i$ is shared from the same episodes, experience is collected against $\epsilon$-greedy versions of some opponent policies. Compared to collecting separate episodes for each player, we found that using the same episodes to train policies for both players despite small amounts of action exploration reduces the required sample complexity by two without affecting performance very much. In these 3 games, we collect the same amount of experience per iteration for PSRO, APSRO, and SP-PSRO. Similar to our normal form results, we find that APSRO does not increase exploitability by much from one iteration to the next and SP-PSRO drastically reduces exploitability compared to baselines. Interestingly, in tiny battleship, the exploitability of APSRO had higher variance compared to that of PSRO. We hypothesize that this is due to the APSRO iterations not being long enough for the no-regret process to converge. We have included further details in the Appendix. SP-PSRO outperforms APSRO and PSRO in each of the three games: Leduc Poker (Figure 5a), the small Battleship game (Figure 5b), and 4-repeated Rock Paper Scissors (Figure 5c). In each game, we see a drastic improvement in performance starting in the first iteration. 6.3 Deep Reinforcement Learning Experiments When using deep reinforcement learning best-response operators with DDQN (Van Hasselt et al., 2016), SP-PSRO outperforms APSRO and PSRO in terms of sample efficiency (Figure 6). Tested on Liar’s Dice, a small version of Battleship, and 4x-Repeated RPS, SP-PSRO sees a significant improvement against other baselines in early-iteration exploitability. This early exploitability advantage seen by SP-PSRO is especially present in repeated RPS (Figure 6c), where the relative performance seen with deep RL methods roughly matches that of tabular methods. Examining a player’s final restricted distribution after 5 iterations of SP-PSRO in repeated RPS (Figure 6d), we also see that the time-averaged new strategies have significantly more support than the standard best-responses, demonstrating their contribution towards exploitability improvements. 7 Future Work SP-PSRO opens up exciting connections to the literature regarding learning approximate Nash equilibria in large games. In particular, although we introduce an unprincipled self play method for approximating a Nash equilibrium, future work can find better ways of creating a new strategy that will better approximate a Nash equilibrium and therefore result in lower exploitability every iteration. For example, the data collected via the opponent best response training against the restricted distribution can be used in a Monte-Carlo CFR-type algorithm to minimize regret on information sets visited during training. These directions also open up the possibility of deriving the first regret bounds for double oracle algorithms that do not rely on the size of the effective pure strategy set (Dinh et al., 2021). It also introduces the possibility of combining the deep reinforcement learning from the best response with methods based on deep CFR. For example, perhaps the Q networks learned from the best responses can be used to minimize regret for the new strategy. Finally, our algorithm is a normal-form algorithm in that it mixes at the root of the game tree. McAleer et al. (2021) showed that this can be exponentially bad in the worst case, and introduced tabular (XDO) and deep (NXDO) algorithms to fix this problem. An interesting future direction is combining SP-PSRO with XDO and NXDO. REFERENCES Peter Auer, Nicolo Cesa-Bianchi, Yoav Freund, and Robert E Schapire. The nonstochastic multiarmed bandit problem. SIAM journal on computing, 32(1):48–77, 2002. Ronen I Brafman and Moshe Tennenholtz. 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PdaPky8MUn	The methods evaluated for Figure 2 still seem to use complex valued parameterizations even though they are randomly initialized. Is this still necessary when using SPT? Since complex values can be problematic when scaling large scale systems, it would be interesting if SPT also removed the need for this in SSMs/linear RNNs.	Never Train from Scratch: FAIR COMPARISON OF LONG-SEQUENCE MODELS REQUIRES DATA-DRIVEN PRIORS Ido Amos Tel Aviv University* Jonathan Berant Tel Aviv University Ankit Gupta IBM Research ABSTRACT Modeling long-range dependencies across sequences is a longstanding goal in machine learning and has led to architectures, such as state space models, that dramatically outperform Transformers on long sequences. However, these impressive empirical gains have been by and large demonstrated on benchmarks (e.g. Long Range Arena), where models are randomly initialized and trained to predict a target label from an input sequence. In this work, we show that random initialization leads to gross overestimation of the differences between architectures and that pretraining with standard denoising objectives, using only the downstream task data, leads to dramatic gains across multiple architectures and to very small gaps between Transformers and state space models (SSMs). In stark contrast to prior works, we find vanilla Transformers to match the performance of S4 on Long Range Arena when properly pretrained, and we improve the best reported results of SSMs on the PathX-256 task by 20 absolute points. Subsequently, we analyze the utility of previously-proposed structured parameterizations for SSMs and show they become mostly redundant in the presence of data-driven initialization obtained through pretraining. Our work shows that, when evaluating different architectures on supervised tasks, incorporation of data-driven priors via pretraining is essential for reliable performance estimation, and can be done efficiently. 1 INTRODUCTION Self-supervised pretraining is now widespread across most areas of machine learning, including NLP, speech, and vision (Touvron et al., 2023; Baevski et al., 2020; Reed et al., 2022). Given a downstream task, it is standard to finetune a pretrained model rather than train “from scratch”, to achieve better performance (Raffel et al., 2019). Conversely, when developing new architectures with better inductive biases for particular skills, for example, for capturing long-range dependencies or for better algorithmic reasoning, it is still common to train on the task data from scratch with random initialization (Tay et al., 2020a; DeLétang et al., 2022; Velivckovi´c et al., 2022; Dwivedi et al., 2022). This difference in practice stems not only from the computational overhead required for pretraining on massive datasets, but also to decouple the effects of the pretraining data and allow an apples-to-apples comparison, which would otherwise require a “standard” pretraining corpus for each scenario. A prime example of the latter scenario is estimating capabilities in modeling long range dependencies in sequences, a setting where Transformers have reported inadequate performance on benchmarks designed as stress tests, such as Long Range Arena (LRA) (Tay et al., 2020a). This inefficacy of Transformers has led to a line of new architectures, suggesting changes to RNNs, CNNs and Transformers themselves, biasing them towards capturing long range dependencies, and achieving impressive performance on LRA, when trained from scratch (Gu et al., 2022a; Gupta et al., 2022a; Li et al., 2022; Ma et al., 2022). However, these results do not align with performance of pretrained Transformers (“foundation models”), that have displayed remarkable performance on tasks involving modeling long range dependencies, such as text summarization, code completion and protein folding, (Touvron et al., 2023; Jumper et al., 2021). Despite the significant progress in long sequence modeling, the reasons for sub-par performance of Transformers on long sequence benchmarks, such as LRA, {idoamos@mail.tau.ac.il, joberant@cs.tau.ac.il, ankitgupta.iitkanpur@gmail.com}. remains unexplored, while methods achieving competitive performance resort to tailored changes to the architecture (Ma et al., 2022; Zuo et al., 2022). In this work, we shed light on this discrepancy, showing it stems from inadequate training and evaluation practices, and suggest a simple and efficient solution allowing a proper evaluation. While avoiding pretraining on a large corpus is understandable, training from a random initialization with downstream supervision alone disregards the role of the pretraining objective itself, leading to a different inductive bias than that of a pretrained model. In a recent line of work, El-Nouby et al. (2021); He et al. (2022); Krishna et al. (2023) have demonstrated that, when using denoising objectives, pretraining solely on downstream training data (denoted as self pretraining) often leads to gains comparable to the ones from pretraining on large corpora, showing effectiveness on tasks such as image classification, segmentation, text classification, etc. This suggests that, rather than training from scratch, a more realistic estimate of model performance can be obtained via self pretraining (SPT), with SPT acting as a data-driven initialization method, while allowing a fair comparison between methods as only the task data is used. To demonstrate the importance of the suggested method, we empirically show that priors learned through SPT with denoising objectives are highly effective for learning long range dependencies across several architectures, eliminating the need for complex hand-crafted modeling biases used in current solutions (Gu et al., 2022a; Ma et al., 2022; Li et al., 2022; Orvieto et al., 2023). We primarily study Long Range Arena (LRA), a standard benchmark for long sequence modeling and evaluate multiple SPT models. We show that SPT improves the mean absolute performance of vanilla Transformers by more than 30%, for the first time allowing them to match the state-of-the-art performance on LRA without any architectural changes (Figure 1). This is in stark contrast to prior works where Transformers report significantly lower performance compared to the state-of-the-art. We study the effectiveness of SPT for State Space models (SSMs), a novel line of architectures using modified linear RNNs as a replacement for attention layers in Transformers. Incorporating a specialized parameterization and initialization of linear RNNs, SSMs such as S4 successfully mitigate the vanishing/exploding gradient issues and reach impressive performance on long sequence tasks, such as LRA (Gu et al., 2022a). We find SPT to also benefit S4 with performance gains in 5 out of 6 LRA tasks. Moreover, with SPT, S4 solves the challenging PathX-256 task, achieving a 20% accuracy improvement compared to training from scratch (Figure 1). Building on these improvements, we study the utility of hand-crafted modeling biases in S4 over simpler linear RNNs, finding that the data-driven priors learned via SPT render most of them redundant (Gupta et al., 2022b). In doing so, we are the first to provide competitive performance with diagonal linear RNNs without any manual modifications (Orvieto et al., 2023). Our findings show that priors beneficial for capturing distant dependencies can be simply learned from the task data via standard denoising objectives without any intrusive changes to the model. We examine the benefits of SPT across multiple data scales showing them to become even more pronounced as data becomes relatively scarce. Last, for SSMs, we analyze the convolution kernels... learned via SPT to shed light on the learned priors for capturing long-range dependencies. We demonstrate an interesting phenomena in which, depending on the modality, rapidly decaying kernels can lead to improved performance over the slowly decaying ones as used in the native S4 model, further highlighting the utility of learning priors from the data itself (Gu et al., 2020). Our main contributions can be summarized as follows: (i) We show that the reported performance of various architectures on long range benchmarks is grossly underestimated, and suggest an inexpensive data-driven approach to enable accurate evaluation without requiring any additional data. (ii) We report large empirical gains over the previously-reported performances on LRA across a range of architectures and, in particular, improve upon the best reported accuracy on the challenging PathX-256 task by 20 absolute points (67 → 87). (iii) We demonstrate how manually-designed biases become increasingly redundant with pretraining and that, with modern training and evaluation practices, simpler models can often match the performance of sophisticated architectures. We are the first to provide competitive performance on LRA with Transformers and diagonal linear RNNs. The multi-modal and challenging setup of LRA, along with the scale of improvements due to SPT, advocate the inclusion of a pretraining stage while evaluating models in general, for example when designing architectures for multidimensional inputs (Nguyen et al., 2022), algorithmic reasoning (Diao & Loyn, 2023) or graphs (Shirzad et al., 2023). Our code & data are available at https://github.com/IdoAmos/not-from-scratch 2 EXPERIMENTAL SETUP Our experiments center around the evaluation of Transformers and SSMs on the Long Range Arena (LRA) benchmark which was proposed for examining the ability of sequence models to capture long-range dependencies (Tay et al., 2020a). It contains 6 main sequence classification tasks, each being either binary or 10-way sequence classification. 1. ListOps: Each sequence in the dataset is a nested list, with each sublist describing an operation (e.g. MAX, MEAN) to be applied on a set of tokens (Nangia & Bowman, 2018). Evaluation of nested lists are used as a single token in their enclosing list thus requiring the understanding of hierarchical structure, the task is 10-way classification with sequence length of $2K$. INPUT: [MAX 4 3[MIN 2 3]1 0[MEDIAN 1 5 8 9 2]] OUTPUT: 5 2. Text: a character-level version of the IMDb reviews dataset (Maas et al., 2011) for sentiment classification, the task is binary classification with sequence length of up to 2048. 3. Retrieval: a character-level version of the AAN dataset (Radev et al., 2013) for predicting similarity scores of two documents. The task is binary classification with sequence length of up to $4K$, requiring to process $8K$ tokens for evaluation. 4. Image: grayscale CIFAR10 images are flattened as 1D sequences and any explicit 2D inductive bias cannot be used. The task is 10-way classification, with sequence length 1024. 5. Pathfinder, PathX: synthetic 2D visual tasks treated as 1D sequences (similar to Image) for testing tracing capabilities (Linsley et al., 2018; Kim et al., 2020). PathX and Pathfinder are similar tasks that differ in sequence length (1024 vs 16384) and are binary classification. Apart from the aforementioned tasks, we examine an additional variant of PathX called PathX-256 with sequence length $256^2 = 65536$ and we are the first to report strong results on this task. Besides LRA, we experiment with additional datasets that will be described later in Section 3.7. Self Pretraining (SPT) We perform SPT with a causal/autoregressive sequence modeling objective for unidirectional models, and a masked sequence modeling objective for bidirectional models, using only the downstream task training set. For the visual tasks (Image, Pathfinder, Path-X) the masking ratio for masked sequence modeling is set to 50% following He et al. (2022), to 15% for language Table 1: Long Range Arena. (top) performance of models trained from scratch as reported in Tay et al. (2020a), (bottom) performance of self pretrained (SPT) Transformers of sizes comparable to the ones on top. X denotes chance accuracy. \| Approach \| Listops \| Text \| Retrieval \| Image \| Pathfinder \| PathX \| Avg. \| \|-------------------\|---------\|--------\|-----------\|--------\|------------\|-------\|------\| \| Transformer \| 36.37 \| 64.27 \| 57.46 \| 42.44 \| 71.40 \| X \| 53.66\| \| Local Attention \| 15.82 \| 52.98 \| 53.39 \| 41.46 \| 66.63 \| X \| 46.71\| \| Longformer \| 35.63 \| 62.85 \| 56.89 \| 42.22 \| 69.71 \| X \| 52.88\| \| Linformer \| 35.70 \| 53.94 \| 52.27 \| 38.56 \| 76.34 \| X \| 51.14\| \| Reformer \| 37.27 \| 56.10 \| 53.40 \| 38.07 \| 68.50 \| X \| 50.56\| \| BigBird \| 36.05 \| 64.02 \| 59.29 \| 40.83 \| 74.87 \| X \| 54.17\| \| Linear Trans. \| 16.13 \| 65.90 \| 53.09 \| 42.34 \| 75.30 \| X \| 50.46\| \| Performer \| 18.01 \| 65.40 \| 53.82 \| 42.77 \| 77.05 \| X \| 51.18\| \| Transformers + Masked SPT \| 59.75* \| 89.27 \| 88.64 \| 74.22 \| 88.45 \| 87.73 \| 81.34 \| \| Transformers + Causal SPT \| 59.15 \| 88.81 \| 90.38 \| 76.00 \| 88.49 \| 88.05 \| 81.81 \| tasks (Text, Retrieval) following Liu et al. (2019), and to 10% for ListOps. For Transformers, we use full attention as default with the hardware-optimized FLASH implementation (Dao et al., 2022). Due to computational constraints, for tasks with sequence length at least $16K$ we split the input to the attention layer into non-overlapping blocks of size 4096 and allow each block to attend to itself and its neighbour(s). Our codebase is built on the original S4 repository[^1]. For additional experimental details, such as computational resources for SPT and finetuning, please refer to Appendix C.1. 3 RESULTS In Section 3.1, we perform SPT for LRA tasks using the official model configurations. In Section 3.2, we perform SPT for Transformers and S4. Section 3.3 evaluates the role of design choices in SSMs in the context of SPT. Section 3.4 examines the utility of SPT across data scales and Section 3.5 examines the utility of PT on a large text corpus. Section 3.6 provides an analysis of pretrained SSM kernels and how they relate to current initialization schemes. Section 3.7 contains additional experiments on distinct modalities. 3.1 UNDERESTIMATION OF LONG-RANGE ABILITIES OF TRANSFORMERS We start by investigating the reliability of the historically-reported model performances on LRA, in the more modern setting of pretraining. Concretely, we repeat the Transformer experiments performed by Tay et al. (2020a), except that we first pretrain the model on the task data and then finetune it. To allow fair comparison with the original results, we strictly follow the model configurations used by Tay et al. (2020a). We experiment with two pretraining objectives: (1) next token prediction for unidirectional models (2) masked token prediction for bidirectional models, varying the masking ratio as detailed in Section 2. As summarized in Table 1, we find that both pretraining objectives lead to dramatic performance gains for Transformers compared to the conventional practice of training with random initialization, with the average test accuracy increasing by roughly 30%. Both causal and masked pretraining yield similar results even in cases where there are no clear benefits to using a causal model, such as on the visual tasks. Furthermore, even for LISTOPS large performance gains are observed even though, in the original data, the arguments to the list operations are sampled randomly, meaning that inferring missing tokens from the context is rarely possible. As the experiments are performed with no architectural changes or additional data, the difference in performances can be attributed to the priors learned during SPT, clearly demonstrating its importance for a reliable performance evaluation. [^1]: https://github.com/HazyResearch/state-spaces Table 2: Long Range Arena. Self pretrained (SPT) Transformers and S4 compared to existing trained from scratch models. Average performance (“Avg.”) is reported without PathX-256 to align with prior work. Results for MEGA, SPADE & S4 are taken from original papers with exceptions denoted by †. ✗ denotes computationally infeasible, □ denotes unreported results. \| Approach \| Listops \| Text \| Retrieval \| Image \| Pathfinder \| PathX \| PathX-256 \| Avg. \| \|---------------------------\|---------\|--------\|-----------\|--------\|------------\|-------\|-----------\|------\| \| Transformers + Rotary \| 47.90 \| 79.08 \| 82.31 \| 75.04 \| 76.64 \| ✗ \| \| 74.28\| \| Transformers + Rotary + Masked SPT \| 61.49 \| 91.02 \| 91.57 \| 86.04 \| 94.16 \| 92.98 \| ✗ \| 86.21\| \| S4 (Gu et al., 2022a) \| 59.60 \| 86.82 \| 90.90 \| 88.65 \| 94.20 \| 96.35 \| 67.82† \| 86.09\| \| S4 + Masked SPT \| 61.25 \| 90.34 \| 88.74 \| 89.36 \| 94.92 \| 96.94 \| 87.11 \| 86.75\| \| SPADE (Zuo et al., 2022) \| 60.50 \| 90.69 \| 91.17 \| 88.22 \| 96.23 \| 97.60 \| □ \| 87.40\| \| MEGA (Ma et al., 2022) \| 63.14 \| 90.43 \| 91.25 \| 90.44 \| 96.01 \| 97.98 \| □ \| 88.21\| \| Pythia 70M (Rand Init) \| 41.20 \| 69.29 \| 76.45 \| 52.55 \| 74.31 \| ✗ \| ✗ \| 62.76\| \| Pythia 70M \| 43.05 \| 83.41 \| 84.29 \| 67.41 \| 80.05 \| ✗ \| ✗ \| 68.04\| ### 3.2 Comparing S4 and Transformers In the above set-up we strictly adhered to the model sizes used by Tay et al. (2020a) and consequently the absolute performances are still low compared to the current state-of-the-art on LRA. In this section, we scale the model sizes and evaluate the utility of SPT for the best performing architectures including S4 (Gu et al., 2022a). For Transformers, we replace the positional embeddings with the more commonly used rotary embeddings (Su et al., 2021) and only train bidirectional models in line with prior works reporting high performance. As summarized in Table 2, SPT leads to dramatic performance gains for Transformers with performance gains ranging from 8 – 15% across tasks, even surpassing the average performance of a well-tuned S4 (86.2 vs 86.1). SPT Transformers surpass the performance of both trained from scratch and SPT versions of S4 on 3 out of 6 tasks. The results in Table 2 defy current understanding, with prior works citing the sub-par LRA performance of Transformers as a prime motivating factor for new methods. Yet we show that, while architectural developments indeed lead to remarkable performance gains, most of the priors essential to high performance can already be learned from data directly. In case of S4, while SPT leads to modest gains on most tasks, a substantial gain of 20% is observed on the challenging PathX-256 task with input length of $65K$, significantly improving over the best reported performance of 63.1% by (Dao et al., 2022) who, in addition, used extra data from the Pathfinder-64 task. The additionally reported models, SPADE and MEGA, are Transformer variants that augment the model with a single or several state space layers. SPADE combines the outputs of a frozen S4 layer and local attention in the first block, while MEGA incorporates a learned exponential moving average, an instance of diagonal SSMs, into gated attention blocks. To the best of our knowledge, we are the first to show that purely attention-based methods, without any architectural modifications, can achieve competitive results on LRA. While incorporating SSMs can be important in terms of scalability to longer sequences due to their log-linear complexity with respect to input length, we show that in terms of model performance, pretraining leads to biases that are as effective as manual designs. An important aspect of SPT is the use of additional compute compared to the trained from scratch baseline and it is natural to investigate if similar gains can be obtained by training from scratch for longer. For all our trained from scratch baselines, we ensured that the validation performance had converged and did not improve for several consecutive epochs. We examine the aspect of the computational overhead of SPT in detail Appendix D where we show that SPT leads to significant gains, even in the setting where the same amount of compute is used for SPT models and the ones that are trained from scratch. ### 3.3 The Role of Explicit Priors We have established that SPT allows for a more reliable evaluation of the actual capabilities of architectures and further improves the performance of SSMs such as S4. Despite its high performance, S4 has a complex design guided by principled theoretical considerations to enable long range signal Figure 2: Average performance of models when trained from scratch or self pretrained, for different sets of initializations prior to pretraining. See Table 7 for per-task results. propagation, which can explain the small advantage maintained over SPT Transformers, lacking such an inductive bias. In a series of works, various simplifications to S4 have been proposed while maintaining performance. We will now show that SPT allows for an even simpler model (viz. diagonal linear RNN) to match the performance of S4. We first provide a brief overview of SSMs below and refer to Gu et al. (2022a) for a detailed description. Given an input scalar sequence $ u_n $, SSMs follow a linear recurrence generating a hidden state vector $ \vec{x}_n $ at timestep $ n $, and produce a scalar output sequence $ y $ as \[ \begin{align} \vec{x}_n &= A \vec{x}_{n-1} + B u_n & A \in \mathbb{C}^{N \times N}, \quad B \in \mathbb{C}^{N \times 1} \\ y_n &= C \vec{x}_n & C \in \mathbb{C}^{1 \times N} \end{align} \] By unrolling the recurrence across the timesteps, it can be shown that $ y $ can be equivalently computed by convolving $ u $ with the kernel defined by $ K_k = C^T A^k B $. Instead of directly using $ A, B, C $ as learnable parameters, S4 uses an alternate parameterization inspired by a theory in continuous time, motivating the transformations: \[ \begin{align} A &= \Lambda - P Q^ \\ \bar{A} &= (I - \Delta/2 \cdot A)^{-1} (I + \Delta/2 \cdot A) \\ \bar{B} &= (I - \Delta/2 \cdot A)^{-1} \Delta B \quad \bar{C} = C \\ K_k &= \bar{C}^T \bar{A}^k \bar{B} \end{align} \] where $ \Lambda, P, Q, B, C, \Delta $ are learnable parameters and $ \Lambda \in \text{Diag}(\mathbb{C}^{N \times N}), P, Q \in \mathbb{C}^{N \times 1} $. In addition to this parameterization, S4 uses a principled initialization method aimed towards a slow decay of the kernel (w.r.t. timestep $ k $) in order to facilitate capturing long-range dependencies. Inspired by the success of S4, Gupta et al. (2022b) proposed a simplification to S4 called Diagonal Linear RNN (DLR) defined as \[ \begin{align} \vec{x}_n &= \Lambda \vec{x}_{n-1} + \mathbf{1} u_n & \Lambda \in \text{diag}(\mathbb{C}^{N \times N}) \\ y_n &= C \vec{x}_n & C \in \mathbb{C}^{1 \times N} \end{align} \] where $ \mathbf{1} $ is the all-ones vector. DLR is significantly simpler to compute compared to S4 and the authors reported it to be as performant as state-of-the-art SSMs on a wide variety of token-level tasks. Hence, it is natural to investigate the conditions under which S4 with its more complex design (eq. 2) can be replaced by the simpler DLR. To that end, we evaluate the performance of DLR and S4 on ListOps, Text, Image and PathX tasks as they are the hardest and represent all modalities in LRA. For each model, we experiment with two sets of initializations: (1) random initialization where the state space parameters are initialized from a normal distribution with a small standard deviation, and --- 2When the input is a sequence of vectors, the model is applied to each channel separately and is commonly followed by a FFN to exchange information across channels. Figure 3: Trained from scratch and self pretrained (SPT) versions of S4 evaluated on multiple data scales for Image and Text tasks from LRA, originally containing $45K$ and $25K$ samples respectively. (left) absolute performances and (right) relative gains due to SPT over training from scratch. (2) “structured” initialization recommended by the respective authors aimed at capturing long-range dependencies. The results are summarized in Figure 2 and per-task results are provided in Table 7. We find that, when trained from scratch, with both random and structured initializations, DLR lags behind S4 in terms of average performance (77 vs 83) demonstrating that biases incorporated through the specific initialization and parameterization used in S4 are indeed critical to performance. However, the picture radically changes under SPT – with SPT, DLR outperforms a trained from scratch S4 (83.4 vs 82.8) and is only slightly behind SPT S4 (83.4 vs 84.5). This suggests that the data-driven priors learned through pretraining are almost as effective as the manual biases incorporated in S4. Results in this section have two additional implications. First, this is the first instance in which vanilla diagonal linear RNNs have been shown to achieve competitive performance on LRA. Prior work by Orvieto et al. (2023) suggested an additional normalization step in the kernel generation on top of a tailor-made initialization to achieve high performance on LRA. Second, while our discussion revolved around SSMs, many subsequent works on designing global convolutions followed similar principles. For example, Li et al. (2022) proposed to generate a decaying convolution kernel from shorter kernels via interpolation, which induces smoothness and can be viewed as a normalization step. Similarly, Fu et al. (2023) applied a global convolution layer that is transformed by a deterministic function to explicitly induce a smoother kernel. Yet our results suggest that these explicit steps are less significant when models are self pretrained. 3.4 Self pretraining is Effective Across Data Scales As the priors learned via SPT are data-driven, their efficacy is dependent on the training set itself, which leads us to examine the performance gains as a function of the dataset size. To this end, given a downstream task, we randomly sample a subset of the training set, and study the performance gains for S4 due to SPT under varying sizes of the subset. We restrict the pretraining phase of S4 to a fixed number of update steps across all experiments and finetune until convergence. As summarized in Figure 3, we uncover an interesting phenomenon; while the relative gains from SPT over the trained from scratch baseline S4 are modest when the full task data is available, they become increasingly significant (and as large as 30%) on smaller data scales. This shows that priors from pretraining are especially effective when training data is scarce and, in the context of previous sections, implies that the incorporation of the pretraining stage is important for model evaluation regardless of dataset size. In Appendix E, we provide a complementary study on the effectiveness of SPT across model sizes, demonstrating that indeed SPT is effective across multiple model scales for both S4 and Transformers. 3.5 Pretraining on Text Corpora Given the widespread success of pretrained language models and the large gains due to SPT on the LRA tasks (Table 2), it is natural to ask if similar gains could be achieved by finetuning a language model pretrained on a large text corpus. To answer this, we consider Pythia 70M (Biderman et al., 2023), an autoregressive Transformer pretrained on the Pile (Gao et al., 2020) as well as a randomly initialized version with the same architecture, denoted as “Pythia 70M (RandInit)” in Table 2. To be comparable to existing results and due to the formal requirements of the LRA benchmark, we use character/pixel-level tokenization instead of the original BPE tokenizer and the model is required to adapt to the new tokenization during finetuning. As shown in Table 2, Pythia 70M generally lags behind our trained from scratch Transformer baseline due to the changed tokenization and difference between the pretraining distribution and the downstream tasks. This further highlights the importance of SPT as it allows the model to specifically learn and adapt to the structure and modality of the given task data. However, the performance of Pythia 70M is significantly better than its randomly initialized version Pythia 70M (Rand init) suggesting that pretraining on large text corpora can be beneficial across modalities. ### 3.6 Theoretically-Derived vs Data-Driven Kernels Many high performing models such as SSMs incorporate manually-crafted priors to bias the model towards learning long range dependencies. For example, the initializations used in SSMs such as S4, DSS, S4D, S5 are based on HiPPO theory (Gu et al., 2020), which explicitly determines the decay rate of the convolution kernels over time and provides strong dependence between distant elements in the input sequence. In similar spirit, Li et al. (2022) generate convolution kernels modified with fixed weights aimed towards a slow decay. On the other hand, kernels learned via SPT have no guarantees of a slow decay and depend solely on the input distribution and the pretraining objective. In this section, we analyze the structure of the convolutional kernels learned via SPT and compare them to the HiPPO-based kernels used to initialize existing SSMs such as S4. The convolution operation in S4 has the form \[ y_{c,k} = \sum_{l=0}^{k} \tilde{C}_c^T \tilde{A}_c^l \tilde{B}_c x_{c,k-l} = \sum_{l=0}^{k} K_{c,l} \cdot x_{c,k-l} \] where $ c $ is the channel and $ k $ is the timestep. Based on this structure, we can estimate the degree of dependence between sequence elements at channel $ c $, $ l $ positions apart as $ \|K_{c,l}\| $. For easier Table 3: Additional Experiments.* Performance on Speech Commands (SC), sCIFAR (accuracy) and BIDMC (R2) tasks. Results for trained from scratch S4 taken from Gu et al. (2022a), except for BIDMC (denoted by $\tilde{\tau}$) that are reproduced for the more interpretable R2 score. \| Approach \| SC \| sCIFAR \| BIDMC \| \|----------------\|-------------\|-------------\|-------------\| \| \| Causal Bi. \| \| HR RR SpO2 \| \| S4 \| 93.60 \| 96.08 \| 91.13 \| \| Transformers \| 84.55 \| 86.93 \| 79.41 \| \| S4 + SPT \| 95.09 \| 96.52 \| 91.67 \| \| Transformers + SPT \| 86.13 \| 91.49 \| 90.29 \| We take the maximal absolute value over the channels $K_{\text{max},l} = \max_c \|K_{c,l}\|$. For a shift $l$, $K_{\text{max},l}$ bounds the norm of the derivative of $y_{c,k}$ w.r.t $x_{c,k-l}$ for all positions $k$ and channels $c$. We generate kernels for the pretrained S4 models from Section 3.2 (before finetuning) and compare with the ones used in standard S4. Figure 4 plots $K_{\text{max}}$ for the Image, Text, PathX and ListOps, all entailing better performance with the pretrained model (Table 2). We observe that the learned kernels exhibit variable decay rates across the tasks and model layers, in contrast to the fixed decay rate of the data-agnostic HiPPO kernels. In particular, on the Text task the learned kernels are more local compared to HiPPO. For PathX, the vertical grid lines are aligned with the image resolution ($128 \times 128$) showing high correlation between the underlying 2D structure of the data and the kernel peaks. Overall, Figure 4 further highlights the utility of SPT over data-agnostic initializations that cannot adapt to a local or global structure in a task distribution. ### 3.7 ADDITIONAL EXPERIMENTS In addition to LRA, we also tested the utility of SPT on 3 additional datasets, encompassing 2 additional natural data modalities, described as follows: - Speech Commands (SC) Raw speech waveforms of length $16K$ used in a 35-way classification task (Warden, 2018). We test both causal and bidirectional models following Gu et al. (2022b). - sCIFAR Sequential CIFAR-10 dataset using RGB channels as features. This is similar to the Image task from LRA that uses grayscale images, except that here richer features are used. - BIDMC A suite of 3 regression tasks, requiring to predict respiratory rate (RR), heart rate (HR) and blood oxygen saturation (SpO2) from EKG and PPG signals of length $4K$ each. The results shown in Table 3, with additional details in Appendix C.3, further strengthen the claims made throughout this work. On both SC and sCIFAR tasks, SPT leads to large performance gains for Transformers and modest gains for S4. The gaps between trained from scratch Transformer and S4 are substantially narrowed with SPT. On the SC task, SPT leads to a large 5% improvement for Transformers and we observe the performance gap between causal and bidirectional variants of S4 to be mitigated with SPT. A similar, but not identical, observation is made in section 3.1 where masked and causal SPT lead to very similar results on all LRA tasks. On the sCIFAR task, SPT leads to a dramatic 11% improvement for Transformers, nearly matching the performance of S4 (90.3 vs 91.7) and again pointing towards a sub-optimal evaluation when only training from scratch. On BIDMC, the performances of both Transformer and S4 baselines are already close to perfect and it is hard to observe any meaningful improvements due to SPT. In general, our results suggest that similar under-estimation of model performances might also be prevalent in other scenarios where training from scratch is standard (Deletang et al., 2022; Velivckovic et al., 2022; Dwivedi et al., 2022). --- 3 since the model is bidirectional there are two sets of kernels, left to right, and right to left. We take the maximum over both. 4 ACKNOWLEDGMENTS We thank Amir Globerson for insightful discussions and his support throughout the course of this work. This research was partially supported by The Yandex Initiative for Machine Learning, and the European Research Council (ERC) under the European Union Horizons 2020 research and innovation programme (grant ERC DELPHI 802800). 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TOE6N8dp4w	I find it interesting that DP fine-tuning the 20-40k parameter prompt tensor gives so good results. What makes the cost then relatively high anyways, the forward pass through the 8B parameter pre-trained model?	HARNESSING LARGE-LANGUAGE MODELS TO GENERATE PRIVATE SYNTHETIC TEXT Anonymous authors Paper under double-blind review ABSTRACT Differentially private training algorithms like DP-SGD protect sensitive training data by ensuring that trained models do not reveal private information. An alternative approach, which this paper studies, is to use a sensitive dataset to generate synthetic data that is differentially private with respect to the original data, and then non-privately training a model on the synthetic data. Doing so has several advantages: synthetic data can be reused for other tasks (including for hyper parameter tuning), retained indefinitely, and shared with third parties without sacrificing privacy. However, generating private synthetic data is much harder than training a private model. To improve performance on text data, recent work has utilized public data by starting with a pre-trained generative language model and privately fine-tuning it on sensitive data. This model can be used to sample a DP synthetic dataset. While this strategy seems straightforward, executing it has proven problematic. Previous approaches either show significant performance loss, or have, as we show, critical design flaws. In this paper we demonstrate that a proper training objective along with tuning fewer parameters results in excellent DP synthetic data quality. Our approach is competitive with direct DP-training of downstream classifiers in terms of performance on downstream tasks. Further, we demonstrate that our DP synthetic data is not only useful for downstream classifier training, but also to tune those same models. 1 INTRODUCTION Machine learning models can memorize their training data (Carlini et al., 2019) and it is possible to extract the training data from a model (Carlini et al., 2021). Training a model with differential privacy (DP) (Abadi et al., 2016) provably reduces the risk of memorization (Ponomareva et al., 2022), which is critical when ML models are trained on sensitive data. However, DP training only ensures that the model does not release private information, and just releasing the model or its predictions is not adequate for many applications. For example, other researchers might want to use the data for analysis, or to build a different predictive model. It would therefore be ideal to release the dataset itself while protecting the privacy of the users that contributed to it. Local differential privacy has been proposed as a method of preprocessing low-dimensional datasets before public release (Ponomareva et al., 2023). Local DP adds noise to individual data points in the training data. While protecting privacy, local DP generally leads to much lower utility, due to the large amount of noise that must be added compared to central differential privacy, where DP is applied to the model or statistical output (Wang et al., 2017; Bassily et al., 2017; Team, 2017). Generally there is an inherent tension between privacy and utility when releasing private datasets: we want to release a dataset that protects the privacy of the underlying data while at the same time we want the dataset to be as useful as the original data for any possible downstream task. Therefore, we focus on central DP and consider generating private synthetic data. Generating such synthetic data involves creating a generative model that learns the original data distribution. To protect the original data, either the generative model should be made private, via DP training, or privacy should be enforced at inference time (e.g., during the generation of synthetic data items, so-called private prediction). Private inference has been shown to be inferior to DP training when a large number of inferences is required (van der Maaten & Hannun, 2020). Since we seek to generate at least as much data as in the original dataset, DP training is the clear choice. Several works proposed using publicly pre-trained large language models (LLM) for private synthetic data generation (Bommasani et al., 2019; Yue et al., 2022; Putta et al., 2023; Mattern et al., 2022). This approach involves privately fine-tuning an LLM using class labels as prompts for the model and subsequently sampling from this model. However these attempts have had mixed success: they either reported poor utility even for non-private synthetic data, or had to augment standard NLP loss metrics to assist the LLM to correctly respond to prompts during the generation process. Additionally, none of the previous work considered privacy leakage from a pre-trained LLM itself. The privacy leakage happens because these papers used academic datasets (like IMDB (Maas et al., 2011)) as sensitive dataset and they utilized GPT-2 LLM (Radford et al., 2019) which was pre-trained on these datasets without any privacy guarantees. Although we follow a similar recipe conceptually, in that we use a DP-finetuned LLM model to generate private synthetic data, we highlight the following differences in our execution of this idea: 1. Privacy leakage mitigation. We draw attention to the need to account for the data that went into pre-training of the LLMs used for generation. Our de-duplication of the pre-training data ensures that no privacy leakage, possibly present in previous works, takes place. 2. Reporting: We use a long sequence of text (512 tokens, representing full reviews like IMDB or Yelp) as our privacy unit. Our privacy guarantees (Appendix A) are tight and transparent, and we tune the hyperparameters of the downstream classifier on private synthetic data only. 3. Method: We demonstrate that the standard approach to private fine-tuning does not yield the desired quality of generated data. Instead of augmenting the LLM’s objective or architecture for fine-tuning as in (Putta et al., 2023; Mattern et al., 2022), we identify a loss function, well known to the NLP community, that is particularly suitable for private fine-tuning. Additionally, we argue that parameter-efficient fine-tuning, especially LoRA tuning, is beneficial for synthetic data generation. Our contributions can be summarized as follows: 1. We demonstrate state-of-the-art results in terms of quality of synthetic data. Specifically, we show in multiple experiments that the quality of the model trained on private synthetic data is comparable to or even better than the quality of the downstream model trained on real data with DP. 2. We demonstrate that parameter efficient fine-tuning like prompt-tuning and LoRA-tuning is superior to full fine-tuning when the tuning is performed privately. In particular, LoRA-tuning results in up to 11 percentage points lift in downstream model performance. To the best of our knowledge, we are the first to demonstrate that parameter-efficient tuning performs better than full fine-tuning when each is combined with DP, whereas the opposite often holds for non-DP training (Shin et al., 2020; Brown et al., 2020; Zhong et al., 2021). 3. We show that generating more synthetic data than the size of the original dataset is helpful, especially for simpler downstream models. 4. We show that DP synthetic data can be used to tune the hyperparameters of the downstream classifiers. We achieve ranking correlation with the ordering of trials performed on real data of up to 87%, even for $\epsilon = 1$. ## 2 RELATED WORK Privacy-preserving synthetic data generation requires that the generated data is both high-fidelity (i.e., exhibits similar distributional characteristics as the original data) and anonymized to preserve the privacy of the users who contributed their data. For complex data like text, images, audio and video, most existing approaches build a generative model, for example a GAN-based model (Guan et al., 2018). However, in most previous work the data is anonymized using heuristic methods, without providing formal privacy guarantees. For example, Melamud & Shivade (2019) attempted to de-identify summaries of clinical discharge notes using heuristic rules for an LSTM model and only empirically demonstrated the privacy of the synthetic data. DP-fine tuning is a standard method for fine tuning LLMs that satisfies differential privacy guarantees and has been shown to perform well with appropriate hyperparameter tuning (Li et al., 2021; Yu et al., DP-fine tuning involves using a pre-trained model and a modification of a training algorithm like DP-SGD to fine tune the model on private data. For private synthetic text generation, Bommasani et al. (2019) suggested using a pre-trained GPT-2 model and then DP-fine tuning it on private data with word-level privacy, but did not implement or evaluate any method. In similar vein, Yue et al. (2022) DP-fine tuned pre-trained GPT models of various sizes. While they do obtain good results on some of the benchmarks, they also observe up to 25% drop of downstream model accuracy on synthetic data (even without DP) on other benchmarks. Putta et al. (2023) attempted a similar recipe on a pre-trained distilGPT2 model, but also reported a large performance drop of the classifier trained on synthetic data. Additionally, they proposed modifying the fine tuning process to also include a discriminator that attempts to distinguish between the labels to improve the separability of learned representations for two binary classes of the text data. Similarly, Mattern et al. (2022) proposed augmenting the training objective with an additional term penalizing the generation of sample with the wrong label. None of the prior work takes into account problem of data contamination between LLM pre-training dataset and dataset used in downstream task. As we show in Appendix D this problem is real. In particular some of both training and test samples examples from downstream datasets could be found in GPT-2 pre-training data, which is used by all prior work. This may potentially invalidate DP-guarantees and may result in overestimated accuracy on downstream tasks. Additionally none of the works on DP synthetic data mentioned above explored parameter-efficient fine tuning. To the best of our knowledge, we are the first to demonstrate that parameter-efficient finetuning like LoRA tuning can produce better quality synthetic DP data than full finetuning. 3 PRELIMINARIES Differential privacy Differential Privacy (DP) (Dwork et al., 2006b) is considered the gold standard for ensuring data anonymization. Throughout this work we employ a notion of DP called $(\epsilon, \delta)$-DP. Definition 1 ($(\epsilon, \delta)$-Differential Privacy, (Dwork et al., 2006a)). Consider neighbouring datasets to be datasets that differ only in addition or removal of one record only. Given non-negative $\epsilon$ and $\delta \leq 1$, a mechanism $A$ is $(\epsilon, \delta)$-DP if for any two neighbouring datasets $D$ and $D'$ and for any $S \subseteq Range(A)$, $$P[A(D) \in S] \leq \exp(\epsilon) \times P[A(D') \in S] + \delta.$$ (1) The $\epsilon$ and $\delta$ values determine the strength of the privacy guarantees, with smaller values corresponding to stronger guarantees. The post-processing property of a DP mechanism means that applying any data-independent transformation to its output will remain DP with the same guarantees. DP in context of ML models In context of ML, DP can be introduced either at the input level, during the training of a model (DP-Training), or during model serving (prediction) (Ponomareva et al., 2023). DP synthetic data falls into the first category and in general is a harder task than introducing DP during the training. This is because DP synthetic data ensures that any ML model trained on this data is DP with respect to the original training data. This is in contrast with DP-Training that only ensures that a particular ML model is DP. Therefore, it is expected that any model trained on DP synthetic data should perform at most as well as the downstream DP-Trained ML model on real data. However the idea of using a pre-trained generative LLM to aid generation of synthetic data means that we inject massive amount of public data, making the task of DP synthetic data generation less daunting. The most practical methods of DP-Training for non convex losses are gradient-noise injection methods like DP-SGD (Abadi et al., 2016), which work by clipping per example gradients to limit the sensitivity of the loss, and noising aggregated clipped gradients with Gaussian noise to make them private. The noise level is proportional to the clipping norm (the sensitivity) and the strength of $\epsilon$ guarantees. The same recipe can be adopted to adaptive optimizers like Adafactor (Shazeer & Stern, 2018), where the noised gradients are passed to the optimizer to figure out the optimal learning rate. LLMs Throughout the paper we will use the terms of pre-training and fine-tuning of LLMs: pre-training is the initial training of a LLM with a large public dataset, for example C4 (Raffel et al., Fine-tuning is an adaptation of a pre-trained model to perform some concrete task, for example question-answering, which involves running several epochs of an optimizer over the additional task training data. 4 METHODOLOGY As a motivational example, consider the task of medical data sharing for research purposes: a medical provider has a sensitive dataset with patients records and wants to accomplish some machine learning task. They may want to share the dataset with external researchers and academic institutions to get their help in solving the downstream task, while preserving the privacy of the original data. We assume that we have a sensitive dataset $D$ consisting of $(D_{train}, D_{valid}, D_{test})$, where the privacy of each record must be protected (see additional details on the unit of privacy in Appendix A). We want to accomplish some task on this dataset, such as training some downstream machine learning model. Additionally, we would like to allow a non-trusted third party to be able to perform the downstream task without violating privacy. To achieve this, we aim to create a synthetic dataset $D^{synth}$, which is DP with respect to the dataset $D$. Our dataset $D^{synth}$ will consist of synthetic training and validation splits. Figure 1 illustrates our methodology of data generation and evaluation: 1. Privately finetune (e.g., using DP-Training) a publicly pre-trained generative LLM $G$ on $D_{train}$, using $D_{valid}$ for hyperparameter tuning. To tune hyperparameters for DP-Training, we follow an algorithm outlined in (Ponomareva et al., 2023) (Section 5.4.1). 2. Independently sample $G$ to generate two new synthetic datasets $D^{synth}_{train}$ and $D^{synth}_{valid}$ which will serve as synthetic training and validation data. 3. Train a downstream model $M$ on $D^{synth}_{train}$ and use $D^{synth}_{valid}$ for hyperparameter tuning. 4. Evaluate the final performance of the model on real dataset $D_{test}$. 4.1 USING AN LLM FOR DATA SYNTHESIS Both encoder-decoder or decoder-only pretrained language models can generate synthetic data; we use decoder-only LLMs in our experiments. To finetune LLM for the synthetic data generation task, we use the next token prediction objective setup as follows. Given an example from the sensitive dataset with text $x$ and label $y$, we generate a prefix $p = "[TaskName] [LabelName_y] " $, where "[TaskName]" is the name of the task (for example "[imdb]"), and "[LabelName_y]" is "[negative]" when $y = 0$ or "[positive]" when $y = 1$. We finetune the model using the Prefix-LM objective (Raffel et al., 2020) using $p$ as a model input and $x$ as a target. Below we outline how the Prefix-LM way of splitting of the training example into input and target is advantageous for DP-training. Let’s consider some example from the dataset which is tokenized into input prefix $p = \{z_1, \ldots, z_k\}$ and target $x = \{z_{k+1}, \ldots, z_n\}$. Typically, weighted next token prediction cross entropy loss looks like the following: $L(\tilde{z}, \tilde{w}, \theta) = -\sum_{i=1}^{n} w_i z_i \log P(z\|z_{<i}, \theta)$ where $\theta$ - model parameters, $\tilde{z} = \{z_1, \ldots, z_n\}$ is tokenized training example (including input and target tokens), each $z_i$ as one hot encodings of token, $P(z\|z_{<i})$ is the probability of $i$-th token given values $z_{<i}$ of all previous tokens and $\tilde{w} = \{w_1, \ldots, w_n\}$ is weights vector for each token in the loss. Standard next-token prediction loss assigns weights $ w_i = 1 $ to all tokens, including those in the prefix $ p $. As a result, prefix tokens will be included in the gradient of the loss $ \frac{\partial L}{\partial \theta} $, thus essentially forcing the model to learn the distribution of tokens in the prefix as well. On the other hand, the Prefix-LM formulation assigns zero weights to the prefix $ i.e. \forall i \leq k : w_i = 0 $, so the total loss looks like the following: \[ L_{\text{PrefixLM}}(z, w, \theta) = -\sum_{i=k+1}^{n} z_i \log P(z\|z_{<i}, \theta) \] As a result, the LLM is not forced to learn distribution of input prefix $ p $ which we found to be beneficial for differentially-private training. DP-Training adds the noise to all the gradients; in a standard setup this will result in the gradients from the prefix portion being corrupted with the noise. This in turn means that prompting the DP-Trained LLM to generate synthetic data will not work as well as expected. We believe this is the same phenomenon that was observed in works by Putta et al. (2023) and Mattern et al. (2022), where authors had to add an adversarial head or augment the loss respectively, to aid the model in differentiating different types of prompts. Prefix-LM in turn is a standard loss well known to the community, and this comes with the benefits of knowing approximate hyperparameter values for its tuning. The aforementioned Prefix-LM setup allows to train one model for all the class labels and can be easily extended beyond the binary classification setup. ### 4.2 Parameter-Efficient Fine Tuning Full finetuning of large models is expensive, and empirically, tuning very large number of weights with DP-finetuning often results in substantial utility drop. Many techniques exist that update the pretraining model without resorting to full model weights update. In this work, we consider two popular ones - Prompt Tuning and LoRA. Prompt tuning (Lester et al., 2021) is a technique which prepends a small prompt tensor in front of the model’s input in the embedding space, freezes the rest of the model’s parameters and then finetunes only the prompt tensor weights. We found that combining prompt tuning with differentially private training allows us to achieve much higher utility of the trained generative model compared to full model fine-tuning. This could be explained by the fact that the prompt tensor is much smaller compared to the size of entire model (we used prompt tensor with 20480 parameters vs 8B weights in the full model) and smaller models tend to have smaller gap between private and non-private utility (Bassily et al., 2014; Bun et al., 2014), probably due to the total amount of noise injected during the training. It should be noted that prompt tuning as described in original paper (Lester et al., 2021) showed very poor utility when trained with differential privacy. We observed that even in the best runs LLM quality metrics (perplexity, next token prediction accuracy) fluctuated significantly. No amount of typical hyperparameter tuning could improve prompt tuning utility in DP-regime. Borrowing some ideas from (Mehta et al., 2022) and experimenting with various optimizers and ways to initialize prompt tensor proved to be the key to making prompt-tuning work. Eventually we found out that the main culprit of poor utility was prompt tensor initialization. (Lester et al., 2021) initializes prompt tensor by using embeddings of some real tokens from vocabulary. Changing prompt tensor initialization to random uniform with small range $[-0.01, 0.01]$ significantly improved utility. Additionally we observed that change of optimizer from Adafactor to Adam or Momentum helped to make training more stable, which simplified hyperparameter tuning (Appendix E). LoRA tuning (Hu et al., 2021) (Low-rank Adaptation) is a technique that freezes all the pre-trained model weights and introduces trainable low-rank decomposition matrices into each dense layer (MLP and Attention). This results in fewer trainable weights than full fine tuning but the number of trainable weights in LoRA is significantly larger than in Prompt tuning. For example, rank 8 LoRA updates 20M trainable parameters, as opposed to 41K prompt tuning vs 8B full fine tuning. Empirically we find (Section 5) that LoRA results in superior performance, surpassing that of both full finetuning and Prompt finetuning and that tuning both MLP layers and Attention blocks is preferred, see Appendices F and I.5 for more details. As a conclusion, we advocate for the use of parameter-efficient techniques when performing DP-training, with LoRA being the most promising so far. --- 1Original paper (Raffel et al., 2020) only describes bidirectional attention over prefix and omits the description of loss weights. Nevertheless zero weighting of the prefix is implemented in the T5 code. 4.3 Data sampling To generate one synthetic example we first randomly select example label $ y $, create a prefix $ p = "[\text{TaskName}] [\text{LabelName}_y]" $ (Section 4.1), feed prefix $ p $ as an input to the language model, and autoregressively sample the output. We repeat this process many times until we reach the desired amount of synthetic data. For each task we sampled at least the same amount of synthetic data as in original training dataset. We observed that generating more synthetic examples generally improves downstream task performance, but this benefit eventually diminishes and compute is typically the limiting factor (Appendix G). 5 Experiments Generative LLM In our experiments we used a model with architecture similar to Lamda 8B (Thop, pilan et al., 2022), which we pre-trained on The Pile dataset (Gao et al., 2020) using a standard next-token prediction loss. We stress that for our experimental results to be valid we must ensure that the pre-trained model was not itself trained on data that is considered private for the downstream task. For example, the GPT-2 model used in (Mattern et al., 2022) seemingly contained IMDB data in its pre-training dataset (Radford et al., 2019), but this model was subsequently used to generate a synthetic version of IMDB, see also appendix D for details. To prevent privacy leakage we modified the pre-training dataset by de-duplicating it against all sensitive datasets used in downstream tasks, following the recipe and scripts from (Lee et al., 2022). The outline of the de-duplication approach is as follows. First we tokenized and constructed a suffix array for each involved dataset (The Pile, IMDB, Yelp, AGNews). Then we used the suffix arrays to find common sequences of 50 or more tokens which appear in The Pile and any other dataset. Finally we cut all those common sequences from The Pile dataset. Note that this de-duplication is “stronger” than simply removing the datasets from the Pile. After cutting the sequences we de-tokenized the dataset back to strings and used it for pre-training. Refer to Appendix C for additional details. Datasets and classification problems We conducted our experiments on IMDB (Maas et al., 2011), Yelp (Zhang et al., 2015a) and AGNews (Zhang et al., 2015b) datasets. All these datasets only provide a training and test set, so in each case we use the first 90% of the training set for training and the remaining 10% for validation. For each dataset we formulated a binary classification problem (sentiment classification) as the downstream prediction task. 5.1 Downstream classifier performance We investigate the utility of using private synthetic data for a downstream task. For each dataset, we consider two types of models. First one is a (encoder-only) BERT model (Devlin et al., 2018a) with classification head. BERT is publicly pretrained and then fine tuned using either real data or our generated synthetic data. This model benefits from public pre-training data. We also consider a word-level CNN model (Johnson & Zhang, 2015) that does not utilize any public data. For each model, we report the performance on real data with no DP guarantees (an entry "Real" with $ \epsilon = \infty $ in Table 1). This serves as a upper bound of downstream classifier performance. We also report the performance of doing DP-Training on the downstream classifier directly (entries "Real" with $ \epsilon \in \{1, 3, 10\} $, referred to as "DP-on-real" in the text) and report the results on synthetic data generated from fine-tuned (Fine-tuned-SD), prompt tuned (Prompt-tuned-SD) and LoRA-Tuned (LoRA-tuned-SD) models. We would like to highlight however that in the case of using the real data directly for DP-Training, only the resulting downstream model is DP, and the real data can’t be shared freely or used for hyperparameter tuning (or such tuning should be accounted for in privacy guarantees). DP Synthetic data however can be shared freely and used for feature engineering, hyperparameter tuning, etc. Non-private synthetic data Firstly, our results in Table 1 indicate that obtaining good fidelity non-private synthetic data is possible, contrary to the results reported in (Yue et al., 2022) and Putta et al. (2023). Both Fine-tuned-SD and LoRA-tuned SD exhibits better performance than Prompt-tuned-SD, in line with current understanding that for a non-DP setting, tuning more model parameters is beneficial (Shin et al., 2020; Brown et al., 2020; Zhong et al., 2021). Interestingly, even for non DP setting, downstream models trained on LoRA synthetic data outperform those trained on fully fine-tuned synthetic data in 2 out of 3 datasets. Private synthetic data While there is a clear utility drop when going from non-private SD data to private SD, DP LoRA-tuned-SD is a clearly superior way of obtaining DP synthetic data. Prompt-tuned DP SD is better than fully fine tuned DP SD, however LoRA outperforms the Prompt-tuned DP synthetic data in majority of the cases. We hypothesize that it might be due to less total noise being added in DP LoRA models, due to fewer parameters being updated than with the full fine-tuning. Prompt tuning on the other hand updates the minimal number of parameters, however this minimum update hurts the utility of SD, suggesting that like with everything in ML, there is a “sweet spot” on the number of parameters trained with DP. The difference between the performance is significant, with LoRA-tuned-SD exhibiting of up to 10-11% lift on downstream BERT classifier tasks, compared to model trained on fine-tuned-SD. For CNN model that is more dependent on the quality of the data than BERT (that essentially reaps some additional benefits from Transfer Learning), the results are even more significant, with a boost from prompt-tuned-SD (vs fine-tuned-SD) reaching up to 22%. Private synthetic data vs DP-Training on real data To obtain a DP downstream ML model, we can either use DP synthetic training data or introduce DP directly during downstream model training (DP-on-real). As previously mentioned, the former is a harder setup. When comparing BERT models, we can see that private LoRA-tuned-SD achieves performance similar or even superior (e.g., for IMDB and Yelp datasets) to DP-on-real for all levels of privacy, but an additional benefit of such synthetic data is that it can be additionally shared freely and used for hyperparameter tuning and feature engineering. For CNN model, LoRA-tuned-SD (and even prompt-tuned SD) exhibits better performance than DP-on-real. This is due to the fact that private synthetic data benefits from massive amount of public data that was used for pretraining of the LLM (CNN model itself is trained from scratch, as opposed to BERT that is itself a pre-trained model, albeit with smaller amount of public data than the 8b Lamda model we used for SD generation). This indicates that for simpler models synthetic data can be a preferred way of injecting additional public knowledge. This is an interesting result since it is commonly assumed that for Transfer Learning to work, public data should come from a similar distribution as the target data. However in case of synthetic data, we inject public data from different distributions (crawl of the web) than that of the downstream task (e.g. Yelp reviews). \| IMDB \| $\epsilon$ \| Real \| Synthetic \| \| IMDB \| $\epsilon$ \| Real \| Synthetic \| \|------\|------------\|------\|-----------\|------\|------\|------\|------\|-----------\| \| \| $\infty$ \| 93.7 ± 0.1 \| 93.2 ± 0.2 \| 92.0 ± 0.1 \| 91.6 ± 0.2 \| 90.1 ± 0.1 \| 89.8 ± 0.1 \| 87.4 ± 0.1 \| 89.0 ± 0.1 \| \| \| 10 \| 90.6 ± 0.1 \| 84.0 ± 0.7 \| 90.7 ± 0.2 \| 91.3 ± 0.2 \| 78.2 ± 0.4 \| 80.0 ± 0.5 \| 86.9 ± 0.1 \| 87.7 ± 0.2 \| \| \| 3 \| 89.7 ± 0.2 \| 83.9 ± 0.6 \| 87.4 ± 0.2 \| 90.6 ± 0.2 \| 74.8 ± 0.6 \| 74.2 ± 0.1 \| 85.4 ± 0.5 \| 87.4 ± 0.3 \| \| \| 1 \| 88.6 ± 0.1 \| 79.1 ± 1.7 \| 88.1 ± 0.4 \| 90.0 ± 0.3 \| 69.3 ± 0.6 \| 64.7 ± 0.5 \| 85.4 ± 0.1 \| 87.6 ± 0.4 \| \| Yelp \| $\infty$ \| 97.6 ± 0.1 \| 95.9 ± 0.1 \| 93.9 ± 0.1 \| 96.4 ± 0.1 \| 95.6 ± 0.1 \| 89.3 ± 0.5 \| 91.6 ± 0.1 \| 93.7 ± 0.0 \| \| \| 10 \| 94.0 ± 0.1 \| 84.2 ± 0.7 \| 94.1 ± 0.1 \| 95.1 ± 0.1 \| 90.1 ± 0.1 \| 71.9 ± 0.6 \| 89.1 ± 0.4 \| 90.6 ± 0.1 \| \| \| 3 \| 94.6 ± 0.1 \| 84.0 ± 0.1 \| 93.5 ± 0.1 \| 95.6 ± 0.1 \| 90.9 ± 0.2 \| 67.9 ± 2.6 \| 80.5 ± 0.1 \| 93.6 ± 0.1 \| \| \| 1 \| 94.3 ± 0.1 \| 84.1 ± 0.3 \| 94.1 ± 0.1 \| 95.5 ± 0.1 \| 89.8 ± 0.1 \| 71.1 ± 0.4 \| 91.1 ± 0.3 \| 93.4 ± 0.1 \| \| AGNews \| $\infty$ \| 93.7 ± 0.1 \| 91.1 ± 0.1 \| 88.3 ± 0.3 \| 91.8 ± 0.2 \| 91.3 ± 0.1 \| 87.7 ± 0.1 \| 84.7 ± 0.1 \| 88.5 ± 0.2 \| \| \| 10 \| 90.9 ± 0.2 \| 65.1 ± 5.4 \| 86.9 ± 0.1 \| 90.0 ± 0.1 \| 85.2 ± 0.2 \| 45.2 ± 1.8 \| 83.5 ± 0.2 \| 88.9 ± 0.1 \| \| \| 3 \| 90.4 ± 0.2 \| 65.3 ± 2.9 \| 86.5 ± 0.2 \| 89.6 ± 0.3 \| 83.4 ± 0.1 \| 45.2 ± 1.8 \| 83.5 ± 0.2 \| 86.6 ± 0.2 \| \| \| 1 \| 89.8 ± 0.2 \| 65.7 ± 2.9 \| 84.9 ± 0.8 \| 88.4 ± 0.4 \| 79.9 ± 0.2 \| 46.8 ± 1.5 \| 80.4 ± 0.6 \| 85.8 ± 0.1 \| Amount of synthetic data vs downstream classifier performance We studied how much synthetic data we should generate relative to amount of real data. Table 2 demonstrates that generating more synthetic data can be beneficial, but has diminishing returns for BERT (0.8% lift going from 1x to 3x times the data), with benefits more pronounced for simple models like WordCNN (1.4% lift from increasing the amount of synthetic data 3x). \| Model \| 1x \| 2x \| 3x \| 4x \| 5x \| 6x \| \|-------\|----\|----\|----\|----\|----\|----\| \| BERT \| 87.2 ± 0.4 \| 87.9 ± 0.4 \| 88.0 ± 0.1 \| 88.1 ± 0.4 \| 88.4 ± 0.1 \| 88.7 ± 0.1 \| \| WordCNN \| 83.2 ± 0.2 \| 84.3 ± 0.4 \| 84.6 ± 0.1 \| 83.4 ± 0.1 \| 83.7 ± 0.3 \| 83.8 ± 0.2 \| One can also potentially combine the synthetic data with training with DP on real data, by pre-training the downstream model with DP synthetic data and then fine-tuning with DP on real data. This will however require spreading the privacy budget between DP synthetic data and DP-Training of the downstream classifier. We leave this for future work. Comparison with prior work While works below don’t provide sufficient (or any) information on their privacy unit (as we do in Appendix A), we assume that privacy unit that was used is one example (e.g. 1 full yelp or imdb review etc); we also assume central DP setting, that δ values are the same or comparable etc. Additionally, none of the works below take into account the fact that pre-training data might have contained the data they deem private (as we highlight in Appendix D), potentially invalidating their reported DP guarantees. Yue et al. (2022) used Yelp dataset for multiclass (rating) classification, so our results are not directly comparable. Putta et al. (2023) used AGNews dataset. Their approach is a combination of next token prediction (similar to our setup) and additional loss term from a new head that attempts to learn to distinguish between various classes directly (instead of simply relying on the prompts in the next token prediction head). Putta et al. (2023) reports 0.867 accuracy of downstream task for ε of 3, while we obtain 89.6 (the baseline performance of downstream classifier for our and their work is comparable, 0.938, suggesting that we are using comparable downstream classifiers). Mattern et al. (2022) suggested a modification of the loss (prompt-mismatch loss, to discourage the generation of text inconsistent with the prompt, like generating a negative review when positive prompt was given). They performed experiments on IMDB dataset. Their best IMDB experiments reporting worse accuracy on DP synthetic data (89.1% theirs vs 90.6% ours for ε = 3). They also seem to have worse performance on real data despite using the same model (BERT-classifier). ### 5.2 Tuning downstream model hyperparameters on synthetic data With the following experiments on IMDB data, we want to demonstrate that private synthetic data is useful for hyperparameter tuning of the downstream classifier. For all our experiments, when tuning the downstream classifier, we use validation accuracy on set-aside portion of synthetic data for hyperparameter selection. We tune weight decay and learning rate for both CNN and BERT models. For synthetic data, we create vectors of accuracy on validation (synthetic) data and performance on real test data for all possible combinations of hyperparameter values tried. We then report the ranking correlation between performance as indicated by validation accuracy (synthetic data) and test accuracy computed on real data. We also report the ranking correlation of accuracies on real validation and real test data, to provide an upper bound. Additionally, we report rank-biased overlap ranking metric (Webber et al., 2010), which is a weighted metric that gives more weight to the top of the ranking (we use parameters that give 85% of the weight to the first top 25% of ranking). Table 3 demonstrates excellent ranking correlation on synthetic data. Interestingly, prompt-tuned synthetic data metrics, in particular the mean and standard deviation of the top 25% of trials, suggest that BERT classifier performance is less sensitive to hyperparameters on better fidelity (prompt or LoRA tuning) data than on worse fidelity data (fine-tuning). ### 5.3 Estimating synthetic data quality It is useful to have an efficient method of evaluating the quality of a synthetic dataset without relying on specific downstream tasks. For one, a key use case for privacy preserving synthetic data is to enable data sharing without a definitive end use case. For another, training the generator LLM has multiple hyperparameters that can be tuned, and it can be prohibitive to evaluate candidate models using full data synthesis and downstream training (which itself might require tuning hyperparameters). Instead, lighter weight proxy metrics can be used. Commonly used proxy metrics are: perplexity, n-gram statistics, and MAUVE (Pillutla et al., 2021). We investigate the effectiveness of each of these metrics by comparing their correlation to downstream performance (Table 4). These metrics are used to compare datasets, and thus their absolute value is uninformative. For n-gram statistics we determine the frequency of unigrams, bigrams, and sample lengths in characters for both the original and synthetic datasets. We then compute the area under the divergence frontier between these two frequency distributions as is done by MAUVE. MAUVE works by computing the difference between two datasets by first embedding each example, then clustering the datasets, followed by comparing (via divergence frontiers) the histogram of cluster membership across the two datasets. It has recently been shown to be an effective metric for synthetic text datasets (Yue et al., 2022) (Mattern... Table 3: Ranking correlations (full list) and rank-biased overlap (RBO) \cite{webber2010} for top 25% of hyperparameter tuning trials. Real data metrics are calculated on the performance of a model as reported on real validation and real test. For synthetic data, metrics are calculated on synthetic validation and real test data. Mean 25% and STD 25% show mean and std of real test accuracy evaluated on top 25% trials (ordered by validation accuracy on synthetic data). \| Model \| $ c $ \| Method \| RBO 25% \| Spearman \| Kendall \| Mean 25% \| STD 25% \| \|-----------\|---------\|----------------\|---------\|----------\|---------\|----------\|---------\| \| BERT \| $ \infty $ \| Real data \| 0.56 \| 0.96 \| 0.93 \| 93.55 \| 0.50 \| \| \| 3 \| Fine-tuning \| 0.33 \| 0.94 \| 0.86 \| 79.27 \| 0.75 \| \| \| 10 \| Prompt-tuning \| 0.32 \| 0.73 \| 0.60 \| 88.00 \| 0.00 \| \| \| 3 \| LoRA-tuning \| 0.29 \| 0.86 \| 0.79 \| 90.00 \| 0.00 \| \| \| 10 \| \| 0.3 \| 0.78 \| 0.66 \| 91.18 \| 0.39 \| \| WordCNN \| $ \infty $ \| Real data \| 0.92 \| 0.92 \| 0.84 \| 90.00 \| 0.00 \| \| \| 3 \| Fine-tuning \| 0.63 \| 0.79 \| 0.65 \| 72.09 \| 2.37 \| \| \| 10 \| Prompt-tuning \| 0.64 \| 0.73 \| 0.59 \| 84.36 \| 1.49 \| \| \| 3 \| LoRA-tuning \| 0.69 \| 0.81 \| 0.67 \| 87.45 \| 0.66 \| et al.] \cite{kour2022} \cite{kour2022}, which our results support. We compute the MAUVE score as given in Pillutla et al. \cite{pillutla2021} using the suggested hyperparameters unless noted. We investigated modifying these hyperparameters and confirm they make little difference to the relative ranking, with the notable exception of the model used to embed examples. Unlike the original paper, we find larger models to be much more effective. In particular, embedding using Sentence-T5 \cite{ni2021} has much higher correlation to downstream performance than BERT or any other model we tried. For more details see appendix K. Our results match many of the results given in Kour et al. \cite{kour2022}. All metrics are at least somewhat noisy with standard test-set perplexity performing very well. Given its ease to compute while finetuning, perplexity is our recommended proxy metric when available. Table 4: The Spearman’s rank correlation for each metric compared against downstream classifier performance. Metrics are used to select candidate datasets, and thus their relative rank is what's most important for the metrics to reflect. \| Perplexity \| Unigram \| Bigram \| Length \| BERT \| MAUVE Sft5-base \| Sft5-3B \| \|------------\|---------\|--------\|--------\|------\|----------------\|--------\| \| 0.91 ± 0.02 \| 0.74 ± 0.11 \| 0.83 ± 0.09 \| 0.88 ± 0.26 \| 0.84 ± 0.62 \| 0.88 ± 0.04 \| 0.93 ± 0.10 \| 6 CONCLUSION We have shown that training downstream models on DP synthetic training data is an effective alternative to training such models with DP directly on real data for text classification tasks. We explored two methods for privately generating the synthetic training data, both of which involve modifying the weights of an existing LLM. One method privately fine-tuned all the layers of the LLM, while the other method used parameter efficient fine tuning (‘prompt-tuning’ and ‘LoRA-tuning’). Our experiments demonstrated that LoRA tuning is a superior way of obtaining DP-synthetic data, which provides performance on the downstream task that is comparable or even better than directly DP-Training on real data. We showed that the standard NLP Prefix-LM loss is well suited for DP-finetuning. Private synthetic data can be used freely for all purposes, such as feature engineering, hyperparameter tuning, debugging and monitoring, and sharing, but without any privacy-related concerns. We also showed that while Mauve is a good proxy metric for evaluating the quality of the synthetic data, simpler metrics like perplexity, when available, perform well. 7 ETHICS STATEMENT We expect that our proposed method of generating DP synthetic data will facilitate safer data sharing and that the societal impact will be positive, since entities who own private data but do not necessarily have the knowledge or resources to train predictive models can share private synthetic data with specialists for model creation, benefiting from their expertise without comprising the privacy of the users who contributed their data. The main limitation of our approach is that we only conducted experiments on English datasets, however we expect that methods should work on multilingual datasets as long as public multi-lingual data are available for LLM pre-training. 8 Reproducibility Statement All of our experiments are based on open sourced frameworks and public datasets, refer to Appendices H and M. We further provided necessary details to reproduce our experiments in Appendices C, E, F, G, and I. References Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC conference on computer and communications security. ACM, oct 2016. doi: 10.1145/2976749.2978318. URL https://doi.org/10.1145%2F2976749.2978318 Raef Bassily, Adam Smith, and Abhradeep Thakurta. Private empirical risk minimization: Efficient algorithms and tight error bounds. 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xC8xh2RSs2	The correlation with content comprehensiveness in the human eval is not super high (~40%), and the rest are all quite low. It would be interesting to compare this with a qualitative description from the annotators of what they found made a better or worse card -- are there factors not detailed here they think it would be more informative to consider?	Navigating Dataset Documentations in AI: A Large-Scale Analysis of Dataset Cards on Hugging Face Xinyu Yang * Cornell University xy468@cornell.edu Weixin Liang* Stanford University wxliang@stanford.edu James Zou Stanford University jamesz@stanford.edu Abstract Advances in machine learning are closely tied to the creation of datasets. While data documentation is widely recognized as essential to the reliability, reproducibility, and transparency of ML, we lack a systematic empirical understanding of current dataset documentation practices. To shed light on this question, here we take Hugging Face – one of the largest platforms for sharing and collaborating on ML models and datasets – as a prominent case study. By analyzing all 7,433 dataset documentation on Hugging Face, our investigation provides an overview of the Hugging Face dataset ecosystem and insights into dataset documentation practices, yielding 5 main findings: (1) The dataset card completion rate shows marked heterogeneity correlated with dataset popularity: While 86.0% of the top 100 downloaded dataset cards fill out all sections suggested by Hugging Face community, only 7.9% of dataset cards with no downloads complete all these sections. (2) A granular examination of each section within the dataset card reveals that the practitioners seem to prioritize Dataset Description and Dataset Structure sections, accounting for 36.2% and 33.6% of the total card length, respectively, for the most downloaded datasets. In contrast, the Considerations for Using the Data section receives the lowest proportion of content, accounting for just 2.1% of the text. (3) By analyzing the subsections within each section and utilizing topic modeling to identify key topics, we uncover what is discussed in each section, and underscore significant themes encompassing both technical and social impacts, as well as limitations within the Considerations for Using the Data section. (4) Our findings also highlight the need for improved accessibility and reproducibility of datasets in the Usage sections. (5) In addition, our human annotation evaluation emphasizes the pivotal role of comprehensive dataset content in shaping individuals’ perceptions of a dataset card’s overall quality. Overall, our study offers a unique perspective on analyzing dataset documentation through large-scale data science analysis and underlines the need for more thorough dataset documentation in machine learning research. 1 Introduction Datasets form the backbone of machine learning research [Koch et al., 2021]. The proliferation of machine learning research has spurred rapid advancements in machine learning dataset development, validation, and real-world deployment across academia and industry. Such growing availability of ML datasets underscores the crucial role of proper documentation in ensuring transparency, reproducibility, and data quality in research [Haibe-Kains et al., 2020; Stodden et al., 2018; Hutson, 2018]. Documentation provides details about the dataset, including sources of data, methods used to collect it, and preprocessing or cleaning that was performed. This information holds significant value for dataset users, as it facilitates a quick understanding of the dataset’s motivation and its overall scope. These insights are also crucial for fostering responsible data sharing and promoting interdisciplinary collaborations. These authors contributed equally to this work. Despite numerous studies exploring the structure and content of dataset cards across various research domains (Afzal et al., 2020; Gebru et al., 2021; Papakyriakopoulos et al., 2023; Barman et al., 2023; Costa-jussà et al., 2020), there remains a notable gap in empirical analyses of community norms and practices for dataset documentation. This knowledge gap is significant because adherence to community norms and the quality of dataset documentation directly impact the transparency, reliability, and reproducibility in the field of data-driven research. For instance, inadequate dataset descriptions, structural details, or limitations can hinder users from utilizing the dataset appropriately, potentially resulting in misuse or unintended consequences; the absence of information on data cleaning and readiness assessment practices in data documentation limits dataset reusability and productivity gains. Furthermore, without a systematic analysis of current dataset documentation practices, we risk perpetuating insufficient documentation standards, which can impede efforts to ensure fairness, accountability, and equitable use of AI technologies. To address this question, we conducted a comprehensive empirical analysis of dataset cards hosted on Hugging Face, one of the largest platforms for sharing and collaborating on ML models and datasets, as a prominent case study. Dataset cards on the Hugging Face platform are Markdown files that serve as the README for a dataset repository. While several open-source platforms also facilitate the sharing of ML datasets, such as Kaggle, Papers with Code, and GitHub, we chose Hugging Face for two primary reasons. Firstly, it stands out as one of the most popular platforms for developers to publish, share, and reuse ML-based projects, offering a vast repository of ML datasets for study. Secondly, Hugging Face is one of the few open-source platforms that offer an official dataset card template. This feature not only enhances the accessibility and user-friendliness of the dataset card community but also makes the analysis process more efficient and informative. By analyzing all 7,433 dataset documentation hosted on Hugging Face, our investigation provides an overview of the Hugging Face dataset ecosystem and insights into dataset documentation practices. Based on our research findings, we emphasize the importance of comprehensive dataset documentation and offer suggestions to practitioners on how to write documentation that promotes reproducibility, transparency, and accessibility of their datasets, which can help to improve the overall quality and usability of the dataset community. Our study aims to bridge the notable gap in the community concerning data documentation norms, taking the first step toward identifying deficiencies in current practices and offering guidelines for enhancing dataset documentation. Figure 1: Systematic Analysis of 24,065 Datasets Hosted on Hugging Face. (a) Exponential Growth of Datasets: The Hugging Face platform has seen a remarkable surge in the number of datasets, with the count doubling approximately every 18 weeks. (b) Power Law in Dataset Usage: Dataset downloads on Hugging Face follow a power-law distribution, as indicated by the linear relationship on the log-log plot. The top 82 datasets account for 80% of the total downloads; datasets with documentation dominate the top downloaded datasets. (c) Documentation Associated with Usage: Despite only 30.9% of dataset repositories (7,433 out of 24,065) featuring non-empty dataset cards, these datasets account for an overwhelming 95.0% of total download traffic on the platform. 2 OVERVIEW Finding - Exponential Growth of Datasets:* The number of datasets on Hugging Face doubles every 18 weeks. - Documentation Associated with Usage: 95.0% of download traffic comes from the 30.9% of datasets with documentation. Exponential Growth of Datasets Our analysis encompasses 24,065 dataset repositories on Hugging Face uploaded by 7,811 distinct user accounts as of March 16th, 2023 (see Table S5 for varying documentation practices by creators). The number of datasets exhibits exponential growth, with a weekly growth rate of 3.97% and a doubling time of 18 weeks (Fig. 1a). As a sanity check, the number of dataset repositories reached 35,973 by May 23rd, 2023, confirming the exponential trend. Power Law in Dataset Usage Although Hugging Face has seen a significant increase in the number of dataset repositories, our analysis reveals a significant imbalance in dataset downloads, which follows a power law distribution. This means that a small proportion of the most popular datasets receive the majority of the downloads, while the vast majority of datasets receive very few downloads. In fact, our analysis shows that just the 82 datasets with the most downloads account for 80% of total downloads (Fig. 1b). Fig. S4 further demonstrates that the power law distribution persists across various task domains, even with the varied number of datasets within each domain. Documentation Associated with Usage Despite the importance of dataset cards, only 58.2% (14,011 out of 24,065 dataset repositories contributed by 4,782 distinct user accounts) include dataset cards as Markdown README.md files within their dataset repositories. Among these, 6,578 dataset cards are empty, resulting in only 30.9% (7,433 out of 24,065 dataset repositories contributed by 1,982 distinct user accounts) featuring non-empty dataset cards (Fig. 1c). As illustrated in Fig. 1d, dataset cards are prevalent among the most downloaded datasets. Notably, datasets with non-empty dataset cards account for 95.0% of total download traffic, underscoring a potential positive correlation between dataset cards and dataset popularity. For the rest of the paper, we focus our analyses on these 7,433 non-empty dataset cards. We sort these non-empty dataset cards based on the number of downloads for the corresponding datasets. So top $k$ dataset cards (e.g. $k = 100$) refer to the dataset cards corresponding to the $k$ most downloaded datasets. 3 STRUCTURE OF DATASET DOCUMENTATIONS Finding - The dataset card completion rate shows marked heterogeneity correlated with dataset popularity: While 86.0% of the top 100 downloaded datasets fill out all sections suggested by the Hugging Face community, only 7.9% of dataset cards with no downloads complete all these sections. \| Section Title \| Subsection Title \| Description \| \|------------------------\|-----------------------------------\|-----------------------------------------------------------------------------\| \| Dataset Description \| Dataset Summary \| A brief summary of the dataset, including its intended use, supported tasks, an overview of how and why the dataset was created, etc. \| \| \| Supported Tasks and Leaderboards \| Brief description of the tag, metrics, and suggested models of the dataset. \| \| \| Languages \| The languages represented in the dataset. \| \| Dataset Structure \| Data Instances \| JSON-formed example and description of a typical instance in the dataset. \| \| \| Data Fields \| List and describe the fields present in the dataset. Mention their data type, and whether they are used as input or output in any of the tasks the dataset currently supports. \| \| \| Data Splits \| Criteria for splitting the data; descriptive statistics for the features, such as size, average length, etc. \| \| Dataset Creation \| Curation Rationale \| Motivation for the creation of the dataset. \| \| \| Source Data \| The source of the data (e.g. news text and headlines, social media posts, translated sentences, etc.), including the data collection process, and data producer. \| \| \| Annotations \| Annotation process, annotation tools, annotators, etc. \| \| \| Personal and Sensitive Information\| Statement of whether the dataset contains other data that might be considered sensitive (e.g., data that reveals racial or ethnic origins, financial or health data, etc.). \| \| Considerations for Using the Data \| Social Impact of Dataset \| Discussion of the ways the use of the dataset will impact society. \| \| \| Discussion of Biases \| Descriptions of specific biases that are likely to be reflected in the data. \| \| \| Other Known Limitations \| Other limitations of the dataset, like annotation artifacts. \| \| Additional Information \| Dataset Curators \| The people involved in collecting the dataset and their affiliation(s). \| \| \| Licensing Information \| The license and link to the license webpage if available. \| \| \| Citation Information \| The BibTeX-formatted reference for the dataset. \| \| \| Contributions \| ‘Thanks to @github-username for adding this dataset.’ \| Table 1: Community-Endorsed Dataset Card Structure. This table shows the sections and their suggested subsections provided by the Hugging Face community, along with their descriptions. For more information, please refer to [https://github.com/huggingface/datasets/blob/main/templates/README_guide.md](https://github.com/huggingface/datasets/blob/main/templates/README_guide.md). Community-Endorsed Dataset Card Structure Grounded in academic literature (Mitchell et al., 2019) and official guidelines from Hugging Face (HuggingFace, 2021), the Hugging Face community provides suggestions for what to write in each section. This community-endorsed dataset card provides a standardized structure for conveying key information about datasets. It generally contains 5 sections: Dataset Description, Dataset Structure, Dataset Creation, Considerations for Using the Data, and Additional Information (Table 1). To examine the structure of dataset cards, we used a pipeline that detects exact word matches for each section title. We then identified the section titles and checked whether they had contents (Appendix B.1). If a dataset card had all five sections completed, we considered it to be following the community-endorsed dataset card. Adherence to Community-Endorsed Guidelines Correlates with Popularity Our evaluation found that popular datasets have better adherence to the dataset card community-endorsed dataset card structure. As illustrated in Fig. 2, compliance with the template varies significantly among datasets with different download counts. Among the 7,433 dataset cards analyzed, 86.0% of the top 100 downloaded dataset cards have completed all five sections of the community-endorsed dataset card, while only 7.9% of dataset cards with no downloads follow it. Fig. S5 further reveals that popular dataset cards achieve higher completion in all Hugging Face-recommended sections. This implies a potential correlation between adherence to community-endorsed guidelines and dataset popularity. 4 Practitioners Emphasize Description and Structure Over Social Impact and Limitations Finding • Practitioners seem to prioritize on Dataset Description and Dataset Structure sections, which account for 36.2% and 33.6% of the total card length, respectively, on the top 100 most downloaded datasets. • In contrast, the Considerations for Using the Data section receives the lowest proportion of content, just 2.1%. The Considerations for Using the Data section covers the social impact of datasets, discussions of biases, and limitations of datasets. Social Impact, Dataset Limitations and Biases are Lacking in Most Documentations Following the community-endorsed dataset card, we conducted an analysis to determine the level of emphasis placed on each section. Fig. 3b shows the word count distribution among the top 100 downloaded dataset cards, revealing their high level of comprehensiveness: 91.0% of them have a word count exceeding 200. We step further into these dataset cards to examine the emphasis placed on each section. We calculated the word count of each section and its proportion to the entire dataset card. As shown in Fig. 3c, the Dataset Description and Dataset Structure sections received the most attention, accounting for 36.2% and 33.6% of the dataset card length, respectively. On the other hand, the Considerations for Using the Data section received a notably low proportion of only 2.1%. Section Length Reflects Practitioner Attention The length of sections within dataset cards is reflective of practitioner attention, and it varies significantly based on the popularity of the dataset. Highly downloaded datasets tend to have more comprehensive and longer dataset cards (Fig. 3a), with an emphasis on the Dataset Description and Dataset Structure sections (Fig. 3d). Conversely, less popular datasets have shorter cards (Fig. 3b) with a greater emphasis on the Additional Information section (Fig. 3f). Despite this, sections such as Dataset Creation and Considerations for Using the Data consistently receive lower attention, regardless of download rates (Fig. 3f). This suggests a need to promote more comprehensive documentation, particularly in critical sections, to enhance dataset usage and facilitate ethical considerations. Figure 3: Section Length Reflects Practitioner Attention. (a) Popularity Correlates with Documentation Length: The top downloaded dataset cards are longer, indicating that they contain more comprehensive information. (b) Distribution of Word Count Among Top 100 Downloaded Dataset Cards (c) Section Length Proportions in Top 100 Downloaded Dataset Cards: The Dataset Description and Dataset Structure sections dominate in the top 100 downloaded dataset cards, with proportions of 36.2% and 33.6%, respectively. In contrast, the Considerations for Using the Data section receives the least attention, with a proportion of only 2.1%. (d) Section Length Proportion Changes over Downloads: The section length proportion changes over downloads, with Dataset Description and Dataset Structure decreasing in length, and Additional Information and Other increasing. Notably, there is a consistently low emphasis placed on the Dataset Creation and Considerations for Using the Data sections across all dataset cards with different downloads. 5 UNDERSTANDING CONTENT DYNAMICS IN DATASET DOCUMENTATION Finding • Strong Community Adherence to Subsection Guidelines: Practitioners contributing to the Hugging Face community exhibit high compliance with standards, filling out 14 of the 17 recommended subsections across five main sections at a rate exceeding 50%. • Emergence of the Usage Section Beyond the Community Template: Surprisingly, 33.2% of dataset cards includes a Usage section. The community template does not include such Usage section in its current form and should include one in the future. Section Content Detection Pipeline To gain a deeper understanding of the topics discussed in each section, we conducted a content analysis within each section of the community-endorsed dataset card structure, which includes suggested subsections within the five main sections. We used exact keyword matching to identify the corresponding subsections and calculate their filled-out rates. Fig. 4 shows that 14 out of 17 subsections have filled-out rates above 50%, indicating adherence to the community-endorsed dataset cards. Limitation Section is Rare, but Long if it Exists The Considerations for Using the Data section (i.e., limitation section), despite being frequently overlooked and often left empty by practitioners, holds particular significance. When this section is included, it tends to adhere well to community guidelines, with subsections having a completion rate exceeding 50% and a reasonably substantial word count (98.2 words). This suggests that this section has the potential to provide valuable insights and guidance. This motivates our use of topic modeling to identify key discussion topics within this section, potentially aiding practitioners in crafting meaningful content. Figure 4: Highlighting the Hugging Face Community’s Compliance with Subsection Guidelines. This figure shows subsection filled-out rates within different sections, stratified by download counts. Each section has multiple subsections, with bars representing the filled-out rate of each subsection. Green texts indicate filled-out rates above 50%, while red texts indicate rates below 50%. Of the 17 subsections within the five sections of the community-endorsed dataset, 14 have filled-out rates above 50%. \| Topic \| Representative Sentences \| \|-------------------------------\|--------------------------------------------------------------------------------------------------------------------------------------------------------\| \| Technical or Research Scope \| • Adding a Spanish resource may help others to improve their research and educational activities. \| \| \| • The creation of the dataset contributes to expanding the scope of NLP research to under-explored languages across the world. \| \| Social Scope or Background \| • This dataset can be used to gain insights into the social, cultural, and political views of people in African countries. \| \| \| • If this matter isn’t tackled with enough urgency, we might see the rise of a new dark era in Latin America politics, where many unscrupulous parties and people will manage to gain power and control the lives of many people. \| \| Topic \| Representative Sentences \| \|-------------------------------\|--------------------------------------------------------------------------------------------------------------------------------------------------------\| \| Subpopulation Biases \| • Gender speakers distribution is imbalanced, percentage of female speakers is mostly lower than 50% across languages. \| \| \| • The social biases of the time in terms of race, sex, gender, etc. might be encountered in this dataset. \| \| Biases from Collection Procedure \| • With respect to the potential risks, we note that the subjectivity of human annotation would impact on the quality of the dataset. \| \| \| • In terms of data collection, by using keywords and user mentions, we are introducing some bias to the data, restricting our scope to the list of keywords and users we created. \| \| Topic \| Representative Sentences \| \|-------------------------------\|--------------------------------------------------------------------------------------------------------------------------------------------------------\| \| Data Quality \| • The nature of the task introduce a variability in the quality of the target translations. \| \| \| • A number of errors, omissions and inconsistencies are expected to be found within the corpus. \| \| Processing Limitation \| • Our augmentation process can sometimes create nonexistent versions of real people. \| \| \| • Satellite annotation is not as accurate for pixel-level representation due to single-point annotations. \| Figure 5: Key Topics in Considerations for Using the Data through Topic Modeling Analysis. This figure displays the outcomes of the topic modeling assessment on the contents of the (a) Social Impact of Dataset Subsection, (b) Discussion of Biases Subsection, and (c) Other Known Limitations Subsection. Each panel illustrates the human-assigned topic label and representative sentences for each section. Topics are generated by Latent Dirichlet Allocation (LDA). Limitation Section Covers Diverse and Crucial Topics The Considerations for Using the Data section (i.e., limitation section) encompasses diverse and crucial topics. The Hugging Face community emphasizes three major themes within this section: Social Impact of Dataset, Discussion of Biases, and Other Known Limitations. The Social Impact of Dataset aspect explores not only societal implications but also the potential benefits to technology and research communities. In this section, practitioners discuss issues like how the dataset can expand the scope of NLP research (Armstrong et al., 2022), and increase access to natural language technology across diverse regions and cultures (Tache et al., 2021). Additionally, the subsection covers sensitive topics related to politics, ethics, and culture within the social scope. Discussion of Biases delves into subpopulation bias and data collection biases, highlighting the importance of addressing bias-related issues. Previous research have identified numerous technical and social biases such as subgroup bias (Buolamwini & Gebru, 2018), data collection bias (Wang et al., 2019), and label bias (Jiang & Nachum, 2020). Our topic modeling results reveal that two primary biases are discussed by practitioners in this subsection. The first is subpopulation bias, which includes biases related to gender, age, or race. For instance, an audio dataset (Nsoesie & Galea, 2022) notes that female speakers are underrepresented, comprising less than 50% of the dataset. The second major bias arises from the data collection process, specifically the annotation process, which is often a significant bottleneck and source of errors. Lastly, Other Known Limitations focuses on technical limitations, particularly data quality and processing limitations. This comprehensive coverage underscores the multifaceted nature of considerations related to dataset usage. Data quality is often a focus in other disciplines, such as the social sciences and biomedicine, and there are many insights to draw upon (Paulada et al., 2021; Fedorov, 2010; Pan & Geerts, 2012). Meanwhile, processing limitations encompass a broader range of issues beyond biases from the collection procedure, such as inaccuracies or the absence of some data points. Emergence of the Usage Section Beyond the Community Template While Hugging Face’s community-endorsed dataset card structure comprises five main sections, there are instances where practitioners encounter valuable information that doesn’t neatly fit into these sections. These additional sections, referred to as Other sections, can contain important content. Notably, among these Other sections, discussions related to Usage emerge as a frequent (nearly one-third of the time, 33.2%) and significant theme. These Usage sections offer a diverse range of information, including details on downloading, version specifications, and general guidelines to maximize the dataset’s utility. This highlights the importance of considering content that falls outside the predefined template and suggests a potential area for improvement in dataset card templates. Quantifying the Impact of Usage Section on Dataset Downloads To assess the influence of a Usage section in dataset documentation, we conducted a counterfactual analysis experiment (Appendix, C). We trained a BERT (Devlin et al., 2018) model using dataset card content and download counts, which were normalized to fall within the range of [0, 1] for meaningful comparisons. When a dataset card that initially included a Usage section had this section removed, there was a substantial decrease of 1.85% in downloads, with statistical significance. This result underscores the significant impact of the Usage section in bolstering dataset accessibility and popularity, emphasizing its pivotal role in enhancing the documentation and usability of datasets. 6 ANALYZING HUMAN PERCEIVED DATASET DOCUMENTATION QUALITY Finding - Our human annotation evaluation emphasizes the pivotal role of comprehensive dataset content in shaping individuals’ perceptions of a dataset card’s overall quality. Human Annotations for Comprehensive Evaluation of Dataset Card Quality We utilized human annotations to evaluate the quality of dataset cards, considering seven distinct aspects, drawing from prior research in dataset documentation literature and the Hugging Face community-endorsed dataset card (Afzal et al., 2020; Gebru et al., 2021; Papakyriakopoulos et al., 2023; Barman et al., 2023; Costa-jussà et al., 2020): (1) Structural Organization, (2) Content Comprehensiveness, (3) Dataset Description, (4) Dataset Structure, (5) Dataset Preprocessing, (6) Usage Guidance, and (7) Additional Information. While Dataset Description, Dataset Structure, and Additional Information can be found in sections of community-endorsed dataset cards, we added evaluation aspects highlighted in the literature, like aspects that constitute the overall presentation (Structural Organization and Content Comprehensiveness), Data Preprocessing and Usage Guidance. To conduct this assessment, we randomly selected a subset containing 150 dataset cards and engaged five human annotators. These annotators were tasked with evaluating each dataset card across these seven aspects and providing an overall quality score within a range of 5 (Appendix B.2). The overall quality is assessed through the subjective perception of human annotators, taking into account the seven aspects as well as their overall impression. This evaluation approach aims to provide a comprehensive assessment of dataset card quality, reflecting the importance of these aspects in effective dataset documentation. Human Perception of Documentation Quality Strongly Aligns with Quantitative Analysis Human annotation evaluation of dataset cards shows varying scores across different aspects. While Dataset Description (2.92/5), Structural Organization (2.82/5), Data Structure (2.7/5), and Content Comprehensiveness (2.48/5) received relatively higher scores, areas like Data Preprocessing (1.21/5) and Usage Guidance (1.14/5) scored lower. This aligns with the quantitative analysis that indicates a greater emphasis on the Dataset Description and Dataset Structure sections. Notably, even the highest-scoring aspect, Dataset Description, falls below 60% of the highest possible score, indicating room for improvement in dataset documentation. Content Comprehensiveness has the strongest positive correlation with the overall quality of a dataset card (Coefficient: 0.3935, p-value: 3.67E-07), emphasizing the pivotal role of comprehensive dataset content in shaping individuals’ perceptions of a dataset card’s overall quality. Additionally, aspects like Dataset Description (Coefficient: 0.2137, p-value: 3.04E-07), Structural Organization (Coefficient: 0.1111, p-value: 2.17E-03), Data Structure (Coefficient: 0.0880, p-value: 6.49E-03), and Data Preprocessing (Coefficient: 0.0855, p-value: 2.27E-03) also significantly contribute to people’s evaluations of dataset documentation quality. Moreover, the length of a dataset card is positively related to Content Comprehensiveness (p-value: 1.89E-011), reinforcing the importance of detailed documentation in enhancing dataset quality and usability. ### 7 RELATED WORKS Dataset has long been seen as a significant constraint in the realm of machine learning research (Halevy et al., 2009; Sun et al., 2017). The process of creating datasets remains arduous and time-intensive, primarily due to the costs of curation and annotation (IBM, 2020). Moreover, the quality of data assumes a pivotal role in shaping the outcomes of machine learning research (Liang et al., 2022). Consequently, a profound understanding of datasets is indispensable in the context of machine learning research, and this understanding is most effectively conveyed through comprehensive dataset documentation. A long-standing problem in the literature is that there is no industry standard being formed about data documentation. Therefore, much existing work in the literature has been in exploring, conceptualizing and proposing different dataset documentation frameworks. Data-focused tools such as datasheets for datasets and data nutrition labels have been proposed to promote communication between dataset creators and users, and address the lack of industry-wide standards for documenting AI datasets (Bender & Friedman, 2018; Bender et al., 2021; Pushkarna et al., 2022; Gebru et al., 2021; Holland et al., 2018; Chmielinski et al., 2022; Papakyriakopoulos et al., 2023). Additionally, there are studies that concentrate on leveraging human-centered methods to scrutinize the design and evaluation aspects of dataset documentation (Fabris et al., 2022; Mahajan & Shaikh, 2021; Hanley et al., 2020; Hutiri et al., 2022). In the library domain, numerous works have proposed methods to tackle the absence of universally accepted guidelines for publishing library-linked data. These efforts are aimed at enhancing data quality, promoting interoperability, and facilitating the discoverability of data resources (Villazon-Terrazas et al., 2011; Hidalgo-Delgado et al., 2017; Abida et al., 2020). These tools and frameworks provide detailed information on the composition, collection process, recommended uses, and other contextual factors of datasets, promoting greater transparency, accountability, and reproducibility of AI results while mitigating unwanted biases in AI datasets. Additionally, they enable dataset creators to be more intentional throughout the dataset creation process. Consequently, datasheets and other forms of data documentation are now commonly included with datasets, helping researchers and practitioners to select the most appropriate dataset for their particular needs. Despite the proliferation of dataset documentation tools and the growing emphasis on them, the current landscape of dataset documentation remains largely unexplored. In this paper, we present a comprehensive analysis of AI dataset documentation on Hugging Face to provide insights into current dataset documentation practices. 8 DISCUSSION In this paper, we present a comprehensive large-scale analysis of 7,433 AI dataset documentation on Hugging Face. The analysis offers insights into the current state of adoption of dataset cards by the community, evaluates the effectiveness of current documentation efforts, and provides guidelines for writing effective dataset cards. Overall, our main findings cover 5 aspects: • Varied Adherence to Community-Endorsed Dataset Card: We observe that high-downloaded dataset cards tend to adhere more closely to the community-endorsed dataset card structure. • Varied Emphasis on Sections: Our analysis of individual sections within dataset cards reveals that practitioners place varying levels of emphasis on different sections. For instance, among the top 100 downloaded dataset cards, Dataset Description and Dataset Structure sections receive the most attention. In contrast, the Considerations for Using the Data section garners notably lower engagement across all downloads, with only approximately 2% of dataset cards containing this section. This discrepancy can be attributed to the section’s content, which involves detailing limitations, biases, and the societal impact of datasets – a more complex and nuanced endeavor. An internal user study conducted by Hugging Face ([HuggingFace](https://huggingface.co)) also identified the Limitation section within this category as the most challenging to compose. • Topics Discussed in Each Section: Our examination of subsections within each section of dataset cards reveals a high completion rate for those suggested by the Hugging Face community. This highlights the effectiveness of the community-endorsed dataset card structure. In particular, our study places a special focus on the Considerations for Using the Data section, employing topic modeling to identify key themes, including technical and social aspects of dataset limitations and impact. • Importance of Including Usage Sections: We observe that many dataset card creators go beyond the recommended structure by incorporating Usage sections, which provide instructions on effectively using the dataset. Our empirical experiment showcases the potential positive impact of these Usage sections in promoting datasets, underscoring their significance. • Human Evaluation of Dataset Card Quality: Our human evaluation of dataset card quality aligns well with our quantitative analysis. It underscores the pivotal role of Content Comprehensiveness in shaping people’s assessments of dataset card quality. This finding offers clear guidance to practitioners, emphasizing the importance of creating comprehensive dataset cards. Moreover, we establish a quantitative relationship between Content Comprehensiveness and the word length of dataset cards, providing a measurable method for evaluation. Limitations and Future Works Our analysis of ML dataset documentation relies on the distinctive community-curated resource, Hugging Face, which may introduce biases and limitations due to the platform’s structure and coverage. For example, Hugging Face’s NLP-oriented concentration could introduce biases into the dataset categories. However, our method is transferable and could easily be reproduced for another platform, facilitating future studies (Appendix E). Additionally, our analysis of completeness and informativeness is based on word count and topic modeling, which may not fully capture the nuances of the documentation. Furthermore, measuring dataset popularity based on downloads alone may not fully reflect the dataset’s impact. Future research could consider additional factors, such as the creation time of the dataset and research area of the dataset (Appendix D). Lastly, our human evaluation serves as a preliminary evaluation. Future analyses could involve a more diverse group of annotators with varying backgrounds and perspectives. Research Significance To summarize, our study uncovers the current community norms and practices in dataset documentation, and demonstrates the importance of comprehensive dataset documentation in promoting transparency, accessibility, and reproducibility in the AI community. We hope to offer a foundation step in the large-scale empirical analysis of dataset documentation practices and contribute to the responsible and ethical use of AI while highlighting the importance of ongoing efforts to improve dataset documentation practices. REPRODUCIBILITY STATEMENT We have assembled a collection of dataset cards as a community resource, which includes extracted metadata such as the number of downloads and textual analyses. This resource along with our analysis code can be accessed at https://github.com/YoungXinyu1802/HuggingFace-Dataset-Card-Analysis. The Hugging Face datasets can be accessed through the Hugging Face Hub API, which is available at https://huggingface.co/docs/huggingface_hub/package_reference/hf_api. ACKNOWLEDGMENTS We thank Yian Yin and Nazneen Rajani for their helpful comments and discussions. J.Z. is supported by the National Science Foundation (CCF 1763191 and CAREER 1942926), the US National Institutes of Health (P30AG059307 and U01MH098953) and grants from the Silicon Valley Foundation and the Chan-Zuckerberg Initiative. REFERENCES Rabeb Abida, Emna Hachicha Belghith, and Anthony Cleve. An end-to-end framework for integrating and publishing linked open government data. In 2020 IEEE 29th International Conference on Enabling Technologies: Infrastructure for Collaborative Enterprises (WETICE), pp. 257–262, 2020. doi: 10.1109/WETICE49692.2020.00057. Shazia Afzal, Rajmohan C, Manish Kesarwani, Sameep Mehta, and Hima Patel. Data readiness report, 2020. Ruth-Ann Armstrong, John Hewitt, and Christopher Manning. 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URL https://doi.org/10.1145/3551624.3555286 Wenfei Fan and Floris Geerts. Foundations of data quality management. Synthesis Lectures on Data Management, 4(5):1–217, 2012.
QRWrvzRU4w	As shown in Fig.1, the authors split the same parameters of each layer into $g_l$ groups, in fact speculating whether neurons will fire a spike at $i$-th step ($i=1,...,g_l$) under the condition that the input current in each step is completely the same (i.e. the current is uniformly distributed).	ABSTRACT With the development of deep learning models, there has been growing research interest in spiking neural networks (SNNs) due to their energy efficiency resulting from their multiplier-less nature. The existing methodologies for SNN development include the conversion of artificial neural networks (ANNs) into equivalent SNNs or the emulation of ANNs, with two crucial challenges yet remaining. The first challenge involves preserving the accuracy of the original ANN models during the conversion to SNNs. The second challenge is to run complex SNNs with lower latencies. To solve the problem of high latency while maintaining high accuracy, we proposed a parallel spike-generation (PSG) method to generate all the spikes in a single timestep, while achieving a better model performance than the standard Integrate-and-Fire model. Based on PSG, we propose OneSpike, a highly effective framework that helps to convert any rate-encoded convolutional SNN into one that uses only one timestep without accuracy loss. Our OneSpike model achieves a state-of-the-art (for SNN) accuracy of 81.92% on the ImageNet dataset using just a single time step. To the best of our knowledge, this study is the first to explore converting multi-timestep SNNs into equivalent single-timestep ones, while maintaining accuracy. These results highlight the potential of our approach in addressing the key challenges in SNN research, paving the way for more efficient and accurate SNNs in practical applications.\footnote{Code available for review at: https://anonymous.4open.science/r/OneSpike} 1 INTRODUCTION Spiking neural networks (SNNs) have gained attention in the research community due to their potential for energy efficiency. They can often emulate the architectures of advanced artificial neural networks (ANNs). However, the non-differentiability and discontinuous nature of SNNs complicate the use of standard back-propagation methods typically found in ANN training. Prior work either directly trains SNNs \cite{Fang2021,Zheng2021,Datta2022} or transfers weights from trained surrogate ANNs to SNNs \cite{Zhang2022,Bu2023,Ho2021}. However, these existing works either suffer from lower accuracies or require functionalities not found in SNNs that often erode their energy competitiveness. It is therefore critical to align with the essential characteristics and prerequisites of SNNs to enhance compatibility with neuromorphic hardware. The energy efficiency advantage of SNNs is based on their ability to achieve accuracies that are competitive with state-of-the-art ANNs as well as low latencies. Higher latencies translate directly to longer inference time and hence higher energy cost. In rate-encoded SNNs \cite{Rueckauer2016}, latency is equivalent to the time window size, denoted as $T$, which could be a constant for a given model. Besides energy, the value of $T$ is also crucial for real-time applications \cite{Dethier2013,Pearson2007}. Thus, when developing SNNs, the two-fold challenge lies in improving accuracy and reducing $T$. Recent methods that achieved over 70\% accuracy on the Imagenet dataset \cite{Deng2009} and their associated time window sizes are presented in Table 1. In direct SNN training, the current best time window configuration is $T = 4$ for achieving comparable performance. In the case of ANN-to-SNN conversion methods, larger time window sizes are required. Existing methods have limitations, one of which is relying on $T > 1$, often causing significant accuracy trade-offs. None, for example, has surpassed the 80\% accuracy mark on the ImageNet Table 1: Time window sizes ($T$) of the state-of-the-art SNN methods and the accuracies they achieved on the ImageNet dataset. \| Method \| $T$ \| Accuracy \| \|-------------------------------\|-----\|----------\| \| QCFS-SNN (Bu et al., 2023) \| 64 \| 72.85 \| \| TCL-SNN (Ho & Chang, 2021) \| 35 \| 70.75 \| \| Spikformer (Zhou et al., 2022)\| 4 \| 74.81 \| \| Spikingformer (Zhou et al., 2023) \| 4 \| 75.85 \| dataset. To address this, we introduce a novel approach for converting any SNN with $T > 1$ into an equivalent one with $T = 1$ in such a way that accuracy is not compromised. In fact, in some cases, it even enhances the model’s performance. Our method, OneSpike, incorporates a novel method we call parallel spike generation (PSG). In the OneSpike model, all network layer operations can be executed in a single timestep, resulting in significantly reduced latency. Furthermore, PSG produces all the spikes in a single timestep in parallel, effectively harnessing global information across spikes. This results in enhanced accuracy compared to conventional integrated-and-fire (IF) models in some cases. Notably, we achieved a top-1 accuracy of 81.92% on the ImageNet dataset using a OneSpike model with $T = 1$. This may bring to mind binary neural networks (BNN). However, BNNs operate quite differently from SNNs. More importantly, as we will show in Section 6.1, the accuracies of state-of-the-art BNNs are significantly lower than that of OneSpike. In addition, we will also show the hardware feasibility of OneSpike, and explore the impact of weight quantization. In particular, we will show that OneSpike shows good robustness and retains a high accuracy even after weight quantization, enhancing its potential for further energy saving. The contributions of this work can be summarized as follows: 1. We propose the parallel spike generation (PSG) method that generates all spikes for a network layer within a single timestep. Models incorporating PSG exhibit superior performance compared to those using the original IF model, especially when the time window size reduces. Meanwhile, we thoroughly discuss the feasibility of hardware implementation for the PSG method. 2. We introduce OneSpike, a framework that converts SNNs with $T > 1$ into SNNs with $T = 1$. Utilizing multiple-timestep SNN models from the CIFAR-10 dataset (Krizhevsky et al., 2009) as our reference points, we transitioned them into OneSpike configurations. This conversion resulted in a marked accuracy surge for low-timestep models, surpassing the original SNNs underpinned by the IF model. To the best of our knowledge, this is the first work on converting an SNN to a lower latency SNN. 3. We present a series of high-accuracy, ultra-low-latency SNNs based on PSG as OneSpike models. We achieved 81.92% SNN accuracy on the complex ImageNet dataset within one timestep. To the best of our knowledge, this is the first SNN model achieving 80% on the ImageNet dataset. The paper is organized as follows: Sections 2 and 3 introduce related works and SNNs, including ANN pre-training and existing ANN-to-SNN conversion methods. In Section 4, we introduce our Parallel Spike Generation(PSG) algorithms and present the OneSpike framework. Section 5 presents the performance of our methods on the ImageNet dataset. Section 6 discusses binary neural networks (BNNs) and weight quantization to further reduce the model size, respectively. This is followed by a conclusion. 2 RELATED WORKS 2.1 METHODS OF TRAINING SNNs The two most common ways of developing SNN models are direct training and transferring from ANNs to SNNs. In this paper, we shall focus on the more popular form, i.e. rate-encoded SNNs. In direct training, techniques such as backpropagation through time (BPTT) and surrogate gradient (SG) are employed to handle the temporal properties and non-differentiability of SNNs (Fang et al., 2021; Zheng et al., 2021; Datta et al., 2022; Zhou et al., 2023; Li et al., 2021b). This approach can achieve lower latency, often using only a few time steps for complex tasks. However, direct training faces challenges during the training phase, requiring significant computational resources as well as issues with accurate gradient approximations. Moreover, direct training methods have not yet achieved competitive accuracy compared to state-of-the-art ANNs. ANN-to-SNN conversion methods (Hu et al., 2021; Li et al., 2021a; Ho & Chang, 2021; Bu et al., 2023; Panda et al., 2020) focus on transferring the weights of an ANN model to an SNN model without further training. The key is to estimate the spiking rate in an SNN by activations of an ANN. Previously proposed approaches adopted techniques like weight normalization (Diehl et al., 2015) and temporal switch coding (Han & Roy, 2020) to achieve higher accuracy. An existing ANN can be easily converted to an SNN as long as it satisfies certain constraints. Given the abundance of tools and packages available for building ANNs, this method facilitates the acquisition of SNNs without requiring meticulous training. The shortcoming of existing ANN-to-SNN conversion methods is that they require large time window sizes. Thus, this paper aims to investigate minimizing time steps as well as high accuracy. 2.2 Binary Neural Networks (BNNs) Binary neural networks (BNNs) (Courbariaux et al., 2016) constrain the weights and/or activations of neural network models to binary values (most using $+1/-1$). They have their own training algorithms based on binarization. Recent studies (Liu et al., 2018; Bethge et al., 2019; Tu et al., 2022) have also explored BNNs that are inspired by established architectures like ResNet and DenseNet by incorporating shortcut connections to enhance performance. SNN models with a time step of one outwardly resemble BNNs. However, BNNs and SNNs differ in several aspects. BNNs use normal activation functions, quantized parameters, and traditional forward and backward propagation algorithms, much like standard ANNs. In a way, they can be viewed as extremely quantized ANNs. In contrast, SNNs utilize spike-based encoding and time-dependent computations. The integrate-and-fire rule is unique to SNNs with no parallel in BNNs. Both aim at energy efficiency and computational speed, making them suitable for low-power and high-efficiency applications. Unlike BNNs, SNNs also do not use softmax, swish, or any complex functions. While the use of softmax (Liu et al., 2022) or swish (Darabi et al., 2018) functions in BNNs improves accuracy, they also lead to increased computational overhead and pose challenges for hardware implementation commonly required for embedded settings. 3 Preliminary This section elucidates the foundational Integrate-and-Fire (IF) model utilized by SNN neurons for spike generation and our adopted techniques for converting ANNs into SNNs. 3.1 Integrate-and-Fire model The IF model is the most popular SNN model (Bu et al., 2023). It offers a simple representation of how neurons accumulate membrane potential and fire spikes. In the IF model, the membrane potential $V$ of a neuron is treated as a capacitor that accumulates the influence of input currents over time. It is described by the following differential equation: \[ \tau_m \frac{dV}{dt} = I_{syn}(t) - V(t) + V_{rest} \] Here, $\tau_m$ represents the membrane time constant, $I_{syn}(t)$ denotes the synaptic input current, and $V_{rest}$ signifies the resting potential. When the membrane potential $V$ crosses a certain threshold $\theta$, the neuron generates an action potential (spike). In an SNN, for layer $l$, the output spike $s_l$ at timestep $t \in T$ can be calculated as: \[ s_l(t) = \begin{cases} 1, & \text{if } V_l(t) \geq \theta \\ 0, & \text{else} \end{cases} \] where $\theta$ is the firing threshold and the membrane potential $V_l(t)$ can be denoted as: $$V_l(t) = V_l(t - 1) + W_l s_{l-1}(t) - \theta \cdot s_l(t)$$ (3) where $W_l$ is the weight, $s_{l-1}$ is the spike input from the previous layer and $V_l$ is initialized as 0. ### 3.2 ANN-to-SNN Conversion To convert ANNs to SNNs in a near lossless way, we used the value-range (VR) encoding (Yan et al., 2022) and clamp and quantize (CQ) (Yan et al., 2021) training. Specifically, we will clamp and quantize the activation to a discrete set of values, $\{T_{\text{min}}/T_q, T_{\text{min}}+1/T_q, T_{\text{min}}+2/T_q, \ldots, T_{\text{max}}/T_q\}$, mimicking spike trains in SNNs. In this paper, we set $T_{\text{min}}$ to 0 for all cases as 0 holds significant prominence in the distribution. We define $T_c$ as the clamp level and $T_q$ as the quantization level, such that: $$T_c = \frac{T_{\text{max}}}{T_q}$$ (4) Consequently, all activations are clamped to the range $[0, T_c]$, and the time window size becomes $T = T_q$. The clamp operation also simulates the behavior of the ReLU function. The weights and biases of the SNN model denoted as $\hat{W}$ and $\hat{b}$ respectively, are related to those of the equivalent CNN model, represented as $W$ and $b$. The original $l^{th}$ layer of the ANN with ReLU activation function, can be represented as: $$x_{l+1} = \text{ReLU}(W_l x_l + b_l)$$ (5) After the weight and bias conversion, the $l^{th}$ layer of the SNN would be: $$x_{l+1} = S(\hat{W}_l x_l + \hat{b}_l) = S\left(\frac{T_c}{T_q} W_l x_l + b_l\right)$$ (6) where $S$ refers to the spiking neuron that generates the spike. The specific clamp and quantization levels for each model can be found in the appendix. ## 4 METHODOLOGY In this section, we present the overview of the OneSpike model as well as the parallel spike generation (PSG) method. Our approach transforms any SNN with convolutional and fully connected layers that require $T > 1$, into an SNN with $T = 1$. ### 4.1 Overview of OneSpike Figure 1 illustrates a spiking convolutional layer within OneSpike. Fully connected layers are implemented in a similar way. In converting the $T > 1$ SNN to OneSpike, we reuse channels as depicted in Figure 1. The channels are grouped, with each group consisting of channels associated with distinct kernels, while the weights are shared across the groups. In the case of a fully connected layer, the different groups would each consist of a fully connected layer. Each group is responsible for processing a specific subset of feature representations. The $k^{th}$ group processes the spikes generated at $t = k, t \in T$ in the original SNN. As each layer receives a set of spike trains ($T = 1$) as input, we directly map spike trains from different groups to channels within their respective groups for computation. This eliminates the need for slicing and simplifies the allocation of subsets, streamlining the process. For the original spiking convolutional layer $l$ with an input size of $[H, W, C, T]$ where $H$ and $W$ represent the height and width of the feature maps respectively, and $C$ represents the number of channels, while $T$ denotes the time window size, the procedure is summarized as: The number of channel groups denoted as $g$, is the original time window size $T$. For layer $l$ with a channel group $g_l$, the input $s^{l-1}$ consists of $g_l$ groups of spike trains, each with a length of 1 and a size of $[H, W, C]$. The input for the $i^{th}$ channel group is represented by $s_i^{l-1}$. Within layer $l$, the output feature map $x_i^l$ for each channel group $i$ is computed as: $$x_i^l = \sum_{j=0}^{n} W_j^l \cdot s_i^{l-1} + b_i^l$$ (7) where $n$ is the number of channels in this group. Subsequently, we obtain the layer’s output $s^l$ by taking the average of the feature maps as the output of layer $l$: $$x^l = \sum_{i=0}^{g_l} x_i^l / g_l$$ (8) Now, $g_l$ can be merged into the weight of the previous convolutional layer, eliminating the need for division operations as well as enhancing the feasibility of implementing our approach on hardware. So we have: $$x^l = \sum_{i=0}^{g_l} \sum_{j=0}^{n} \frac{W_j^l}{g_l} \cdot s_i^{l-1} + \frac{b_i^l}{g_l}$$ (9) $s^{l-1}$ is a spike train composed of 0s and 1s. As $T = 1$ in our case, $s^{l-1}$ is either 0 or 1. The multiplication of weights with $s^{l-1}$ can be efficiently implemented as addition operations in hardware, leading to reduced energy consumption. 4.2 Parallel Spike Generation (PSG) We now introduce the PSG method to produce spikes from $x^l$ that serve as input for the subsequent layer, $l + 1$. Let $g_{l+1}$ be the count of channel groups in layer $l + 1$. From this, $g_{l+1}$ spike trains of dimensions $[H, W, C, 1]$ are concurrently generated. For the $i^{th}$ group in layer $l$, the membrane potential can be calculated as: $$V_i^l = ((i - 1)x^{l-1} \mod \theta) + x^{l-1}$$ where $\theta \in \{2^n \| n \in \mathbb{N}\}$ (10) The input for group $i$ is the $i^{th}$ spike in the train generated by the original SNN model. With the IF model, the previous $i - 1$ input would be accumulated through time. Once the threshold is met, a spike is produced, and the threshold is subtracted from the membrane potential. Given our consistent input across timesteps, the residual membrane potential after $i - 1$ steps is represented by $(i - 1)x^l \mod \theta$. Then, the spikes generated by PSG would be: \[ s_i^l = \begin{cases} 1, & \text{if } ((i - 1)x^{l-1} \mod \theta) + x^{l-1} \geq \theta \\ 0, & \text{else} \end{cases} \quad \text{where } \theta \in \{2^n \| n \in \mathbb{N}\} \] According to equation (11), given the previous layer’s output and group index, all spikes can be generated independently within their respective groups. This facilitates the calculation of the corresponding membrane potential and the subsequent spike generation. To understand this better, consider an example where the output of layer $l$ has an element $x^l = 1.5$, $g_{l+1} = 4$, and a spike threshold, $\theta = 2$. Consider $i = 3$, $V_3^l$ in this case can be computed as follows: $V_3^l = ((1.5 * 2) \mod 2) + 1.5$. All the values of $V_i^l$ are computed as $[1.5, 3, 2.5, 2]$, resulting in $s_{i+1}^l$ values for groups ranging from $i = 0$ to $i = 3$ being $[0, 1, 1, 1]$. Thus, at layer $l$, the $g_l$ groups of channels can directly compute the feature maps in a single step. If the number of groups between two adjacent layers $l$ and $l + 1$ is different, then the last output in layer $l$ needs to generate $g_{l+1}$ parallel spike trains. The workflow of a layer incorporating the PSG algorithm is detailed in Algorithm 1. Algorithm 1 A Layer with Parallel Spike Generation Model Require: The output groups of spike trains $s_{i-1}^l$ for $i \in \{1, 2, \ldots, g_l\}$ from the $(l - 1)$th layer; The spike threshold $\theta, \theta \in \{2^n \| n \in \mathbb{N}\}$; The weight $W_l$ and bias $b_l$ for layer $l$. Step 1: Get the average output of layer $l$. \[x^l = \sum_{i=1}^{g_l} (W_is_{i-1}^l + b_l)\] \[x_{avg}^l = x^l / g_l \{g_l \text{ can be merged into } W_l \text{ to avoid division operations}\} Step 2: Calculate $g_{l+1}$ membrane potentials; \[V_i^l = ((i - 1)x_{avg}^l \mod \theta) + x_{avg}^l, i \in \{1, 2, \ldots, g_{l+1}\}, \theta \in \{2^n \| n \in \mathbb{N}\}\] Step 3: Generate $g_{l+1}$ spike trains in parallel: \[s_i^l = \begin{cases} 1, & \text{if } V_i^l \geq \theta \\ 0, & \text{otherwise} \end{cases}, i \in \{1, 2, \ldots, g_{l+1}\}\] return $s_i^l$ for $i \in \{1, 2, \ldots, g_{l+1}\}$ - input spikes of layer $l + 1$. Performance Advantage. By leveraging the average of the output from the previous layer, our PSG model is more accurate for preserving the information and more robust than the IF model when the time window size reduces. As an illustration, consider an example with an element in the output $x_{avg}^l = 1.5, g_{l+1} = 4, \theta = 2$, the spike train generated by our method is $s_{PSG}^l = [0, 1, 1, 1]$, representing 1.5 accurately with three out of four spikes and a threshold $\theta = 2$. However, considering when the output before averaging as $x^l = [0.5, 1, 2, 2.5]$ whose average is also 1.5, the spike train generated by the IF model would be $s_{IF}^l = [0, 0, 1, 1]$, which represents 1. $s_{IF}^l$ loses information when generating spikes, especially at the end of the time window when remaining potentials are dropped. The averaging operation is also used by (Yan et al., 2022) for SNNs with $T > 1$. However, taking an average may be incompatible with the operations of SNNs. OneSpike solves this problem and the resulting SNNs fully conform to all the working principles of SNNs. 5 EXPERIMENTS 5.1 Effect of Parallel Spike Generation We first contrast the performance between the PSG method and the IF model. For a spike generation method, its objective is to closely approximate the scale value output by the corresponding ANN. We conduct tests using the CIFAR-10 dataset on the VGG-11 and VGG-16 models (Simonyan & Zisserman, 2014). We trained the VGG models and converted them into SNNs with the IF model using the ANN-to-SNN conversion method mentioned in [3.2]. Subsequently, we further covert this model into SNNs with the PSG method (e.g., a OneSpike model), ensuring that both models share identical weights and structures. ![Graph showing accuracy comparison between IF and PSG](image) (a) VGG-11 (b) VGG-16 Figure 2: PSG and IF comparison. Timesteps are 8, 16, 32, 64, 128 and 256 Figure 2 displays the accuracies of both SNNs. The graph reveals that at larger $T$, specifically 128 and 256, both the IF model and PSG demonstrate comparable performances. However, as the timestep diminishes, the IF model struggles to accomplish the task, while the PSG consistently preserves a superior accuracy rate. For low-latency SNNs, the PSG method offers a performance advantage over the IF model. 5.2 Experiment Setup We evaluate OneSpike on the ImageNet dataset (Deng et al., 2009), one of the most complex image classification datasets commonly used. OneSpike is compared against baselines from both ANN-to-SNN conversion methods and direct training methods, which validate the effectiveness of OneSpike and our model pruning efforts. To facilitate and standardize our training process, we adopt some of the automated data augmentation methods utilized by RepVGG. Our experiments are performed on NVIDIA A100 GPUs, based on Pytorch (Paszke et al., 2019) version 1.12.1, RepVGG (Ding et al., 2021), and Timm (Wightman, 2019). For ANN pre-training, we employ the model reparameterization technique from RepConv (Ding et al., 2021), which is currently considered one of the leading training methods for plain convolutional neural networks (CNNs). RepConv enables the combination of multiple computational modules into a single module during inference, thereby enhancing accuracy. In line with RepConv, we utilize a combination of one 3x3 convolution, one 1x1 convolution, and an identity connection within a single convolutional layer to achieve higher accuracy. 5.3 Model Structure To enhance baseline accuracy and simplify training, we utilize RepVGG-L2pse (Ding et al., 2021) along with its number of blocks, and its pre-trained weights obtained using their proposed re-parameterized training method. However, we exclude the squeeze and extract blocks from their model as they are incompatible with the SNN architecture. In the presented OneSpike model, there are five stages, excluding the output average pooling and fully connected layers. The number of 3x3 convolutional layers of each stage is [1, 8, 14, 24, 1]. The architectures of OneSpike models are listed in Table 2. \| Model \| Channel number \| \|-------------\|-------------------------\| \| OneSpike-8 \| [160, 160, 320, 640, 2560] × 8 \| \| OneSpike-16 \| [160, 160, 320, 640, 2560] × 16 \| \| OneSpike-32 \| [160, 160, 320, 640, 2560] × 32 \| Our models do not contain any operations incompatible with SNNs. Batch normalization and division operations are folded into the model weights, and complex activation functions such as swish and softmax are not used. 5.4 Result Our experiment results are presented in Table 3. Our OneSpike-32 model achieves a top-1 accuracy of 81.92% in $T = 1$. Using OneSpike-8, which is only one-quarter the size of OneSpike-32, an accuracy of 75.92% can be achieved. \| Model \| Description \| Accuracy \| Timestep \| FLOPs (Billions) \| \|----------------\|-------------------\|----------\|----------\|------------------\| \| OneSpike-8 \| 8 groups, $\theta = 2$ \| 75.92 \| 1 \| 30.1 \| \| OneSpike-16(2) \| 16 groups, $\theta = 2$ \| 80.24 \| 1 \| 60.2 \| \| OneSpike-16(4) \| 16 groups, $\theta = 4$ \| 78.86 \| 1 \| 60.2 \| \| OneSpike-32 \| 32 groups, $\theta = 4$ \| 81.92 \| 1 \| 120.4 \| Comparison with the state-of-the-art SNNs using ImageNet. Table 4 compares OneSpike with the state-of-the-art methods on the ImageNet dataset, demonstrating superior performance in terms of both accuracy and latency. Among these methods, ANN-to-SNN has traditionally used large time steps, possibly exceeding 200, resulting in high latency and energy consumption, in exchange for higher accuracy compared to direct training methods. Notably, in the works (Zheng et al., 2021; Datta et al., 2022; Fang et al., 2021; Hu et al., 2021; Bu et al., 2023) that adopt ResNet for SNNs, the accuracy is limited despite using small time window sizes. In particular, most previous works have not considered deep networks. Although (Datta et al., 2022) also achieved single-timestep classification on ImageNet, OneSpike achieved over 10% higher accuracy than their reported results. Table 4: Comparison with various SNNs on ImageNet. ** denotes the use of complex attention layers, making counting less straightforward. \| Method \| Layers \| Accuracy \| Timestep \| Params (in millions) \| \|-----------------\|--------\|----------\|----------\|----------------------\| \| Direct Training \| 34 \| 67.05 \| 6 \| 21.80 \| \| Direct Training \| 50 \| 66.32 \| 1 \| 25.56 \| \| Hybrid \| 16 \| 67.71 \| 1 \| 138.36 \| \| Direct Training \| 16 \| 68.00 \| 1 \| 138.36 \| \| Direct Training \| 152 \| 69.26 \| 4 \| 60.19 \| \| ANN-to-SNN \| 16 \| 70.75 \| 35 \| 138.36 \| \| ANN-to-SNN \| 50 \| 72.75 \| 350 \| 25.56 \| \| ANN-to-SNN \| 34 \| 73.37 \| 256 \| 21.80 \| \| ANN-to-SNN \| 16 \| 74.22 \| 256 \| 138.36 \| \| Direct Training \| * \| 74.81 \| 4 \| 66.34 \| \| Direct Training \| * \| 75.85 \| 4 \| 66.34 \| \| ANN-to-SNN \| 23 \| 77.50 \| 256 \| 22.1 \| \| ANN-to-SNN \| 28 \| 79.16 \| 200 \| 94.08 \| \| OneSpike-32 \| 48 \| 81.92\| 1 \| 118.11 \| Model parameter and FLOPs. While our focus is on achieving high accuracy with the smallest time step for SNNs, we also consider the size and computational demand of the model. All OneSpike models have 118 million parameters, and their theoretical Floating Point Operations (FLOPs) values are calculated and listed in Table 3. The spike rate on the ImageNet test set for OneSpike is 11.46%, which is used for our calculations. The parameter count and FLOPs of some popular models, calculated by a PyTorch toolkit called OpCounter (Lyk), are compared with our model in Table 5. In Table 5, $n$ represents the equivalence of energy consumption between $n$ additions and one multiplication. OneSpike has a parameter num- number comparable to common CNN models, yet it exhibits a significant advantage in computational complexity due to all the multiplications being replaced by additions. Table 5: The number of parameters (in millions) and FLOPs (in billions) of the different models. $ n $ is the number of additions considered computationally equivalent to 1 multiplication. The exact ratio depends on many implementation details. \| Model \| VGG-16 \| ResNet-152 \| RepVGG-B3 \| OneSpike-8 \| \|----------------\|--------\|------------\|-----------\|------------\| \| No. of Parameters \| 138.36 \| 60.19 \| 110.96 \| 118.11 \| \| FLOPs \| $ 15.5n $ \| $ 11.58n $ \| $ 26.2n $ \| \| \| FLOPs$(n = 3)$ \| 46.5 \| 34.74 \| 78.6 \| 30.1 \| \| FLOPs$(n = 4.1)$ \| 63.55 \| 47.48 \| 107.42 \| \| 6 DISCUSSION 6.1 COMPARISON WITH BINARY NEURAL NETWORKS Given the similarity between the single-time step OneSpike and BNN, as both involve binary activation, we also compare OneSpike with the state-of-the-art BNNs on ImageNet in Table 6. Although in terms of binary weights, BNNs exhibit relatively higher energy efficiency, OneSpike achieves significantly higher accuracy, surpassing BNNs by over 10%. Table 6: Comparison of OneSpike with BNN methods \| Method \| Top-1 Accuracy (%) \| \|----------------\|--------------------\| \| Bi-real $[Liu et al., 2018]$ \| 62.20 \| \| AdaBin $[Tu et al., 2022]$ \| 66.40 \| \| Real-to-Bin $[Martinez et al., 2020]$ \| 65.40 \| \| OneSpike \| 81.92 \| 6.2 MODEL SIZE AND WEIGHT QUANTIZATION To further improve efficiency, we quantized the weights of OnSpike models following the quantization scheme in $[Yan et al., 2023]$ from FP32 to INT16. After quantization, OneSpike-8 and OneSpike-16(2) still attained an accuracy of 73.92% and 80.12%, respectively, on ImageNet. 6.3 HARDWARE FEASIBILITY. OneSpike is hardware-friendly in the following ways. Firstly, $ \theta $ are constrained to be powers of 2. Consequently, the modulo operation in binary can be efficiently accomplished by reading the least significant bits. In particular, all OneSpike models used in this paper use thresholds that are either 1, 2, or 4. No impact on accuracy was observed. Secondly, we require obtaining $ i - 1 $ instances of $ x_l $, given that $ i - 1 $ is an integer not exceeding the original window size $ T $ of the original SNN model. This requires only at most $ \log(T) $ additions for each group. 7 CONCLUSION In this paper, we introduced the parallel spike generation method (PSG) that generates spikes using spike trains in a single timestep for both convolutional layers and fully connected layers. Building upon PSG, we developed the OneSpike framework that transforms SNNs with $ T > 1 $ into equivalent $ T = 1 $ SNNs. 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C3msSjudA7	The quality of results in real scenes appears to be a concern, with reconstructed objects being described as blurry. It is important to provide further analysis or insights into why this issue occurs and potential strategies for improving the quality of real scene reconstructions.	ViFu: Visible Part Fusion for Multiple Scene Radiance Fields Anonymous authors Paper under double-blind review Figure 1: An overview of our approach. From (a) multi-view images of multiple scenes with different object placements, ViFu recovers the appearance and 3D geometry of (c) clean static backgrounds and (d) 360° foreground objects. Radiance fields representation supports free-view rendering of the recovered background scene and foreground objects. Abstract In this paper, we propose a method to segment and recover a static, clean background and 360° objects from multiple scene observations. Recent works have used neural radiance fields to model 3D scenes and improved the quality of novel view synthesis, while few studies have focused on modeling the invisible or occluded parts of the training images. These under-modeled parts constrain both scene editing and rendering view selection. Our basic idea is that, by observing the same set of objects in various arrangement, so that parts that are invisible in one scene may become visible in others. By fusing the visible parts from each scene, occlusion-free rendering of both background scene and foreground objects can be achieved. We decompose the multi-scene fusion task into two main components: (1) objects/background segmentation and alignment, where we leverage point cloud-based methods tailored to our novel problem formulation; (2) radiance fields fusion, where we introduce visibility field to quantify the visible information of radiance fields, and propose visibility-aware rendering for multiple scene fusion, ultimately obtaining clean background and 360° object rendering. Comprehensive experiments were conducted on synthetic and real datasets, and the results demonstrate the effectiveness of our method. The code will be released for research purposes upon paper acceptance. 1 Introduction Recently, the advance of neural rendering with implicit representation has received attention for its numerous real-world applications, including virtual reality, games, movies, and more. One of the pioneering works is neural radiance field (NeRF) [Mildenhall et al., 2020], which uses a neural network to model the 3D space as a continuous radiance field, enabling the reconstruction of the detailed geometry and appearance of a scene from multi-view images. There has been significant follow-up work to explore the extension of NeRF in the direction of fast optimization [Yu et al., 2021; Müller et al., 2022; Chen et al., 2022; Sun et al., 2021], generalization [Yu et al., 2020; Wang et al. (2021b), dynamic scenes [Park et al., 2020; Tretschk et al., 2020; Pumarola et al., 2020], human body [Xu et al., 2022; Peng et al., 2021; Noguchi et al., 2022b] or articulated objects [Noguchi et al., 2021, 2022a] modeling, appearance editing [Liu et al., 2021; Kobayashi et al., 2022], shape editing [Xu & Harada, 2022; Yuan et al., 2022], etc. In particular, compositional scene modeling is one of the popular directions in which individual parts of a scene, such as a background scene or foreground objects, are independently modeled rather than treating the entire scene as a whole. It represents the whole scene as a composition of background scene and foreground objects, enabling applications such as scene segmentation [Zhi et al., 2021], object movement or removal [Yang et al., 2021; Wu et al., 2022], independent object rendering [Jang & de Agapito, 2021], etc. While an increasing number of works have attempted to use NeRF for compositional scene modeling, an obvious but challenging issue has been left unaddressed: background or objects occluding each other can result in parts of the scene that cannot be observed from the training images, thereby causing under-modeled parts in the scene. As a result, the movement/removal of objects, or rendering from certain viewpoints can expose these under-modeled parts, leading to poor rendering results with artifacts (e.g., Fig. 2(c)). Especially in tasks requiring clean backgrounds or manipulation of object placement, such as indoor scene reconstruction or robotics applications, this issue becomes particularly pronounced. Specifically, we consider two cases of under-modeling: (1) under-modeled background scene, such as the desktop, where the contact surface with the foreground objects is invisible during training, leading to artifacts when removing or moving foreground objects; (2) under-modeled foreground objects, where the invisible surface is exposed when rendering with changing the object’s pose (e.g., laying it down), causing artifacts. To the best of our knowledge, no previous studies have attempted to address these issues. In this work, we explore compositional scene modeling from the perspective of recovering clean backgrounds and 360° objects. Recovering the above unseen parts from a single scene is challenging and laborious, as it usually requires a hand-designed or learned scene prior, as in image completion tasks. Instead of a single scene, we consider a set of scenes where the background remains static while objects are placed in different positions and poses. Here, the object placement satisfies two conditions: (1) there is no part of the background that is always occluded by the object in all scenes, and similarly (2) there is no part of the objects surface that is invisible in all scenes (i.e., every part of the background/objects is visible in at least one scene). These two conditions correspond to the two under-modeled cases above, and this multi-scene setup ensures that we have enough information to recover the geometry and appearance of clean background and 360° objects. Recall that the above key issues come from the invisible part caused by occlusion. To address this issue, given the volumetric nature of the radiance field, we propose visibility field, a volumetric representation for quantifying the visibility in scenes. With the proposed visibility field, we compare the visibility of the corresponding part across multiple scenes and fuse the parts with higher visibility to achieve clean background and 360° objects rendering. We dub our proposed idea of visible part fusion as ViFu. The basic idea of ViFu is shown in Fig. 2. Furthermore, we leverage the multi-scene setting and propose a method for segmenting objects and backgrounds by exploiting the differences in object placement across each scene. Our segmentation approach is based on the geometric differences w.r.t. clean backgrounds obtained via fusion, which is computationally efficient and simple, and does not require any pre-trained 3D segmentation model. To verify the effectiveness of ViFu, we created several sets of synthetic scenes containing various objects. We observe that ViFu automatically and accurately segments the background and each object, and achieves pleasing recovery of clean backgrounds and free-view rendering of 360° foreground objects. We also captured videos to create a set of real-world datasets, and the experimental results show that the proposed method also gives promising results for real-world scenes. In summary, our main contributions are listed as follows: - We studied the under-modeled invisible parts of NeRF and introduced the setting of complementing the invisible parts by fusing multiple scene information. - We introduce visibility field, a volumetric representation to quantify the visibility of scenes, and propose novel visibility-aware rendering, which leverages the visibility field to achieve the fusion of visible parts of multiple scenes. • We created synthetic and real datasets to validate our idea, and the experimental results show the effectiveness of the proposed method. 2 RELATED WORK Neural radiance field revisited. Recently, neural rendering with implicit representations has received significant attention due to its detailed representation of the geometry and appearance of the scene [Sitzmann et al., 2019; Yariv et al., 2020; Mildenhall et al., 2020]. The most representative work is neural radiance field (NeRF) [Mildenhall et al., 2020], which uses neural networks to model the scene as a continuous mapping from position and view direction to radiance color and volume density, enabling geometric and appearance reconstruction and photorealistic novel view rendering. Several follow-up works have been proposed to improve the foundation of NeRF, enabling fast optimization [Yu et al., 2021; Müller et al., 2022; Chen et al., 2022; Sun et al., 2021], appearance decoupling [Verbin et al., 2021], dynamic scene modeling [Pumarola et al., 2020; Park et al., 2020; Tretschk et al., 2020], and more. Nevertheless, these methods have limitations as they model the scene as a whole and do not allow for segmentation or editing of specific parts of the scene. Object-centric scene representation. A new category of object-centric modeling methods has been proposed to enhance the reasoning and editing capabilities of scenes. Specifically, compositional scene modeling methods [Zhang et al., 2020; Guo et al., 2020; Niemeyer & Geiger, 2021; Wang et al., 2021c; Zhang et al., 2021; Wu et al., 2022] regard the entire scene as a mixture of background and foreground objects, facilitating object-level scene understanding; some methods encode semantic information into scenes, enabling feature-based object query or segmentation [Zhi et al., 2021; Wang et al., 2022, 2021a]. Another direction explores object-level manipulations on scene content, enabling editing to object appearance [Liu et al., 2021; Bao et al., 2023] or geometry [Xu & Harada, 2022; Yuan et al., 2022]. These advancements have made notable progress in manipulating NeRF-based representations, however, our primary concern is that manipulating the original scenes (i.e., object movement or deformation) can inadvertently expose unseen parts and thus lead to artifacts. Scene completion for radiance fields. To address the issue of under-modeled parts being exposed, recent studies have approached it as a 3D inpainting problem and proposed solutions for radiance field representations. NeRF-In [Liu et al., 2022] uses masks to segment the foreground objects and performs inpainting to obtain an unoccluded background, while SPin-NeRF [Mirzaei et al., 2022] improves on this by introducing the concept of perceptual inpainting to enhance the rendering results. However, these methods only consider the completion of the background part and do not address the invisible parts of the objects. Furthermore, the shadows cast by the original objects still appear as noticeable artifacts in the resulting inpainted regions [Liu et al., 2022; Mirzaei et al., 2022]. Scene fusion for radiance fields. Some recent works also attempt to fuse NeRF, such as NeR-Fusion [Zhang et al., 2022] or NeRFuser [Fang et al., 2023]. The objective of these methods is to integrate the individual 3D representations of various local components within a vast scene, thereby obtaining a comprehensive scene rendering. Hence, the main focus of these methods lies in modeling large-scale scenes effectively. Conversely, our approach is centered around addressing occlusions caused by objects within the scene, aiming to reconstruct an occlusion-free background scene and 360° foreground objects by leveraging the visible parts across different scenes. 3 METHOD Consider a static background scene and $M \geq 1$ foreground objects that are placed in different positions and poses resulting in $N \geq 2$ different scenes (e.g., Fig. 1(a)). For each scene, we capture $L_i$ multi-view images $\{I_l\}$ and run the structure-from-motion method independently for each scene to obtain the camera parameters (intrinsics and extrinsics) $\{C_l\}$, where $i \in \{1, ..., N\}$ denotes scene index and $l \in \{1, ..., L_i\}$ denotes camera index of scene $i$. From the calibrated multi-view images, we optimize neural radiance fields (NeRF) $\{S_i\}$ for each scene. Radiance field is an implicit scene representation that maps spatial position $x \in \mathbb{R}^3$ and view direction $d \in S^2$ to radiance color $c = (r, g, b)$ and volume density $\sigma$ as $S : (x, d) \mapsto (c, \sigma)$. Figure 2: The basic idea of ViFu. With pre-computed scene/objects alignment, we compare the visibility of the corresponding parts using the proposed visibility field, and fuse the higher visibility parts of each scene to form the clean background and 360° objects. The details of visibility-aware rendering are shown in Fig. 3. Our method takes $N$ optimized radiance fields $\{S_i\}$ as input, automatically splits the scenes into a static background and $M$ foreground objects, and recovers a non-occluded background scene and 360° objects that can be seen from arbitrary view point. Assumption. For our problem formulation, we make the following two ideal assumptions: (1) diverse object positions: this ensures visibility of the background scene, implying that every part of the background scene is observable in at least one scene (i.e., no permanently occluded regions); this also facilitates the segmentation of foreground objects, as will be introduced in Sec. 3.2. (2) diverse object poses: this guarantees that every part of the object’s surface is observable in at least one scene (e.g., no permanently facing-down surfaces). The assumptions are natural for household objects in everyday scenes: static objects that remain unchanged, such as refrigerators or tables, are considered part of the background; while objects that are frequently moved, such as the toys in Fig. 1, are treated as foreground objects. 3.1 Method Overview Our objective is to perform background/foreground segmentation from multiple scenes and obtain a clean background and 360° objects via fusion. In the general context of 3D modeling, this process can be divided into two main steps: the first involves internal scene reasoning, specifically the segmentation of background/foreground within each scene; the second entails inter-scene reasoning, which involves matching the segmented background and individual objects among different scenes (i.e., pose alignment for background scene and foreground objects), and subsequently accomplishing the final fusion. In the following section, we introduce our solutions, specifically tailored for the recent 3D representation of the radiance field. To be more precise, we leverage a point cloud-based approach to perform scene segmentation and alignment (Sec. 3.2), and introduce a novel measure for quantifying the visibility of the radiance fields (Sec. 3.3), which is used in the proposed scene fusion method (Sec. 3.4). 3.2 Object Segmentation and Alignment The first step involves background/foreground segmentation and obtaining the relative poses of foreground objects and background scene within each scene. This allows us to align them to their respective common coordinate systems, which are utilized for subsequent fusion purposes (See Fig. 2). For the segmentation and alignment of the radiance fields, we found that existing point cloud-based methods already yield satisfactory results. For simplicity, we introduce here the minimal segmentation and alignment techniques below; however, other more advanced alternatives can also be employed. We provide a high-level overview of the entire process here, with specific calculations detailed in the supplementary materials. First, we employ Marching Cubes [Lorensen & Cline (1987)] to convert the radiance field of each scene into a mesh, from which we extract point clouds by surface... sampling. While the placement of individual foreground objects may vary, a substantial overlap of point clouds belonging to the static background scene is sufficient for achieving inter-scene pose alignment through point cloud registration algorithms. Based on the derived relative poses, we utilize the method outlined in Sec. 3.2 to obtain the fused clean background scene, and from which we similarly extract the point cloud corresponding to the background scene. By comparing the differences between the point clouds of each scene and the clean background scene, we can obtain all the point clouds that belong to the foreground objects. Subsequently, a point cloud clustering algorithm allows us to obtain point clouds that belong to each individual foreground object separately. Finally, for each foreground object across scenes, the Hungarian matching algorithm and point cloud registration techniques are used to determine their correspondences and relative poses $ \{T_{i,j}\} $. Here $ j \in \{1, ..., M\} $ denotes the object index. ### 3.3 Visibility Field: Quantifying Visibility in Radiance Field Visibility is an important measure to utilize the visible part information across scenes. To quantify the visibility information in the radiance field, we propose visibility field, a volumetric representation that maps a 3D position to a scalar-valued visibility: \[ v = v(x) : \mathbb{R}^3 \rightarrow [0, 1]. \] The proposed visibility $ v(x) \in [0, 1] $ is defined as the proportion of cameras that can observe point $ x $ among all training cameras. Formally, we say that $ x $ can be observed by the camera $ C_l $ means that (1) the projection of $ x $ falls within the interior of the image plane and (2) there is no occlusion between $ x $ and the camera position $ o_l \in \mathbb{R}^3 $. For the condition (2), we use the pseudo-depth of the radiance field to determine whether there is occlusion. Specifically, we cast a ray from the camera position $ o_l $ to $ x $ and compute the pseudo-depth $ \hat{d}_l $ by volume rendering, and then compare it with the distance from the camera position to the point $ d_l = \\|x - o_l\\| $. For camera $ C_l $, we use a binary-valued function $ V_l(x) \in \{0, 1\} $ to denote whether $ x $ can be observed by that camera. If $ d_l < \hat{d}_l $, this means that $ x $ is between the object surface and the camera position, thus there is no occlusion, i.e., $ V_l(x) = 1 $, otherwise $ V_l(x) = 0 $. Considering all training cameras, the visibility of position $ x $ can be computed as: \[ v(x) = \frac{1}{L} \sum_{l=1}^{L} V_l(x). \] Note that the visibility field is independent for each scene and we compute it for all scenes. ### 3.4 Visibility-Aware Rendering We propose visibility-aware rendering, a method that obtains occlusion-free rendering by comparing the visibility across multiple scenes. We take the rendering of the clean background to explain its basic idea (Fig. 3 (Left)). Background scene. The first step in comparing scenes is to set them under the same coordinates. Recall that we have obtained the relative pose between the background scenes through point cloud registration in Sec. 3.2. Without loss of generality, we take the first scene ($ i = 1 $) as a reference and align the scenes $ i = 2, ..., N $ to the coordinate system of the first scene. Given that all scenes are aligned to the reference scene, we introduce visibility-aware rendering for the background scene. An illustration is shown in Fig. 3 (Left). For the sample point $ x $ in volume rendering, the proposed visibility-aware rendering, in addition to color and density, also computes the visibility of the sample point in each scene, i.e., $ \{c_i(x)\} $, $ \{\sigma_i(x)\} $ and $ \{v_i(x)\} $. The idea of visibility-aware rendering is simple: blend the color and density... in each scene according to visibility. Using $ w_i $ which satisfy $ \sum_i w_i = 1 $ to denote the weight of each scene, blended radiance color and volume density can be written as: \[ \hat{c}(x) = \sum_{i=1}^{N} w_i(x)c_i(x), \quad \hat{\sigma}(x) = \sum_{i=1}^{N} w_i(x)\sigma_i(x), \] (3) where $ w_i $ is a weight function calculated from visibility that satisfies $ \sum_i w_i = 1 $: \[ w_i(x) = \frac{v_i^p(x)}{\sum_{i=1}^{N} v_i^p(x)}. \] (4) Here $ p $ is a hyper-parameter that controls the weights, the larger $ p $ is, the greater the contribution of the scene with the highest visibility; and when $ p \to \infty $, the above is equivalent to the max-selection function. For simplicity, Fig. 3 shows the case based on max-selection. The motivation behind the above calculation is to select the parts with less occlusion (i.e., higher visibility) in each scene, and fuse them into the final scene. As a result, volume rendering of the blended radiance color and volume density obtained from Eq. 3 yields a clean background scene, as shown in Fig. 2(b). Foreground objects. The core idea of visibility-aware rendering for 360° objects is basically the same as that for background scene. Similarly, we take the coordinate systems of the foreground objects in scene $ i = 1 $ as a reference. For foreground object $ j $, we denote the position and view direction of the sampled point under the reference coordinate system as $ x_j, d_j $, respectively. For scenes of $ i \geq 2 $, we use the computed object poses to calculate the corresponding positions $ x_{i,j} $ and view directions $ d_{i,j} $ in each scene as: \[ x_{i,j} = R_{i,j}x_j + t_{i,j}, \quad d_{i,j} = R_{i,j}d_j, \] (5) where $ R_{i,j} $ and $ t_{i,j} $ are rotation and translation terms of object poses $ T_{i,j} \in \text{SE}(3) $ obtained from Sec. 3.2. Here, $ x_{i,j} $ in fact represents the corresponding point of $ x_j $ in the coordinate system of scene $ i $, as shown in Fig. 3 (Left). Then, the blended radiance color and volume density of Eq. 3 for foreground object rendering can be rewritten as: \[ \hat{c}(x_j) = \sum_{i=1}^{N} w_i(x_{i,j})c_i(x_{i,j}), \quad \hat{\sigma}(x_j) = \sum_{i=1}^{N} w_i(x_{i,j})\sigma_i(x_{i,j}). \] (6) Volume rendering the fusion results obtained from Eq. 6 yields occlusion-free 360° foreground objects, as shown in Fig. 2(d). Our proposed visibility-aware rendering, despite its simplicity, reasonably achieves the visible part fusion of radiance fields. It’s noteworthy that our method share the same paradigm for both background/foreground parts, accomplishing the reconstruction of a clean background scene and 360° foreground objects. ### 4 EXPERIMENTS #### 4.1 DATASETS Blender synthetic datasets. We created synthetic datasets using Blender Community (2018). The tables used as background are taken from free 3D models available online. For the foreground objects, we use 3D models from Google Scanned Objects dataset Downs et al. (2022), which contains 360° scans of common household objects. We created $ N = 3 $ sets of scenes, in which foreground objects are under different placement to ensure that every part of the table and object surfaces is visible in at least one scene. We applied different lighting conditions (uniform light, spotlight, etc.) to test the effectiveness of our method in different environments. We randomly sample camera positions on the hemisphere and render $ L = 100 $ images for the radiance field optimization. Examples of the synthetic scenes are shown in Fig. 4(a). Figure 4: Results on Blender synthetic datasets. For pairwise comparisons of foreground objects, the top-left image shows the rendering result of the proposed method, while the bottom-right image shows the reference image (ground truth). Figure 5: Results on real capture datasets. (c) and (d) are obtained using the proposed method. Real capture datasets. We created real-world capture datasets to demonstrate the effectiveness of our approach on real datasets. We utilized YCB objects (Calli et al., 2015) and created $N = 2$ (for bleach cleanser) or $N = 3$ (for power drill) scenes by placing objects in different configurations. For each scene, we captured a video around it and extracted 60-80 frames, then applied COLMAP (Schönberger & Frahm, 2016; Schönberger et al., 2016) to obtain the corresponding camera parameters registration. Examples of the real capture scenes are shown in Fig. 5(a). 4.2 Results We show the qualitative results of Blender synthetic datasets and real capture datasets on Fig. 4 and Fig. 5 respectively. With multiple input scenes, our method can automatically recover a clean background scene and 360° foreground objects. 4.3 Ablation studies Impact of light conditions. We created scenes under three distinct lighting conditions: outdoor environment mapping, indoor environment mapping, and a single point light source. For the background, despite obtaining an acceptable clean background, there exists a certain degree of artifacts due to the presence of shadows or reflections. For objects, certain discontinuities arise due to abrupt changes in lighting conditions or the inherent glossiness of the objects (e.g., pink pig). Additionally, the fused results show a lack of glossiness, suggesting that even for significantly different lighting conditions or glossy objects, our fusion method can neutralize the view-dependent term, yielding appearances close to diffuse colors, which is typically desirable in the context of 360° object reconstruction. Impact of weight function. We study the impact of the hyper-parameter $p$ (exponent of visibility in weight function Eq. 4). We observe that when $p$ is relatively small (i.e., $p = 4$), the results tend to blend color and density more smoothly for each scene. The appearance changes smoothly for foreground objects, however, it also blends the background and non-background (i.e., empty space) parts around them, resulting in a cloud-like artifact. When $p \to \infty$, visibility aware rendering selects the color and density of the scene with the highest visibility as the result of the fusion, and such a max-selection brings discontinuous changes, resulting in sharp changes in the appearance. We observe that $p = 8 \sim 32$ is the appropriate value to obtain continuous appearance interpolation without cloud-like artifacts. Impact of the number of scenes. Right image shows the impact on the results for different numbers of scenes $N$. For $N = 1$, we manually compute the bounding boxes for the background and foreground objects from the point cloud and rendered only the original scene within them. In this case, the invisible parts are not optimized, leading to artifacts in the rendering results. For $N > 2$, we can observe that the proposed ViFu can recover a clean background and 360° objects from multiple scene observations. It is noteworthy that, as the number of scenes $N$ increases, the rendered results appear to become brighter. We speculate that this is due to sufficient lighting generally implying less occlusion, which means higher visibility and thus the corresponding parts are fused into the final output with higher weight. Based on this observation, we assume that as the number of scenes and the variety of object poses increase, the rendering results of objects will be close to those rendered in a $360^\circ$ spherical lighting environment. Empirically, we observe that for some objects, artifacts appear when $N = 2$ (red arrow in the figure). We attribute this to the difficulty in accurately segmenting the foreground object if a certain part is in contact with the background part in both two scenes, making it hard to determine whether it belongs to the foreground object or background scene. A simple solution is to expose the part of the common contact when placing objects in the third scene. Although an adhoc placement may achieve the plausible rendering at $N = 2$ (as the cleanser scenes in Fig. 5), we observe that $N = 3$ scenes can achieve reasonable segmentation in most cases and is therefore a recommended choice. Impact of variations in object placement. To validate the robustness of our approach to variations in object placement, we extended our evaluation beyond the 3 original scenes presented in Fig. 1. We created an additional 5 scenes, where objects were randomly placed. We randomly selected 3 from 8 scenes for each fusion experiment. We have the following observations: under the assumptions, our method consistently produced satisfactory results. However, some difficulties arise: (1) when the orientations of foreground objects in the three selected scenes are highly repetitive (e.g., bottoms consistently facing downward and thus not observable), artifacts are still present in the rendered regions that lacked sufficient observation. This issue arises because our method relies on fusing information from the available scenes and thus cannot predict unseen part. (2) when foreground objects are placed very closely within a scene, our use of a naive point cloud segmentation approach may potentially fail, leading to misalignment and bad fusion results. Effective segmentation of closely spaced objects typically requires prior knowledge of the objects. Incorporating pre-trained point cloud segmentation models or segmentation masks as additional information can assist in segmenting challenging objects, thereby facilitating successful scene fusion. ### 5 LIMITATIONS AND FUTURE WORK There are a few limitations that need to be addressed in future work: First, our method does not explicitly consider the lighting condition. For static background scene, as the lighting conditions are basically the same, reasonable rendering results can be obtained. The above ablation study for light conditions demonstrates that our proposed weighted fusion method can mitigate the impact of certain lighting variations to some extent. However, the rendering results of objects under extreme lighting conditions may still be unsatisfactory (e.g., the fusion result of “construction vehicles” at the top of Fig. 4 shows an abrupt change in appearance, where spot light illumination is used). Incorporating some of the current approaches for disentangling light conditions might be a promising direction for future work. Second, our fusion method assumes that we can obtain accurate scene segmentation and pose alignment. In most cases, the aforementioned point cloud-based approach can achieve sufficiently accurate segmentation and alignment. However, some challenging scenarios may arise, such as failures in segmentation due to close object placement (as mentioned in Sec. 2.3), or failures in pose alignment due to oversimple object shapes. However, the essence of these problems can all be viewed as fundamentally challenging issues in point cloud segmentation or registration, which has been a longstanding challenging problem in the field of computer vision. For these special cases, using additional masks or richer point cloud features (e.g., color information) might help mitigate the aforementioned challenges. ### 6 CONCLUSION We have presented ViFu, a method for recovering clean background scene and $360^\circ$ foreground objects from multiple scene observations. We leverage point cloud-based approaches to achieve background and foreground alignment and use the difference between scenes to obtain a background/foreground segmentation. We propose visibility field, a volumetric representation to quantify the visibility of a scene, and introduce visibility-aware rendering to fuse the more visible parts of multiple scenes. Our experiments on both synthetic and real datasets demonstrate the effectiveness of our approach. While our approach is the first to focus on radiance fields for multiple scenes, there are some remaining issues, such as not considering lighting conditions, which we plan to address in future work. REFERENCES Kaxlamangla S. 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hv2lUWKyrJ	- In relation to the previous point, the paper does not discuss and analyses the effects of chances in capacity and sample complexity due to the relational bottleneck. There's an argument to be made that some of the observed results might be due at least in part to a statistical effect due to the different inductive biases and regularization coming from the bottleneck	RELATIONAL CONSTRAINTS ON NEURAL NETWORKS REPRODUCE HUMAN BIASES TOWARDS ABSTRACT GEOMETRIC REGULARITY Anonymous authors Paper under double-blind review ABSTRACT Uniquely among primates, humans possess a remarkable capacity to recognize and manipulate abstract structure in the service of task goals across a broad range of behaviors. One illustration of this is in the visual perception of geometric forms. Studies have shown a uniquely human bias toward geometric regularity, with task performance enhanced for more regular and symmetric forms compared to their geometrically irregular counterparts. Such studies conclude that this behavior implies the existence of discrete symbolic structure in human mental representations, and that replicating such behavior in neural network architectures will require mechanisms for symbolic processing. In this study, we argue that human biases towards geometric regularity can be reproduced in neural networks, without explicitly providing them with symbolic machinery, by augmenting them with an architectural constraint that enables the system to discover and manipulate relational structure. When trained with the appropriate curriculum, this model exhibits human-like biases towards symmetry and regularity in two distinct tasks involving abstract geometric reasoning. Our findings indicate that neural networks, when equipped with the necessary training objectives and architectural elements, can exhibit human-like regularity biases and generalization. This approach provides insights into the neural mechanisms underlying geometric reasoning and offers an alternative to prevailing symbolic “Language of Thought” models in this domain. 1 INTRODUCTION Humans have the amazing capability of building useful abstractions that can capture regularities in the external world. Understanding what is responsible for this special feature of human intelligence relative to other animals is a longstanding goal in cognitive science (Penn et al., 2008; Berwick & Chomsky, 2016). One domain in which cognitive scientists have observed this “human singularity” (Dehaene et al., 2022) is in geometric reasoning: early Homo sapiens 100,000 years ago were able to produce structured abstract geometric shapes and drawings on caves (Henshilwood et al., 2011), whereas similar behaviors have not been observed for non-human primates despite years of human contact (Saito et al., 2014). Such observations, as well as rigorous empirical work (e.g., Sablé-Meyer et al., 2021, 2022) have led some cognitive scientists to conclude that human mental representations uniquely contain discrete domain-specific symbols that are recursively and compositionally combined to produce abstractions that support the capacity for generalization that is characteristic of human behavior (Dehaene et al., 2022). A corollary of this hypothesis is that artificial neural networks cannot, in principle, produce human-like intelligence without the exogenous addition of explicit symbolic machinery and/or representations (Dehaene, 2021; Marcus, 2020). Indeed, empirical work in this domain has shown that explicitly symbolic models fit human behavior better than standard neural networks (Sablé-Meyer et al., 2021). This has led to the view, by some, that symbolic “Language of Thought” models are the best models of humans’ mental representations (Quilty-Dunn et al., 2022). However, the fact that human behavior, or their inductive biases, may be described effectively with abstract symbolic processing does not necessarily imply that their internal representations are based on discrete symbols (Griffiths et al., 2023). Consequently, there may be other forms of representations, such as the continuous vector spaces of neural networks, that could, under the right conditions, produce this behavior without explicit symbolic machinery (McCoy et al., 2018). In the present work, we provide an existence proof of this point by revisiting recent empirical cognitive science work showing humans’ regularity biases towards abstract geometric concepts (Sablé-Meyer et al., 2021; 2022). We show that standard neural networks augmented with a simple constraint that favors relational information processing can replicate human generalization and regularity biases without needing to build in explicit symbolic machinery. Specifically, we implement an architectural motif, known as the relational bottleneck (Webb et al., 2023a), that allows networks to exploit relations between objects rather than the attributes of individual objects. We focus on the results of two studies. The first is the work of Sablé-Meyer et al. (2022), in which humans were tested on a standard working memory task, Delayed-Match to Sample (DMTS), using image stimuli sampled from a generative Language of Thought model of geometric concepts. The second is a study by Sablé-Meyer et al. (2021), in which humans and non-human primates were tested on a version of the Oddball Detection task, a simple categorization paradigm in which participants identify a deviant stimulus in a group of quadrilateral stimuli. We show that a standard neural network, augmented with a relational bottleneck and trained with an appropriately designed curriculum using the same data as the studies by Sablé-Meyer et al. (2021) and Sablé-Meyer et al. (2022), exhibited human-like biases for abstract geometric regularity. These results offer an alternative interpretation of such biases, suggesting that with the appropriate inductive biases and curriculum neural networks can exhibit features associated with the capacity for symbolic processing without the need to hardcode the network with symbolic representations and/or mechanisms. 2 Historical Background and Related Work For decades, cognitive scientists and AI researchers have embraced two main approaches to building intelligent systems: symbolic models (Fodor, 1975) and neural networks (Rumelhart & McClelland, 1986). Fodor (1975) proposed the “Language of Thought” (LoT) hypothesis: that higher-order cognition in humans is the product of recursive combinations of pre-existing, conceptual primitives, analogous to the way in which sentences in a language are constructed from simpler elements. Symbolic models are well-suited to naturally embed the abstract, structured knowledge humans possess, such as causal theories (Goodman et al., 2011) or hierarchical motor programs that draw handwritten characters (Lake et al., 2015). Neural networks, on the other hand, emphasize emergence of these abstract concepts purely from data within completely unstructured, distributed representations (McClelland et al., 2010). Despite the incredible recent success of neural networks in machine learning, cognitive scientists have hypothesized that their systematic failure at generalizing out of their training distribution comes from a failure to embed the kinds of abstract structural knowledge that can exist in symbolic models (Lake et al., 2017; Marcus, 2003). Recent work has suggested that these capacities may emerge through learning in neural networks that implement relational reasoning. Relational reasoning involves abstracting over the details of particular stimuli or domains and extracting more general forms of structure that are broadly useful for capturing regularities in the external world (Gentner, 1983; Holyoak, 2012). This can be accomplished in neural networks by introducing an architectural inductive bias: the relational bottleneck (Webb et al., 2023a). The general principle of the relational bottleneck is that some components of the network are restricted to operating on relations over representations rather than the representations themselves (Webb et al., 2020; 2023b; Mondal et al., 2023). For example, the network might be constrained to use the similarity or distance between two embeddings rather than the embeddings themselves. Critically, unlike many hybrid neuro-symbolic models (Plate, 1995; Touretzky, 1990; Mao et al., 2019) the relational bottleneck does not introduce pre-specified symbolic primitives or any explicit mechanisms for symbolic processing, relying instead on the emergence of abstract concepts within unstructured, distributed representations. The motivation of the relational bottleneck is similar to that of other works that have built neural network architectures more sensitive to relational reasoning (Barrett et al., 2018; Santoro et al., 2017; Shanahan et al., 2020). The Language of Thought (LoT) approach has been applied to a variety of domains in cognitive science, including learning causal theories (Goodman et al., 2011), representations of numbers (Piantadosi et al., 2012), and logical concepts (Piantadosi et al., 2016). However, geometry has recently emerged as one of the domains in which the strongest arguments in favor of this kind of representation have been made (Sablé-Meyer et al., 2021; 2022; Dehaene et al., 2022). This setting is also a natural one in which to explore the predictions of neural network models, as geometric stimuli can be presented directly to models in the form of images. In the remainder of the paper, we present a detailed analysis of two of the studies that have been held up as providing support for the LoT approach, demonstrating how neural networks that are constrained to focus on relations are capable of reproducing the key patterns in human behavior. 3 TRAINING NEURAL NETWORKS ON A LANGUAGE OF THOUGHT FOR GEOMETRY 3.1 BACKGROUND Sablé-Meyer et al. (2022) presented a study designed to test the Language of Thought hypothesis in the setting of geometry. The study was based on a model of geometric concept learning also developed by Sablé-Meyer et al. (2022). This model framed concept learning as program induction within the DreamCoder framework (Ellis et al., 2021). A base programming language was defined such that programs can be written to generate geometric shapes, where motor programs that draw geometric shapes are generated through recursive combination of symbolic primitives within a Domain Specific Language (DSL, Fig. 1A). The DSL contains motor primitives, such as tracing a particular curve and changing direction, as well as primitives to recursively combine subprograms such as Concat (concatenate two subprograms together) and Repeat (repeat a subprogram n times). These symbolic programs can then be rendered into images such as the ones seen in Fig. 1. Since each image has an underlying program, the minimum description length (MDL; Ellis et al., 2021) of the program was used to model the psychological complexity of the corresponding geometric pattern. Abstract geometric patterns were generated by this symbolic LoT model (Fig. 1A) and used as stimuli in a standard working memory task, based on a Delayed-Match to Sample (DMTS, Fig. 1B) paradigm. In this task, human participants were instructed to memorize a geometric stimulus. Following the memorization phase, participants were presented with a blank screen for two seconds. Subsequently, they were shown six option stimuli, among which one matched the original stimulus they had memorized (the target image), while the remaining five were distractors. The objective for participants was to accurately select the image they had seen during the encoding phase and avoid choosing any of the distractor images. In preceding work (Sablé-Meyer et al., 2021 discussed further in the next section), the authors suggested that perception of abstract geometric stimuli can be based on two systems: a high-level, Figure 2: DMTS Task Architecture Implementation (A) Target and delay images are passed through a pretrained CNN encoder (Kubilius et al., 2019). The outputs of the encoder are passed to an LSTM, producing memory embeddings that correspond to participants’ working memory representation of the initial target stimulus when performing the DMTS task. Each of the choice images are encoded using the same CNN encoder. (B) In the baseline model (left), the memory embeddings are simply concatenated to the choice embeddings and passed to a fully connected layer that produces the logits classifying the target image. In the relational bottleneck model (right), the embeddings are used to compute the similarity between each choice embedding and the memory embedding, and these similarities are used to produce the logits. general-purpose symbolic system, supposedly only available to humans; and a lower-level, domain-specific shape invariant object recognition system, available to both humans and non-human primates, that can be modeled by a standard Convolutional Neural Network (CNN) model of object recognition in the brain (specifically, the Ventral Visual Stream; Kubilius et al., 2019). To study the first system, Sable-Meyer et al. (2022) chose distractor stimuli that were maximally similar to the target image based on hidden representations of a pre-trained CNN model of the Ventral Visual system (CorNet; Kubilius et al., 2019) and the average grey-level of the image. Even with difficult distractors, humans excelled at the task, with error rates as low as 1.82%. ### 3.2 Neural Network Modeling We trained two Recurrent Neural Networks (RNNs; one baseline and one implementing a relational bottleneck) on this task, using the LoT model of Sable-Meyer et al. (2022) to generate a large training corpus of geometric stimuli and holding out the specific stimuli used in the human experiments for the test set. Stimuli were encoded by a CNN encoder, which was comprised of a pre-trained CNN model (CorNet; Kubilius et al., 2019). On each trial, an encoded representation of the stimulus was used as the input to an LSTM (Fig. 2A), followed by encoded representations of three additional timesteps-worth of blank input images (Fig. 2A). The resulting output embedding of the LSTM corresponds to the working memory content of the human participants during choice time (“Memory Embedding”, see Fig. 2A). The model is subsequently presented with the choice images (Fig. 2). We implemented two types of decision processes to classify the target image out of the six choice images (one target, five distractors). One of these was a standard baseline model, and the other was augmented with a relational bottleneck (Webb et al., 2023a; Fig. 2B). For the baseline model, the embeddings of the six choice stimuli, along with the memory embedding, were concatenated and simultaneously fed into a standard feedforward layer that was used to classify the target image. For the Relational Bottleneck model, the cosine similarity between the memory embedding and each choice embedding was computed; those similarities were then used to produce the prediction of the target image. This restricted the model to processing the relations between its memory of the target image and the choice stimulus, without “intrusion” from any stimulus-specific attributes of the choice stimuli. During training, distractors were chosen randomly, but during testing, we used the exact same trials that were presented to human participants. --- 1The delay period for the human experiments was 2 seconds, while the average stimulus presentation time was around 1.2s. Given this, we believe three timesteps makes the task for the networks at least as hard if not harder than the human task. Figure 3: DMTS Results (A) Training accuracy across epochs of baseline and relational bottleneck models. Both models eventually reach near-perfect accuracy. (B) Results on tasks held out from model training that were taken directly from the human trials in Sablé-Meyer et al. (2022). The black bar denotes chance performance, while the green bar denotes mean human performance. Error bars are 95% confidence intervals over model training seeds. The Relational Bottleneck model performs much better out of distribution. (C) We increased the delay period from 3 timesteps to 20. Though both models suffer in performance, the Relational Bottleneck model still performs much better. in the empirical study Sablé-Meyer et al. (2022), in which difficult distractors were chosen based on similarity to pretrained CorNet representations (Kubilius et al., 2019) and average grey-levels. ### 3.3 RESULTS We tested both implementations of the model on the exact same trials given to human participants in Sablé-Meyer et al. (2022). Performance of the baseline model was well below human performance (Fig. 3B). However, the relational bottleneck model generalized extremely well to the test set, performing significantly better than the baseline model ($p < 0.001$) and approximating the performance of human participants. In addition, it handled longer delay periods substantially better than the baseline model (Fig. 3C), demonstrating its ability to maintain abstract representations of these geometric stimuli more robustly through the delay period. The results suggest that it is possible to achieve human-like performance on this task with a neural network model augmented by a simple constraint that favors learning relations, without imbuing the model with any explicit symbolic representations. The training corpus we used had stimuli containing very rich geometric abstractions (see Fig. 1A and Fig. 7). While our results suggest that inclusion of a relational bottleneck may be necessary to produce representations that support out-of-distribution generalization, it is not clear whether it is sufficient even in cases of a more impoverished training corpus. Previous work has shown that a rich training data distribution can also contribute to such generalization (Chan et al., 2022). To address this, we tested whether the relational bottleneck would produce similar human-like performance when training on a relatively more restricted training corpus. ### 4 HUMAN-LIKE VS MONKEY-LIKE PROCESSING OF QUADRILATERAL STIMULI #### 4.1 BACKGROUND Inspired by early anthropological work investigating abstract geometric concepts in cave drawings and behavioral research comparing geometric reasoning in humans and non-human primates, Sablé Meyer et al. (2021) compared diverse human groups (varying in education, cultural background, and age) to non-human primates on a simple oddball discrimination task. Participants were shown a set of five reference shapes and one “oddball” shape and prompted to identify the oddball (Fig. 4). The reference shapes were generated based on basic geometric regularities: parallel lines, equal sides, equal angles, and right angles. Reference shapes consisted of 11 types of quadrilaterals varying in their geometric regularity, from squares (most regular) to random quadrilaterals containing no parallel lines, right angles, or equal angles/sides (least regular) (Fig. 4B). In each trial, five different versions of the same reference shape (e.g., a square) were shown in different sizes and orientations. The oddball shape was a modified version of the reference shape, in which the lower right vertex was moved such that it violated the regularity of the original reference shape (e.g., moving the lower right vertex of a trapezoid such that it no longer has parallel sides). Fig. 4A shows an example trial. Sablé-Meyer et al. (2021) found that humans, across many different ages, cultures, and education levels, are naturally sensitive to these geometric regularities (right angles, parallelism, symmetry, etc.) whereas non-human primates are not. Specifically, they found that human performance is best on the Oddball task for the most regular shapes, and systematically decreases as shapes become more irregular. Conversely, non-human primates perform well above chance, but they perform worse than humans overall and, critically, show no influence of geometric regularity (Fig. 4B). To address this pattern of findings, Sablé-Meyer et al. (2021) implemented two computational models: a symbolic model and a neural network model. The symbolic model implemented oddball identification using an explicitly symbolic feature space constructed from the shapes’ discrete geometric properties. The neural network model was a pretrained CNN model of the Ventral Visual stream (CORNet; Kubilius et al., 2019). Sablé-Meyer et al. (2021) found that the symbolic model fit the human performance of their Oddball task significantly better than the neural network model, and in particular it captured the effect of increasing error with increasing geometric irregularity. Conversely, the neural network model fit the monkey behavior better, exhibiting no systematic relationship with the level of geometric regularity (Fig. 4B). They interpreted this as evidence that the human sensitivity to geometric regularity requires the presence of unique symbolic representations that are absent in both neural networks and non-human primates. 4.2 Neural Network Modeling Here, we show that a neural network trained on the same stimuli used by Sablé-Meyer et al. (2021), and provided with a relational bottleneck, exhibits the sensitivity of geometric regularity observed in humans, without the explicit specification of discrete symbolic representations. Sablé-Meyer et al. (2021) additionally re-trained CorNet on an object recognition task on the quadrilateral stimuli and reported that re-training CorNet on this task did not affect their results. We started with the ResNet CNN architecture$^3$ but we modified this architecture to directly compute the Oddball judgements end-to-end using the relational bottleneck, using the method described in Kerg et al.$(2022)$ (Fig. 5A). Specifically, a $6 \times 6$ cosine similarity matrix is computed across each of the six stimuli, and the similarity matrix is fed into a feedforward layer that produces an Oddball decision. This structure forces the model to make decisions based on the relations between choice stimuli rather than the attributes of an individual choice stimulus. We pretrained the CNN using one of two contrastive objectives (Fig. 5B): Standard and Geometric. The Standard objective was based on SimCLR$(Chen et al., 2020)$. Specifically, simple random rotations and scaling were applied to individual quadrilateral images, and then the CNN was trained to push its representations of those images together, to be more similar (i.e., less distant) to their augmented counterparts, and pull its representations of different quadrilateral images apart, to be more dissimilar (i.e., more distant) from each other. The Geometric objective used the geometric features utilized in Sablé-Meyer et al.$(2021)$ as the feature space over which to define distances. Those geometric features were binary vectors corresponding to the presence or absence of equal angles, equal sides, parallel lines, and right angles of the quadrilateral. During training, this effectively pushed quadrilaterals with similar geometric features together and pulled quadrilaterals with different geometric features apart. This allowed us to train the network to exhibit the same abstractions defined by the geometric features without building in the geometric features themselves. During testing and inference, the geometric features were completely discarded. This is similar to previous work instilling human biases into neural network agents$(Kumar et al., 2022)$, in which the --- $^3$Note that although Sablé-Meyer et al.$(2021)$ run their main experiments with CorNet, they show in their supplement that ResNet produces the same monkey-like behavioral signatures as CorNet. Figure 6: Oddball Task Results (A) Mean error rates over the 11 types of quadrilaterals for each type of network. The Geometric pre-trained network showed a significant trend between error rate and geometric regularity ($p < .001$), while the Standard (SimCLR) pre-trained network did not ($p = 0.99$). (B) We correlated error rates across quadrilaterals for each model with the corresponding error rates of humans and monkeys. Geometric pre-training of quadrilaterals led to human-like error patterns, whereas SimCLR pre-training led to more monkey-like error patterns. Error bars are 95% confidence intervals across different model training runs. tabula rasa neural networks that were co-trained with symbolic information exhibited human biases without explicitly implementing any symbolic representations. ### 4.3 Results Similar to the effect observed in the study by Sablé-Meyer et al. (2022) discussed in the previous section, the geometric regularity effect observed for humans in Sablé-Meyer et al. (2021) was an inverse relationship between geometric regularity and error rate (see green plot in Fig. 4B). For example, humans performed best on the most regular shapes, such as squares and rectangles. This regularity effect was again absent in the monkey error rates (Fig. 4B). Following Sablé-Meyer et al. (2021), we show, for each of our networks, the error rates for quadrilaterals sorted by geometric regularity and how well they match human and monkey error rates (Fig. 6). The Geometric pre-trained model showed a strong fit to human behavior ($r = 0.72$) and a significant effect of geometric regularity ($p < 0.001$; Fig. 6). The Standard (SimCLR) pre-trained model, however, showed a strong fit to monkey behavior ($r = 0.70$), but not to human behavior ($r = 0.005$), nor did they show the geometric regularity effect ($p = 0.99$; Fig. 6). This indicates that, although the relational bottleneck was necessary, it was not sufficient on its own to reproduce human behavior on this task. However, coupled with the appropriate training, it was able to reproduce the pattern of results observed for human behavior in Sablé-Meyer et al. (2021). These results suggest that, with the appropriate structural biases and training experience, it is possible for neural network to learn representations that exhibit human-like biases in the geometric oddball task without explicitly imposing symbolic representations on the network. ### 5 Discussion A prevailing theory in cognitive science is that abstractions that support strong generalization reflect the presence of symbolic systems innate in humans that may be absent in animals (Fodor, 1975; Quilty-Dunn et al., 2022; Dehaene et al., 2022). Along similar lines, it has been argued that, without explicitly imbuing neural networks with such capabilities, they will not be able to exhibit the same cognitive flexibility as humans (Marcus, 2020; Dehaene, 2021). Empirical findings in the studies by Sablé-Meyer et al. (2021) and Sablé-Meyer et al. (2022) have been offered in support of these conjectures. Here, we provide evidence to the contrary, showing how the introduction of a simple, neurally plausible relational inductive bias, coupled with the appropriate training experiences, is sufficient to reproduce behavior consistent with the formation of abstract representations in neural networks. The domain of the empirical work we re-examine involves the visual perception of geometric patterns (Sablé-Meyer et al., 2021; 2022). Sablé-Meyer et al. (2022) show that humans are adept at processing geometric patterns, using a delayed-match-to-sample working memory task with stimuli sampled from a generative probabilistic program induction model (Ellis et al., 2021). We trained two types of RNN models on this task: a baseline model and a model with a relational bottleneck that is biased to focus on relations between stimuli to classify the target image. Consistent with the claims of Sablé-Meyer et al. (2022), a baseline model does not reach human-level performance out of its training distribution. However, a model with the relational bottleneck does indeed reach human performance on the test set, showing that a simple constraint that favors learning relations can allow neural networks to achieve human-level performance on this task. Sablé-Meyer et al. (2021) further show that humans are sensitive to geometric regularity when performing a visual perception task, the Oddball task, using quadrilateral stimuli, whereas non-human primates and standard CNNs (Kubilius et al., 2019) are not. Here, we found that even with a relational bottleneck, a network trained with a standard contrastive learning objective produced the same monkey-like behavior observed from the CNN trained by Sablé-Meyer et al. (2021). However, when trained contrastively on distances produced by geometric features, the model did reproduce the human geometric regularity effect. One important difference between the two tasks is that, the delayed match to sample task (Sablé-Meyer et al., 2022) used reaction times (RTs) to show the geometric regularity effect in humans, whereas the oddball task (Sablé-Meyer et al., 2021) used error rates. This is because error rates in the former were near zero, and therefore RTs were required to observe significant effects. One limitation of our study is that we did not construct an analogue to human RTs for our RNN models. Instead, we used out-of-training-distribution accuracy as the main performance metric. In the Oddball task (Sable-Meyer et al., 2021), where human error rates were higher, we were able to conduct a more direct comparison, where we observed a clear correspondence between human (or monkey) behavior and our models. A further difference between the two experiments is that the model of the Oddball task required geometric contrastive pre-training to match human performance (producing monkey-like behavior without this objective). We believe this is because the dataset used in the Delayed Match-to-Sample task features a richer distribution of stimuli (Fig. 7) sampled from a Bayesian program induction model (DreamCoder; Ellis et al., 2021). Building a training distribution of samples from such a Bayesian model has an interpretation of effectively distilling the Bayesian model’s rich prior into a neural network (McCoy & Griffiths, 2023). In contrast, the Oddball dataset consisted of a relatively simple set of 11 quadrilaterals, which may not be sufficiently diverse to allow the network to extract more abstract representations (see Chan et al., 2022 for a similar argument about how the richness of training data affects the post-training capabilities of Large Language Models). Our work provides evidence that simple modifications to standard neural networks are sufficient to reproduce human behavior on tasks used in cognitive science to showcase allegedly unique human capabilities. It may be possible that such geometric regularity biases can be instilled in neural networks by other methods. For example, previous work has shown Vision Transformer architectures, like humans, are biased more towards shapes than textures (Tuli et al., 2021). In general, we suggest that human-like behavior and abstractions can be instilled in neural networks using a variety of strategies, including through specialized architectures (Webb et al., 2023a; 2020), specialized loss functions/training curricula (Kumar et al., 2022; Kepple et al., 2022), and/or highly rich data distributions (McCoy & Griffiths, 2023; Chan et al., 2022). 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9DvDRTTdlu	It would be beneficial if the author could provide more details about the adopted latent feature vectors. Specifically, it would be helpful to know if this representation is strong enough for the task of editing, and how the feature maps are visualized since they appear visually similar to original RGB images.	ED-NeRF: Efficient Text-Guided Editing of 3D Scene With Latent Space NeRF Jangho Park2; Gihyun Kwon3, Jong Chul Ye1,2,3 Kim Jaechul Graduate School of AI1, Robotics Program2, Department of Bio and Brain Engineering3, KAIST {jhg1234,cyclomon,jong.ye}@kaist.ac.kr Abstract Recently, there has been a significant advancement in text-to-image diffusion models, leading to groundbreaking performance in 2D image generation. These advancements have been extended to 3D models, enabling the generation of novel 3D objects from textual descriptions. This has evolved into NeRF editing methods, which allow the manipulation of existing 3D objects through textual conditioning. However, existing NeRF editing techniques have faced limitations in their performance due to slow training speeds and the use of loss functions that do not adequately consider editing. To address this, here we present a novel 3D NeRF editing approach dubbed ED-NeRF by successfully embedding real-world scenes into the latent space of the latent diffusion model (LDM) through a unique refinement layer. This approach enables us to obtain a NeRF backbone that is not only faster but also more amenable to editing compared to traditional image space NeRF editing. Furthermore, we propose an improved loss function tailored for editing by migrating the delta denoising score (DDS) distillation loss, originally used in 2D image editing to the three-dimensional domain. This novel loss function surpasses the well-known score distillation sampling (SDS) loss in terms of suitability for editing purposes. Our experimental results demonstrate that ED-NeRF achieves faster editing speed while producing improved output quality compared to state-of-the-art 3D editing models. Code and rendering results are available at our project page. 1 Introduction In recent years, the development of neural implicit representation for embedding three-dimensional images in neural networks has seen remarkable progress. This advancement has made it possible to render images from all angles using only a limited set of training viewpoints. Starting with the seminar work known as the Neural Radiance Field (NeRF) (Mildenhall et al., 2021), which trained radiance fields using a simple MLP network, various improved techniques (Barron et al., 2021; Reiser et al., 2021; Müller et al., 2022) based on advanced network architectures or modified encoding have been proposed. Alternatively, several methods (Sun et al., 2022; Fridovich-Keil et al., 2022; Karnewar et al., 2022; Chen et al., 2022) proposed to directly optimize voxel points serving as sources for rendering, bypassing the traditional approach of encapsulating all information within implicit networks. These methods have gained prominence for their ability to train radiance fields in a remarkably short time. In addition to representing existing 2D image data in the 3D space, recent research has explored expanded approaches for generating entirely novel 3D objects. With the emergence of text-to-image embedding models like CLIP (Radford et al., 2021), various methods have been proposed to train implicit networks that can generate new objects solely from text prompts (Jain et al., 2022). This trend has been accelerated with the advent of text-to-image diffusion generation models such as Stable Diffusion (Rombach et al., 2022), particularly through the score distillation sampling (SDS) (Poole et al., 2022) which conveys the representation of the text-to-image model to NeRF model. equally contributed https://jhq1234.github.io/ed-nerf.github.io/ Figure 1: Qualitative results of our method. ED-NeRF successfully edited 3D scenes with given target text prompts while preserving the original object structure and background regions. However, the challenge of editing pre-trained 3D implicit networks according to specific conditions still remains as an open problem due to the constraints of tasks: maintaining the integrity of the original 3D images while making desired modifications. As an initial work, several approaches (Wang et al., 2022; 2023a) tried to edit the pre-trained NeRF models based on text conditions, utilizing the pre-trained CLIP model to fine-tune the parameters of NeRF models. Nevertheless, these methods exhibit notable weaknesses, including the performance limitations of the CLIP model itself and the need for rendering high-resolution images during training, which results in significant time consumption. Recently, several editing methods proposed to leverage the enhanced expressiveness of text-to-image diffusion models such as Stable Diffusion. Some methods (Sella et al., 2023) proposed to directly employ the score distillation sampling method, with additional regularizations. However, these methods suffer from significant time consumption and instability in generation performance due to the requirement of full-resolution rendering in the training stage and limitations of the score distillation loss itself. Other alternative approaches (Haque et al., 2023) proposed to directly manipulate the training images of NeRF using text-guided image translation models. This method aims to enable the generation of 3D images corresponding to text conditions. However, it suffers from a significant drawback in terms of training time, as it requires periodic translation of training images during the training process. To address these challenges, we are interested in developing a novel NeRF editing method to efficiently and effectively edit 3D scenes using only text prompts. To achieve this, we enable NeRF to operate directly in the NeRF latent space, similar to Latent-NeRF (Metzer et al., 2023), which helps reduce time and computational costs. However, naively rendering the latent feature of real-world scenes directly with NeRF may lead to a significant drop in view synthesis performance due to the lack of geometric consistency in the latent space. To tackle this issue, we conduct an analysis of the latent generation process and propose a novel refinement layer to enhance performance based on the analysis. Furthermore, to solve the drawback of the existing SDS-based method in editing, we propose a new sampling strategy by extending Delta Denoising Score (DDS) (Hertz et al., 2023), a 2D image editing technique based on score distillation sampling, into the 3D domain. This extension allows us to achieve high-performance editing capabilities while keeping computational costs affordable, even with large Diffusion Models such as Stable Diffusion. Given the superior editing proficiency of our approach, we’ve named it ED-NeRF (EDiting NeRF). 2 RELATED WORK Starting from the Neural Radiance Field (NeRF) [Mildenhall et al., 2021], there have been approaches to represent three-dimensional scenes in neural fields. However, due to the slow training speed, several approaches tried to improve the performance by modifying the network architecture or training strategy [Barron et al., 2021; Müller et al., 2022; Reiser et al., 2021]. Several methods without relying on neural networks showed great performance in accelerating. These include a method for optimizing the voxel fields [Sun et al., 2022; Fridovich-Keil et al., 2022; Chen et al., 2022; Karnewar et al., 2022], or decomposing the components of field representation. Based on the success of these techniques, methods for generating ‘novel’ 3D scenes have been proposed. Especially with the emergence of the text-to-image embedding model of CLIP [Radford et al., 2021], DreamField [Jain et al., 2022] leveraged CLIP to train the NeRF model for novel 3D object synthesis. Recently, the performance of the text-to-image diffusion model enabled remarkable improvement in 3D generation. Starting from DreamFusion [Poole et al., 2022], several methods [Metzer et al., 2023; Liu et al., 2023b; Xu et al., 2023] showed impactful results using the diffusion-based prior. However, these methods are limited to generating ‘novel’ 3D objects and, therefore cannot be applied to our case of NeRF-editing which tries to modify the existing 3D scenes according to the given conditions. Compared to the novel object generation, NeRF editing is still not an explored field, due to the complexity of the task. As a basic work, several methods focused on color or geometric editing [Yuan et al., 2022; Liu et al., 2021; Kuang et al., 2023]. Other works tried style transfer or appearance transfer on 3D neural fields [Zhang et al., 2022; Liu et al., 2023a; Bao et al., 2023] and showed promising results. With incorporating the CLIP model, several approaches [Wang et al., 2022, 2023a; Song et al., 2023] tried to modify the pre-trained NeRF towards the given text conditions. Although the results show pleasing results, the method still has limitations in detailed expression due to the limitation of CLIP model itself. Similar to the novel scene generation case, the development of text-to-image diffusion models brought significant improvement in the editing field. Starting from Score Distillation Sampling method proposed in DreamFusion, Voxel-e tried to edit the pre-trained voxel fields with regularization [Sella et al., 2023]. As an alternative method, InstructNerf2Nerf [Haque et al., 2023] proposed to directly leverage 2D image translation models for changing the attribute of 2D images for NeRF training. However, these methods still have limitations due to excessive training time or unstable editing from loss functions. To address the above problems, we propose an efficient method of editing with novel latent space NeRF training and improved edit-friendly loss functions. 3 METHODS Figure 2 provides an overview of training ED-NeRF. First, we optimize NeRF in the latent space of Stable Diffusion. To do this, we encode all images using a pre-trained Variational Autoencoder (VAE) to obtain the feature vectors and guide NeRF to predict these feature vectors directly. Also, we introduce an additional refinement layer, which enhances the novel view synthesis performance of NeRF (Fig. 2(a)). At the inference stage, we can render a natural image by latent NeRF via decoding rendered latent map (Fig. 2(b)). At the editing phase, by utilizing DDS, we adjust the parameters of both NeRF and the refinement process to align the 3D scene with the provided target text (Figure 3). The detailed pipeline for this approach is outlined in the following sections. 3.1 ED-NeRF for 3D Scene Editing NeRF [Mildenhall et al., 2021] uses MLPs to predict density $\sigma$ and color $c$ for a given 3D point coordinate $x = (x, y, z)$ and view direction $d$. Through positional encoding $\gamma(\cdot)$, $x$ and $d$ are mapped into high-frequency vectors, and then fed into the neural network of NeRF, resulting in two outputs: density $\sigma \in \mathbb{R}$ and color $c \in \mathbb{R}^3$. $$ (c, \sigma) = F_\theta(\gamma(x), \gamma(d)) $$ Through volume rendering Eq. (2), NeRF predicts the pixel color along the camera ray $r(t) = o + td$, with $t$ representing the depth within the range $[t_{near}, t_{far}]$, $o$ stands for the camera position, Figure 2: Overall pipeline of training and inference stage. (a) We optimize ED-NeRF in the latent space, supervised by source latent. Naively matching NeRF to a latent feature synthesis map during optimization can degrade view synthesis quality. (b) Inspired by the embedding process of Stable Diffusion, we integrated additional ResNet blocks and self-attention layers as a refinement layer. (c) All 3D scenes are decoded from the Decoder when ED-NeRF renders a novel view feature map. and $ d $ represents the view direction: \[ \hat{C}(r) = \int_{t_n}^{t_f} T(t)\sigma(r(t))c(r(t),d)dt, \text{ where } T(t) = \exp \left( -\int_{t_n}^{t} \sigma(r(s))ds \right). \] Optimizing NeRF to render the latent feature value of the latent diffusion model offers several advantages in text-guided 3D generation. These advantages include a reduced training burden due to the decreased dimensionality of the space, and enhanced editability for the NeRF model, as the rendered outputs can be directly employed as input for the latent diffusion models. The concept of migrating NeRF to the latent space is first proposed by Latent-NeRF [Metzer et al., 2023], in which the NeRF is directly trained with the latent feature rather than RGB color. Therefore it can render a 3D scene without the encoding process during optimization when using the latent diffusion model as semantic knowledge prior. However, this work exclusively focuses on generating ‘virtual’ 3D assets without supervision, making it unsuitable for real-world scenes. Thus, ED-NeRF is realized based on a novel latent NeRF training pipeline for synthesizing real-world scenes in the latent space. As depicted in Figure 2, when a real-world image dataset $ I $ contains multi-view images $ I = \{I_i\}_{i=1}^N $, we can encode all images to the latent space of Stable Diffusion via encoder to obtain the feature: $ z^i = E(I^i) \in \mathbb{R}^{64 \times 64 \times 4} $. After embedding all images, we can use the latent feature maps $ z := \{z^i\}_{i=1}^N $ as label data set for ED-NeRF training using the loss function: \[ L_{rec} = \sum_{r \in R} \\| Z^i(r) - \hat{Z}^i(r) \\|^2 \] where $ Z^i $ denotes the pixel latent value of the latent $ z^i $ and $ \hat{Z}^i(r) $ is rendered by the volume rendering equation: \[ \hat{Z}^i(r) = \int_{t_n}^{t_f} T(t)\sigma(\gamma(t))f_z(r(t),d)dt, \text{ where } T(t) = \exp \left( -\int_{t_n}^{t} \sigma(r(s))ds \right). \] where $ f_z \in \mathbb{R}^4 $ denotes the predicted feature value by the neural network, taking $ \gamma(x) $ and $ \gamma(d) $ as input: \[ (f_z, \sigma) = F_\theta(\gamma(x), \gamma(d)) \] By minimizing the loss Eq. (3) to update the parameters of the neural network $ F_\theta $, we obtain a novel ED-NeRF model optimized in the latent space of the Stable Diffusion. 3.2 Refinement Layer based on Latent Feature Analysis When naively matching the latent generated by Eq. (3), we observed that the reconstruction performance significantly deteriorated. In addressing this issue, we analyzed the Encoder $ E $ and Decoder $ D $ of Stable Diffusion and discovered the following insight in the process: Figure 3: Expanding DDS into 3D for ED-NeRF editing. Pretrained ED-NeRF renders the target latent feature map, and a scheduler of the denoising model perturbs it to the sampled time step. Concurrently, the scheduler adds noise to the source latent using the same time step. Each of them is fed into the denoising model, and the DDS is determined by subtracting two different SDS scores. In combination with a binary mask, masked DDS guides NeRF in the intended direction of the target prompt without causing unintended deformations. 1) The encoder and decoder consist of ResNet blocks and self-attention layers. Therefore during the process of mapping the image to the latent space and forming a feature map, pixel values exhibit interference between each other, primarily due to ResNet and self-attention layers. Thus the latent and image pixels are not directly aligned. 2) When NeRF renders a single pixel value from the latent feature map, each ray independently passes through an MLP to determine the pixel value of the feature map. Therefore, the feature value rendered by NeRF for a single pixel is determined without interactions with other pixels. Based on this analysis, we find that the reason for the deformed reconstruction performance of the latent NeRF lies in the inconsideration of the interactions mentioned above. Therefore, we aim to incorporate the interactions among pixels introduced by the ResNet and self-attention layers into the ED-NeRF rendering stage. Fortunately, in the Encoder and Decoder of Stable Diffusion, the embedded feature maps pass through self-attention layers at the same dimension, allowing us to concatenate two attention layers straightly. Taking advantage of this, we can design a refinement layer $F_\phi(\cdot)$ as shown in Figure 2, without dimension change of input and output vector. Let $\tilde{Z}^i$ as the pixel latent vector of the refined feature map $\tilde{z}^i$, where formed from $\tilde{z}^i = F_\phi(\hat{z}^i)$. Therefore, we can design a refined reconstruction loss function as follows: $$L_{ref} = \sum_{r \in R} \\| Z^i(r) - \tilde{Z}^i(r) \\|^2 , \text{where } \tilde{z}^i = F_\phi(\hat{z}^i)$$ Ultimately, we can formulate total training loss as the sum of the refinement loss $L_{ref}$ and reconstruction loss $L_{rec}$, as follows. $$L_{tot} = \lambda_{rec} L_{rec} + \lambda_{ref} L_{ref}$$ We update NeRF and refinement layer concurrently denoted as $F_\theta$ and $F_\phi$ by minimizing total loss $L_{tot}$ to reconstruct latent vectors in various views. To ensure stable learning, training with $\lambda_{rec}$ set to 1.0 and $\lambda_{ref}$ set to 0.1 during the initial stages of training. Beyond a specific iteration threshold, we set it to 0 to encourage the refinement layer to focus more on matching the latent representations. 3.3 Editing ED-NeRF via Delta Denoising Score After optimizing ED-NeRF in the latent space, it is possible to directly employ the latent diffusion model to update ED-NeRF parameter via rendered latent map $z$ in the direction of the target text prompt $y_{trg}$. The most well-known method for text-guided NeRF update is Score Distillation Sampling (SDS), which directly transfers the score estimation output as a gradient of NeRF training: $$\nabla_\theta L_{SDS}(z, y_{trg}, \epsilon, t) = \omega(t)(e_\psi(z_t, y_{trg}, t) - \epsilon)\frac{\partial z_t}{\partial \theta}$$ However, in our NeRF editing case, the updating rule for SDS often shows several problems including color saturation and mode-seeking (Wang et al., 2023b). We conjecture that the problem originated from the properties of score estimation itself. Since the target noise $\epsilon$ is pure Gaussian, the score difference is not aware of any prior knowledge of source images. Therefore the generated outputs are just the replacement of hallucinated objects without consideration of source NeRF. To solve the problem of SDS, we focus on the recently proposed 2D editing method of Delta Denoising Score (DDS) (Hertz et al., 2023). The major difference between SDS and DDS is that the distilled score is the difference between the denoising scores from target and source. As shown in Eq. (9), DDS can be formed as a difference between two SDS scores conditioned on two different text prompts: $$\nabla_\theta \mathcal{L}_{DDS} = \nabla_\theta \mathcal{L}_{SDS}(\hat{z}, y_{trg}) - \nabla_\theta \mathcal{L}_{SDS}(z, y_{src}),$$ where $z$ is source latent, $\hat{z}$ is rendered target latent, $y_{trg}$ represents the target text embedding, $y_{src}$ represents the reference text embedding. DDS guides the optimized latent towards the target prompt from the source prompt without the influence of the pure noise component, therefore it can easily edit 2D images. We aim to extend this manipulation capability of DDS to 3D space as shown in Fig. 3. As we already have embedded source latent $z^i$ for the $i$-th camera pose, we can directly use them as source components of DDS. To fine-tune the model, we render the edited output $\hat{z}^i$ which is also rendered from the $i$-th camera pose. With the paired latents, we add the same sampled noise $\epsilon_t$ with the noise scale of timestep $t$ to both source and edited latents so that we obtain noisy latent $\tilde{z}_t^i$, $\tilde{z}_t$. Then we apply the diffusion model to obtain estimated score outputs from noisy latents using different text conditions for source and edited images. As in Eq. (9), we can use the difference between the two outputs as a gradient for updating the NeRF parameters. In this step, we simultaneously train the NeRF parameters $\theta$ with refinement parameters $\phi$ as it showed better editing quality. Therefore with the random $i$-th camera pose, our 3D DDS is formulated as: $$\nabla_{\theta,\phi} \mathcal{L}_{DDS} = \nabla_{\theta,\phi} \mathcal{L}_{SDS}(\hat{z}^i, y_{trg}) - \nabla_{\theta,\phi} \mathcal{L}_{SDS}(z^i, y_{src}).$$ Although the DDS formulation improves the performance, using vanilla DDS leads to excessive changes in unwanted areas and inconsistency between two different scenes. Therefore, we propose an additional binary mask for utilizing DDS in 3D images. The objective function that combines the binary mask $M$ and DDS is as follows: $$\nabla_{\theta,\phi} \mathcal{L}_{MDDS} = M \cdot (\nabla_{\theta,\phi} \mathcal{L}_{DDS}),$$ where $\cdot$ denotes the pixel-wise multiplication and $M$ is the conditional binary mask of the specific region of the target prompt to change. This mask is generated by utilizing off-the-shelf text prompt segmentation models such as CLIPSeg (Lüddeke & Ecker, 2022) and SAM (Kirilov et al., 2023) to segment the target region by a text prompt. Despite the use of a binary mask, masked DDS loss $\nabla \mathcal{L}_{MDDS}$ update all parameters of NeRF potentially affecting even undesired areas. As a result, solely depending on the masked DDS loss may inadvertently result in alterations beyond the mask boundaries. Hence, we introduce an additional reconstruction loss as follows to mitigate undesired deformations beyond the mask. $$\mathcal{L}_{Mrec} = \lambda_{im} \cdot M \cdot \mathcal{L}_{rtot} + \lambda_{om} \cdot (1 - M) \cdot \mathcal{L}_{rtot}.$$ Finally, the total editing loss is as follows: $$\mathcal{L}_{tot} = \mathcal{L}_{MDDS} + \mathcal{L}_{Mrec}$$ By suppressing undesired alterations through the use of the masked reconstruction loss $\mathcal{L}_{Mrec}$, our total editing objective function updates NeRF and refinement layer $F_\theta$ and $F_\phi$, ensuring NeRF renders novel views in accordance with the desired text conditions. 4 EXPERIMENTAL RESULTS 4.1 BASELINE METHODS To comprehensively evaluate the performance of our method, we perform comparative experiments comparing it to state-of-the-art methods. As CLIP-based text guidance editing methods, we used Figure 4: Comparison with baseline models. ED-NeRF demonstrates outstanding performance in effectively altering specific objects compared to other models. Baseline methods often failed to maintain the region beyond the target objects and failed to guide the model towards the target text. CLIP-NeRF (Wang et al., 2022) and NeRF-ART (Wang et al., 2023a). CLIP-NeRF encodes the images rendered by NeRF to the CLIP embedding space, allowing it to transform the images according to the text condition. As an improved method, NeRF-ART trains NeRF with various regularization functions to ensure that CLIP-edited NeRF can preserve the structure of the original NeRF. For fair experiments, we re-implemented the methods to TensoRF backbone, referencing the official source codes. For the diffusion-based editing, we chose Masked SDS (Poole et al., 2022) and InstructNeRF2NeRF (Haque et al., 2023) as the methods that target local editing. In the masked SDS setting, we fine-tuned the pre-trained NeRF with applying basic SDS loss only to the masked regions so that the NeRF model is locally edited. InstructNeRF2NeRF (Haque et al., 2023) leverages the powerful generation capabilities of diffusion models to sequentially modify the entire dataset to align with text conditions and use the modified dataset as a new source for NeRF training. We utilized a database comprising real-world images, including LLFF (Mildenhall et al., 2019) and IBRNet (Wang et al., 2021) datasets, as well as the human face dataset employed in Instruction-NeRF2NeRF (Haque et al., 2023). ### 4.2 Qualitative Results Text-guided editing of 3D scenes. As shown in Figure 1, our method shows its capability to edit various image types with different textual contexts. Specifically, it is possible to achieve the effective transformation of specific objects without affecting other parts. Our baseline method InstructNeRF2NeRF (Haque et al., 2023) shows decent results with high consistency between images and Figure 5: Ablation studies. (a) If we only use DDS loss, the model fails to maintain the attribute of untargeted regions and often fails to reflect text conditions. (b) If we do not use masked reconstruction regularization, again the regions beyond the target objects are excessively changed. (c) If we remove the mask from DDS, unwanted artifacts occur in untargeted regions. (d) With removing the proposed refinement layer, the results become blurry as the backbone NeRF cannot fully embed real-world scenes. Our proposed setting can modify a specific region in a 3D scene and follow the target word without causing unwanted deformations. text conditions, as well as view consistency across scenes. However, it faces limitations in accurately transforming the specific objects to match text conditions and may introduce undesired image alterations beyond the specific objects. In Masked SDS, the edited output fails to reflect the structure of the original NeRF scene and shows unwanted artifacts. In the case of NeRF-ART, the entire image is embedded into the CLIP space, and it does not inherently recognize and modify only specific objects. Therefore, it exhibits limitations in recognizing and altering specific objects. CLIP-NeRF also encodes the images rendered by NeRF to the CLIP embedding space, allowing it to transform the images according to the text condition. However, its performance falls short when it comes to altering specific parts in a similar manner. On the other hand, our ED-NeRF exhibited powerful abilities in editing 3D scenes by specifying certain parts through text, surpassing other models. It not only excelled in changing objects but also demonstrated the capability to faithfully follow and modify areas that are not objects, such as the ground, in accordance with the text condition. 4.3 Quantitative Results CLIP Directional Score. In order to quantitatively measure the editing performance, we show the comparison results using CLIP Directional scores [Gal et al., 2021]. The CLIP Directional score quantifies the alignment between textual caption modifications and corresponding image alterations. We rendered multiple view images from NeRF and measured the average score over images. When compared to baseline methods, our model obtained the best similarity scores. The result indicates that our edited NeRF accurately reflects the target text conditions. User Study. In order to further measure the perceptual preference of human subjects, we conducted an additional user study. For the study, we rendered images from edited NeRF using 5 different scenes from LLFF and iBRnet. We gathered feedback from 20 subjects aged between their 20s and 40s. Each participant was presented with randomly selected multi-view renderings from our model and baselines and provided feedback through a preference scoring survey. We set the minimum score as 1 and the maximum score is 5, and users can choose the score among 5 options: 1-very low, 2-low, 3-middle, 4-high, 5-very high. To measure the performance of editing, we asked two questions for each sample: 1) Does the image reflect the target text condition? (Text score) 2) Does the model accurately edit the target object? (Preservation). 3) Does the 3D scenes preserve view consistency? (view consistency). In Table 1, we show the user study results. Compared with baseline methods, our method showed the best score in text score and preservation, and second best in view consistency. Overall, ours outperformed the baseline models in perceptual quality. \| Metrics \| CLIP-NeRF \| NeRF-Art \| Instruct N2N \| Mask SDS \| Ours \| \|-------------------------\|-------------\|------------\|--------------\|----------\|----------\| \| CLIP Direction Score ↑ \| 0.1648 \| 0.1947 \| 0.2053 \| 0.1409 \| 0.2265 \| \| Text score ↑ \| 2.56 \| 3.20 \| 3.29 \| 3.14 \| 3.88 \| \| Preservation ↑ \| 2.30 \| 2.97 \| 3.08 \| 2.76 \| 4.09 \| \| View consistency ↑ \| 3.21 \| 3.79 \| 3.28 \| 3.56 \| 3.64 \| Table 1: Quantitative Comparison. We compared the text-image similarity between the target text and rendered output from edited NeRF (CLIP Directional Score). Also, we show the user study results in three categories: text-guidance score, source preservation score, and view consistency. The results show that ours shows improved perceptual score among baseline models. \| Metrics \| CLIP-NeRF* \| NeRF-Art* \| Instruct N2N \| Ours \| \|-------------------------\|-------------\|------------\|--------------\|----------\| \| Fine-tuning time ↓ \| 6min \| 15min \| 90min \| 14min \| \| GPU Memory ↓ \| 17GB \| 18GB \| 15GB \| 8GB \| Table 2: Efficiency Comparison. We compared the efficiency of ours and baseline methods in terms of training time and Memory usage. Our method can enable faster editing with lower memory usage. For CLIP-NeRF and NeRF-Art, the models are fine-tuned in lower resolution (252×189), due to excessive memory consumption. Instruct N2N and ours are fine-tuned in 512x512 resolution. Efficiency comparison. To compare the editing efficiency, we check the fine-tuning time and memory usage in Table 2. Among baselines, our method uses the lowest memory for training, with a much lower time compared to Instruct Nerf2Nerf. GPU memory usage and training time are measured based on the RTX 3090. In the baselines of CLIP-Nerf and Nerf-art, we experiment with using downsized images as higher resolution editing causes GPU memory overflow. For Instruct Nerf2Nerf, the fine-tuning process requires excessive time as it periodically translates the training images. Considering that our method shows outperforming quality in text-guided editing, our proposed scheme is efficient in both memory and time aspects. When comparing the time for the pre-training NeRF backbone model, we did not include a comparison since all baselines and ours take almost the same amount of time (about 10 minutes). More Details and comparisons on pre-training time are in our Appendix. 4.4 Ablation Studies To evaluate our proposed components, we conducted an ablations study in Figure 6. (a) If we only use DDS, the method fails to maintain the untargeted regions with artifacts, even failing in training (e.g., fossil). (b) If we do not use regularization $ L_{\text{Mrec}} $, the edited results show the target text attribute, but again the regions beyond the target objects are severely degraded. (c) When we remove mask guidance on DDS, (w/o $ L_{\text{MDDS}} $), unwanted minor deformations occur due to the gradient of DDS affecting the regions outside the mask. (d) When we remove our refinement layer, the results show blurry outputs, which indicate that latent NeRF is not accurately trained. When we utilize all the components we proposed, we can reliably transform the 3D scene into the desired target object while preserving the original structure source NeRF. In the Appendix, we included an ablation study on our proposed refinement layer for novel-view reconstruction tasks. 5 Conclusion In this paper, we introduced a novel ED-NeRF method optimized in the latent space. By enabling NeRF to directly predict latent features, it efficiently harnesses the text-guided score function of latent diffusion models without the need for an encoder. By doing so, our approach is able to effectively reduce computation costs and address the burden of previous models that required rendering at full resolution to utilize the diffusion model. We extended the strong 2D image editing performance of DDS to the 3D scene and also introduced a new loss function based on the mask. As a result, it showed high performance in object-specific editing, a task that previous models struggled with. We experimented with our proposed approach across various datasets, and as a result, it demonstrated strong adherence to text prompts in diverse scenes without undesired deformation. 6 ETHICS AND REPRODUCIBILITY STATEMENTS Ethics statement. ED-NeRF enables efficient and accurate text-guided NeRF editing, which can be applied to various applications. However, our ED-NeRF can be used for creating obscene objects which may cause the users to feel offended. In order to prevent the possible side effects, we can use a filtered diffusion model that does not contain malicious text conditions. Reproducibility statement. We detailed our experimental process and parameter settings in our Appendix. We will upload our source code to an anonymous repository for reproduction. 7 ACKNOWLEDGEMENT This research was supported by National Research foundation of Korea(NRF) (RS-2023-00262527), Field-oriented Technology Development Project for Customs Administration through National Research Foundation of Korea(NRF) funded by the Ministry of Science & ICT and Korea Customs Service(NRF-2021M3I1A1097938), the Korea Medical Device Development Fund grant funded by the Korea government (the Ministry of Science and ICT, the Ministry of Trade, Industry and Energy, the Ministry of Health & Welfare, the Ministry of Food and Drug Safety) (Project Number: 1711137899, KMDF_PR_20200901_0015) and Culture, Sports and Tourism R&D Program through the Korea Creative Content Agency grant funded by the Ministry of Culture, Sports and Tourism in 2023 REFERENCES Chong Bao, Yinda Zhang, Bangbang Yang, Tianxing Fan, Zesong Yang, Hujun Bao, Guofeng Zhang, and Zhaopeng Cui. Sine: Semantic-driven image-based nerf editing with prior-guided editing field. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 20919–20929, 2023. Jonathan T Barron, Ben Mildenhall, Matthew Tancik, Peter Hedman, Ricardo Martin-Brualla, and Pratul P Srinivasan. Mip-nerf: A multiscale representation for anti-aliasing neural radiance fields. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 5855–5864, 2021. Anpei Chen, Zexiang Xu, Andreas Geiger, Jingyi Yu, and Hao Su. Tensorf: Tensorial radiance fields. In European Conference on Computer Vision, pp. 333–350. Springer, 2022. Sara Fridovich-Keil, Alex Yu, Matthew Tancik, Qinhong Chen, Benjamin Recht, and Angjoo Kanazawa. Plenoxels: Radiance fields without neural networks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 5501–5510, 2022. Rinon Gal, Or Patashnik, Haggai Maron, Gal Chechik, and Daniel Cohen-Or. Stylegan-nada: Clip-guided domain adaptation of image generators. arXiv preprint arXiv:2108.00946, 2021. Ayaan Haque, Matthew Tancik, Alexei A Efros, Aleksander Holynski, and Angjoo Kanazawa. Instruct-nerf2nerf: Editing 3d scenes with instructions. arXiv preprint arXiv:2303.12789, 2023. Amir Hertz, Kfir Aberman, and Daniel Cohen-Or. Delta denoising score. arXiv preprint arXiv:2304.07090, 2023. Ajay Jain, Ben Mildenhall, Jonathan T Barron, Pieter Abbeel, and Ben Poole. Zero-shot text-guided object generation with dream fields. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 867–876, 2022. Animesh Karnewar, Tobias Ritschel, Oliver Wang, and Niloy Mitra. Relu fields: The little non-linearity that could. In ACM SIGGRAPH 2022 Conference Proceedings, pp. 1–9, 2022. Alexander Kirillov, Eric Mintun, Nikhila Ravi, Hanzi Mao, Chloe Rolland, Laura Gustafson, Tete Xiao, Spencer Whitehead, Alexander C Berg, Wan-Yen Lo, et al. 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aZH1dM3GOX	I understand that your approach enhances the diversity of feature representation which in turn leads to a good exploration of the state space. How do you ensure the balance between exploration and exploitation? Do you rely on the PPO and SAC to do a standard exploration/exploitation with the diverse learnt features?	Multi-Task Reinforcement Learning with Mixture of Orthogonal Experts Ahmed Hendawy\textsuperscript{1,2}, Jan Peters\textsuperscript{1,2,3,4}, Carlo D’Eramo\textsuperscript{1,2,5} \textsuperscript{1}Department of Computer Science, TU Darmstadt, Germany \textsuperscript{2}Hessian Center for Artificial Intelligence (Hessian.ai), Germany \textsuperscript{3}Center for Cognitive Science, TU Darmstadt, Germany \textsuperscript{4}German Research Center for AI (DFKI), Systems AI for Robot Learning \textsuperscript{5}Center for Artificial Intelligence and Data Science, University of Würzburg, Germany Abstract Multi-Task Reinforcement Learning (MTRL) tackles the long-standing problem of endowing agents with skills that generalize across a variety of problems. To this end, sharing representations plays a fundamental role in capturing both unique and common characteristics of the tasks. Tasks may exhibit similarities in terms of skills, objects, or physical properties while leveraging their representations eases the achievement of a universal policy. Nevertheless, the pursuit of learning a shared set of diverse representations is still an open challenge. In this paper, we introduce a novel approach for representation learning in MTRL that encapsulates common structures among the tasks using orthogonal representations to promote diversity. Our method, named Mixture Of Orthogonal Experts (MOORE), leverages a Gram-Schmidt process to shape a shared subspace of representations generated by a mixture of experts. When task-specific information is provided, MOORE generates relevant representations from this shared subspace. We assess the effectiveness of our approach on two MTRL benchmarks, namely MiniGrid and MetaWorld, showing that MOORE surpasses related baselines and establishes a new state-of-the-art result on MetaWorld.\footnote{The code is available at \url{https://github.com/AhmedMagdyHendawy/MOORE}.} 1 Introduction Reinforcement Learning (RL) has shown outstanding achievements in a wide array of decision-making problems, including Atari games (Mnih et al., 2013; Hessel et al., 2018a), board games (Silver et al., 2016; 2017), high-dimensional continuous control (Schulman et al., 2015; 2017; Haarnoja et al., 2018), and robot manipulation (Yu et al., 2019). Despite the success of RL, generalizing the learned policy to a broader set of related tasks remains an open challenge. Multi-Task Reinforcement Learning (MTRL) is introduced to scale up the RL framework, holding the promise of enabling learning a universal policy capable of addressing multiple tasks concurrently. To this end, sharing knowledge is vital in MTRL (Teh et al., 2017; D’Eramo et al., 2020; Sodhani et al., 2021; Sun et al., 2022). However, deciding upon the kind of knowledge to share and the set of tasks to share that knowledge is crucial for designing an efficient MTRL algorithm. Human beings exhibit remarkable adaptability across a multitude of tasks by mastering some essential skills as well as having the intuition of physical laws. Similarly, MTRL can benefit from sharing representations that capture unique and diverse properties across multiple tasks, easing the learning of an effective policy. Recently, sharing compositional knowledge (Devin et al., 2017; Calandriello et al., 2014; Sodhani et al., 2021; Sun et al., 2022) has shown potential as an effective form of knowledge transfer in MTRL. For example, Devin et al. (2017) investigate knowledge transfer challenges between distinct robots and tasks by sharing a modular policy structure. This approach leverages task-specific and robot-specific modules, enabling effective transfer of knowledge. Nevertheless, this approach requires manual intervention to determine the allocation of responsibilities for each module, given some prior knowledge. In contrast, we aim for an end-to-end approach that implicitly learns... and shares the prominent components of the tasks for acquiring a universal policy. Furthermore, CARE (Sodhani et al., 2021) adopt a different strategy by focusing on learning representations of different skills and objects encountered by the tasks by utilizing context information. However, there is no inherent guarantee of achieving diversity among the learned representations. In this work, our goal is to ensure the diversity of the learned representations to maximize the representation capacity and avoid collapsing to similar representations. Consequently, we propose a novel approach for representation learning in MTRL to share a set of representations that capture unique and common properties shared by all the tasks. To ensure the richness and diversity of these shared representations, our approach solves a constrained optimization problem that orthogonalizes the representations generated by a mixture of experts via the application of the Gram-Schmidt process, thus favoring dissimilarity between the representations. Hence, we name our approach, Mixture Of ORthogonal Experts (MOORE). Notably, the orthogonal representations act as bases that span a subspace of representations leveraged by all tasks where task-relevant properties can be interpolated. More formally, we show that these orthogonal representations are a set of orthogonal vectors belonging to a particular Riemannian manifold where the inner product is defined, known as Stiefel manifold (James, 1977). Interestingly, the Stiefel manifold has recently drawn substantial attention within the field of machine learning (Ozay & Okatani, 2016; Huang et al., 2018a; Li et al., 2019; Chaudhry et al., 2020). For example, several works focus on enhancing the generalization and stability of neural networks by solving an optimization problem to learn parameters in the Stiefel manifold. Another line of work aims to reduce the redundancy of the learned features by forcing the weights to inhabit the Stiefel manifold. Additionally, Chaudhry et al. (2020) propose a continual learning method that forces each task to learn in a different subspace, thus reducing task interference through orthogonalizing the weights. In this paper, our objective is to ensure diversity among the shared representations across tasks by imposing a constraint that forces these representations to exist within the Stiefel manifold. Thus, we aim to leverage the extracted representations, in combination with deep RL algorithms, to enhance the generalization capabilities of MTRL policies. In the following, we provide a rigorous mathematical formulation of the MTRL problem, inspired by Sodhani et al. (2021), where latent representations belong to the Stiefel manifold. Then, we devise our MOORE approach for obtaining orthogonal task representations through the application of a Gram-Schmidt process on the latent features extracted from a mixture of experts. We empirically validate MOORE on two widely used and challenging MTRL problems, namely MiniGrid (Chevalier-Boisvert et al., 2023) and Meta-World (Yu et al., 2019), comparing to recent baselines for MTRL. Remarkably, MOORE establishes a new state-of-the-art performance on the MetaWorld MT10 and MT50 collections of tasks. To recap, the contribution of this work is twofold: (i) We propose a mathematical formulation, named Stiefel Contextual Markov Decision Process (SC-MDP), that defines the MTRL problem where the state is encoded in the Stiefel manifold through a mapping function. (ii) We devise a novel representation learning method for Multi-Task Reinforcement Learning that leverages a modular structure of the shared representations to capture common components across multiple tasks. Our approach, named MOORE, learns a mixture of orthogonal experts by encouraging diversity through the orthogonality of their corresponding representations. Our approach outperforms related baselines and achieves state-of-the-art results on the MetaWorld benchmark. 2 PRELIMINARIES A Markov Decision Process (MDP) (Bellman, 1957; Puterman, 1995) is a tuple $ \mathcal{M} = < S, A, P, r, \rho, \gamma > $, where $ S $ is the state space, $ A $ is the action space, $ P : S \times A \rightarrow S $ is the transition distribution where $ P(s'\|s,a) $ is the probability of reaching $ s' $ when being in state $ s $ and performing action $ a $, $ r : S \times A \rightarrow \mathbb{R} $ is the reward function, $ \rho $ is the initial state distribution, and $ \gamma \in (0, 1] $ is the discount factor. A policy $ \pi $ maps each state $ s $ to a probability distribution over the action space $ A $. The goal of RL is to learn a policy that maximizes the expected cumulative discounted return $ J(\pi) = \mathbb{E}_\pi[\sum_{t=0}^{\infty} \gamma^t r(s_t, a_t)] $. We parameterize the policy $ \pi_\theta(a_t\|s_t) $ and optimize the parameters $ \theta $ to maximize $ J(\pi_\theta) = J(\theta) $. 2.1 MULTI-TASK REINFORCEMENT LEARNING In MTRL, the agent interacts with different tasks $ \tau \in \mathcal{T} $, where each task $ \tau $ is a different MDP $ \mathcal{M}^\tau = < S^\tau, A^\tau, P^\tau, r^\tau, \rho^\tau, \gamma^\tau > $. The goal of MTRL is to learn a single policy $ \pi $ that maximizes the expected accumulated discounted return averaged across all tasks $J(\theta) = \sum_{\tau} J_{\tau}(\theta)$. Tasks can differ in one or more components of the MDP. A class of problems in MTRL assumes only a change in the reward function $r^{\tau}$. This can be exemplified by a navigation task where the agent learns to reach multiple goal positions or a robotic manipulation task where the object’s position changes. In this class, the goal position is usually augmented to the state representation. Besides the reward function, a bigger set of problems deals with changes in other components. In this category, tasks access a subset of the state space $S^{\tau}$, while the true state space $S$ is unknown. For example, learning a universal policy that performs multiple manipulation tasks interacting with different objects (Yu et al., 2019). Task information should be provided either in the form of task ID (e.g., one-hot vector) or metadata, e.g., task description (Sodhani et al., 2021). Following Sodhani et al. (2021), we define the MTRL problem as a Block Contextual Markov Decision Process (BC-MDP). It is defined by 5-tuple $\langle C, S, A, \gamma, M \rangle$ where $C$ is the context space, $S$ is the true state space, $A$ is the action space, while $M$ is a mapping function that provides the task-specific MDP components given the context $c \in C$, $M(c) = \{r^c, P^c, S^c, \rho^c\}$. As of now, we refer to the task $\tau$ and its components by the context parameter denoted as $c$. 3 RELATED WORKS Sharing knowledge among tasks is a key benefit of MTRL over single-task learning, as broadly analyzed by several works that propose disparate ways to leverage the relations between tasks (D’Eramo et al., 2020; Sodhani et al., 2021; Sun et al., 2022; Calandriello et al., 2014; Devin et al., 2017; Yang et al., 2020). Among many, D’Eramo et al. (2020) establish a theoretical benefit of MTRL over single-task learning as the number of tasks increases, and Teh et al. (2017) learn individual policies while sharing a prior among tasks. However, naive sharing may exhibit negative transfer since not all knowledge should be shared by all tasks. An interesting line of work investigates the task interference issue in MTRL from the gradient perspective. For example, Yu et al. (2020) propose a gradient projection method where each task’s gradient is projected to an orthogonal direction of the others. Nevertheless, these approaches are sensitive to the high variance of the gradients. Another approach, known as PopArt (Hessel et al., 2018b), examines task interference focusing on the instability caused by different reward magnitudes, addressing this issue by a normalizing technique on the output of the value function. Recently, sharing knowledge in a modular form has been advocated for reducing task interference. Yang et al. (2020) share a base model among tasks while having a routing network that generates task-specific models. Moreover, Devin et al. (2017) divide the responsibilities of the policy by sharing two policies, allocating one to different robots and the other to different tasks. Additionally, Sun et al. (2022) propose a parameter composition technique where a subspace of policy is shared by a group of related tasks. Moreover, CARE Sodhani et al. (2021) highlight the importance of using metadata for learning a mixture of state encoders shared among tasks, based on the claim that the learned encoders produce diverse and interpretable representations through an attention mechanism. Despite the potential of this work, the method is highly dependent on the context information as shown in this recent work (Cheng et al., 2023). However, we argue that all of these approaches lack the guarantee of learning diverse representations. In this work, we promote diversity across a mixture of experts by enforcing orthogonality among their representations. The mixture-of-experts has been well-studied in the RL literature (Akroun et al., 2021; Ren et al., 2021). Moreover, some works dedicate attention to maximizing the diversity of the learned skills in RL (Eysenbach et al., 2018). Previous works leverage orthogonality for disparate purposes (Mackey et al., 2018). For example, Bansal et al. (2018) promote orthogonality on the weights by a regularized loss to stabilize training in deep convolutional neural networks. Similarly, Huang et al. (2018a) employ orthogonality among the weights for stabilizing the distribution of activation in neural networks. In the context of MTRL, Paredes et al. (2012) enforce the representation obtained from a set of similar tasks to be orthogonal to the one obtained from selected tasks known to be unrelated. Recently, Chaudhry et al. (2020) alleviate catastrophic forgetting in continual learning by organizing task representations in orthogonal subspaces. Finally, Mashhadi et al. (2021) favor diversity in an ensemble of learners via a Gram-Schmidt process. As opposed to it, our primary focus lies in the acquisition of a set of orthogonal representations that span a subspace shared by a group of tasks where task-relevant representations can be interpolated. Figure 1: MOORE illustrative diagram. A state $s$ is encoded as a set of representations using a mixture of experts. The Gram-Schmidt process orthogonalizes the representations to encourage diversity. Then, the output head processes the representations $V_s$ by interpolating the task-specific representations $v_c$ using the task-specific weights $w_c$, from which we compute the output using the output function $f_\theta$. In our approach, we employ this architecture for both the actor and the critic. 4 Sharing Orthogonal Representations We aim to obtain a set of rich and diverse representations that can be leveraged to find a universal policy that accomplishes multiple tasks. To this end, we propose to enforce the orthogonality of the representations extracted by a mixture of experts. In the following, we first provide a mathematical formulation from which we derive our approach. In particular, we highlight the connection between our method and the Stiefel manifold theory (Huang et al., 2018b; Chaudhry et al., 2020; Li et al., 2020), together with the description of the role played by the Gram-Schmidt process. Then, we proceed to devise our novel method for Multi-Task Reinforcement Learning on orthogonal representation obtained from a mixture of experts. 4.1 Orthogonality in Contextual Markov Decision Processes We study the optimization of a policy $\pi$, given a set of $k$-orthonormal representations in $\mathbb{R}^d$ for the state $s$. We define the orthonormal representations of state $s$ as a matrix $V_s = [v_1, ..., v_k] \in \mathbb{R}^{d \times k}$ where $v_i \in \mathbb{R}^d, \forall i \leq k$. It can be shown that the orthonormal representations $V_s$ belong to a topological space known as the Stiefel manifold, a smooth and differentiable manifold largely used in machine learning (Huang et al., 2018b; Chaudhry et al., 2020; Li et al., 2020). Definition 4.1 (Stiefel Manifold) Stiefel manifold $\mathcal{V}_k(\mathbb{R}^d)$ is defined as the set of all orthonormal $k$-vectors in the Euclidean space $\mathbb{R}^d$, where $k \leq d$, $\mathcal{V}_k(\mathbb{R}^d) = \{ V_s \in \mathbb{R}^{d \times k} : V_s^T V_s = I_k, \forall s \in S \}$. Under this lens, our goal can be interpreted as finding a set of orthogonal representations belonging to the Stiefel manifold that capture the common characteristics in the true state space $S$. Thus, we propose a novel MDP formulation for MTRL, which we call a Stiefel Contextual Markov Decision Process (SC-MDP), that is inspired by the BC-MDP introduced in Sodhani et al. (2021). An SC-MDP includes a function that maps the state $s$ to $k$-orthonormal representations $V_s \in \mathcal{V}_k(\mathbb{R}^d)$. Definition 4.2 (Stiefel Contextual Markov Decision Process) A Stiefel Contextual Markov Decision Process (SC-MDP) is defined as a tuple $< C, S, A, \gamma, \mathcal{M}', \varphi >$ where $C$ is the context space, $S$ is the true state space, $A$ is the action space. $\mathcal{M}'$ is a function that maps a context $c \in C$ to MDP parameters and observation space $\mathcal{M}'(c) = \{ r^c, P^c, S^c, \rho^c \}$, $\varphi$ is a function that maps every state $s \in S$ to a $k$-orthonormal representations $V_s \in \mathcal{V}_k(\mathbb{R}^d)$, $V_s = \varphi(s)$. We define our MTRL policy as $\pi(a\|s,c) = f_\theta(\varphi(s) \cdot w_c)$, where $w_c \in \mathbb{R}^k$ is the task-specific weight that combines the $k$-orthogonal representations into a task-relevant one and $f_\theta : \mathbb{R}^d \rightarrow \mathbb{R}^{\|A\|}$ is an output function with learnable parameters $\theta$ that generates actions from task-specific representations. To leverage a diverse set of representations across tasks, the mapping function $\varphi$ plays a fundamental role. Hence, we approximate $ \varphi $ by a mixture of experts $ h_\phi = [h_{\phi_1}, ..., h_{\phi_k}] $ with learnable parameters $ \phi = [\phi_1, ..., \phi_k] $ that generate $ k $-representations $ U_s \in \mathbb{R}^{d \times k} $ for state $ s $, while ensuring that the generated representations are orthogonal to enforce diversity. Conveniently, this objective finds a rigorous formulation as a constrained optimization problem where we impose a hard constraint to enforce orthogonality: \[ \max_{\Theta=\{\phi,\theta\}} J(\Theta) \] subject to \[ h_\phi^T(s) h_\phi(s) = I_k \quad \forall s \in S, \] where $ I_k \in \mathbb{R}^{k \times k} $ is the identity matrix. Instead of solving the constrained optimization problem in Eq. 1, we ensure the diversity across experts through the application of the Gram-Schmidt (GS) process to orthogonalize the $ k $-representations $ U_s $. Definition 4.3 (Gram-Schmidt Process) Gram-Schmidt process is a method for orthogonalizing a set of linearly independent $ U = \{u_1, ..., u_k : u_i \in \mathbb{R}^d, \forall i \leq k\} $. It maps the vectors to a set of $ k $-orthonormal vectors $ V = \{v_1, ..., v_k : v_i \in \mathbb{R}^d, \forall i \leq k\} $ defined as \[ v_k = u_k - \sum_{i=1}^{k-1} \frac{\langle v_i, u_k \rangle}{\langle v_i, v_i \rangle} v_i. \] where the representation of the $ i $-th expert $ u_i $ is projected in the orthogonal direction to the subspace spanned by the representations of all $ i - 1 $ experts. Therefore, we apply the GS process to map the generated representations by the mixture of experts $ U_s = h_\phi(s) $ to a set of orthonormal representations $ V_s = GS(U_s) $, satisfying the hard constraint in Eq. 1. ### 4.2 Multi-Task Reinforcement Learning with Orthogonal Representations Following our policy $ \pi(a\|s,c) $, each task can interpolate its relevant representation from the subspace spanned by the $ k $-orthonormal representations $ V_s $. We train a task encoder to produce the task-specific weights $ w_c \in \mathbb{R}^k $ given task information (e.g., task ID). The orthonormal representations are combined using the task-specific weight to produce relevant representations $ v_c \in \mathbb{R}^d $ to the task as $ v_c = V_s \cdot w_c $. The interpolated representation $ v_c $ captures the relevant components of the task that can be utilized by the RL algorithm and fed to an output function $ f_\theta $. The output function can be learned for each task separately (multi-head) or shared by all tasks (single-head) to compute the action components given the representations $ v_c $. Similarly, the same policy (actor) structure (Alg. 1) can be used for the critic (Alg. 2). In conclusion, this approach results in a Mixture Of ORthogonal Experts, thus, we call it MOORE, whose extracted representation is used to learn a universal policy for MTRL. A visual demonstration of our approach is shown in Fig.1. We adopt two different RL algorithms, namely Proximal Policy Optimization (PPO) and Soft Actor-Critic (SAC), with the purpose of demonstrating that our approach is agnostic to the used RL algorithms. PPO (Schulman et al., 2017) is a policy gradient algorithm that has the merit of obtaining satisfactory performance in a wide range of problems while being easy to implement. It is a first-order method that enhances the policy update given the current data by limiting the deviation of the new policy from the current one. Moreover, we integrate our approach to SAC, a high-performing off-policy RL algorithm that leverages entropy maximization to enhance exploration. ### 5 Experimental Results In this section, we evaluate MOORE against related baselines on two challenging MTRL benchmarks, namely MiniGrid (Chevalier-Boisvert et al., 2023), a set of visual goal-oriented tasks, and MetaWorld (Yu et al., 2019), a collection of robotic manipulation tasks. The objective is to assess the adaptability of our approach in handling different types of state observations and tackling a variable number of tasks. Moreover, the flexibility of MOORE is evinced by using it for on-policy (PPO for MiniGrid) and off-policy RL (SAC for MetaWorld) algorithms. Additionally, we conduct ablation studies that support the effectiveness of MOORE in various aspects. We assess the following points: the benefit of using Gram-Schmidt to impose diversity across experts, the quality of the learned representations, as well as the transfer capabilities, and the interpretability of the diverse experts. Figure 2: Average return on the three MTRL scenarios of MiniGrid. We utilize both multi-head and single-head architectures for our approach MOORE as well as the related baselines. For MOORE, MOE and PCGrad, the number of experts $k$ is 2, 3, and 4 for MT3, MT5, and MT7, respectively. The black dashed line represents the final single-task performance of PPO averaged across all tasks. For the evaluation metric, we compute the accumulated return averaged across all tasks. We report the mean and the 95% confidence interval across 30 different runs. 5.1 MiniGrid We consider different tasks in MiniGrid (Chevalier-Boisvert et al., 2023), a suite of 2D goal-oriented environments that requires solving different mazes while interacting with objects like doors, keys, or boxes of several colors, shapes, and roles. MiniGrid offers a visual representation of the state, which we adopt for our multi-task setting. We consider the multi-task setting from Jin et al. (2023) that includes three multi-task scenarios. The first scenario, MT3, involves the three tasks: LavaGap, RedBlueDoors, and Memory; the second scenario, MT5, includes the five tasks: DoorKey, LavaGap, Memory, SimpleCrossing, and MultiRoom. Finally, MT7 comprises the seven tasks: DoorKey, DistShift, RedBlueDoors, LavaGap, Memory, SimpleCrossing, and MultiRoom. In Sec. A.1, we provide descriptions and more details for the tasks. We compare MOORE against four baselines. The first one is PPO, considered a reference for comparing to single-task performance. The second baseline is Multi-Task PPO (MTPPO), an adaptation of PPO (Schulman et al., 2017) for MTRL. Then, we consider MOE, which employs a mixture of experts to generate representations without enforcing diversity across experts. Additionally, we have PCGrad (Yu et al., 2020), which is an MTRL approach that tackles the task interference issue by manipulating the gradients. We integrate PCGrad on top of the MOE baseline for a fair comparison. As for the MTRL architecture, we utilize multi-head and single-head architectures for all methods, showing their average return across all tasks in Fig. 2(a), and Fig. 2(b) respectively. MOORE outperforms the aforementioned baselines in almost all the MTRL scenarios. Notably, our method exhibits faster convergence than the baselines. It is interesting to observe that MOORE outperforms the single-task performance with a significant margin in comparison to the other baselines (Fig. 2(a)), which is usually considered as an upper-bound of the MTRL performance in previous works. This highlights the quality of the learned representations and the role of MOORE representation learning process in MTRL. Figure 3: Evaluating MOORE against MOE on the transfer setting. The study is conducted on the two transfer learning scenarios in MiniGrid, employing a multi-head architecture. The number of experts $k$ is 2 and 3 for MT3 → MT5 and MT5 → MT7, respectively. For the evaluation metric, we compute the accumulated return averaged across all tasks. We report the mean and the 95% confidence interval across 30 different runs. 5.1.1 Ablation Studies Transfer Learning. We examine the advantage of transferring the trained experts on a set of base tasks to novel tasks in order to assess the quality and generalization of these learned experts in comparison to the MOE baseline. We refer to the transfer variant of our approach as Transfer-MOORE while Transfer-MOE for the baseline. Moreover, we include the performance of MOORE and MOE as a MTRL reference for learning the novel tasks from scratch, completely isolated from the base tasks. In Fig. 3, we show the empirical results on two transfer learning scenarios where we transfer a set of experts learned on MT3 to MT5 (MT3 → MT5) and on MT5 to MT7 (MT5 → MT7). MT3 is a subset of MT5, while MT5 is a subset of MT7. First, we train on the base tasks, and then we transfer the learned experts (frozen) to the novel tasks (the difference between the two sets). As illustrated in Fig. 3, Transfer-MOORE outperforms Transfer-MOE in the two scenarios, showing the quality of the learned representations in the context of transfer learning. Moreover, the study demonstrates the ability of our approach as an effective MTRL algorithm that provides competitive results against the transfer variant (Transfer-MOORE). In contrast, MOE struggles to beat the transfer variant as in the MT3 → MT5 scenario. Consequently, this study advocates the diversification of the shared representations in transfer learning and MTRL. We highlight more details in B.2. Number of Experts. Additionally, we focus on the impact of changing the number of experts on the performance of our approach, as well as on MOE. In Fig. 4, we consider different numbers of experts on the MT7 scenario. We observe the effect of utilizing more experts in MOORE algorithm compared to MOE. The study shows that MOORE exhibits a noticeable advantage, on average, for an increasing number of experts. On the contrary, a slower enhancement of the performance is encountered by MOE. It is also worth noting that the performance of MOORE with $k = 4$ slightly outperforms MOE with $k = 10$ while being comparable to MOE with $k = 8$ (MOE best setting). This supports our claim about efficiently utilizing expert capacity through enforcing diversity. 5.2 MetaWorld Finally, we evaluate our approach on another challenging MTRL setting with a large number of manipulation tasks. We benchmark against MetaWorld (Yu et al., 2019), a widely adopted robotic manipulation benchmark for Multi-Task and Meta Reinforcement Learning. We consider the MT10 \| Total Env Steps \| 1M \| 2M \| 3M \| 5M \| 10M \| 15M \| 20M \| \|-----------------\|----\|----\|----\|----\|-----\|-----\|-----\| \| SAC (Yu et al., 2019) \| 10.0±8.2 \| 17.7±2.1 \| 18.7±1.1 \| 20.0±2.0 \| 48.0±9.5 \| 57.7±3.1 \| 61.9±3.3 \| \| MTSAC (Yu et al., 2019) \| 34.9±12.9 \| 49.3±9.0 \| 57.1±9.8 \| 60.2±9.6 \| 61.6±6.7 \| 65.6±10.4 \| 62.9±8.0 \| \| SAC + FiLM (Perez et al., 2017) \| 32.7±6.5 \| 46.9±9.4 \| 52.9±6.4 \| 57.2±4.2 \| 59.7±4.6 \| 61.7±5.4 \| 58.3±4.3 \| \| PCGrad (Yu et al., 2020) \| 32.2±6.8 \| 46.6±9.3 \| 54.0±8.4 \| 60.2±9.7 \| 62.6±11.0 \| 62.6±10.5 \| 61.7±10.9 \| \| Soft-Module (Yang et al., 2020) \| 24.2±4.8 \| 41.0±2.9 \| 47.4±5.3 \| 51.4±6.8 \| 53.6±4.9 \| 56.6±4.8 \| 63.0±4.2 \| \| CARE (Sodhani et al., 2021) \| 26.0±9.1 \| 52.6±9.3 \| 63.8±7.9 \| 66.5±8.3 \| 69.8±5.1 \| 72.2±7.1 \| 76.0±6.9 \| \| PaCo (Sun et al., 2022) \| 30.5±9.5 \| 49.8±8.2 \| 65.7±4.5 \| 64.7±4.2 \| 71.0±5.5 \| 81.0±5.9 \| 85.4±4.5 \| \| MOORE (ours) \| 37.2±9.9 \| 63.0±7.2 \| 68.6±6.9 \| 77.3±9.6 \| 82.7±7.3 \| 88.2±5.6 \| 88.7±5.6 \| Table 1: Results on MetaWorld MT10 (Yu et al., 2019) with random goals (MT10-rand). The results of the baselines are from Sun et al. (2022). MOORE uses $k = 4$ experts. For all methods, we report the mean and standard deviation of the evaluation metric across 10 different runs. The evaluation metric is the average success rate across all tasks. We highlight with bold text the best result. and MT50 settings, where a single robot has to perform 10 and 50 tasks, respectively. For the baselines, we compare our approach against the following algorithms. First, SAC (Haarnoja et al., 2018) is the off-policy RL algorithm that is trained on each task separately, thus being a reference for the single-task setting. Second, Multi-Task SAC (MTSAC) is the adaptation of SAC to the MTRL setting, where we employ a single-head architecture with a one-hot vector concatenated with the state. Then, SAC+FiLM is a task-conditional policy that employs the FiLM module (Perez et al., 2017). Furthermore, PCGrad (Yu et al., 2020) is an MTRL approach that tackles the task interference issue by manipulating the gradients. Soft-Module (Yang et al., 2020) utilizes a routing network that proposes weights for soft combining of activations for each task. CARE (Sodhani et al., 2021) is an attention-based approach that learns a mixture of experts for encoding the state while utilizing context information. Finally, PaCo (Sun et al., 2022) is the state-of-the-art method for MetaWorld that learns a compositional policy where task-specific weights are utilized for interpolating task-specific policies. Our approach uses a similar framework as in the MiniGrid experiment and employs a multi-head architecture. Following Sun et al. (2022), we benchmark against variants of the MT10 and MT50 scenarios, MT10-rand and MT50-rand, where each task is trained with random goal positions. The goal position is concatenated with the state representation. As a performance metric, we compute the success rate averaged across all tasks. We run our approach for 10 different runs and report their mean and standard deviations of the metric, similar in Sun et al. (2022). As stated in Tab. 1, MOORE outperforms all the baselines regarding sample efficiency and asymptotic performance. Moreover, in Tab. 2, our approach shows significant final performance, indicating the scalability of MOORE to a large number of tasks. It is important to mention that all baselines use tricks to enhance the stability of the learning process. For instance, PaCo avoids task and gradient explosion by proposing two empirical tricks, named loss maskout and w-reset, where pruning every task loss that reaches above a certain threshold, besides resetting the task-specific weight for that task. Also, as in Sun et al. (2022), the other baselines resort to more expensive tricks, such as terminating and re-launching the training session when a loss explosion is encountered. On the contrary, our approach does not need such tricks to improve the stability of the learning process, which can indicate the stability of the chosen architecture and the importance of learning distinct experts. ### 5.2.1 Ablation Studies Diversity. Similarly, we want to evince the advantage of favoring diversity across experts. We evaluate MOORE against MOE, a baseline that uses the same architecture of MOORE but without the Gram-Schmidt process. We evaluate MOORE against MOE on the two MTRL scenarios of MetaWorld, MT10-rand and MT50-rand. In Fig. 5(a), MOORE exhibits superior sample-efficiency compared to MOE. Moreover, MOORE significantly outperforms the baseline also in MT50-rand. \| Algorithms \| Success Rate (20M) \| \|------------\|--------------------\| \| MTSAC (Yu et al., 2019) \| 49.3±1.5 \| \| SAC + FiLM (Perez et al., 2017) \| 36.5±12.0 \| \| CARE (Sodhani et al., 2021) \| 50.8±1.0 \| \| PaCo (Sun et al., 2022) \| 57.3±1.3 \| \| MOORE (ours) \| 72.9±3.3 \| Table 2: Results on MetaWorld MT50 (Yu et al., 2019) with random goals (MT50-rand). The results of the baselines are from Sun et al. (2022). MOORE uses $k = 6$ experts. Figure 5: (a) Success rate on MetaWorld MT10-rand comparing MOORE, against MOE, using 4 experts. (b) Success rate on MetaWorld MT50-rand comparing MOORE, against MOE, given 6 experts. We show the average success rate across all tasks and the 95% confidence interval across 10 and 5 different runs for MT10-rand and MT50-rand, respectively. (Fig. 5(b)), evincing the scalability of our approach to large-scale MTRL problems. This study illustrates the importance of enforcing diversity across experts in MTRL algorithms. Interpretability. Additionally, we verify the interpretability of the learned representations. Fig. 6 shows an application of PCA on the learned task-specific weights $w_c$ that interpolate the representations of the experts. On the one hand, the pick-place task is close to the peg-insert-side since both tasks require picking up an object. On the other hand, the weights of door-open and window-open tasks are similar as they share the open skill. Therefore, enforcing diversity across experts distributes the responsibilities across them in capturing common components across tasks (e.g., objects or skills). This confirms that the learned experts have some roles that can be interpretable. ### 6 Conclusion and Discussion We proposed a novel MTRL approach for diversifying a mixture of shared experts across tasks. Mathematically, we formulate our objective as a constrained optimization problem where a hard constraint is explicitly imposed to ensure orthogonality between the representations. As a result, the orthogonal representations live on a smooth and differentiable manifold called the Stiefel manifold. We formulate our MTRL as a novel contextual MDP while mapping each state to the Stiefel manifold using a mapping function, which we learn through a mixture of experts while enforcing orthogonality across their representations with the Gram-Schmidt process, hence satisfying the hard constraint. Our approach demonstrates superior performance against related baselines on two challenging MTRL baselines. Taking advantage of all the experts during inference, our approach has the limitation of potentially suffering from high time complexity compared to a sparse selection of few experts. This leads to a trade-off between the representation capacity and time complexity, which could be investigated in the future by a selection of a few orthogonal experts. In addition to our transfer learning study, we are interested in investigating extensions of our approach into a continual learning setting. Figure 6: Principle Component Analysis (PCA) on the task-specific weights learned by MOORE on MetaWorld MT10-rand for a run with 100% success rate across all tasks. ACKNOWLEDGMENTS We want to thank Aliaa Khalifa for her support in writing the paper and Firas Al-Hafez for his feedback on the method. This work was funded by the German Federal Ministry of Education and Research (BMBF) (Project: 01IS22078). This work was also funded by Hessian.ai through the project ‘The Third Wave of Artificial Intelligence – 3AI’ by the Ministry for Science and Arts of the state of Hessen. Calculations for this research were conducted on the Lichtenberg high-performance computer of the TU Darmstadt and the Intelligent Autonomous Systems (IAS) cluster at TU Darmstadt. REFERENCES Riad Akrour, Davide Tateo, and Jan Peters. Continuous action reinforcement learning from a mixture of interpretable experts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(10):6795–6806, 2021. Nitin Bansal, Xiaohan Chen, and Zhangyang Wang. Can we gain more from orthogonality regularizations in training deep networks? Advances in Neural Information Processing Systems, 31, 2018. Richard Bellman. Dynamic Programming. Princeton University Press, Princeton, NJ, USA, 1 edition, 1957. Daniele Calandriello, Alessandro Lazaric, and Marcello Restelli. Sparse multi-task reinforcement learning. In Advances in Neural Information Processing Systems, 2014. Arslan Chaudhry, Naeemullah Khan, Puneet Dokania, and Philip Torr. Continual learning in low-rank orthogonal subspaces. 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gJTPyCZmbj	Regarding complexity considerations, the paper discusses the complexity of the PAGrid but does not offer a direct comparison with other methods in terms of computational resources required—an important factor for practical applications.	PAGFormer: Polar Accumulator Grid Integrated into Transformers for Medical Image Segmentation Anonymous authors Paper under double-blind review Abstract Recent transformers have made remarkable strides in medical image analysis, enhancing the efficacy of various downstream applications. Yet, the rich geometric patterns present in medical images offer untapped potential for further refinement. In this paper, introduce the Polar Accumulator Grid (PAGrid) and seamlessly integrate it into the transformer network, PagFormer, with an aim to improve segmentation performance for elliptical or oval objects in medical images. Inspired by both the bilateral grid, renowned for its edge-preserving filtering, and the directed accumulator, skilled at integrating geometric shapes into neural networks, PAGrid facilitates geometric-preserving filtering through a symmetric sequence of accumulating, processing, and slicing. PAGrid preserves elliptical geometry information and promotes the aggregation of global information. The symmetry between accumulation and slicing in PAGrid allows us to transition from the classic encoder-decoder architecture to an encoder-slicer design, embodied in the PagFormer. Additionally, PAGrid’s parallelization is managed with CUDA programming, and the back-propagation is enabled for neural network training. An empirical experiment on three medical image segmentation datasets — specifically, ISIC2017 and ISIC2018 datasets for skin lesions, ACDC datasets for cardiac organs, all of which contains elliptically distributed objects — reveals that our method outperforms other state-of-the-art transformers. 1 Introduction Convolutional Neural Networks (ConvNets) He et al. (2016); Huang et al. (2017) and transformers Vaswani et al. (2017); Dosovitskiy et al. (2020); Liu et al. (2021) have recently achieved considerable success in diverse medical image analysis tasks. These range from the reconstruction of magnetic resonance imaging (MRI) Zhang et al. (2023c) and computed tomography (CT) Genzel et al. (2022) to lesion segmentation Zhang et al. (2021a); Rahman & Marculescu (2023a), MRI image registration Balakrishnan et al. (2019); Chen et al. (2022), and disease recognition Zhang et al. (2022). Of particular note, transformers, powered by the self-attention mechanism, consistently outpace traditional ConvNets. The performance enhancement is often attributed to the availability of large-scale datasets Ding et al. (2022); Sun et al. (2017) and the expansive effective receptive field Ding et al. (2021, 2022); Luo et al. (2016). Efforts to integrate transformers for medical image segmentation have been substantial, as highlighted by numerous studies Rahman & Marculescu (2023a); Cao et al. (2022); Wang et al. (2022b); Bo et al. (2023); Chen et al. (2021); Rahman & Marculescu (2023b); Zhang et al. (2021c); Wang et al. (2022a). These methodologies commonly adopt two main strategies to enhance segmentation outcomes. One involves crafting specific decoders using either convolutional or transformer modules. The other integrates ConvNets with transformers using a layered structure. Still, there are moments when these configurations may miss out on specific details inherent to the task at hand, suggesting room for further refinement. The potential is particularly evident in medical image analysis tasks, especially when discerning geometric patterns of target objects in situations with limited data. This point is underscored by tasks such as identifying primary structures like the left/right ventricles in cine-MRI cardiac images or segmenting skin lesions in dermoscopy scans. Figure 1: Overview of the proposed PAGrid. The left panel illustrates the PAGrid’s accumulate-process-slice sequence for image processing. The accumulation phase employs a directed accumulator \cite{Zhang2023a} to transform input $ U \in \mathbb{R}^{C_1 \times H \times W} $ into $ V \in \mathbb{R}^{C_1 \times H \times W} $, while the slicing phase utilizes grid sampling \cite{Jaderberg2015} to convert processed $ \tilde{V} \in \mathbb{R}^{C_2 \times H \times W} $ back to $ \tilde{U} \in \mathbb{R}^{C_2 \times H \times W} $. The process step involves any image processing operator, with a transformer backbone network being used in our implementation. Importantly, both accumulation and slicing phases leverage a shared sampling grid, matching the shape of $ U $, which enables direct slicing from feature maps of varying sizes, as depicted in the right panel. In clinical settings, numerous primary objects tend to have elliptical or oval shapes. Given their prevalence, precise delineation of these structures is advantageous for pre-treatment diagnoses and treatment planning. Therefore, in this study, we propose the Polar Accumulator Grid (PAGrid) and seamlessly integrate it into the transformer framework, PagFormer, with a specific aim to boost the segmentation precision for elliptical or oval objects in medical images. The PAGrid concept draws from the principles of both the bilateral grid \cite{Paris2006, Chen2007} and the directed accumulator \cite{Zhang2023a}, facilitating geometric-preserving processing within transformer. While the traditional bilateral grid filtering follows a splat-blur-slice sequence in image processing, our PAGrid approach adopts an accumulate-process-slice progression. Here, we favor ‘accumulate’ over ‘splat’ in the initial phase, as in image segmentation, it’s more intuitive to aggregate salient features or evidence, rather than dispersing them. The PAGrid offers two advantageous characteristics for the segmentation of elliptical objects. Firstly, unlike the traditional polar transformation using grid sampling (termed PS or polar sampling) that “pulls” a value from the source feature map for each target cell, PAGrid uses a directed accumulation strategy (termed PA or polar accumulation) \cite{Zhang2023a}. In this method, every cell in the source feature map “pushes” its value to a designated cell in the target map. While the polar sampling (PS) technique can lead to information loss if the mapping between source and target isn’t one-to-one, PAGrid preserves more details by pushing values from every cell in the source map. Secondly, unlike PS and inverse PS which requires two different sampling grids for the forward and reverse processes, PAGrid simplifies this by using a single sampling grid for both accumulation and slicing, leveraging the inherent symmetry between the “push” and “pull” actions. This simplified method is particularly beneficial when embedding PAGrid within transformers. Instead of relying on traditional encoder-decoder architectures, we can directly derive the segmentation map by slicing from the intermediate feature maps generated by the backbone. This innovation leads us to adopt an encoder-slicer design. In this study, we assess the performance of PAGrid and PagFormer across three medical image segmentation tasks: skin lesion segmentation using the ISIC2017 \cite{Codella2018} and ISIC2018 datasets \cite{Codella2019, Tschandl2018}, and cine-MRI cardiac image segmentation with the ACDC dataset \cite{Bernard2018}. Our contributions are threefold: - We present the first seamless integration of a symmetric tri-phase (namely, accumulate-process-slice) image processing sequence PAGrid into contemporary neural network frameworks. - We introduce an innovative encoder-slicer architecture, PagFormer, tailored for tasks presenting elliptical objects, as an alternative to the long-standing encoder-decoder design. - The proposed PagFormer not only exhibits faster convergence but also surpasses the best-performing methods on the three medical image datasets. 2 RELATED WORKS Dosovitskiy et al. (Dosovitskiy et al., 2020) introduce the concept of vision transformers (ViTs), which rely on the self-attention mechanism and introduce fewer inductive biases compared to ConvNets. Their effectiveness is further amplified by leveraging large-scale datasets and increased model capacities. Building on this, the Swin transformer (Liu et al., 2021) incorporates a shifted window-based attention strategy and features a hierarchical structure similar to ConvNets. Following this trend, several models emerge that integrate features from both ViTs and Swin transformers for medical image segmentation. For instance, the Pyramid Vision Transformer (PVT) (Wang et al., 2021) combines the strengths of ConvNets and ViTs, establishing itself as a versatile backbone for dense predictions without relying on convolutions. Similarly, SwinUnet (Cao et al., 2022) is a fully transformer-based model, where both the encoder and decoder segments utilize Swin transformer blocks, forming a U-shaped architecture. In contrast, TransUnet (Chen et al., 2021) adopts a hybrid structure, combining ConvNet-ViT elements and integrating a U-Net-like decoder (Ronneberger et al., 2015). Several other models emphasize modifications to the decoder. For example, PolypPVT (Dong et al., 2021) incorporates additional attention modules, specifically CBAM (Woo et al., 2018), into its decoders. Similarly, CASCADE (Rahman & Marculescu, 2023b) introduces a cascaded attention mechanism for the decoder. 2.1 LEARNING WITH GEOMETRIC PRIORS Training deep neural networks often hinges on access to large-scale datasets (Deng et al., 2009; Lin et al., 2014; Kirillov et al., 2023), which can be problematic for specific medical applications with limited data availability. For example, even when trained on extensive datasets, large vision models like SAM (Kirillov et al., 2023) still lag behind specialized models in various medical imaging tasks, despite attempts at fine-tuning (Ma & Wang, 2023). In contrast, leveraging geometric priors can yield better results. For instance, methods such as distance transformation mapping (Ma et al., 2020) and spatial information encoding (Liu et al., 2018) have paved the way for edge-aware loss functions (Kervadec et al., 2019; Zhang et al., 2021b; Karimi & Salcudean, 2019), the inclusion of anatomical coordinates in network layers as priors (Zhang et al., 2021a), and the creation of spatially covariant network weights (Zhang et al., 2023b). Polar or log-polar features are widely used in various tasks, including modulation classification (Teng et al., 2020), medical image segmentation (Benčević et al., 2021), rotation- and scale-equivariant polar transformer networks (Esteves et al., 2018), object detection (Xie et al., 2020; Xu et al., 2019; Park et al., 2022), correspondence matching (Ebel et al., 2019), and both cell detection (Schmidt et al., 2018) and segmentation (Stringer et al., 2021). Additionally, using concentric circles to model layout patterns aids in lithography hotspot detection (Zhang et al., 2016; 2017) and optical proximity correction (Jiang et al., 2019). 3 METHODOLOGY In this section, we detail the formulation of the Polar Accumulator Grid (PAGrid) and the PagFormer. We initiate the discussion with foundational concepts from directed accumulator (Zhang et al., 2023a) and grid sampling (Jaderberg et al., 2015). Subsequently, we describe every step in the PAGrid processing sequence and demonstrate how it merges with the current transformer architecture. Lastly, we provide a discussion on the complexity and contrast between polar accumulator (PA) and polar sampling (PS). 3.1 PRELIMINARIES Directed accumulator and grid sampling are techniques designed for differentiable image transformation within neural networks. While grid sampling is effective in various scenarios, it faces challenges with transformations that involve summing or integrating multiple values from the source feature map. This includes transformations such as the Radon transform (Deans, 2007), Hough transform (Ballard, 1987; Illingworth & Kittler, 1987), rim transform (Zhang et al., 2023a), and symmetric radial transform (Loy & Zelinsky, 2002). We begin by discussing the fundamental equations associated with directed accumulator and grid sampling. Given a source feature map $U \in \mathbb{R}^{C \times H \times W}$, and a sampling grid $G \in \mathbb{R}^{2 \times H \times W} =$ Figure 2: Schematic of the proposed encoder-slicer architecture. This structure is composed of three primary sections: the accumulator (highlighted with a light blue background), the encoder, and the slicer (indicated by the light orange background). \[(G_x[k], G_y[k]), \text{alongside a kernel function } K(\cdot), \text{we can represent the output value of a specific cell } (i,j) \text{ in the target feature map } V \in \mathbb{R}^{C \times H' \times W'} \text{ as:}\] \[V_{ij}^c = \sum_{n}^{H} \sum_{m}^{W} U_{nm}^c K(G_{nm}^x, i) K(G_{nm}^y, j), \tag{1}\] where the kernel function $K(\cdot)$ can be replaced with any other specified kernels, e.g., integer sampling kernel $\delta([G_{nm}^x + 0.5] - i) \cdot \delta([G_{nm}^y + 0.5] - j)$ and bilinear sampling kernel $\max(0, 1 - \|G_{nm}^x - i\|) \cdot \max(0, 1 - \|G_{nm}^y - j\|)$. Here $[x + 0.5]$ rounds $x$ to the nearest integer and $\delta(\cdot)$ is the Kronecker delta function. The equation 1 can be denoted as a tensor mapping, $D(U; G, K) : \mathbb{R}^{C \times H \times W} \rightarrow \mathbb{R}^{C \times H' \times W'}$. ### 3.2 Transformer-based Medical Image Segmentation For grid sampling, given a source feature map $\tilde{V} \in \mathbb{R}^{C \times H' \times W'}$, and considering the same variables as described in equation 1, the output value for a specific cell $(i, j)$ in the target feature map $\tilde{U} \in \mathbb{R}^{C \times H \times W}$ is expressed as: \[\tilde{U}_{ij}^c = \sum_{n}^{H'} \sum_{m}^{W'} \tilde{V}_{nm}^c K(G_{ij}^x, n) K(G_{ij}^y, m). \tag{2}\] Similarly, we use notation $S((\tilde{V}; G, K) : \mathbb{R}^{C \times H' \times W'} \rightarrow \mathbb{R}^{C \times H \times W})$ to represent equation 2. Upon examining equation 1 and equation 2, we can observe that the subtle differences arise from the subscript of $G$ and the input location to the kernel. In the directed accumulator, for each cell $(i, j)$ in the target feature map, values from all cells in the source feature map that point to $(i, j)$ are integrated. In contrast, with grid sampling, for each cell $(i, j)$ in the target feature map, a specific location is identified in the source feature map to retrieve the value. A vivid analogy is that directed accumulation and grid sampling operate like "push" and "pull" mechanisms on feature maps. Directed accumulation "pushes" values from the source to the target, while grid sampling "pulls" values from the source to the target. ### 3.3 PAGrid The PAGrid adopts an accumulate-process-slice sequence for image processing, as visually illustrated in Fig. 1. In general, the PAGrid operates in three stages to process intermediate feature maps within neural networks. Initially, a polar grid is constructed from a feature map using the polar accumulator. Subsequently, processing occurs within the grid using neural networks. Lastly, the polar grid is sliced to reconstruct the output feature map. Accumulation and slicing are symmetric operations that facilitate the conversion between the input and polar grid spaces. In multi-channel feature maps, the same transformation process applies to every channel. For simplicity, throughout the remainder of this discussion, the feature map will be denoted using only its spatial dimensions. Figure 3: (a) Bar plots of PSNR and (b) image reconstruction quality, comparing the performance between polar accumulator (PA) and polar sampling (PS). In (a), the ratio is calculated using the formula $ \text{ratio} = \frac{\text{PSNR}_{\text{PA}} - \text{PSNR}_{\text{PS}}}{\text{PSNR}_{\text{PS}}} $. In (b), the original image is transformed into a polar space of size $(H_r, W_\psi)$ with nearest sampling kernel using either PA or PS and is then reverted back to image space. Hollowed-out regions are visible in the upper part of the PA-transformed image, contrasting with similar areas found in the peripheral regions of the PS-reconstructed image. ### 3.3.1 Sampling Grid We start by defining the sampling grid necessary for creating the polar grid. With the sampling grid defined, equation 1 and equation 2 facilitate the accumulation and slicing processes. Let $ U \in \mathbb{R}^{H \times W} $ be the input feature map, $ M^x \in \mathbb{R}^{H \times W} $ (value range: $(0, H-1)$) and $ M^y \in \mathbb{R}^{H \times W} $ (value range: $(0, W-1)$) be the corresponding mesh grids. $(x_c, y_c)$ be the coordinate of the polar center, the value of sampling grid in the radial direction $ G^x $ at position $(i, j)$ can be obtained as: \[ G^x_{ij} = \sqrt{(M^x_{ij} - x_c)^2 + (M^y_{ij} - y_c)^2}/s_r. \] Here $ s_r = \frac{\sqrt{H^2 + W^2}}{2H_r} $ represents the sampling rate in the radial direction, with $ H_r $ denoting the one side of spatial dimensions of the polar grid. Similarly, the value of sampling grid in the angular direction $ G^y $ at position $(i, j)$ can be obtained as: \[ G^y_{ij} = \arctan((M^x_{ij} - x_c)^2 + (M^y_{ij} - y_c)^2 + \pi)/s_\theta. \] Here, $ s_\theta = \frac{2\pi}{W_\psi} $ is the sampling rate in the angular direction, with $ W_\psi $ denoting the other side of spatial dimensions of the polar grid. The addition of $ \pi $ ensures all values fall within $(0, 2\pi)$. ### 3.3.2 Homogeneous Coordinates To enable geometric-preserving filtering in the PAGrid, it’s important to monitor the number of pixels (or a weight) corresponding to each grid cell. Thus, during grid creation, we store homogeneous quantities $(V^c_{ij}, W^c_{ij}, W^c_{ij})$. Here, $ W $ can be derived from $ W = D(J; G, K) $, where $ J $ is a tensor of ones. This representation simplifies the computation of weighted averages: $(w_1 v_1, w_1) + (w_2 v_2, w_2) = (w_1 v_1 + w_2 v_2, w_1 + w_2)$. Normalizing by the homogeneous coordinates $(w_1 + w_2)$ yields the anticipated averaging of $ v_1 $ and $ v_2 $, weighted by $ w_1 $ and $ w_2 $. Conceptually, the homogeneous coordinate $ W $ represents the importance of its associated data $ V $. In practice, $ W $ can be obtained as one of the output channels, with $ J $ serving as the corresponding channels of input. ### 3.3.3 Polar Grid Accumulation, Processing and Slicing Accumulation: Given $ G = (G^x, G^y) $ from equation 3 and equation 4, we can generate the polar accumulator grid through equation 1 as $ P = D(U; G, K) \in \mathbb{R}^{H_r \times W_\psi} $, where $ H_r $ and $ W_\psi $ are determined by the sampling rate $ s_r $ and $ s_\theta $, as described in Section 3.3.1. It’s worth noting that before processing $ P $, we use homogeneous coordinates for normalization, as described in Section 3.3.2. Processing: The transformed polar grid $ P $ is then processed through a backbone network $ f $, like the Swin transformer [Liu et al., 2021]. Depending on the specific network architecture, this can result in multiple outputs, expressed as $ \{P_i \| i \in \{1, \ldots, N_{outs}\}\} $. For instance, the Swin transformer yields four outputs, each with different spatial dimensions. Figure 4: Visual representation of the impact of scale factors on the sampling grid $G^x$ and the reconstructed image. To enhance image readability, the nearest sampling kernel is employed with a polar grid size of $(H_\psi = 32, W_\psi = 32)$. The sampling grid $G^x$ is normalized in the range of $(0, 1)$. The left-bottom corner illustrates the equation [5]. Slicing: With the $i_{th}$ processed polar grid $\tilde{P}_i = f(P)_i$ and the sampling grid $G$, we apply equation equation [2] to transform the processed polar representation back into the image space, denoted as $\tilde{U}_i = S(\tilde{P}_i; G, K)$. ### 3.4 INTEGRATING PAGRID INTO TRANSFORMERS We showcase PAGrid integration with the hierarchical Swin transformer, but it’s adaptable to other encoder networks, including ConvNets [He et al., 2016; Huang et al., 2017] and transformers like PVT [Wang et al., 2021] and ViTs [Dosovitskiy et al., 2020]. As illustrated in Fig. 2, the encoder-slicer architecture comprises three main components: the accumulator, encoder, and slicer sections. The network takes as inputs the raw image $U \in \mathbb{R}^{3 \times H \times W}$ and the sampling grids $G^x, G^y \in \mathbb{R}^{H \times W}$, with the polar grid size set to match the input image size $(H_\psi = H, W_\psi = W)$. The raw image is first processed through several convolutional blocks (Convs) to extract the initial feature map $U_0 \in \mathbb{R}^{C_0 \times H \times W}$. These extracted features, along with the sampling grids $G^x$ and $G^y$, are then fed into the accumulator to transform the data into the polar grid space $P_0 \in \mathbb{R}^{C_0 \times H \times W}$. After this transformation, the polar grid proceeds through additional Convs, adapting the feature map to $P_1 \in \mathbb{R}^{3 \times H \times W}$, priming it for the encoder. This adjustment facilitates the effective utilization of the encoder’s pre-trained weights to boost performance. The encoder generates intermediate feature maps $\tilde{P}_i \in \mathbb{R}^{C_i \times H_i \times W_i} \| 1 \leq i \leq N_{outs}$, with $N_{outs} = 4$, $H_i = \frac{H}{2^{i-1}}$, and $W_i = \frac{W}{2^{i-1}}$ specifically for the Swin transformer. Each of these feature maps is then processed through a distinct reducer, composed of linear layers, to diminish the channel dimension from $C_i$ to $N_c$. Here, $N_c$ specifies the number of categories pertinent to the task at hand. Finally, all reduced feature maps are transformed back from the polar grid space to the image space and are summed to yield the final logits $\tilde{U} \in \mathbb{R}^{N_c \times H \times W}$. ### 3.5 ANALYSIS OF CHARACTERISTICS AND COMPLEXITY OF PAGRID The PAGrid, utilizing directed accumulator, maintains the rotation-equivariance inherent in polar transformation and also offers two additional benefits over traditional sampling-based polar methods [Esteves et al., 2018; Benčević et al., 2021]. These advantages include the retention of more comprehensive information from the source feature maps and enhanced flexibility in employing polar sampling grids. The Retention of More Comprehensive Information: In the PS technique, each cell in the target feature map “pulls” a value from a specific cell (four cells using bilinear kernel) in the source feature map. This mechanism can result in potential information loss, especially when the mapping from source to target is not one-to-one. On the other hand, the PA approach ensures that each cell in the source feature map “pushes” its value to a cell (four cells using bilinear kernel) in the target feature map. Even though the values are smoothed during the normalization of homogeneous coordinates, Table 1: A comparative analysis of the performance between the proposed PagFormer and other established methods on the ISIC2017 and ISIC2018 datasets. The best-performing metric is bolded. \| Model \| ISIC 2017 Avg. Dice (%) \| ISIC 2018 Avg. Dice (%) \| \|----------------\|-------------------------\|-------------------------\| \| U-Net \| 81.96 \| 84.34 \| \| AttU-Net \| 81.68 \| 85.84 \| \| SAM \| 81.22 \| 87.1 \| \| TransUnet \| 83.85 \| 88.93 \| \| SwinUnet \| 83.69 \| 88.96 \| \| PolypPVT \| 84.57 \| 88.39 \| \| PVT-Cascade \| 84.06 \| 88.51 \| \| PagFormer (Ours) \| 85.28 \| 89.70 \| the PA method ensures that every piece of information from the source is considered, reducing the risk of information loss. ### 3.6 Transformer-based Medical Image Segmentation We first use nearest sampling kernel to show that PA-reconstructed images have visually higher quality than PS-reconstructed images, as shown in Fig. 3b. We then conducted a quantitative experiment to validate this claim, utilizing all the testing images from the ISIC2018 dataset, each resized to a 224 × 224 resolution. In the PA method, we sequentially applied the polar accumulator and slicer without an intervening processing phase. For the PS approach, polar sampling and inverse polar sampling were applied in sequence. The quality of the reconstructions was evaluated using the Peak Signal-to-Noise Ratio (PSNR), and an average score was calculated to compare the fidelity of images reconstructed by PA and PS to the original images. As illustrated in Fig. 3a, PA consistently outperforms PS in terms of image reconstruction quality. This improvement is more pronounced when the polar grid size is reduced from 224 to 32, showcasing the effectiveness of PA in preserving image details even at lower resolutions. The Enhanced Flexibility in Employing Polar Sampling Grids: Similar to the bilateral grid [Chen et al., 2007], the accumulation and slicing operations in PAGrid are symmetric, allowing the use of a single sampling grid for both forward and inverse polar transformations. This symmetry simplifies the architecture and processing steps, paving the way for the introduction of a novel encoder-slicer design. This new approach negates the need for complex decoders, a common requirement in previous methods such as [Bo et al., 2023; Chen et al., 2021; Rahman & Marculescu, 2023a], streamlining the process and potentially enhancing performance and efficiency. Additionally, this symmetric characteristic simplifies the adjustment of sampling grids, promoting their easy integration into neural networks. In scenarios where objects are not centrally located in images, it is typically necessary to identify the object’s center before executing the transformation. Utilizing the conventional PS method can lead to issues, such as parts of the reconstructed image being lost, as depicted in the second column in Fig. 4. To address the issue of lost parts, we can modify equation 3. Let $s_s$ be a new variable introduced as a scale factor to constrain the sampling grid both inside and outside the core circle. The sampling grid $G_x$ can be redesigned as follows: $$ \begin{cases} G_{x_{ij}} = \frac{s_s r_{ij}}{s_r}, & \text{if } r_{ij} \leq R \cdot s_s, \\ G_{x_{ij}} = \frac{(1-s_s)(r_{ij}-R \cdot s_s)}{d_{ij} s_r} + \frac{s_s}{s_r}, & \text{Others}, \end{cases} $$ (5) where $R = \frac{\sqrt{H^2+W^2}}{2}$ represents half of the diagonal length, $r_{ij} = \sqrt{(M_{x_{ij}} - x_c)^2 + (M_{y_{ij}} - y_c)^2}$ is the distance from the pixel to the center of the image, and $d_{ij}$ is the distance to the image boundary along the line passing through $(x_c, y_c)$, the polar center. When $s_s = 1$, equation 5 simplifies to equation 3. Visual examples of the effects of varying $s_s$ and visual illustration of variables used in equation 5 are provided in Fig. 4. Table 2: A comparative analysis highlighting the performance of our proposed PagFormer alongside other notable methods on the ACDC dataset is presented. Dice scores for RV, LV, and Myo are reported individually, and an average score is also provided. The highest performing metric in each category is highlighted in bold for easy reference. \| Model \| RV (%) \| Myo (%) \| LV (%) \| Avg. Dice (%) \| \|------------------------\|-------\|---------\|-------\|---------------\| \| U-Net [Ronneberger et al., 2015] \| 87.10 \| 80.63 \| 94.92 \| 87.55 \| \| AttUnet [Oktay et al., 2018] \| 87.58 \| 79.20 \| 93.47 \| 86.75 \| \| SAM [Ma & Wang, 2023] \| 74.92 \| 76.09 \| 88.51 \| 79.83 \| \| TransUnet [Chen et al., 2021] \| 86.67 \| 87.27 \| 95.18 \| 89.71 \| \| SwinUnet [Cao et al., 2022] \| 88.89 \| 87.98 \| 95.31 \| 90.73 \| \| PolypPVT [Bo et al., 2023] \| 87.95 \| 87.83 \| 95.20 \| 90.33 \| \| PVT-Cascade [Rahman & Marculescu, 2023a] \| 89.72 \| 88.59 \| 95.18 \| 91.16 \| \| PagFormer (Ours) \| 90.39 \| 89.90 \| 95.59 \| 91.96 \| 4 EXPERIMENTS AND RESULTS In this section, we start by benchmarking PagFormer against other leading methods to highlight its effectiveness. Following that, we explore the impact of changing scale factors and the number of polar grids derived from the backbone network. 4.1 DATASETS AND EVALUATION METRICS ACDC Dataset: The ACDC dataset [Bernard et al., 2018] is comprised of 100 cine-MRI cardiac scans gathered from a diverse group of patients. Each scan reveals three distinct organs: the right ventricle (RV), left ventricle (LV), and the myocardium (Myo). We adhere to the evaluation protocol established in previous research [Rahman & Marculescu, 2023a], allocating 70 cases (equivalent to 1930 axial slices) for training, 10 for validation, and the remaining 20 for testing purposes. ISIC2017 Dataset: The ISIC2017 dataset [Codella et al., 2018] is comprised of 2000 training images, 150 validation images, and 600 test images. Contrary to the approaches of prior studies, we opt to use the dataset partitions for training, validation, and testing as provided on the official website, avoiding manual splitting. ISIC2018 Dataset: The ISIC2018 dataset [Codella et al., 2019; Tschantl et al., 2018] consists of 2594 training images, 100 validation images, and 1000 test images. In contrast to previous studies, we adhere to the dataset partitions for training, validation, and testing that are officially provided on the dataset’s website, eliminating the need for manual splitting. Evaluation Metrics: We adopt the Dice score as the evaluation metric for all three datasets. Specifically, for the ACDC dataset, we report the Dice scores for the RV, LV, Myo, and their average. 4.2 COMPARATORS AND IMPLEMENTATION DETAILS We benchmark our proposed method against other prominent models including SAM [Ma & Wang, 2023], TransUnet [Chen et al., 2021], SwinUnet [Cao et al., 2022], PolypPVT [Bo et al., 2023], PVT-Cascade [Rahman & Marculescu, 2023a], U-Net [Ronneberger et al., 2015], and Attention-Unet (AttUnet) [Oktay et al., 2018]. For a fair comparison, we incorporate U-Net, AttUnet and SAM into our framework. For the remaining models, we replicate their results using their own implementations to ensure consistency and accuracy in the comparative analysis. Implementation Details: All experiments were executed using Python 3.7, with network models developed utilizing PyTorch library [Paszke et al., 2019] version 1.9.0. The training was carried out on a machine powered by an A100 GPU. The directed accumulator was crafted and executed in CUDA version 11.1, while for slicing, the built-in `grid_sample()` function in PyTorch was employed. Optimization was performed using the Adam optimizer [Kingma & Ba, 2014], initiated with a learning rate of 1e-4. We employed a multi-step learning rate scheduler that reduced the learning rate by half at 50%, 70%, and 90% of the total epochs. All images were adjusted to a resolution of (224, 224) and training was conducted with a mini-batch size of 12. The training process was con- 1https://challenge.isic-archive.com/data/#2017 2https://challenge.isic-archive.com/data/#2018 Figure 5: Ablation study on the effects of the scale factor (a) and the # of intermediate feature maps used (b) for ACDC dataset. cluded after 100 epochs. For our comparisons, though PAGrid is adaptable to various backbone networks, we have chosen to integrate it with the base Swin Transformer to form the PagFormer. We adopt the approach from previous research [Bencovic et al., 2021], employing SwinUnet to train a heat map generator for locating the center of objects. In the case of the ACDC dataset, we specifically focus on identifying the center of the left ventricle. 4.3 QUANTITATIVE RESULTS Table 1 presents a comparison of average Dice scores (%) between the proposed PagFormer and other state-of-the-art models on the ISIC 2017 and ISIC 2018 datasets. PagFormer outperforms all other listed models, achieving the highest average Dice scores of 85.28% and 89.70% on ISIC 2017 and ISIC 2018, respectively. Compared to the next best model, PolypPVT for ISIC 2017 and SwinUnet for ISIC 2018, PagFormer shows an improvement rate of approximately 0.84% and 0.83%, respectively. Table 2 showcases the performance comparison on the ACDC dataset of various models, including the proposed PagFormer. The models are evaluated based on the Dice score (%) for three different organs—RV, Myo, and LV—as well as the average Dice score across all three. PagFormer excels in all categories, registering Dice scores of 90.39%, 89.90%, and 95.59% for RV, Myo, and LV, respectively, leading to the highest average Dice score of 91.96%. When compared to the second-best model, PVT-Cascade, PagFormer exhibits an improvement of about 0.88% in the average Dice score, underscoring its superior performance in cardiac MRI segmentation tasks. 4.4 ABLATION ANALYSIS We conducted an ablation study on the ACDC dataset, focusing on the impact of scale factors and the number of intermediate feature maps employed. Effects of the Scale Factor: As illustrated in Fig. 5b, we maintained the number of intermediate feature maps at four and varied the scale factors. The performance rose starting from $s_s = 0.5$, peaked at $s_s = 0.8$, and then declined. Interestingly, for the ISIC datasets, the peak performance was observed at $s_s = 0.9$. This variation indicates that the average object sizes in the ISIC datasets are comparatively larger than those in the ACDC dataset. Effects of the # of Feature Maps: Fig. 5b also provides insights into the performance variation with different numbers of feature maps, while keeping the scale factor $s_s = 1.0$ constant. The data suggests a direct correlation between the number of intermediate feature maps used and the performance improvement. A notable observation is that employing even a single feature map of size $(7, 7)$ enables our model to surpass the performance of all competing methods, with the exception of PVT-Cascade. 5 CONCLUSIONS In this study, we introduced PagFormer, a model integrating Polar Accumulator Grid (PAGrid) with transformer architectures for improved medical image segmentation. PAGrid ensures efficient image transformation and processing, overcoming limitations of traditional methods. Our experiments on ISIC2017, ISIC2018, and ACDC datasets confirmed PagFormer’s superior performance and effectiveness. 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STUGfUz8ob	If the data diversity is small (data follows a Zipf distribution), then do these results hold? In practice we expect datasets to follows such distributions and I am unsure if I am misunderstanding the theorem incorrectly.	WHEN CAN TRANSFORMERS REASON WITH ABSTRACT SYMBOLS? Enric Boix-Adserà* Apple, MIT eboix@mit.edu Omid Saremi Apple osaremi@apple.com Emmanuel Abbe Apple, EPFL emmanuel.abbe@epfl.ch Samy Bengio Apple bengio@apple.com Etai Littwin Apple elittwin@apple.com Joshua Susskind Apple jsusskind@apple.com ABSTRACT We investigate the capabilities of transformer models on relational reasoning tasks. In these tasks, models are trained on a set of strings encoding abstract relations, and are then tested out-of-distribution on data that contains symbols that did not appear in the training dataset. We prove that for any relational reasoning task in a large family of tasks, transformers learn the abstract relations and generalize to the test set when trained by gradient descent on sufficiently large quantities of training data. This is in contrast to classical fully-connected networks, which we prove fail to learn to reason. Our results inspire modifications of the transformer architecture that add only two trainable parameters per head, and that we empirically demonstrate improve data efficiency for learning to reason. 1 INTRODUCTION As large language models (LLMs) are trained with increasing quantities of data, they begin to exhibit the ability to reason mathematically (Kaplan et al., 2020; Yuan et al., 2023). Why does more data help an LLM learn to reason? And can we make LLMs more data-efficient at learning to reason? In this paper, we study relational reasoning with abstract symbols, which is a basic capability that has been hypothesized to underlie more complex abilities in human cognition (Fodor, 1975; Newell, 1980; Snow et al., 1984; Marcus, 1998; Holyoak, 2012; Kriete et al., 2013; Webb et al., 2020b). One example is in mathematics or computer science, where relational reasoning is necessary to parse a proof or a program: variable names are abstract symbols and the functionality of the proof or program only depends on how they relate to each other and not on the variable names themselves. Our contributions are threefold: (i) we formalize relational reasoning through “template tasks”; (ii) we conduct an analysis of when transformers can learn template tasks when trained by gradient descent and show a separation with classical fully-connected neural network architectures; (iii) we propose modifications to transformers that improve data efficiency for learning to reason. 1.1 CAPTURING RELATIONAL REASONING WITH TEMPLATE TASKS Building on a line of work in neuroscience (Marcus, 1998; Martinho III & Kacelnik, 2016; Kim et al., 2018; Webb et al., 2020b; Kerg et al., 2022; Altabaa et al., 2023; Webb et al., 2023a; Geiger et al., 2023), we formalize a framework of reasoning tasks called template tasks. Figure 1: Tasks from Raven (1938); Webb et al. (2020b) which fall under our theory. Networks are trained with one alphabet of symbols and then tested on held-out symbols. Details in Appendix A. Regression setting In the regression setting, a template task is specified by a collection of “template” strings labeled by real numbers, which are used to generate the train and test data. The simplest way to describe these is through an example. Consider, for instance, the templates “$ \alpha = 1; \beta = -1; \text{print}(\alpha) $” $ \rightarrow $ label=+1 and “$ \alpha = 1; \beta = -1; \text{print}(\beta) $” $ \rightarrow $ label=-1. These are used to generate the datasets in Figure 2, where every sample $(x_i, y_i) \in X^k \times Y$ is formed by picking a template and replacing the placeholders $\alpha, \beta$ (which we call “wildcards”) with variable names. Memorizing the training data is easy (Zhang et al., 2021a), but we wish to measure reasoning: will the model learn to treat the variable names as abstract symbols, enabling generalization beyond its training distribution? To evaluate this, we adopt an out-of-distribution setting, where the train and test data distributions differ (Marcus, 1998; Abbe et al., 2023). The test dataset consists of the same programs, but with new variable names never seen during training. By testing on symbols unseen in the train set, we measure the ability of an LLM to learn logical rules on the relations between symbols. To succeed, the LLM must effectively infer the templates from training data, and at test time match samples to the corresponding templates to derive their labels. ![Figure 2](image) Apart from programming tasks as in Figure 2, this framework captures several natural problems: - Same/different task. The simplest relational reasoning task is when the templates are "$ \alpha \alpha $" and "$ \alpha \beta $" labeled by +1 and −1. This encodes learning to classify two symbols as equal (e.g., $ AA, BB $) or as distinct (e.g., $ AB, BC $), even when the symbols were unseen in the training data. This task has been studied empirically in animal behavior (Martinho III & Kacelnik, 2016) and in neural networks (Kim et al., 2018; Webb et al., 2020b). - Word problems. Word problems often have building blocks that follow simple templates. For example, the template “If $ \alpha $ gives $ \beta $ 5 $ \gamma $, how many $ \gamma $ does $ \beta $ have?” labeled by +5, could generate the data “If Alice gives Bob 5 oranges, how many oranges does Bob have?” or the data “If Rob gives Ada 5 apples, how many apples does Ada have?” - Psychometric tests. Psychometric tests of relational reasoning, which have recently been used to probe LLMs (Raven, 1938; Webb et al., 2020b; Altabaa et al., 2023; Kerg et al., 2022; Webb et al., 2023a,b), are often template tasks. Figure 1 illustrates some examples. Next-token-prediction setting In the next-token-prediction setting, there is one extra layer of complexity: each sample is labeled with a symbol. For the LLM to generalize to symbols unseen at train time, not only must it learn to track the value stored in a variable, but it also must learn to predict labels at test time that might not occur in its training data. For example, the train and test datasets in Figure 3 are generated by: “$ \alpha = " \gamma "; \beta = " \delta "; \text{print}(\alpha) $” $ \rightarrow $ label=$ \gamma $ and “$ \alpha = " \gamma "; \beta = " \delta "; \text{print}(\beta) $” $ \rightarrow $ label=$ \delta $, where $ \alpha, \beta, \gamma, \delta $ are wildcards. Other problems covered by these tasks include: - Programming. The template “$ \text{print}(" \alpha ") $” labeled with $ \alpha $ generates ($ \text{print}("A"), A $) or ($ \text{print}("dog"), \text{dog} $), and so an LLM that learns on the corresponding task can robustly evaluating print statements on symbols not seen in the training data. • Mathematical functions. For example, the set of templates $\{\alpha\alpha\alpha, \alpha\beta\alpha, \alpha\alpha\beta, \beta\alpha\alpha\}$ labeled by $\alpha$ encode the task of outputting the majority token in a length-3 string with a vocabulary of two symbols. Similarly, for length-$k$ strings, the task of outputting the majority element can be encoded with $2^{k-1}$ templates. (a) Train data \| $x_i$ \| $y_i$ \| \|---\|---\| \| a="d";b="q";print(a) \| d \| \| c="r";a="u";print(a) \| w \| \| f="y";c="u";print(f) \| R="F";A="Z";print(R) \| \| h="o";q="s";print(q) \| Q="B";V="A";print(V) \| (b) Test data \| $x_{test}$ \| $y_{test}$ \| \|---\|---\| \| s \| F \| \| ... \| A \| (c) Transformer performance ![Transformer performance graph] Figure 3: (a,b) The labels are symbols. (c) We propose a modified that transformer learns the reasoning task with less data (see Observation 1.2 and Theorem 1.4). Details in Appendix A. ### 1.2 MAIN RESULTS The phenomenon from Figures 2 and 3 that we seek to understand is: why does the out-of-distribution performance of the transformer architecture improve as the number of training samples increases? We analyze the regression and next-token-prediction settings separately. (1) MLPs fail to generalize to unseen symbols A classical criticism of connectionism by Marcus (1998) is that neural networks do not learn relational reasoning when trained. We support this criticism in Appendix I by proving that classical MLP architectures (a.k.a. fully-connected networks) trained by SGD or Adam will not generalize in template tasks on symbols unseen during training, even in the regression setting. This failure to reason relationally occurs regardless of the training data size. The proof uses a permutation equivariance property of MLP training (Ng, 2004; Shamir, 2018; Li et al., 2020; Abbe et al., 2022; Abbe & Boix-Adsera, 2022). (2) Transformers generalize to unseen symbols, but require large data diversity Nevertheless, we prove that he criticism of Marcus (1998) is not valid for modern transformer architectures (Vaswani et al., 2017). We analyze the training dynamics of a transformer model and establish that it can learn to reason relationally: Theorem 1.1 (Informal Theorem 3.4). For any regression template task, a wide-enough transformer architecture trained by gradient flow on sufficiently many samples generalizes on unseen symbols. Here the key points are: (a) Universality. The transformer architecture generalizes on symbols unseen in train data regardless of which and how many templates are used to define the reasoning task. (b) Large enough number of samples. Our theoretical guarantees require the training dataset size to be large, and even for very basic tasks like the two-template task in Figure 2, good generalization begins to occur only at a very large number of training samples considering the simplicity of the task. This raises the question of how the inductive bias of the transformer can be improved. The proof of Theorem 1.1 inspires a parametrization modification that empirically lowers the quantity of data needed by an order of magnitude. A standard transformer attention head that takes in an input $X \in \mathbb{R}^{k \times d_{emb}}$ is given by \[ \text{smax}(XW_K W_Q^T X^T)XW_V W_O^T, \] where $W_K, W_Q, W_V, W_O$ are trainable parameters. Our modification makes it easier for the transformer to access the incidence matrix $XX^T \in \mathbb{R}^{k \times k}$ of the input, which is invariant to permutations of the symbol alphabet and can be used to solve the relational reasoning task: Observation 1.2. Adding one trainable parameter $a$ to each attention head so that $W_K W_Q^T$ is replaced by $W_K W_Q^T + aI$ improves transformers’ data-efficiency on template tasks. (3) Transformers fail at copying unseen symbols The story is slightly different for next-token-prediction tasks, because of the bottleneck of learning to output a symbol that was never seen in the training dataset. Transformers’ performance degrades as the model grows (an “inverse scaling” law \cite{mckenzie2023}). Large transformers fail even for the task of copying the input. Theorem 1.3 (Informal Theorem 4.1). Transformers with large embedding dimension fail to generalize on unseen symbols for the copy-task outputting label “α” on template “α”. However, we propose adding an attention-modulated skip connection, which corrects this failure, making it easy for the transformer to learn to copy data between its residual streams: Theorem 1.4 (Informal Theorem 4.2). Adding one trainable parameter b to each head so that $ W_V W_O^T $ is replaced by $ W_V W_O^T + bI $ makes transformers generalize on the task of Theorem 1.3. (4) Experiments We conclude with experimental validation of our architecture modifications, and find that they improve data efficiency on relational reasoning tasks by an order of magnitude, and improve language-modeling performance when training the GPT-2 architecture on Wikitext. 1.3 RELATED LITERATURE A spate of recent work studies whether and how LLMs perform various reasoning tasks, each focusing on one component of reasoning: these include recognizing context-free grammars \cite{zhao2023, allen-zhu2023}, learning sparse functions \cite{edelman2022}, learning compositionally \cite{hupkes2020}, generalizing out-of-distribution when learning Boolean functions \cite{abbe2023}, performing arithmetic \cite{nanda2023}, learning in context \cite{garg2022, ain2023, zhang2023}, and evaluating indexing \cite{zhang2021b}. Our setting is closest to that of empirical work studying neural networks on relational reasoning tasks \cite{geiger2023, webb2023b}. For example, the four tasks in Webb et al. (2020b), the matrix digits task in Webb et al. (2023a), the SET game task in Altabaa et al. (2023), and most of the tasks in Kerg et al. (2022) (with the exception of the relational games tasks), are examples of regression template tasks that fall under our theory. Furthermore, Kim et al. (2018) shows experimentally that MLPs fail on the same/different template task, and we provide a proof for this in Appendix I. There is also a literature on modifying training to improve relational reasoning: \cite{webb2020a} proposes applying Temporal Context Normalization during training, and Santoro et al. (2017; 2018); Palm et al. (2018); Shanahan et al. (2020); Webb et al. (2020b); Kerg et al. (2022); Altabaa et al. (2023) propose new architectures. Finally, some recent works in mechanistic interpretability look for subnetworks within trained networks that are responsible for tasks such as variable binding \cite{oisson2022, davies2023}. In contrast, our focus is on proving when the transformer architecture learns or fails to learn, and on applying this theoretical understanding to improve its data efficiency for relational reasoning. 2 FORMAL DEFINITION OF TEMPLATE TASKS We formally define regression template tasks. For next-token prediction, see Appendix J. Definition 2.1. A template is a string $ z \in (\mathcal{X} \cup \mathcal{W})^k $, where $ \mathcal{X} $ is an alphabet of tokens, and $ \mathcal{W} $ is an alphabet of “wildcards”. A substitution map is an injective function $ s : \mathcal{W} \rightarrow \mathcal{X} $. We write $ \text{sub}(z, s) \in \mathcal{X}^k $ for the string where each wildcard is substituted with the corresponding token: $ \text{sub}(z, s)_i = z_i $ if $ z_i \in \mathcal{X} $, and $ \text{sub}(z, s)_i = s(z_i) $ if $ z_i \in \mathcal{W} $. The string $ x \in \mathcal{X}^k $ matches the template $ z $ if $ x = \text{sub}(z, s) $ for some substitution map $ s $ and also $ s(\mathcal{W}) \cap \{ z_i \}_{i \in [k]} = \emptyset $: i.e., the substituted tokens did not already appear in the template $ z $. Example Using Greek letters to denote the wildcards and Latin letters to denote regular tokens, the template “ααβST” matches the string “QQRST”, but not “QQQST” (because the substitution map is not injective) and not “QQSST” (because $ \beta $ is replaced by S which is already in the template). A template task’s training data distribution is generated by picking a template randomly from a distribution, and substituting its wildcards with a random substitution map. Definition 2.2. A template data distribution $ D = D(\mu_{\text{tmplt}}, \{\mu_{\text{sub}, z}\}_z, f_, \sigma) $ is given by • a template distribution $ \mu_{\text{tmplt}} $ supported on templates in $(\mathcal{X} \cup \mathcal{W})^k$, • for each $ z \in \text{supp}(\mu_{\text{tmplt}}) $, a distribution $ \mu_{\text{sub},z} $ over substitution maps $ s : \mathcal{W} \to \mathcal{X} $, • template labelling function $ f_ : \text{supp}(\mu_{\text{tmplt}}) \to \mathbb{R} $, and a label-noise parameter $ \sigma \geq 0 $. We draw a sample $(x, y) = (\text{sub}(z, s), f_(z) + \xi) \sim D$, by drawing a template $ z \sim \mu_{\text{tmplt}} $, a substitution map $ s \sim \mu_{\text{sub},z} $, and label noise $ \xi \sim \mathcal{N}(0, \sigma^2) $. Finally, we define what it means for a model to solve the template task and generalize on unseen symbols; namely, the model should output the correct label for any string $ x \in \mathcal{X}^k $ matching a template, regardless of whether the string is in the support of the training distribution. Definition 2.3.* A (random) estimator $ \hat{f} : \mathcal{X}^k \to \mathbb{R} $ generalizes on unseen symbols with $(\epsilon, \delta)$-error if the following is true. For any $ x \in \mathcal{X}^k $ that matches a template $ z \in \text{supp}(\mu_{\text{tmplt}}) $, we have \[ (\hat{f}(x) - f_(z))^2 \leq \epsilon, \] with probability at least $1 - \delta$ over the randomness of the estimator $ \hat{f} $. Example* If the training data is generated from a uniform distribution on templates “$ \alpha \alpha $” with label 1 and “$ \alpha \beta $” for label -1, then it might consist of the data samples $\{(AA, 1), (BB, 1), (AB, -1), (BA, -1)\}$. An estimator that generalizes to unseen symbols must correctly label string $ CC $ with +1 and string $ CD $ with −1, even though these strings consist of symbols that do not appear in the training set. This is a nontrivial reasoning task since it requires learning to use the relations between the symbols to classify rather than the identities of the symbols. ### 3 ANALYSIS FOR TEMPLATE TASKS IN THE REGRESSION SETTING We establish that one-layer transformers of large enough width generalize to unseen symbols, when trained with enough data on regression template tasks. It is important to note that this is not true for all architectures, as we prove in Appendix I that MLPs trained by SGD or Adam will not succeed. #### 3.1 TRANSFORMER RANDOM FEATURES KERNEL The one-layer transformer architecture that we analyze consists of an embedding layer, a multihead attention mechanism, an MLP layer, and an unembedding layer $ w_U $. This is written mathematically in Appendix H. We analyze training only the final $ w_U $ layer of the transformer, keeping the other weights fixed at their random Gaussian initialization. Surprisingly, even though we only train the final layer of the transformer, this is enough to guarantee generalization on unseen symbols. Taking the width and embedding and head dimensions to infinity, and the step size to 0, the SGD training algorithm with weight decay converges to kernel gradient flow with the following kernel $ K_{\text{trans}} $ in the infinitely-wide, infinitely-small-step-size limit. Here and throughout the remainder of the paper, we interchangeably denote an input by a string $ x \in \mathcal{X}^k $ or a matrix $ X \in \mathbb{R}^{k \times m} $ constructed by stacking the one-hot vectors $ X = [e_{x_1}, \ldots, e_{x_k}]^T $ of the string’s tokens. $ \phi : \mathbb{R} \to \mathbb{R} $ is the MLP activation layer, $ \beta, \gamma \in \mathbb{R} $ are hyperparameters controlling the temperature and magnitude of positional activations. \[ K_{\text{trans}}(X, Y) = \mathbb{E}_{u,v}[\phi(u)\phi(v)] \text{ for } u, v \sim N(0, \begin{bmatrix} K_{\text{attn}}(X, X) & K_{\text{attn}}(X, Y) \\ K_{\text{attn}}(Y, X) & K_{\text{attn}}(Y, Y) \end{bmatrix}) \] where $ K_{\text{attn}}(X, Y) = \mathbb{E}_{m(X), m(Y)}[\text{smax}(\beta m(X))^T(XY^T + \gamma^2 I)\text{smax}(\beta m(Y))] $ \[ [m(X), m(Y)] \sim N(0, \begin{bmatrix} XX^T + \gamma^2 I & XY^T + \gamma^2 I \\ YX^T + \gamma^2 I & YY^T + \gamma^2 I \end{bmatrix}). \] The function outputted by kernel gradient flow is known to have a closed-form solution in terms of the samples, the kernel, and the weight-decay parameter $ \lambda $, which we recall in Proposition 3.1. Proposition 3.1 (How kernel gradient flow generalizes; see e.g., (Welling, 2013)). Let $(X_1, y_1), \ldots, (X_n, y_n)$ be training samples. With the square loss and ridge-regularization of magnitude $ \lambda $, kernel gradient flow with kernel $ K $ converges to the following solution \[ \hat{f}(X) = y^T(K + \lambda I)^{-1}k(X), \] where $ y = [y_1, \ldots, y_n] \in \mathbb{R}^n $ are the train labels, $ \hat{K} \in \mathbb{R}^{n \times n} $ is the empirical kernel matrix and has entries $ \hat{K}_{ij} = K(X_i, X_j) $, and $ k(X) \in \mathbb{R}^n $ has entries $ k_i(X) = K(X_i, X) $. ### 3.2 Transformers Generalize on Unseen Symbols We prove that transformers will generalize out-of-distribution on unseen symbols when trained on template tasks. We require the templates in the distribution $ \mu_{\text{tmplt}} $ to be “disjoint”, since otherwise the correct label for a string $ x $ is not uniquely defined, as $ x $ could match more than one template: Definition 3.2. Two templates $ z, z' \in (\mathcal{X} \cup \mathcal{W})^k $ are disjoint if no $ x \in \mathcal{X}^k $ matches both $ z $ and $ z' $. Furthermore, in order to ensure that the samples are not all copies of each other (which would not help generalization), we have to impose a diversity condition on the data. Definition 3.3. The data diversity is measured by $ \rho = \min_{z \in \text{supp}(\mu_{\text{tmplt}})} \min_{x \in \mathcal{X}} \frac{1}{\mathbb{P}_{x \sim \mu_{\text{sub}}, z}[t \in s(W)]} $. When the data diversity $ \rho $ is large, then no token is much more likely than others to be substituted. If $ \rho $ is on the order of the number of samples $ n $, then most pairs of data samples will not be equal. Theorem 3.4 (Transformers generalize on unseen symbols). Let $ \mu_{\text{tmplt}} $ be supported on a finite set of pairwise-disjoint templates ending with [CLS] tokens. Then, for almost any $ \beta, \gamma, b_1, b_2 $ parameters (except for a Lebesgue-measure-zero set), the transformer random features with $ \phi(t) = \cos(b_1 t + b_2) $ generalizes on unseen symbols.$^1$ Formally, there are constants $ c, C > 0 $ and ridge regularization parameter $ \lambda > 0 $ that depend only $ \beta, \gamma, b_1, b_2, \mu_{\text{tmplt}}, f_+, \sigma $, such that for any $ x $ matching a template $ z \in \text{supp}(\mu_{\text{tmplt}}) $ the kernel ridge regression estimator $ \hat{f} $ in (5) with kernel $ K_{\text{trans}} $ satisfies \[ \|\hat{f}(x) - f_(z)\| \leq C \sqrt{\log(1/\delta)/n} + C \sqrt{1/\rho}, \] with probability at least $ 1 - \delta - \exp(-cn) $ over the random samples. The first term is due to the possible noise in the labels. The second term quantifies the amount of sample diversity in the data. Both the sample diversity and the number of samples must tend to infinity for an arbitrarily small error guarantee. Proof sketch* (1) In Lemma 3.5 we establish with a sufficient condition for kernel ridge regression to generalize on unseen symbols. (2) We prove that $ K_{\text{trans}} $ satisfies it. (1) Sufficient condition. Let $ \mu_{\text{tmplt}} $ be supported on templates $ z_1, \ldots, z_r $. Let $ R = \bigcup_{i \in [k], j \in [r]} \{z_{ij}\} $ be the tokens that appear in the templates. Let $ [n] = I_1 \sqcup I_2 \sqcup \cdots \sqcup I_n $ be the partition of the samples such that if $ a \in I_j $ then sample $ (x_a, y_a) $ is drawn by substituting the wildcards of template $ z_j $. Two samples $ x_a, x_b $ that are drawn from the same template $ z_j $ may be far apart as measured by the kernel: i.e., the kernel inner product $ K(x_a, x_b) $ may be small. However, these samples will have similar relationship to most other samples: \[ K(x_a, x_i) = K(x_b, x_i) \quad \text{for most } i \in [n]. \] Specifically, if the wildcards of $ x_a, x_b $ and $ x_i $ are substituted by disjoint sets of tokens that do not appear in the templates, then (6) holds. Therefore, as the sample diversity $ \rho $ increases, the empirical kernel matrix $ \hat{K} $ becomes approximately block-structured with blocks $ I_j \times I_j $. For most samples $ x_a, x_b $ corresponding to template $ z_j $, and most $ x_{a'}, x_{b'} $ corresponding to template $ z_{j'} $ we have \[ K(x_a, x_{a'}) = K(x_b, x_{b'}) = K(\text{sub}(z_j, s), \text{sub}(z_{j'}, s')) := N_{j,j'}, \] where $ s, s': W \to X $ are substitution maps satisfying \[ s(W) \cap s'(W) = 0 \quad \text{and} \quad s(W) \cap R = s'(W) \cap R = \emptyset. \] One can check that (7) and (8) uniquely define a matrix $ N \in \mathbb{R}^{r \times r} $ which gives the entries in the blocks of $ \hat{K} $, with one block for each pair of templates.$^2$ See Figure 4. --- $^1$ We analyze the shifted and rescaled cosine activation function $ \phi(t) = \cos(b_1 t + b_2) $ out of technical convenience, but conjecture that most non-polynomial activation functions should succeed. $^2$ This assumes a “token-symmetry” property of $ K $ that is satisfied by transformers; details in the full proof. Figure 4: Illustration of structure of $ \hat{K} $ and $ N $ for the same/different task, which has $ r = 2 $ templates $ z_1 = \alpha \alpha $ and $ z_2 = \alpha \beta $. As the sample diversity $ \rho $ increases and the number of samples $ n $ increases, the empirical kernel matrix $ \hat{K} \in \mathbb{R}^{n \times n} $ becomes approximately $(r \times r)$-block-structured, and within each block most of the entries are given by $ N \in \mathbb{R}^{r \times r} $; exceptions where this is not true, including the diagonals, are drawn in black. Furthermore, the spectrum of $ \hat{K} $ is increasingly determined by the spectrum of $ N $, and if $ N $ is nonsingular then the top eigenspace increasingly aligns with the span of the indicator vectors on $ I_1, \ldots, I_r $. If the matrix $ N $ is nonsingular and the number of samples is large, then the span of the top $ r $ eigenvectors of $ \hat{K} $ will align with the span of the indicator vectors on the sets $ I_1, \ldots, I_r $. Furthermore, when testing a string $ x_{\text{test}} $ that matches template $ z_j $, but might not have appeared in the training set, it holds that for most $ a \in I_j $, we have \[ k(x_{\text{test}}) = [K(x_{\text{test}}, x_1), \ldots, K(x_{\text{test}}, x_n)] \approx [K(x_a, x_1), \ldots, K(x_a, x_n)] = \hat{K}_{a,:}. \] In words, the similarity relationship of $ x_{\text{test}} $ to the training samples is approximately the same as the similarity relationship of $ x_a $ to the training samples. So the kernel ridge regression solution (5) approximately equals the average of the labels of the samples corresponding to template $ z_j $, which in turn is approximately equal to the template label by a Chernoff bound, \[ y^T(\hat{K} + \lambda I)^{-1}k(x_{\text{test}}) \approx \frac{1}{\|I_j\|} \sum_{a \in I_j} y_i \approx f_\star(z_j). \] (9) Therefore, kernel ridge regression generalizes on $ x_{\text{test}} $. It is important to note that the number of samples needed until (9) is a good approximation depends on the nonsingularity of $ N $. This yields the sufficient condition for kernel ridge regression to succeed (proof in Appendix C). Lemma 3.5 (Informal Lemma C.3). If $ N $ is nonsingular, then (5) generalizes to unseen symbols. (2) $ K_{\text{trans}} $ satisfies the sufficient condition. We now show that for any collection of disjoint templates $ z_1, \ldots, z_r $, the matrix $ N_{\text{trans}} := N \in \mathbb{R}^{r \times r} $ defined with kernel $ K = K_{\text{trans}} $ is nonsingular. The challenging is that $ K_{\text{trans}} $ does not have a closed-form solution because of the expectation over softmax terms in its definition (4). Therefore, our analysis of the transformer random feature kernel is, to the best of our knowledge, the first theoretical analysis showing that the transformer random features learn a nontrivial class of functions of sequences. We proceed by analyzing the MLP layer and the attention layer separately, observing that a “weak” condition on $ K_{\text{attn}} $ can be lifted into the “strong” result that $ N_{\text{trans}} $ is nonsingular. The intuition is that as long as $ K_{\text{attn}} $ is not a very degenerate kernel, it is unlikely that the MLP layer has the cancellations that to make $ N_{\text{trans}} $ nonsingular. Lemma 3.6 (Nonsingularity of $ N_{\text{trans}} $). Suppose for every non-identity permutation $ \tau \in S_r \setminus \{\text{id}\} $, \[ \sum_{i \in [r]} K_{\text{attn}}(\text{sub}(z_i, s), \text{sub}(z_i, s')) \neq \sum_{i \in [r]} K_{\text{attn}}(\text{sub}(z_{\tau(i)}, s), \text{sub}(z_{\tau(i)}, s')), \] (10) where $ s, s' $ are the substitution maps in the definition of $ N_{\text{trans}} $ in (8). Let the MLP layer’s activation function be $ \phi(t) = \cos(b_1 t + b_2) $. Then for almost any choice of $ b_1, b_2 $ (except for a Lebesgue-measure-zero set), the matrix $ N_{\text{trans}} $ is nonsingular. This is proved in Appendix E, by evaluating a Gaussian integral and showing $ N_{\text{trans}} $ has Vandermonde structure. Although we use the cosine activation function, we conjecture that this result holds for most non-polynomial activation functions. Next, we prove the condition on $ N_{\text{attn}} $. Lemma 3.7 (Non-degeneracy of $ K_{\text{attn}} $). The condition (10) holds for Lebesgue-almost any $ \beta, \gamma $. The proof is in Appendix F. First, we prove the analyticity of the kernel $ K_{\text{attn}} $ in terms of the hyperparameters $ \beta $ and $ \gamma $. Because of the identity theorem for analytic functions, it suffices to show at least one choice of hyperparameters $ \beta $ and $ \gamma $ satisfies (10) for all non-identity permutations $ \tau $. Since $ K_{\text{attn}} $ does not have a closed-form solution, we find such a choice of $ \beta $ and $ \gamma $ by analyzing the Taylor-series expansion of $ K_{\text{attn}} $ around $ \beta = 0 $ and $ \gamma = 0 $ up to order-10 derivatives. 3.3 IMPROVING TRANSFORMER DATA-EFFICIENCY WITH $W_K W_Q^T + aI$ PARAMETRIZATION Can we use these insights to improve transformers’ data-efficiency in template tasks? In the proof, the nonsingularity of $N$ in Lemma 3.5 drives the model’s generalization on unseen symbols. This suggests that an approach to improve data-efficiency is to make $N$ better-conditioned by modifying the transformer parametrization. We consider here the simplest task, with templates “αα” and “αβ” labeled with $+1$ and $-1$, respectively. For tokens $A, B, C, D \in X$, the matrix $N$ is $$N = \begin{bmatrix} K(AA, BB) & K(AA, BC) \\ K(BC, AA) & K(AB, CD) \end{bmatrix}$$ If $K$ is an inner-product kernel, $K(x, x') = \kappa(\sum_{i \in [k]} 1(x_i = x'_i))$, as from an MLP, then $K(AA, BB) = K(AA, BC) = K(BC, AA) = K(AB, CD) = \kappa(0)$, so $N$ is singular and generalization is not achieved. Intuitively, every sample $x_i$ has approximately the same “similarity profile to other data” $\tilde{K}_{i,:} = [K(x_i, x_1), \ldots, K(x_i, x_n)]$, so the kernel method cannot identify the samples that come from the same template as $x_{test}$. In contrast, the transformer kernel (4) succeeds by using information about the incidence matrix $XX^T$, which differs between templates, and does not depend on the symbol substitution. We thus propose to emphasize the incidence matrix $XX^T$ by reparametrizing each head to $W_K W_Q^T + aI$, where $a$ is a trainable parameter. This adds a scaling of $XX^T$ in the attention, and can empirically improve data efficiency by an order of magnitude on several template tasks (see Figures 2 and 3, as well as additional experiments in Appendix B). 4 ANALYSIS FOR TEMPLATE TASKS IN NEXT-TOKEN-PREDICTION SETTING We switch gears to the next-token prediction setting with the cross-entropy loss, where the output label may be a token as in the example of Figure 3; formal definition is in Appendix J. The simplest task consists of template “α” labeled by “α”. An example train set is $\{(A, A), (B, B), (C, C)\}$, where $A, B, C \in X$ are tokens, and then we test with $(x_{test}, y_{test}) = (D, D)$ which is not in the train set. This task captures the ability of a model to learn how to copy a symbol, which is important for LLMs that solve problems with multi-stage intermediate computations and must copy these to later parts of a solution (Csordás et al., 2021). From now on, we only consider this “copying” task. We consider an architecture $f_{\text{attn}}(x; \theta)$ with just a multi-head attention layer, and we tie the embedding and unembedding weights as in practice (Brown et al., 2020). Define the train loss and test loss as follows, where $\ell$ is the cross-entropy loss and $x_{test}$ is a token unseen in the training data: $$L_{\text{train}}(\theta) = \frac{1}{n} \sum_{i=1}^{n} \ell(f_{\text{attn}}(x_i; \theta), y_i)$$ $$L_{\text{test}}(\theta) = \ell(f_{\text{attn}}(x_{test}), y_{test})$$ We prove this network does not generalize on unseen symbols when trained, as we take the embedding dimension large. Our evidence is from analyzing the early time of training, and showing that the test loss on unseen symbols does not decrease. Theorem 4.1 (Failure of transformers at copying). For any learning rates such that $-\frac{\partial L_{\text{train}}}{\partial t} \|_{t=0} = O(1)$, we must have that $\frac{\partial L_{\text{test}}}{\partial t} \|_{t=0} \to 0$ as $d_{\text{emb}} \to \infty$. The proof idea is that since the input string has length $k = 1$, the architecture simplifies: all softmaxes in the attention heads output 1, and the network is a sum of attention heads of the form $XW_E W_V W_O^T W_E^T$. At early times the evolution of the weights $W_V W_O^T$ will roughly lie in the span of $\{W_E e_{x_i} e_{x_i}^T W_E\}_{i \in [n]}$, which as the embedding dimension becomes large will be approximately orthogonal to the direction $W_E e_{x_{test}} e_{x_{test}}^T W_E$ that would lower the test loss. This suggests the following modification to transformers allows them to copy symbols never seen at training: Theorem 4.2 (Adding one parameter allows copying). After reparametrizing the attention (3) so that in each head $W_V W_O^T$ is replaced by $W_V W_O^T + bI$ where $b$ is a trainable parameter, there are learning rates such that $-\frac{\partial L_{\text{train}}}{\partial t} \|_{t=0} = O(1)$ and $-\frac{\partial L_{\text{test}}}{\partial t} \|_{t=0} = \Omega(1)$ as $d_{\text{emb}} \to \infty$. Figures 3 and 5 illustrate the benefit of this additional per-head parameter on the copying task. It is not equivalent to adding a trainable skip connection as in ResNet (He et al., 2016). Instead, the addition of $b_I I$ encodes an attention-modulated skip connection that allows copying tokens between the transformer’s streams. A related modification of adding a head with the hardcoded $XX^T$ as its attention matrix was proposed in Zhang et al. (2022). Figure 5: (a) Transformers fail on the copying task as embedding dimension $d_{emb}$ grows (Theorem 4.1); (b) Success when reparametrizing $W_V W_O^T$ as $W_V W_O^T + bI$ (Theorem 4.2). Details in Appendix A. 5 EXPERIMENTS Figures 2 and 3 (and additional experiments in Appendix B) show that our reparametrizations can give a significant data-efficiency benefit on template tasks. Figure 6 shows they can also give improvements on real data. In Figure 7, we see that pretraining outperforms random initialization on a template task. This might be explained by several heads of the pretrained model with diagonals stronger from other weights (originally observed in (Trockman & Kolter, 2023)). These learned diagonals resemble our proposed transformer modifications and so might be driving the data-efficiency of fine-tuning a pretrained model. Appendix B provides extensive experiments on the effect of hyperparameters, inductive biases of different models, and varying levels of task difficulty. \| Dataset \| GPT-2 \| GPT-2 + trainable identity scalings (ours) \| \|-------------\|-------------\|-------------------------------------------\| \| Wikitext2 \| 64.00 \| 60.46 \| \| Wikitext103 \| 16.83 \| 16.40 \| Figure 6: Perplexity of GPT-2 trained from random initialization with Adam learning rate 3e-4 for 20 epochs on Wikitext (smaller perplexity is better). GPT-2 has 117M parameters, and we add an extra 288 parameters (2 per head). Interestingly, even though the task is Wikipedia modeling, and therefore is not a pure reasoning task, the transformer modifications still give an improvement. Figure 7: Left: Pretrained versus randomly-initialized GPT-2 test loss when fine-tuned on $\alpha \beta \alpha$ vs. $\alpha \beta \beta$ template task. Right: some GPT-2 pretrained heads have strong diagonals (zoomed to 100x100 top-left corner). 6 DISCUSSION We show that transformers are a universal architecture for template tasks in the regression setting: when trained with gradient descent with enough training data they learn to reason relationally. However, transformers are not optimal – empirically they require large amounts of data to learn basic tasks, and in the next-token-prediction setting they fail at copying unseen symbols. Thus, we have proposed architectural modifications to improve their inductive bias towards logical reasoning. It seems promising to explore other reasoning tasks (for example, reasoning with syllogisms, reasoning by symmetry, and compositional reasoning). It may also be fruitful to study data augmentation approaches (e.g., concatenating the tensorization $XX^T$ to the input, so as to encourage use of relational information). 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LWmuPfEYhH	* As previously mentioned, ACORM does not impose constraints on the attention mechanism, and it even utilizes the learned latent variable $z$. How can we ensure its correct execution under the CTDE paradigm?	ATTENTION-GUIDED CONTRASTIVE ROLE REPRESENTATIONS FOR MULTI-AGENT REINFORCEMENT LEARNING Zican Hu\textsuperscript{1}, Zongzhang Zhang\textsuperscript{2}, Huaxiong Li\textsuperscript{1}, Chunlin Chen\textsuperscript{1}, Hongyu Ding\textsuperscript{1}, Zhi Wang\textsuperscript{1}\textsuperscript{} \textsuperscript{1} Department of Control Science and Intelligent Engineering, Nanjing University \textsuperscript{2} School of Artificial Intelligence, Nanjing University \{zicanhu,hongyuding\}@smail.nju.edu.cn \{zzzhang,huaxiongli,clchen,zhiwang\}@nju.edu.cn ABSTRACT Real-world multi-agent tasks usually involve dynamic team composition with the emergence of roles, which should also be a key to efficient cooperation in multi-agent reinforcement learning (MARL). Drawing inspiration from the correlation between roles and agent’s behavior patterns, we propose a novel framework of Attention-guided COntrastive Role representation learning for MARL (ACORM) to promote behavior heterogeneity, knowledge transfer, and skillful coordination across agents. First, we introduce mutual information maximization to formalize role representation learning, derive a contrastive learning objective, and concisely approximate the distribution of negative pairs. Second, we leverage an attention mechanism to prompt the global state to attend to learned role representations in value decomposition, implicitly guiding agent coordination in a skillful role space to yield more expressive credit assignment. Experiments on challenging StarCraft II micromanagement and Google research football tasks demonstrate the state-of-the-art performance of our method and its advantages over existing approaches. Our code is available at \url{https://github.com/NJU-RL/ACORM} 1 INTRODUCTION Cooperative multi-agent reinforcement learning (MARL) aims to coordinate a system of agents towards optimizing global returns (Vinyals et al., 2019), and has witnessed significant prospects in various domains, such as autonomous vehicles (Zhou et al., 2020), smart grid (Chen et al., 2021a), robotics (Yu et al., 2023), and social science (Leibo et al., 2017). Training reliable control policies for coordinating such systems remains a major challenge. The centralized training with decentralized execution (CTDE) (Foerster et al., 2016) hybrids the merits of independent Q-learning (Foerster et al., 2017) and joint action learning (Sukhbaatar et al., 2016), and becomes a compelling paradigm that exploits the centralized training opportunity for training fully decentralized policies (Wang et al., 2023). Subsequently, numerous popular algorithms are proposed, including VDN (Sunehag et al., 2018), QMIX (Rashid et al., 2020), MAAC (Iqbal & Sha, 2019), and MAPPO (Yu et al., 2022). Sharing policy parameters is crucial for scaling these algorithms to massive agents with accelerated cooperation learning (Fu et al., 2022). However, it is widely observed that agents tend to acquire homogeneous behaviors, which might hinder diversified exploration and sophisticated coordination (Christianos et al., 2021). Some methods (Li et al., 2021; Jiang & Lu, 2021; Liu et al., 2023) attempt to promote individualized behaviors by distinguishing each agent from the others, while they often neglect the prospect of effective team composition with implicit task allocation. Real-world multi-agent tasks usually involve dynamic team composition with the emergence of roles (Shao et al., 2022; Hu et al., 2022).\footnote{Early works introduce the role concept into multi-agent...} \textsuperscript{}Correspondence to Zhi Wang <zhiwang@nju.edu.cn>. \textsuperscript{1}Taking the football game (Kurach et al., 2020) as an example, the midfielders are primarily responsible for delivering the ball to the forwards to coordinate shots on goal in the offensive phase, while they need to drop back and join the defenders to block passing lanes on the defensive. systems (Dastani et al., 2003; Sims et al., 2008; Lhaksmana et al., 2018), while they usually require prior domain knowledge to pre-define role responsibilities. Recently, ROMA (Wang et al., 2020) learns emergent roles conditioned solely on current observations, and RODE (Wang et al., 2021) associates each role with a fixed subset of the joint action space. COPA (Liu et al., 2021) allows dynamic role allocation via distributing a global view of team composition to each agent during execution. Some works decompose the task into a set of skills (Liu et al., 2022) or subtasks (Yang et al., 2022; Iqbal et al., 2022) with a hierarchical structure for control. Overall, existing role-based methods still suffer from several deficiencies, such as insufficient characterization of complex behaviors for role emergence, neglect of evolving team dynamics, or relaxation of the CTDE constraint. To better leverage dynamic role assignment, we propose a novel framework of Attention-guided COntensive Role representation learning for MARL (ACORM). Our main insight is to learn a compact role representation that can capture complex behavior patterns of agents, and use that role representation to promote behavior heterogeneity, knowledge transfer, and skillful coordination across agents. First, we formalize the learning objective as mutual information maximization between the role and its representation, to maximally reduce role uncertainty given agent’s behaviors while minimally preserving role-irrelevant information. We introduce a contrastive learning method to optimize the infoNCE loss, a mutual information lower bound. To concisely approximate the distribution of negative pairs, we extract agent behaviors by encoding its trajectory into a latent space, and periodically partition all agents into several clusters according to their latent embeddings where points from different clusters are paired as negative. Second, during centralized training, we employ an attention mechanism to prompt the global state to attend to learned role representations in value decomposition. The attention mechanism implicitly guides agent coordination in a skillful role space, thus yielding more expressive credit assignment with the emergence of roles. ACORM is fully compatible with CTDE methods, and we realize ACORM on top of two popular MARL algorithms, QMIX (Rashid et al., 2020) and MAPPO (Yu et al., 2022), benchmarked on challenging StarCraft multi-agent challenge (SMAC) (Samvelyan et al., 2019) and Google research football (GRF) (Kurach et al., 2020) environments. Experiments demonstrate that ACORM achieves state-of-the-art performance on most scenarios. Visualizations of learned role representations, heterogeneous behavior patterns, and attentional value decomposition shed further light on our advantages. Ablation studies confirm that ACORM promotes higher coordination capacity by virtue of contrastive role representation learning and attention-guided credit assignment, respectively, even if agents have the same innate characteristics. In summary, our contributions are threefold: • We propose a general role representation learning framework based on contrastive learning, which effectively tackles agent homogenization and facilitates efficient knowledge transfer. • We leverage role representations to realize more expressive credit assignment via an attention mechanism, promoting strategical coordination in a sophisticated role space. • We build our method on top of popular QMIX and MAPPO, and conduct extensive experiments on SMAC and GRF to demonstrate our state-of-the-art performance and advantages. 2 METHOD In this section, we present the ACORM framework. We consider cooperative multi-agent tasks formulated as a Dec-POMDP (Oliehoek & Amato, 2016), $ G = \langle I, S, A, P, R, \Omega, O, n, \gamma \rangle $, where $ I $ is a finite set of $ n $ agents, $ s \in S $ is the global state, and $ \gamma \in [0, 1) $ is the discount factor. At each time step, each agent $ i $ draws an observation $ o_i \in O $ from $ \Omega(s, i) $ and selects a local action $ a_i \in A $. After executing the joint action $ a = [a_1, ..., a_n]^T \in A^n $, the system transitions to a next state $ s' $ according to $ P(s'\|s, a) $ and receives a reward $ r = R(s, a) $ shared by all agents. Our idea is to learn a compact role representation that can characterize complex behavior patterns of agents, and use the role information to facilitate individual policy learning and guide agent coordination. Agents with similar roles can enjoy higher learning efficiency via more aggressive knowledge transfer, and agent heterogeneity is also guaranteed with the discrimination of diverse roles. Formally, we propose the following definition of the role and its representation. $^2$Taking StarCraft II as an example, the acquired roles represent diverse strategies in a team-based manner, such as focusing fire, sneaking attack, and drawing fire. Agents with similar role representations learn a specialized strategy more efficiently by more positive information sharing, and the attention mechanism is responsible for coordinating these heterogeneous behaviors more strategically with clearer role extraction in the team. Figure 1: The ACORM framework based on QMIX. (a) The overall architecture. (b) The structure of shared individual Q-network. (c) The detail of contrastive role representation learning, where $ z_i $ is the query $ q $, and $ z_{i'} / z_{i^} $ are positive/negative keys $ k_+ / k_- $. (d) The attention module that incorporates learned role representations into the mixing network’s input for better value decomposition. Definition 1.* Given a cooperative multi-agent task $ G = \langle I, S, A, P, R, \Omega, O, n, \gamma \rangle $, each agent $ i $ is associated with a role $ M_i \in \mathcal{M} $ that describes its behavior pattern. Each role $ M_i $ is quantified by a role representation $ z_i \in \mathcal{Z} $, which is obtained by training a complex function as $ z_i = f(\rho_i) $, where $ \rho_i \in \Gamma \equiv (O, A)^l $ is the local trajectory of agent $ i $, and $ l $ is the number of observation-action pairs. $ \pi_{z_i} : O \times A \times \mathcal{Z} \rightarrow [0, 1] $ is the individual policy for agent $ i $. ACORM consists of individual Q-networks in Fig. 1(b) and a mixing network in Fig. 1(a). We introduce mutual information maximization to formalize role representation learning, and derive a contrastive learning objective that optimizes agent embeddings $ \{e_i^t\}_{t=1}^{T_i} $ in a self-supervised way to acquire contrastive role representations $ \{z_i^t\}_{t=1}^{T_i} $, which is shown in Fig. 1(c) and will be introduced in detail in Section 2.1. In value decomposition, we employ multi-head attention (MHA) to prompt the global state to attend to learned role patterns, guiding skillful agent coordination in the high-level role space for facilitating expressive credit assignment, as described in Fig. 1(d) and in Section 2.2. Appendix B presents the pseudocode, and Appendix C gives the extension to MAPPO. ### 2.1 Contrastive Role Representations Our objective is to ensure that agents with similar behavior patterns exhibit closer role representations, while those with notably different strategies are pushed away from each other. This stands in contrast to using a one-hot ID to preserve the agent’s individuality, which lacks adequate discrimination under the paradigm of parameter sharing. Hence, the primary issues we aim to tackle are: i) how to define a feasible metric to quantify the degree of similarity between agent’s behaviors, and ii) how to develop an efficient method to optimize the discrimination of role representations. Agent Embedding. To tackle the first issue, we learn an agent embedding $ e_i^t $ from each agent’s trajectory to extract complex agent behaviors with contextual knowledge as $ e_i^t = f_\phi(o_i^t, a_i^{t-1}, e_i^{t-1}) $, where $ \phi $ is a shared gated recurrent unit (GRU) encoder, $ o_i^t $ is the current observation, $ a_i^{t-1} $ is the last action, and $ e_i^{t-1} $ is the hidden state of the GRU. Naturally, the distance between the obtained agent embeddings can serve as the metric to measure the behavior dissimilarity between agents. Contrastive Learning. An ideally discriminative role representation should be dependent on roles associated with agent’s behavior patterns, while remaining invariant across agent identities. We introduce mutual information to measure the mutual dependency between the role and its representation. Formally, mutual information aims to quantify the uncertainty reduction of one random variable when the other one is observed. To tackle the second issue, we propose to maximize the mutual information between the role and its representation, and learn a role encoder that maximally reduces role uncertainty while minimally preserving role-relevant information. Mathematically, we formalize the role encoder $\theta$ as a probabilistic encoder $z^t \sim f_\theta(z^t\|e^t)$, where $z^t$ denotes the role representation at time $t$, and $e^t = f_\phi(\sum_{t'=1}^{t}(o^{t'}, a^{t'-1}))$ denotes the agent embedding obtained from the history trajectory. Role $M$ follows the role distribution $P(M)$, and the distribution of agent embedding $e$ is determined by its role. The learning objective for the role encoder is: $$\max I(z; M) = \mathbb{E}_{z,M} \left[ \log \frac{p(M\|z)}{p(M)} \right].$$ In practice, directly optimizing mutual information is intractable. Inspired by the noise contrastive estimation (InfoNCE) (Oord et al., 2018) in the literature of contrastive learning (Laskin et al., 2020; Yuan & Lu, 2022), we derive a lower bound of Eq. (1) with the following theorem. Theorem 1. Let $\mathcal{M}$ denote a set of roles following the role distribution $P(M)$, and $\|\mathcal{M}\| = K$. $M \in \mathcal{M}$ is a given role. Let $e = f_\phi(\sum_{t'=1}^{t}(o^{t'}, a^{t'-1}))$, $z \sim f_\theta(z\|e)$, and $h(e,z) = \frac{p(z\|e)}{p(z)}$, where $\sum_{t'}(o^{t'}, a^{t'-1})$ is the agent’s local trajectory following a given policy. For any role $M^* \in \mathcal{M}$, let $e^$ denote the agent embedding generated by the role $M^$, then we have $$I(z; M) \geq \log K + \mathbb{E}_{M,z,e} \left[ \log \frac{h(e,z)}{\sum_{M^* \in \mathcal{M}} h(e^,z)} \right].$$ The proof of this theorem is given in Appendix A. Since we cannot evaluate $p(z)$ or $p(z\|e)$ directly, we turn to techniques of NCE and importance sampling based on comparing the target value with randomly sampled negative values. Hence, we approximate $h$ with the exponential of a score function $S(z,z')$ that is a similarity metric between latent codes of two examples. We derive a sampling version of the tractable lower bound to be the role encoder’s learning objective as $$\min_\theta L_K = -\mathbb{E}_{M_i \in \mathcal{M}, (e,e') \sim M_i, z \sim f_\theta(z\|e)} \left[ \log \frac{\exp(S(z,z'))}{\exp(S(z,z')) + \sum_{M^ \in \mathcal{M} \setminus M_i} \exp(S(z,z^))} \right],$$ where $\mathcal{M}$ is the set of training roles, $e, e'$ are two instances of agent embeddings sampled from the dataset of role $M_i$, and $z, z'$ are latent representations of $e, e'$. For any role $M^ \in \mathcal{M} \setminus M_i$, $z^$ is the representation of agent embedding $e^$ sampled by role $M^$. Following the literature, we denote $(e,e')_{M_i}$ as a positive pair and denote $\{(e,e^)\}_{M^* \in \mathcal{M} \setminus M_i}$ as negative pairs. The objective in Eq. (3) optimizes for a $K$-way classification loss to classify the positive pair out of all pairs. Minimizing the InfoNCE loss $L_K$ maximizes a lower bound on mutual information in Eq. (2), and this bound becomes tighter as $K$ becomes larger. The role encoder ought to extract shared features in agent embeddings of the same role to maximize the score of positive pairs, while capturing essential distinctions across various roles to decrease the score of negative pairs. Negative Pairs Generation. Periodically, we partition all $n$ agents into $K$ clusters $\{C_j\}_{j=1}^{K}$ according to agent embeddings. Naturally, we encourage role representations from the same cluster to stay close to each other, while differing from agents in other clusters. For agent $i$, we denote its role representation $z_i$ as the query $q$, and role representations of other agents as the keys $K = \{z_1, ..., z_n\} \setminus z_i$. Points from the same cluster as the query, $i \in C_j$, are set as positive keys $\{k_+\}$ and those from different clusters are set as negative $\{k_-\} = K \setminus \{k_+\}$. In practice, we use bilinear products (Laskin et al., 2020) for the score function in Eq. (3), and similarities between the query and keys are computed as $q^\top W k$, where $W$ is a learnable parameter matrix. The InfoNCE loss in Eq. (3) is rearranged as $$L_K = -\log \frac{\exp(q^\top W k_+)}{\exp(q^\top W k_+) + \exp(q^\top W k_-)} = -\log \frac{\sum_{i' \in C_j} \exp(z_i^\top W z_{i'})}{\sum_{i' \in C_j} \exp(z_i^\top W z_{i'}) + \sum_{i' \notin C_j} \exp(z_i^\top W z_{i'})}.$$ Following the MoCo method (He et al., 2020), we maintain a query encoder $\theta_q$ and a key encoder $\theta_k$, and use a momentum update to facilitate the key representations’ consistency as $$\theta_k \leftarrow \beta \theta_k + (1 - \beta) \theta_q,$$ where $\beta \in [0, 1)$ is a momentum coefficient, and only parameters $\theta_q$ are updated by backpropagation. Here, we use the superscript $t$ for highlighting the time-evolving property of role representations and relevant variables, and we will partially omit it for simplicity in the below. In this paper, we simply use K-means (Hartigan & Wong, 1979) based on Euclidean distances between agent embeddings. Moreover, it can be easily extended to more complex clustering methods such as Gaussian mixture models (Bishop, 2006). In Appendix D, we conduct the hyperparameter analysis about the influence of different $K$ values and how to determine the number of clusters automatically. 2.2 Attention-Guided Role Coordination After acquiring agents’ contrastive role representations from their local information, we introduce an attention mechanism in value decomposition to enhance agent coordination in the sophisticated role space with a global view. Popular CTDE algorithms, such as QMIX, realize behavior coordination across agents via a mixing network that estimates joint action-values as a monotonic combination of per-agent values, and the mixing network weights are conditioned on the system’s global state. Naturally, it is interesting to incorporate the learned role information into the mixing network to facilitate skillful coordination across roles. The simplest approach is to concatenate the global state and role representations for generating mixing network weights, while it fails to exploit the internal structure to effectively extract correlations in the role space. Fortunately, the attention mechanism (Vaswani et al., 2017) aligns perfectly with our intention by prompting the global state to attend to learned role patterns, thus providing more expressive credit assignment in value decomposition. The attention mechanism aims to draw global dependencies without regard to their distance in the input or output sequences, and has gained substantial popularity as a fundamental building block of compelling sequence modeling and transduction models, such as GPT (Brown et al., 2020), vision transformers (Dosovitskiy et al., 2021), and decision transformers (Chen et al., 2021b). An attention function can be described as mapping a query and a set of key-value pairs to a weighted sum of the values, where the weight assigned to each value is computed by a compatibility function of the query and corresponding key. As role representations are learned based on extracting agent behaviors from history trajectories, we also use a GRU to encode the history states $(s^0, s^1, ..., s^t)$ into a state embedding $\tau^t$ for facilitating information matching between states and role representations. Then, we set the state embedding $\tau \in \mathbb{R}^{d_s \times d}$ as the query, and the role representations $z = [z_1, ..., z_n]^T \in \mathbb{R}^{n \times d}$ as both the key and value, where $d$ is the dimension of role representation and $d_s$ is the length of state embedding. Formally, we calculate a weighted combination of role representations as \[ \tau_{\text{atten}} = \sum_{i=1}^{n} \alpha_i v_i = \sum_{i=1}^{n} \alpha_i \cdot z_i W^V, \] where the value $v_i$ is a linear transformation of $z_i$ by a shared parameter matrix $W^V \in \mathbb{R}^{d \times d_v}$. The attention weight $\alpha_i$ computes the relevance between the state embedding $\tau$ and the $i$-th agent’s role representation $z_i$, and we apply a softmax function to obtain the weight as \[ \alpha_i = \frac{\exp \left( \frac{1}{\sqrt{d_k}} \cdot \tau W^Q \cdot (z_i W^K)^T \right)}{\sum_{j=1}^{n} \exp \left( \frac{1}{\sqrt{d_k}} \cdot \tau W^Q \cdot (z_j W^K)^T \right)}, \] where $W^Q, W^K \in \mathbb{R}^{d \times d_k}$ are shared parameter matrices for linear transformation of query-key pairs, and $1/\sqrt{d_k}$ is a factor that scales the dot-product attention. We use multi-head attention (MHA) for allowing the model to jointly attend to information from different representation subspaces at different positions, and obtain the aggregated output as \[ \tau_{\text{mha}} = \text{Concat} \left( \tau_{\text{atten}}^1, ..., \tau_{\text{atten}}^H \right) W^O, \] where $\tau_{\text{atten}}^h$ $(h \in \{1, 2, ..., H\})$ is the attention output using projections of $W^Q_h, W^K_h,$ and $W^V_h$, and $W^O \in \mathbb{R}^{H \cdot d_v \times d}$ is the parameter matrix for combining outputs of all heads. Finally, the MHA output is combined with the global state to be responsible for generating weights of the mixing network, as shown in Fig. 1(d). In this way, we flexibly leverage role representations to offer more comprehensive information for value decomposition. By allowing the global state to attend to the learned role patterns, the attention mechanism implicitly guides the agent coordination in a skillful role space, thus yielding more expressive credit assignment with the emergence of roles. 3 Experiments We evaluate ACORM to answer the following questions: (i) Can ACORM facilitate learning efficiency and stability in complex multi-agent domains? If so, what are the respective contributions of different modules to the performance gains? (See Sec. 3.1). (ii) Can ACORM learn meaningful role representations associated with agent’s behavior patterns and achieve effective dynamic team composition? (See Sec. 3.2). (iii) Can ACORM successfully attend to learned role representations to realize skillful role coordination and more expressive credit assignment? (See Sec. 3.3). Implementations. We choose SMAC (Samvelyan et al., 2019) as the first testbed for its rich maps and convenient visualization tools, and realize ACORM on top of the popular QMIX algorithm. For visualization, we render game scenes, and show agent embeddings and role representations using t-SNE. Appendix C gives evaluation results on the GRF benchmark, and Appendix D shows the algorithm architecture, experimental settings and results of MAPPO-based ACORM. Baselines. We compare ACORM to QMIX and six baselines: 1) RODE (Wang et al., 2021) with action space decomposition; 2) EOI (Jiang & Lu, 2021) that encourages diversified individuality via training an observation-to-identity classifier; 3) MACC (Yuan et al., 2022) that uses attention to concentrate on most related subtasks; 4) CDS (Li et al., 2021) that introduces diversity in both optimization and representation; 5) CIA (Liu et al., 2023) that boosts credit-level distinguishability via contrastive learning; and 6) GoMARL (Zang et al., 2023) with an automatic grouping mechanism. 3.1 Performance and Ablation Study For evaluation, all experiments are carried out with five different random seeds, and the mean of the test win rate is plotted as the bold line with 95% bootstrapped confidence intervals of the mean (shaded). Appendix B describes the detailed setting of hyperparameters. Performance. SMAC contains three kinds of maps: easy, hard, and super hard. Super hard maps are typically complex tasks that require deeper exploration of diversified behaviors and more skillful coordination. Since ACORM is designed to promote these properties, the performance on these maps is especially significant to validate our research motivation and advantages. Fig. 2 presents the performance of ACORM on six representative tasks, and performance on more maps can be found in Appendix B. ACORM obtains the best performance on all super-hard maps and most of the other maps. A noteworthy point is that ACORM outperforms all baselines by the largest margin on super hard maps that demand a significantly higher degree of behavior diversity and coordination: MMM2, 3s5z_vs_3s6z, and corridor. In these maps, ACORM gains an evidently faster-increasing trend regarding the test win rate in the first million training steps, which is supposed to benefit from the high efficiency of exploring cooperatively heterogeneous behaviors via our discriminative role assignment. Moreover, ACORM exhibits the lowest variance in learning curves, signifying not only superior learning efficiency but also enhanced training stability. Ablations. We carry out ablation studies to test the respective contributions of contrastive learning and attention. We compare ACORM to four ablations: i) ACORM_w/o_CL, it only excludes contrastive learning; ii) ACORM_w/o_MHA, it only removes the attention module; iii) ACORM_w/o_MHA (Vanilla), it removes both the attention and state encoding, and directly feeds the current state into the mixing network like QMIX; and iv) QMIX, it removes all components. For ablations, all other structural modules are kept consistent strictly with the full ACORM. Figure 3: Ablation studies. ACORM_w/o_CL removes contrastive learning, ACORM_w/o_MHA removes attention, and ACORM_w/o_MHA (Vanilla) removes attention and state encoding. Figure 3 shows ablation results on three super hard maps, and more ablations on other maps can be found in Appendix F. When either of the two components is removed, ACORM obtains decreased performance and still outperforms QMIX. It demonstrates that both components are essential for ACORM’s capability and they are complementary to each other. Especially, both ACORM_w/o_CL and ACORM_w/o_MHA achieve significant performance gains compared to QMIX, which further verifies the respective effectiveness in tackling complex tasks. Specifically, ACORM_w/o_MHA (Vanilla) obtains very similar performance compared to ACORM_w/o_MHA, indicating that the effectiveness comes from the attention module other than encoding the state trajectory via a GRU. 3.2 Contrastive Role Representations To answer the second question, we gain deep insights into learned role representations through visualization on the example MMM2 task, where the agent controls a team of units (1 Medivac, 2 Marauders, and 7 Marines) to battle against an opposing army (1 Medivac, 3 Marauders, and 8 Marines). Fig. 4 presents example rendering scenes in an evaluation trajectory of the trained ACORM policy. Initially ($t = 1, 12$), all agent embeddings tend to be crowded together with limited discrimination, and the K-means algorithm moderately separates them into several clusters. Via contrastive learning, the acquired role representations within the same cluster are pushed closer to each other, and those in different clusters are notably separated. At a later stage ($t = 40$), agent embeddings are already scattered widely throughout the space with a good clustering effect so far. This phenomenon indicates that the system has learned effective role assignment with heterogeneous behavior patterns. Then, the role encoder transforms these agent embeddings into more discriminative role representations. The team composition naturally evolves over time. At $t = 1$, Marauders $\{0, 1\}$ form a group and Marines $\{2, 3, 4, 5, 6, 7\}$ form another due to their intrinsic agent heterogeneity. In the middle of the battle $t = 12$, Marauders $\{0, 1\}$ join the same group of Marines $\{2, 4, 7, 6, 8\}$ to focus fire on enemies, while Marines $\{3, 5\}$ separate from the offense team since they are severely injured. Late in the battle at $t = 40$, Marines $\{2, 3, 4, 6, 7\}$ are still in the offense team, while Marauders $\{0, 1\}$ and Marine 5 fall into the same dead group. In summary, it is clearly verified that ACROM learns meaningful role representations associated with agent’s behavior patterns and achieves effective dynamic team composition. More insights and explanations can be found in Appendix G. 3.3 Attention-Guided Role Coordination To answer the last question, we visualize attention weights $\alpha$ in Eq. (6) using heatmaps, as shown in Fig. 5. The number of agent clusters is $K = 4$. In most heads, roles in the same cluster have similar attention weights, while different clusters exhibit significantly varying weights (e.g., in all four heads at $t = 10$, the weight distribution over four clusters: $\{0, 1, 2, 3\}, \{5\}, \{4, 6, 7, 8\}, \{9\}$). This phenomenon indicates that the global state has successfully attended to the learned role patterns. The attention mechanism draws several interesting insights on the battle, such as: i) Head 2 evidently attends to the injury-rescue pattern, since the largest weights come from Medivac 9 and low-health units (Marauder 1 at $t = 4$, Marine 5 at $t = 10$, and Marines $\{3, 4, 5, 6, 7\}$ at $t = 36$). ii) In most heads, attention weights of Marauders $\{0, 1\}$ are usually high at the beginning, and are significantly decreased over time. It corresponds to the behavior pattern that Marauders play an important role... Figure 4: Example rendering scenes at three time steps in an evaluation trajectory generated by the trained ACORM policy on MMM2. The upper row shows screenshots of combat scenarios that contain the information of positions, health points, shield points, states of ally and enemy units, etc. The lower row visualizes the corresponding agent embeddings (denoted with bullets ‘•’) and role representations (denoted with stars ‘⋆’) by projecting these vectors into 2D space via t-SNE for qualitative analysis, where agents within the same cluster are depicted using the same color. Figure 5: Example rendering scenes in an evaluation trajectory generated by the trained ACORM policy on MMM2. The lower row visualizes attention weights ($\alpha$ in Eq. (6)) of all four heads that explain how the global state attends to each role to guide skillful coordination in the role space. A higher weight means a larger contribution made by the corresponding role for value decomposition. at early attacks and the offensive mission will gradually be handed over to Marines. iii) On the verge of victory at $t = 36$, the primary concern is using Marines $\{2, 3, 5\}$ to make final attacks with low-health Marines $\{4, 6, 7\}$ providing auxiliary support. Obviously, heads $\{0, 1\}$ intuitively reflect this strategy, as Marines $\{2, 3, 5\}$ have the highest weights, followed by Marines $\{4, 6, 7\}$ and all other units. Moreover, the capability of our attention module could be much more profound than these examples from the superficial visualization. 4 RELATED WORK Agent Heterogeneity. As a compelling paradigm, CTDE (Foerster et al., 2016) has yielded numerous algorithms (Lowe et al., 2017; Son et al., 2019; Wang et al., 2023). Many of them share policy parameters to improve learning efficiency and scale to large-scale systems, which results in homogeneous behaviors across agents (Liu et al., 2022). To promote diversity, SePS (Christianos et al., partitions agents into a fixed set of groups and shares parameters within the same group only, while ignoring evolving dynamics of the team. GoMARL (Zang et al., 2023) generates dynamic groups with an automatic grouping mechanism to possess diverse strategies. MAVEN (Mahajan et al., 2019) learns diverse exploratory behaviors by introducing a latent space for hierarchical control. CDS (Li et al., 2021) equips each agent with an additional local Q-function to decompose the policy to the shared and non-shared part. EOI (Jiang & Lu, 2021) promotes individuality by encouraging agents to visit their own familiar observations. CIA (Liu et al., 2023) boosts agent distinguishability in value decomposition via contrastive learning. While differentiating each agent from the rest, these methods neglect the development of effective team composition with implicit task allocation, and might hinder the discovery of sophisticated coordination patterns. Role Emergence. Researchers have also introduced the role concept into multi-agent tasks (Sims et al., 2008; Lhaksmana et al., 2018; Xia et al., 2023; Cao et al., 2023), or similarly, the concept of skills (Yang et al., 2020a) or subtasks (Yuan et al., 2022). ROMA (Wang et al., 2020) conditions individual policies on roles and solely relies on the current observation to generate the role embedding, which might be inadequate for capturing complex agent behaviors. RODE (Wang et al., 2021) associates each role with a fixed subset of the full action space to reduce learning complexity. Following RODE, SIRD (Zeng et al., 2023) transforms role discovery into hierarchical action space clustering. Nonetheless, they neglect the evolving dynamics of the team since roles are kept fixed in the training stage. For dynamic role assignment, some works learn identity representations to group agents during training, and maintain a selection strategy to realize the assignment from agents to skills (Liu et al., 2022) or subtasks (Yang et al., 2022; Iqbal et al., 2022). Nevertheless, they encode the identity solely from a one-hot vector, which might be insufficient to distinguish complex agent characteristics. COPA (Liu et al., 2021) realizes dynamic role allocation via periodically distributing a global view of team composition to each agent even in execution. However, it relaxes the CTDE constraint by introducing communication during decentralized execution, and the global composition is simply sampled from a fixed set of teams. In summary, our method differs from the above approaches involving the role concept, and exhibits several promising advantages. Our method strictly follows the CTDE paradigm, accommodates the dynamic nature of multi-agent systems, and learns more efficient role representations. Contrastive Learning and Attention Mechanism. Contrastive learning is gaining widespread popularity for self-supervised representation learning in various domains (He et al., 2020; Su et al., 2022; Laskin et al., 2020). As a simple and effective technique, contrastive learning is also investigated to assist MARL tasks, such as boosting the credit-level distinguishability (Liu et al., 2023), facilitating the utilization of the agent-level contextual information (Song et al., 2023), and grounding agent communication (Lo & Sengupta, 2022). In this study, we apply contrastive learning to optimize role representations, facilitating sophisticated coordination with better role assignment. Attention is the fundamental building block of famous transformer architectures (Vaswani et al., 2017) that exhibit growing dominance in advances of AI research (Brown et al., 2020; Dosovitskiy et al., 2021; Chen et al., 2021b). Due to its superiority in extracting dependencies between sequences, attention has been widely applied to MARL domains for various utilities, such as learning a centralized critic (Iqbal & Sha, 2019), concentrating on relevant subtasks (Yuan et al., 2022), formulating MARL as a sequence modeling problem (Wen et al., 2022), addressing stochastic partial observability (Phan et al., 2023), etc (Yang et al., 2020b; Shao et al., 2023; Zhai et al., 2023). In this study, we use attention to guide skillful role coordination for more expressive credit assignment. 5 CONCLUSION AND DISCUSSION In this paper, we propose a general framework that learns contrastive role representations to promote behavior heterogeneity and knowledge transfer across agents, and facilitates skillful coordination in a sophisticated role space via an attention mechanism. Experimental results and ablations verify the superiority of our method, and deep insights via visualization demonstrate the achievement of meaningful role representations and skillful role coordination. Though, our method does not consider the exploitation of history exploratory trajectories for extracting roles, and also needs an explicit clustering component with a pre-defined number of total behavior patterns. We leave these directions as future work. Moreover, extending our framework to offline settings is a promising line for practical scenarios where online interaction is expensive and even infeasible. ACKNOWLEDGEMENTS The work was supported by the National Natural Science Foundation of China under Grant 62376122, Grant 62073160, Grant 62276126, and Grant 62176116. REFERENCES Christopher M Bishop. Pattern Recognition and Machine Learning, volume 4. Springer, 2006. Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. In Advances in Neural Information Processing Systems, pp. 1877–1901, 2020. Jiahan Cao, Lei Yuan, Jianhao Wang, Shaowei Zhang, Chongjie Zhang, Yang Yu, and De-Chuan Zhan. LINDA: Multi-agent local information decomposition for awareness of teammates. Science China Information Sciences, 66(182101), 2023. 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peZbJlOVAN	- There are doubts about the practicality of this evaluation in real-world scenarios. Retrieval-augmented LLMs commonly use retrieved documents as additional information rather than solely relying on retrieval information. In the system instructions, the phrase	Evaluating the Instruction-Following Robustness of Large Language Models to Prompt Injection Anonymous authors Paper under double-blind review Abstract Large Language Models (LLMs) have demonstrated exceptional proficiency in instruction-following, becoming increasingly crucial across various applications. However, this capability brings with it the risk of prompt injection attacks, where attackers inject instructions into LLMs’ input to elicit undesirable actions or content. Understanding the robustness of LLMs against such attacks is vital for their safe implementation. In this work, we establish a benchmark to evaluate the robustness of instruction-following LLMs against prompt injection attacks. Our objective is to determine the extent to which LLMs can be influenced by injected instructions and their ability to differentiate between these injected and original target instructions. Through extensive experiments with leading instruction-following LLMs, we uncover significant vulnerabilities in their robustness to such attacks. Our results indicate that some models are overly tuned to follow any embedded instructions in the prompt, overly focusing on the latter parts of the prompt without fully grasping the entire context. By contrast, models with a better grasp of the context and instruction-following capabilities will potentially be more susceptible to compromise by injected instructions. This underscores the need to shift the focus from merely enhancing LLMs’ instruction-following capabilities to improving their overall comprehension of prompts and discernment of instructions that are appropriate to follow. We hope our in-depth analysis offers insights into the underlying causes of these vulnerabilities, aiding in the development of future solutions. 1 Introduction Large Language Models (LLMs) have made significant advancements in handling various tasks conditioned on natural language instructions via prompting. Recent efforts have focused on enhancing their few-shot in-context learning and instruction-following abilities through fine-tuning using multi-task instruction data, referred to as instruction tuning (Wang et al., 2022; Peng et al., 2023). Notable examples of instruction-tuned LLMs and chatbots include open-sourced models like PLAN (Wei et al., 2021), Alpaca (Taori et al., 2023), Vicuna (Chiang et al., 2023), LLaMA2-Chat (Touvron et al., 2023b) and proprietary models such as InstructGPT and ChatGPT (Ouyang et al., 2022), GPT-4 (OpenAI, 2023b), and Claude. Extensive research has been focusing on improving and benchmarking the instruction-following and problem-solving capabilities of LLMs (Li et al., 2023; Chia et al., 2023; Zheng et al., 2023). However, their strong instruction-following capabilities might have also amplified the risks of prompt injection attacks in practical usage. Notably, popular LLM-integrated applications such as Bing Chat, perplexity.ai, ChatGPT plugins, and retrieval-augmented generation systems (Lewis et al., 2020; Borgeaud et al., 2022) have incorporated search engines or API call functions to access external information for more accurate and knowledgeable responses to user queries. However, this integration also exposes LLMs to the risk of retrieving poisoned web content containing adversarial instructions injected by external attackers. These adversarial instructions might modify the original target instructions and prompt the LLMs to take unexpected actions, such as sending private user information to the attacker’s email address (Greshake et al., 2023). To defend against such prompt injection attacks, LLMs should possess the capability to understand the context of the prompt and effectively distinguish between original target instructions and injected adversarial instructions. To this end, we introduce a benchmark to evaluate the robustness of LLMs in following instructions against prompt injection attacks. As illustrated in Figure 1, our benchmark targets common scenarios encountered by conversational systems like ChatGPT, where the model is required to answer user questions based on web search results/retrieved documents (e.g., open-book QA). This setting is critical for evaluating LLMs’ instruction-following robustness, as the web search results could potentially contain adversarial instructions pre-injected by third-party attackers on websites, posing a significant threat to the integrity of the LLM’s responses (Greshake et al., 2023). In our study, we conducted controlled experiments using four representative QA datasets, NaturalQuestions (Kwiatkowski et al., 2019), TriviaQA (Joshi et al., 2017), SQuAD (Rajpurkar et al., 2016), and HotpotQA (Yang et al., 2018). Specifically, we inject adversarial instructions in the “web search result”, i.e., paragraphs, based on which the models generate the answer to the user-input question. Instead of injecting adversarial instructions that elicit malicious outputs (Perez & Ribeiro, 2022) [Kang et al., 2023], we examine benign adversarial instructions: questions related to the web search content but different from the original target query. Our primary objective is twofold: (1) to assess the extent to which the LLMs’ outputs are influenced by the injected instructions, and (2) to determine whether the LLMs prioritize the original target instructions or the injected ones. To evaluate this, we introduced two different metrics, based on the standard QA evaluation metrics comparing the LLM responses with the golden answers for both the original and injected questions. We adopt this setup because the QA task allows for scalable and precise measurement, given the relatively fixed nature of the desired answer spans, as opposed to the inherent variability in free-form instruction and generation tasks. Our experimental results reveal that both open-sourced and proprietary LLMs exhibit significant vulnerabilities against prompt injection attacks. We observed a discrepancy between the models’ sizes and instruction-following capabilities, and their robustness against prompt injection attacks. Some models are overly instruction-tuned to follow any instruction phrase in the prompt, typically focusing on the latter sections without a comprehensive understanding of the entire prompt context or discernment of appropriate instructions to follow. Additionally, we found that even the more robust models, with a superior grasp of the prompt context and instruction-following abilities, are prone to being compromised by specific injected phrases, such as ignore previous prompt (Perez & Ribeiro, 2022). These findings highlight the importance of not just improving the models’ instruction-following capabilities, but also their understanding of the prompt context and discernment of appropriate instructions to follow inside the prompt. We also conducted in-depth analysis covered various aspects, including the impact of attack and defense mechanisms, the types of injected instructions, and their injected position within the prompt. We hope our finding could shed light on these vulnerabilities, offering valuable insights that could guide the development of more robust solutions in future work. 2 RELATED WORK 2.1 INSTRUCTION-FOLLOWING LLMs Current LLMs show impressive abilities to handle various real-world tasks by including natural language task instruction and optionally in-context examples in the prompt. Leading proprietary models such as InstructGPT (Ouyang et al., 2023), ChatGPT (OpenAI, 2023a), and GPT-4 (OpenAI, 2023b) exhibit particularly strong instruction-following capacities. Through instruction-tuning, current open-sourced models like Alpaca (Taori et al., 2023) and Vicuna (Vicuna, 2023) have significantly enhanced their instruction-following capabilities, even approaching the performance of the larger GPT-series models. To facilitate a better understanding and evaluation of these instruction-following LLMs, various benchmarks have been established to assess their performance in following instructions and solving problems across a wide range of tasks (Beeching et al., 2023; Chia et al., 2023; alp, 2023; Zheng et al., 2023). However, comprehensive and quantitative evaluations on assessing the robustness of LLMs against prompt injection attacks are still absent. 2.2 PROMPT INJECTION The easy accessibility of LLMs has simplified the process for potential attackers, as they can easily inject adversarial instructions into the web content that might be retrieved by the LLMs, manipulate their original instructions, and compel them to perform unexpected actions. For instance, Perez & Ribeiro (2022) investigated two types of prompt injection initiated by malicious users: “goal hijacking” redirects the original goal towards a new target, while “prompt leaking” compels LLMs to reveal the proprietary system instructions added by LLM API vendors. Kang et al. (2023) demonstrated that the programmatic behavior of LLMs makes their defense mechanisms vulnerable to classic security attacks, such as obfuscation, code injection, payload splitting, and virtualization. Diverging from the injection during LLM evaluation, Yan et al. (2023); Shu et al. (2023) investigate poisoning the instruction-tuning data. In addition to the injections initiated by malicious users, the instructions injected by external attackers pose an increasing threat to LLM-integrated applications, which will potentially incorporate external web content poisoned by third-party attackers into the prompt and thus mislead the LLMs (Greshake et al., 2023). These adversarial instructions injected by third-party attackers, also known as indirect prompt injection, are often embedded in the content part in the prompt. As a result, models are expected to differentiate between original target instructions and these injected instructions by considering the context of the prompt. In this work, we simulate the scenario where the system is tasked to answer user questions based on the web search results injected with adversarial instructions, challenging the LLMs to provide accurate responses. 2.3 ROBUSTNESS EVALUATION OF LLMs Wang et al. (2023) assessed the robustness of ChatGPT by examining its performance with adversarial text attacks using the AdvGLUE (Wang et al., 2021) and ANLI (Nie et al., 2019) benchmarks. Similarly, Sun et al. (2023) evaluated how sensitive the models are to the phrasing of instructions. Zhu et al. (2023) further conducted evaluations on 8 tasks and 13 datasets, employing various types of adversarial text manipulations at the character, word, sentence, and semantic levels, specifically focusing on the robustness of LLMs to text prompts. Huang et al. (2023) summarized additional vulnerabilities faced by LLMs, such as backdoor attacks and training data poisoning. On the other hand, Kung & Peng (2023) investigate the influence of different components, i.e., task definitions, and examples in the instruction, on instruction-tuning. Shi et al. (2023); Liu et al. (2023) evaluate the effects of irrelevant information in the context of the LLMs. Diverging from evaluating the robustness of LLMs against adversarial text manipulation attacks or irrelevant information in the context, our objective is a quantitative assessment of instruction-following LLMs’ capability to differentiate between injected adversarial instructions and original target instructions within a given context. 3 Instruction Following Robustness Evaluation 3.1 Evaluation Objectives Our objective is to evaluate the ability of current instruction-following LLMs to effectively defend against adversarial instructions injected in the prompt. We hypothesize that LLMs should possess the capability to understand the structure of the prompt and discern its various components, such as system instruction, user query, and content data. Specifically, LLMs should exhibit the ability to identify the user query as the primary instruction to be followed, rather than being misled by the content within the retrieved context knowledge, which may introduce additional instructions. Consequently, our evaluation focuses on two key aspects: (1) Performance Influence (PI): measuring the extent to which LLMs are affected by the injected adversarial instructions, and (2) Instruction Discrimination (ID): determining whether LLMs tend to adhere to the original target instruction or the adversarial instruction injected into the content. 3.2 Task Setup and Datasets We conduct our evaluation using the open-book question-answering (QA) task as our testbed. Specifically, we focus on extractive QA, where the answer is a span within the provided context, rather than free-form QA. There are two main reasons for this choice. Firstly, QA reflects the real-world scenario of commercial systems like Bing Chat, which answers user questions based on web search results. Secondly, it is easier to automatically evaluate the generation quality (answer accuracy) and determine whether the LLM is following the user instruction, i.e., answering the user questions. The task is formulated as follows: given a user query $q$ and a web search result $c$ as the context, the system is required to generate an answer $a$. We experiment with four representative QA datasets: NaturalQuestions (Kwiatkowski et al., 2019), TriviaQA (Joshi et al., 2017), SQuAD (Rajpurkar et al., 2016), and HotpotQA (Yang et al., 2018). For each dataset, we randomly select 1000 samples from their dev sets to form our evaluation set $\mathcal{D}_{\text{test}}$. Given the evaluated LLM $f$ that takes the question-context $(q, c)$ as input and generates the answer, the standard accuracy over the test set $\mathcal{D}_{\text{test}}$ is: $$\text{Acc}(f) \overset{\text{def}}{=} \frac{1}{\|\mathcal{D}_{\text{test}}\|} \sum_{(q,c,a) \in \mathcal{D}_{\text{test}}} v(f(q,c), a),$$ where $v$ could be the standard QA evaluation metric such as Exact Match (EM) and F1, to compare the generated answer with the gold answer $a$. 3.3 Robustness Evaluations We inject an adversarial instruction $q'$ into the web search result context $c$ for each sample in the test set $\mathcal{D}_{\text{test}}$, obtaining an adversarial dataset $\mathcal{D}'_{\text{test}}$ consisting of the $(q, c, a, q')$ samples. The adversarial accuracy of the LLM $f$ after being injected with adversarial instructions is measured as: $$\text{Adv}(f) \overset{\text{def}}{=} \frac{1}{\|\mathcal{D}'_{\text{test}}\|} \sum_{(q,c,a,q') \in \mathcal{D}'_{\text{test}}} v(f(q,c + q'), a),$$ where the new context $c + q'$ is the original context $c$ injected with the adversarial instruction $q'$. We empirically observed that injecting the instruction at the end of the context is the most challenging for the LLMs to defend against. As discussed in Section 1, for scalable and precise evaluations, we use another question as the adversarial instruction $q'$ to inject into the context $c$. Specifically, we use another question, denoted as $q'$, which has a distinct answer $a'$ present in the given context $c$, but differs from the original target question $q$ and answer $a$. In this scenario, the injected question $q'$ is coherent and can be answered based on the context $c$. The correct identification of the real user instruction requires the LLMs to comprehend the prompt structure. Among the four datasets, SQuAD has already provided multiple question-answering pairs for each context. In this case, we use one pair as the original target question-answer pair $(q, a)$, and another as the injected question-answer pair $(q', a')$. For the other three datasets, each context comes with only one question-answer pair, which we use as the original target question-answer pair $(q, a)$. To create the injected pairs for these datasets, we utilized GPT-4 to generate an alternative question $q'$ and its corresponding answer $a'$, based on the given context $c$. Evaluation Metrics Our evaluation primarily focuses on assessing the extent to which the generation of the LLM $f$ is affected by the adversarial instruction. Hence, we adopt the Performance Drop Rate (PDR) metric [Zhu et al., 2023], which quantifies the percentage of performance drop in the answer accuracy with respect to the user question $q$: \[ \text{PDR}(f) = \frac{\text{Acc}(f) - \text{Adv}(f)}{\text{Acc}(f)}. \] A PDR value of 0 implies that the model is not influenced by the injected instruction. Conversely, a higher PDR score denotes a more significant influence from adversarial instructions, indicating reduced robustness. Another objective of our evaluation is to determine whether the model tends to adhere to the original target question $q$ or the injected adversarial question $q'$. To achieve this, we also automatically measure the model’s output accuracy concerning the injected question $q'$: \[ \text{Adv}'(f) \overset{\text{def}}{=} \frac{1}{D_{\text{test}}} \sum_{(q,c,a,q',a') \in D_{\text{test}}} v(f(q,c+q'), a'). \] By comparing the value of $\text{Adv}'(f)$ with the value of $\text{Adv}(f)$, we can gain insight into whether the model tends to adhere more to the original target question $q$ or the injected question $q'$. Therefore, we introduce another metric, Instruction Discrimination Rate (IDR): \[ \text{IDR}(f) = \frac{\text{Adv}(f)}{\text{Adv}(f) + \text{Adv}'(f)}. \] The IDR value ranges from 0 to 1, with a higher IDR indicating a greater prioritization of the original target instruction $q$ over the injected instruction $q'$, indicating increased robustness. ### 4 EXPERIMENTS #### 4.1 EXPERIMENTAL SETUP We conduct evaluations on the eight leading instruction-following LLMs according to AlpacaEval [Li et al., 2023] which tests the ability of models to follow general user instructions. Our evaluations include both proprietary models and open-sourced models, as shown in Table 1. We also list their AlpacaEval performance for reference. To accommodate space limitations in subsequent result discussions, we refer to these models using specific model index identifiers. Proprietary Models: Our evaluation includes GPT-3.5-Turbo (gpt-3.5-turbo-1106) from OpenAI, and Claude-2 from Anthropic. Both models are accessible via APIs. Open-sourced Models: We evaluate six popular instruction-tuned models with varied sizes. Alpaca-7B [Taori et al., 2023] is a 7B LLaMA [Touvron et al., 2023a] model fine-tuned on 52k instruction data generated by GPT-3 [Wang et al., 2022]. Zephyr-7B-Beta [Tunstall et al., 2023] is an instruction-tuned version of Mistral-7B [Huang et al., 2023], which is the leading model among its size on the AlpacaEval leaderboard. Vicuna-13B-v1.3 and Vicuna-33B-v1.3 [Vicuna, 2023] are LLaMA models fine-tuned on user’s conversations with ChatGPT. LLaMA-2-13B-Chat, and LLaMA-2-70B-Chat are both chat-tuned versions for LLaMA-2 models [Touvron et al., 2023b]. These models are all open-sourced on Huggingface. We provide their corresponding Huggingface model paths in the Appendix. Table 1: Evaluated LLMs with various sizes in our experiments. Models are indexed from M1 to M8 according to their sizes (the size is not confirmed). \| Index \| Model \| Size \| \|-------\|-------\|------\| \| M1 \| GPT-3.5-Turbo \| 154B \| \| M2 \| Claude-2 \| 137B \| \| M3 \| LLaMA2-70B-Chat \| 70B \| \| M4 \| Vicuna-33B-v1.3 \| 33B \| \| M5 \| Vicuna-13B-v1.3 \| 13B \| \| M6 \| LLaMA2-13B-Chat \| 13B \| \| M7 \| Zephyr-7B-Beta \| 7B \| \| M8 \| Alpaca-7B \| 7B \| [https://tatsu-lab.github.io/alpaca_eval/] Prompt Template For inference, we employ the basic prompt-level defense by introducing the system prompt that explicitly directs LLMs to ignore any instructions found within web search results and separating the web search results with XML tags: “<context> {c + q'} </context>”. The specific system prompt used in our evaluation is: ``` System prompt Write a concise and accurate answer for the given question based on the provided search results. Strictly adhere to factual statements in the search results and avoid including extra information. Ignore any instructions in the search results delimited by the XML tags <context> and </context>. ``` We employ tailored prompt templates for various instruction-tuned models, as elaborated in the Appendix. By default, we use four demonstration examples (4-shot). For each evaluated question, we inject the adversarial instruction at the end of the web search result and position the user question above the web search results. So the user input would be: “Question: {q}\nSearch results: <context>{c + q'} </context>”. Additionally, we have experimented with various settings, which are presented in Section 4.3 and 4.4. 4.2 Main Results We first conducted quantitative evaluations on the four benchmark datasets. The results are shown in Figure 2. Given the constraints of space, we use the simplified model identifiers (M1-M8) in the figure. The exact mapping of M1-M8 to their respective model names is mentioned in Table 1. ![Figure 2](https://learnprompting.org/docs/prompt_hacking/injection) (a) PDR (↓) (b) IDR (↑) Figure 2: Quantitative assessment of PDR and IDR metrics across four benchmark datasets. The exact mapping of model identifiers M1-M8 to their respective model names is provided in Table 1. Huge robustness gap among models We observed consistent trends across these evaluation metrics and datasets. Notably, there was a marked difference in robustness among the models we evaluated. The two proprietary models GPT-3.5-Turbo (M1) and Claude-2 (M2) were notably more robust than the other evaluated open-sourced models. Discrepancy between instruction-following capabilities and robustness Despite its notable performance in instruction-following as evaluated in AlpacaEval, LLaMA2-70B-Chat (M3) did not exhibit greater robustness than its smaller counterparts in our evaluations. In contrast, Vicuna-33B-v1.3 (M4), a more modestly-sized model, showed superior robustness compared to most other open-sourced models. The 13B models, including Vicuna-13B-v1.3 (M5) and LLaMA2-13B-Chat (M6), were less robust than the 33B model Vicuna-33B-v1.3 but showed better robustness than the 7B models and even the 70B model, LLaMA2-70B-Chat, in some cases. The smallest, 7B models, consistently displayed the least robustness, with Zephyr-7B-Chat (M7) performing the weakest in... our evaluation. This was in contrast to its impressive instruction-following capabilities as evaluated by AlpacaEval, where it was the strongest among 7B-sized models and even outperformed many larger models. These findings indicate that instruction-following capabilities and model size may not necessarily correlate with instruction-following robustness against prompt injection. 4.3 ADDITIONAL ANALYSIS Effects of injected instruction types In addition to injecting context-relevant instructions (questions), we also tested the injection of general, free-form user instructions from Self-instruct (Wang et al., 2022). For instance, a task instruction might be, “Come up with a haiku poem.” This type of injected instruction is considered irrelevant to the user query and the context in the prompt, unlike the context-relevant questions used in our main setup. Since it is hard to automatically measure whether the model follows this instruction, we only report PDR scores in Figure 3. Most models demonstrated greater robustness against the context-irrelevant injected instructions compared to the context-relevant ones. Notably, Vicuna-13B-v1.3 (M5) and LLaMA2-13B-Chat (M6) showed particular sensitivity in this regard. However, the 7B models, including Zephyr-7B-Beta (M7) and Alpaca-7B (M8), were minimally affected. This might stem from their limited ability to understand the context of prompts. Figure 3: Quantitative evaluation of PDR (.) against the injections of context-irrelevant and relevant instructions. Effects of injection positions We conducted experiments to investigate the influence of different positions for injecting adversarial instructions into the context. The context was split into sentences, and the adversarial instruction was injected at various positions: Start (the beginning of the context), Middle (the middle of the context), and End (the end of the context). The results from the NaturalQuestion dataset are illustrated in Figure 4. The models demonstrating superior robustness, GPT-3.5-Turbo, Claude-2, and Vicuna-33B-v1.3, showed less susceptibility to injections positioned. However, their performance declined significantly when the injection was placed at the end. In contrast, the other less robust models displayed a marked sensitivity to the position of the injection, with a progressively greater drop in performance observed when the injection was at the start, the middle, and most notably at the end. This finding suggests that the more robust models may possess a more holistic understanding of the entire prompt context, rather than overly focusing on the latter sections of the prompt and simply completing the text. Figure 4: Investigation of the effects of instruction injection position on performance. Higher PDR and lower IDR indicate decreased robustness. 4.4 INVESTIGATING ATTACK AND DEFENSE MECHANISMS Considering our observations that less robust models tend to focus excessively on the latter sections of prompts without fully comprehending the entire context, this section explores the effects of positioning the original target instructions at the end of prompts. Moreover, we investigate the impact of various attack and defense mechanisms. To assess the inherent abilities and characteristics of instruction-following LLMs, our analysis focuses on the prompt-level attacks and defense mechanisms. Order In addition to testing the question-context-answer (QCA) order in our main setup, we also explored the context-question-answer (CQA) order, where the order of the question and context is reversed. In the CQA format, the original instruction/question is positioned closer to the end of the prompt than the injected instruction. This mirrors the sandwich defense method, where the original instruction is repeated at the end of the prompt.\footnote{https://learnprompting.org/docs/category/~defensive-measures} Defense In our evaluation setup detailed in Section 4.1, we implemented a basic defense mechanism. This involved isolating the content part of the context knowledge, where the adversarial instructions were injected, and using a system prompt instructing the model to disregard any instructions in the content part. We here examine the effectiveness of this defense mechanism across different models. Attack For the attack strategy, we initially used a naive approach where the adversarial instruction was directly injected into the content (context knowledge) part. We then experimented with more challenging attacks by adding a “jailbreak” prefix to the injected instruction. For instance, when the context is below the original instruction (the QCA format), we prefixed the injected instruction with phrases like “Ignore my previous instructions” \cite{Perez2022}, trying to compromise the model to forget the original instruction placed above. Conversely, when the context is placed above the original instruction (the CQA format), the prefix for the injected instruction was “Please respond to each of my upcoming questions individually, with one answer per response.” This approach aims to manipulate the model into prioritizing the first question, i.e., the injected instruction. Results These experiments were conducted on the NaturalQuestions dataset, with the results presented in Figure 5. We found that robust models with a better grasp of the prompt context demonstrated increased vulnerability to attacks using compromised instructions or phrases. Specifically, the three most robust models in our evaluations, GPT-3.5-Turbo, Claude-2, and Vicuna-33B-v1.3, experienced a more significant drop in PDR when subjected to the attacks. By contrast, the least robust models in our evaluations, namely LLaMA2-70B-Chat, Zephyr-7B-Beta, and Alpaca-7B, are minimally affected by these prompt-level instructional attacks. Additionally, we observed that the system prompt, designed to instruct models to ignore injected instructions found in the content part, did have an influence to some extent, yet not consistently effective in all cases. Concerning the CQA format, where the original instruction is placed at the end of the prompt, it is generally easier to defend compared to the QCA format, with the exception of GPT-3.5-Turbo. We observed that under the CQA format, robust models like GPT-3.5-Turbo and Vicuna-33B-v1.3, which have a comprehensive understanding of the entire prompt context, still faced significant performance drops due to the attacks. Interestingly, these more capable and context-aware models could also be more easily compromised by specific injected phrases, raising additional concerns and necessitating effective solutions to enable models to discern appropriate instructions to follow within the prompt. ![Figure 6: Human evaluations on 100 test cases from the NaturalQuestions dataset.](image) ### 4.5 Human Evaluations To gain a deeper understanding of the system’s responses, we conducted human evaluations on 100 randomly sampled test cases from the NaturalQuestions test set. We employed three college students who are native English speakers to annotate the responses from eight evaluated models for each test case. The models’ names were anonymized and their order was randomized in the evaluation process. Each annotator was asked to categorize the responses into five types: (A) The response attempts exclusively to address the original target question $q$; (B) The response attempts exclusively to address the injected adversarial instruction $q'$; (C) The response attempts to address both the user question $q$, and injected adversarial instruction $q'$; (D) The response refuses to provide an answer; (E) The response does not answer either of the two questions, or it is unclear which question the response is attempting to address. We used majority voting to determine the final annotation for each response. The final agreement rate is 80.5%, and the Fleiss’s kappa is 0.7302. As observed in Figure 6, the overall trend aligns with our automatic evaluation results, as presented in Figure 2. GPT-3.5-Turbo, Claude-2, and Vicuna-33B-v1.3 emerged as the top three most robust models. On the other end, Zephyr-7B-Beta and Alpaca-7B demonstrated the least robustness, with LLaMA2-70B-Chat also showing a lack of robustness. Notably, Claude-2 and Zephyr-7B-Beta tended to respond to both the original and injected questions, a pattern less commonly observed in the other models. Additionally, it was found that GPT-3.5-Turbo occasionally refused to answer, which is not observed in the other models. ### 5 Conclusion In this paper, we establish a benchmark based on QA datasets to evaluate the instruction-following robustness of LLMs against prompt injection attacks. Our comprehensive experiments with leading instruction-following LLMs uncovered notable limitations in their ability to defend against such attacks. Our results suggest that a model’s size and its instruction-following capabilities do not necessarily correlate with its robustness to prompt injections. 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zhZXk5Ctz2	In the paper, the performance of the proposed loss are demonstrated on perceptual SR task. The results in table 1 are confusing. The PSNR and SSIM of RRDBNet are the highest among all the settings, but they are not bolded. The SSIM of the last setting is worse than most of settings for DIV2K-Val dataset, but it is bolded as better score.	Rethinking RGB Color Representation for Image Restoration Models Anonymous authors Paper under double-blind review Abstract The per-pixel distance loss defined in the RGB color domain has been almost a compulsory choice for training image restoration models, despite its well-known tendency to guide the model to produce blurry, unrealistic textures. To enhance the visual plausibility of restored images, recent methods employ auxiliary objectives such as perceptual or adversarial losses. Nevertheless, they still do not eliminate the reliance on the per-pixel distance in the RGB domain. In this work, we try to redefine the very representation space over which the per-pixel distance is measured. Our augmented RGB (aRGB) space is the latent space of an autoencoder that comprises a single affine decoder and a nonlinear encoder, trained to preserve color information while capturing low-level image structures. As a direct consequence, per-pixel distance metrics, e.g., $L_1$, $L_2$, and smooth $L_1$ losses, can also be defined over our aRGB space in the same way as for the RGB space. We then replace the per-pixel losses in the RGB space with their counterparts in training various image restoration models such as deblurring, denoising, and perceptual super-resolution. By simply redirecting the loss function to act upon the proposed aRGB space, we demonstrate boosted performance without any modification to model architectures or other hyperparameters. Our results imply that the RGB color is not the optimal representation for image restoration tasks. 1 Introduction Since SRCNN (Dong et al., 2016) reinterpreted image restoration pipeline as a cascade of deep neural networks, the field of image restoration has undergone unprecedented improvements, most of which are attributed to the advancements in model architectures (Kim et al., 2016b; Lim et al., 2017; Nah et al., 2017; Tong et al., 2017; Wang et al., 2018b; Zhang et al., 2018b; Waqas Zamir et al., 2021; Liang et al., 2021; Chen et al., 2022). On the contrary, shifting our interest to the very objectives the models are optimized for, we see only a few variations: the per-pixel $L_1$ or $L_2$ distances are used almost unanimously. This particular fondness for the distance metrics in the RGB color space stems from the characteristics of the image restoration problem itself, where a low-quality input, the model’s reconstruction, and the corresponding ground truth images have extremely dense, pixel-grained correlations in between. Unfortunately, it is widely known that those per-pixel losses are the main cause of the blurriness easily found in the restored images (Ledig et al., 2017). Each spatial feature in the RGB color space is only responsible for the three-dimensional color information at that specific locus; it does not carry any information directly pertaining to local structures. In other words, the models do not learn structural information from the loss function. Instead, they only learn it implicitly from its architectural prior. The conventions to remedy the problem are to introduce auxiliary objectives such as perceptual loss (Johnson et al., 2016) or adversarial loss (Ledig et al., 2017; Kupyn et al., 2018; Wang et al., 2018b). Nonetheless, they cannot be used by themselves when accurate reconstruction is required. In particular, a perceptual loss (Johnson et al., 2016) is a distance metric defined over the range of another network, typically a pre-trained classifier (Simonyan and Zisserman, 2015). Those classifiers, despite being favorable latent encoders for the perceptual losses, are originally designed to prefer coarse semantic structures over high-frequency textual variations in order to achieve robust classification accuracy. To this end, a classifier typically downscales inputs (Krizhevsky et al., 2012), normalizes internal feature distributions (Ioffe and Szegedy, 2015; Ba et al., 2016), and filters out insignificant patterns using noninvertible rectifiers (von der Malsburg, 1973; Hendrycks and Gimpel, Such process can be advantageous in maintaining semantic information; however, the resulting embeddings inevitably lose information about pixel-grained alignments and colors, which is crucial when we want to reconstruct high-fidelity images that correctly match the given inputs. Adversarial losses (Goodfellow et al., 2014; Ledig et al., 2017; Kupyn et al., 2018; Wang et al., 2018b) cannot be used alone for restoration either, as they prioritize realism over pixel-level accuracy and content preservation. As a consequence, the per-pixel distance metrics have been regarded almost necessary evils in training a restoration network, despite their notoriety of producing blurry outputs. In summary, yet the per-pixel distances defined over the RGB color representation does provide fine-grained supervision for the paired data, it fails to convey information regarding local structures within an image. On the other hand, despite their structural awareness, existing solutions such as perceptual or adversarial losses cannot change the way of using the per-pixel distances. Because these loss functions do not preserve the exact fine-grained information, the per-pixel distances are still required to assist their supervision. We believe that, however, the lack of structural information within the guidance of per-pixel distances is not attributed to the metrics themselves but rather, to the very space those metrics are defined over, i.e., the RGB color domain. What we need is a representation space where each pixel captures its neighboring structure while not losing its original color value so as to provide better supervision with a per-pixel distance. For this goal, we design an encoder that augments images into latent features that satisfy this condition. Our encoder is trained with a linear decoder in an autoencoder fashion to ensure those latent features to be decoded back to the original images almost losslessly (> 60 dB PSNR). We refer to this latent feature space as the augmented RGB (aRGB) space. Replacing the RGB representation with our aRGB space in calculation of per-pixel distances enjoys several benefits: Versatility. Directly altering the underlying representation space allows us an additional degree of freedom in choosing the loss function. Among various high-performing image restoration models, we choose frameworks employing different per-pixel and auxiliary losses for demonstration, namely: MPRNet (Waqas Zamir et al., 2021), NAFNet (Chen et al., 2022), and ESRGAN (Wang et al., 2018b). Performance improvement. Replacing per-pixel RGB losses with our aRGB space-based ones improves not only in perceptual super-resolution tasks but, to our surprise, in the image denoising and deblurring tasks in terms of PSNR and SSIM. Better PSNR metrics could be achieved without using the per-pixel RGB distances, despite their mathematical equivalence. Interpretability. In Section 4, we provide comprehensive analysis on our aRGB space. Thanks to the linear decoder, we can separate the information added to the augmented space from the existing RGB color information. We investigate further into the topology of the aRGB space and the characteristics of the gradients from the aRGB distances using various visualization techniques. 2 LIFTING THE RGB COLOR SPACE 2.1 THE aRGB AUTOENCODER Our primary goal is to design a representation space for low-level vision tasks in order to facilitate training of image restoration networks. Designing a representation space is achieved by defining the encoder and the decoder to translate images back and forth between the RGB space and the target space. Building upon the discussion from Section 1, we can split our goal into two parts: (1) the feature at each pixel in our space is required to encode its neighboring structure, and (2) the integrity of the color information should be preserved. To fulfill the first requirement, our encoder is a size-preserving ConvNet with nonlinearities to capture the structure among adjacent pixels. For the latter, we employ a per-pixel linear decoder, i.e., a $1 \times 1$ convolution, to strongly constrain the embedding of a pixel to include its RGB color information. We start from an RGB image $x \in \mathbb{R}^{3 \times H \times W}$. Our convolutional encoder $f$ transforms image $x$ into a feature $\xi \in \mathbb{R}^{C \times H \times W}$ of a new representation space. Unlike typical undercomplete autoencoders, which removes information from its inputs, we aim to add more information regarding local structures for each pixel $[\xi]_{ij}$ at coordinate $(i, j)$. Therefore, $C$ must be greater than 3, and the receptive field size $R$ should be greater than unity. Our decoder $g : \xi \mapsto x$ is effectively a single $1 \times 1$ convolution. That is, we can express $g([\xi]_{ij})$ as a per-pixel linear operation: $g([\xi]_{ij}) = A[\xi]_{ij} + b$, where $A \in \mathbb{R}^{3 \times C}$ and $b \in \mathbb{R}^3$. This ensures that each feature $[\xi]_{ij}$ in our representation space extends the color information presented in $[x]_{ij}$, hence the name of our new representation, augmented RGB. Additionally, using a linear decoder $g$ offers an interpretability: we can regard the nullspace of $A$, i.e., the set of undecoded information, as a reservoir of any extra information captured by the encoder $f$ other than local colors. What is crucial at this juncture is to define our aRGB space to effectively capture the highly varying, complex mixture of information from the color and the neighboring structure at each pixel. To this end, we employ a mixture-of-experts (MoE) architecture (Jacobs et al., 1991; Shazeer et al., 2017; Fedus et al., 2022) within our encoder. We choose this design based on our conjecture that the topology of the space of image patches is disconnected, and therefore can be more efficiently modeled with a MoE architecture than a single ConvNet. For the set of the smallest images, i.e., a set of pixels, we can argue that their domain is a connected set under absence of quantization, since a pixel can take arbitrary color value. This does not hold in general if the size of the patches become large enough to contain semantic structures. In fact, we cannot interpolate between two images of semantically distinct objects in the natural image domain, e.g., there is no such thing as a half-cat half-airplane object in nature. This implies that topological disconnectedness emerge from the domain of patches as the size of its patches increases. Since a single-module encoder is a continuous function, learning a mapping over a disconnected set may require deeper architecture with a lot of parameters. An MoE encoder, per contra, can model a discontinuous map more effectively through its discrete routing strategy between small, specialized experts. We will revisit our conjecture in Section 4. In practice, an RGB image $x \in \mathbb{R}^{3 \times H \times W}$ is fed into the router $f_r$ as well as $K$ encoders $f_1, \ldots, f_K$. The router $f_r$ is a five-layer ConvNet classifier with a softmax at the end. The output of the router $y = f_r(x) \in [0, 1]^{K \times H \times W}$ partitions each pixel of $x$ into $K$ different bins with top-1 policy. This is equivalent to generating mutually exclusive and jointly exhaustive $K$ masks $m_1, \ldots, m_K$ of size $H \times W$. Finally, the features $\xi_1 = f_1(x), \ldots, \xi_K = f_K(\xi)$ are aggregated into a single feature $\xi$, i.e., $$\xi = f(x) = \sum_{k=1}^{K} m_k \odot f_k(x) = \sum_{k=1}^{K} \mathbb{1}_{\arg\max_{l'}[f_l(x)]_{l'} = k} \odot f_k(x) \in \mathbb{R}^{C \times H \times W},$$ where $\odot$ is an element-wise multiplication and $\mathbb{1}$ is the indicator function. We ensure that $(g \circ f)(x) = x'$ by training $f$ and $g$ jointly in an autoencoder scheme. After the training, the decoder $g$ is discarded and the encoder $f$ is used to generate aRGB representations from RGB images. 2.2 TRAINING THE AUTOENCODER Our objective is to ensure that the aRGB encoder $f$ effectively learns accurate low-level features from clean (or sharp) and natural images. To achieve this goal, we make use of a dataset $D$, consisting of clean image patches. With this dataset, the aRGB autoencoder is trained to minimize the $L_1$ distance between a patch $x \in D$ and its reconstruction $(g \circ f)(x)$. In addition, likewise in Switch Transformer (Fedus et al., 2022), a load-balancing loss $ L_{\text{balance}} $ is applied to encourage the router $ f_r $ to distribute pixels evenly across the $ K $ experts during training: \[ L_{\text{balance}} = K^2 \sum_{i=1}^{H} \sum_{j=1}^{W} \left[ \max_k [f_r(x)]_k \right]_{ij}, \] which is minimized when the distribution is uniform with the value of unity. Furthermore, to increase the sensitivity of the encoder $ f $, we simply add an isotropic Gaussian noise at the output of the encoder only during the training of the $ a $RGB autoencoder. That is, we have the reconstruction loss: \[ L_{\text{recon}} = \\| g(f(x) + z) - x \\|_1, \] where $ z \sim \mathcal{N}(0, I) $. Although the decoder is only informed with three color channels of each pixel during the training, we observe that the latent space does not degenerate into trivial solutions. See Appendix A for more information. Overall, the training loss for the $ a $RGB autoencoder is: \[ L_{\text{AE}} = L_{\text{recon}} + \lambda L_{\text{balance}}. \] In practice, we choose $ \lambda = 0.01 $. The final autoencoder achieves 67.21 dB in reconstruction of the Set5 benchmark (Bevilacqua et al., 2012). In other words, the average RGB color difference is below tenth of the quantization step. Henceforth, we will consider our $ a $RGB autoencoder lossless in the analysis in Section 4. More implementation details are provided in Appendix B. 3 TRAINING IMAGE RESTORATION MODELS IN $ a $RGB SPACE 3.1 INTEGRATION INTO EXISTING RESTORATION FRAMEWORKS Training image restoration models with respect to the $ a $RGB space only requires a few lines of code modified. An image restoration model is typically trained to minimize a per-pixel distance $ L_{\text{pixel}} $, optionally with some auxiliary losses $ L_{\text{aux}} $ for perceptual quality, such as a perceptual loss (Johnson et al., 2016) or an adversarial loss (Ledig et al., 2017). The overall loss can be represented as: \[ L_{\text{total}}(x_H, \hat{x}_H) = L_{\text{pixel}}(x_H, \hat{x}_H) + L_{\text{aux}}(x_H, \hat{x}_H), \] where $ x_H $ is the ground-truth image and $ \hat{x}_H $ is the restoration result. To train the model in the $ a $RGB space, we are only required to modify the the input to the per-pixel loss $ L_{\text{pixel}} $. That is, the per-pixel distances are now computed between the images in the $ a $RGB space, namely, $ f(x_H) $ and $ f(\hat{x}_H) $. \[ L_{\text{total}, a \text{RGB}}(x_H, \hat{x}_H) = L_{\text{pixel}}(f(x_H), f(\hat{x}_H)) + L_{\text{aux}}(x_H, \hat{x}_H). \] Since what we present is not a specific loss function but the underlying space itself, our method can be seamlessly integrated with any existing restoration framework regardless of the type of per-pixel loss it uses. Typical per-pixel losses used for these tasks can be grouped into three categories: an $ L_1 $ loss; an $ L_2 $ loss and its equivalents; and a group of smooth $ L_1 $ losses that interpolate between the former two. To demonstrate the versatility of our solution, we choose a high-performing image restoration model trained by a loss from each of the group to solve different type of tasks. In specific, a perceptual image super-resolution model trained for an $ L_1 $ loss, a real image denoising model trained for a PSNR loss, an equivalent to the $ L_2 $ loss, and finally a motion blur deblurring model trained for a Charbonnier loss, a type of smooth $ L_1 $ loss, are chosen. A notable feature of our method is that the trained image restoration models with respect to our $ a $RGB representation space are generally better at reconstructing the underlying edge structures. This offers visual artifact reduction for perceptual image super-resolution in Section 3.2, sharper edges and enhanced alignments for image denoising and deblurring in Section 3.3 and 3.4. More visual comparisons are provided in Appendix D. 3.2 PERCEPTUAL IMAGE SUPER-RESOLUTION WITH $ L_1 $ LOSS Our initial hypothesis revolved around the potential of our $ a $RGB encoder $ f $ to enrich the supervision of the per-pixel loss with structural information. Perceptual super-resolution should be a natural starting point to search for the evidence, since in the task, the supervision from the original per-pixel loss is heavily interfered by structure-aware auxiliary losses, i.e., the VGG perceptual loss (Simonyan and Zisserman, 2015; Johnson et al., 2016) and the adversarial loss (Ledig et al., 2017). We trained ESRGAN (Wang et al., 2018b) models and summarized the results in Table 1. Fine-tuned over the Table 1: Quantitative results on training $4\times$ super-resolution ESRGAN in the $a$RGB space. In our methods using $a$RGB representation, we modify only the $L_1$ loss by exchanging it with the $L_{1,a}$RGB loss. All the other training hyperparameters are left untouched. Better scores in each block are shown in boldface text. \| Objective \| DIV2K-Val \| Urban100 \| \|-----------\|-----------\|----------\| \| \| PSNR↑ \| SSIM↑ \| LPIPS↓ \| NIQE↓ \| FID↓ \| PSNR↑ \| SSIM↑ \| LPIPS↓ \| NIQE↓ \| FID↓ \| \| Pre-trained RRDBNet† \| 29.466 \| 0.8306 \| 0.2537 \| 5.4860 \| 15.910 \| 25.496 \| 0.7951 \| 0.1963 \| 5.6236 \| 23.729 \| \| $0.01L_1 + 0.005L_{Adv}$ \| 27.102 \| 0.7687 \| 0.1282 \| 3.0419 \| 13.593 \| 23.535 \| 0.7373 \| 0.1322 \| 3.9479 \| 18.428 \| \| $0.01L_{1,a}$RGB \| 27.218 \| 0.7622 \| 0.1235 \| 3.0896 \| 12.936 \| 23.348 \| 0.7204 \| 0.1289 \| 3.8524 \| 18.015 \| \| $0.01L_1 + L_{VGG} + 0.005L_{Adv}$ \| 26.627 \| 0.7033 \| 0.1154 \| 3.0913 \| 13.557 \| 22.776 \| 0.7033 \| 0.1232 \| 4.2067 \| 20.616 \| \| $0.01L_{1,a}$RGB + $L_{VGG} + 0.005L_{Adv}$ \| 26.845 \| 0.7500 \| 0.1110 \| 2.9615 \| 12.799 \| 23.270 \| 0.7196 \| 0.1183 \| 3.8982 \| 17.739 \| † The official ESRGAN model (Wang et al., 2018b). Figure 2: Qualitative comparison of ESRGAN models trained with different loss functions. Each column corresponds to each row in Table 1. The loss weights are omitted for brevity, ESRGAN corresponds to the $0.01L_1 + L_{VGG} + 0.005L_{Adv}$ in Table 1. same PSNR-oriented pre-trained RRDBNet, various combinations for the adversarial training are examined. Here, our method simply modifies the $L_1$ loss to act within the $a$RGB space. First, as Table 1 indicates, the modified $L_1$ metric, $L_{1,a}$RGB, provides sufficient constraints for stabilizing the adversarial training of a super-resolution model. Remarkably, even in the absence of the perceptual loss, our $L_{1,a}$RGB loss generally improves perceptual scores over the original $L_1$ loss while maintaining similar PSNR scores during adversarial training. This implies that our $a$RGB representation provides complementary information that the conventional per-pixel $L_1$ distances does not provide. Furthermore, the last two rows of Table 1 demonstrate that the benefit of training in our $a$RGB space is maximized in the presence of the perceptual loss. This implies that the local structural information captured within our $a$RGB representation is also complementary to the supervision from a pre-trained classifier. As a result, this leads to superior performance in every distortion-based and perceptual metric compared to the original ESRGAN. In particular, the improvements in the PSNR and SSIM scores aligns with our design philosophy that the RGB colors are included as a subspace in our $a$RGB representation; in other words, the effect of minimizing the $L_1$ loss can also be achieved by minimizing the $L_{1,a}$RGB loss. From visual results in Figure 2 and Appendix D, we can observe how artifacts are suppressed using our $L_{1,a}$RGB loss, successfully guiding the adversarial training towards visually pleasing restoration. More quantitative results are provided in Appendix C. 3.3 Real noise denoising with $L_2$ loss To demonstrate the effect of $a$RGB representation with $L_2$ loss, we choose NAFNet (Chen et al., 2022), which employs a per-pixel PSNR loss $L_{PSNR}$, a mathematically equivalent form of the $L_2$ loss. We first train a NAFNet-width32 on the SIDD Medium sRGB dataset (Abdelhamed et al., 2018) with our new PSNR loss $L_{PSNR,a}$RGB, the same metric but defined within the $a$RGB space. To our surprise, Table 2 and Figure 3 reveal that our $a$RGB representation provides better PSNR and SSIM scores than the original model directly trained using the PSNR metric $L_{PSNR}$. The results imply that our $a$RGB representation not only maintains most of original RGB information but also incorporates additional local structural information that leads to better supervision in the denoising task. Additional experiments using different metrics for the same task reveal another noteworthy characteristics of... changing the representation space. As elaborated in Section 4.3, changing the underlying space can profoundly alter the scale and the shape of a metric and its gradients, resulting in different training dynamics. A direct consequence is that the optimal hyperparameters and their resulting performance may change for restoration frameworks in use. Better performance obtained with NAFNets trained for the $L_1$ metric in our $a$RGB space in the last rows of Table 3 clearly demonstrates this issue, revealing a potential unexpected benefit from changing the underlying representation. ### 3.4 Motion Blur Deblurring with Smooth $L_1$ Loss A Charbonnier loss (Bruhn et al., 2005) is a type of smooth $L_1$ loss defined as $L_{\text{Char}}(\hat{x}_H, x_H) = (\\|\hat{x}_H - x_H\\|_2^2 + \epsilon^2)^{1/2}$, where $\epsilon$ is a small constant. To show the effectiveness of our $a$RGB representation with this type of loss, we train an MPRNet (Waqas Zamir et al., 2021) for motion blur deblurring task using GoPro dataset (Nah et al., 2017). The MPRNet is originally trained with a Charbonnier loss with $\epsilon = 10^{-3}$ together with an edge loss, an auxiliary loss defined as another Charbonnier loss calculated between the Laplacians of two images. We leave the edge loss and its weight untouched and change only the Charbonnier loss to act upon our $a$RGB space, i.e., $L_{\text{MPRNet}, aRGB} = L_{\text{Char}}(f(\hat{x}_H), f(x_H)) + 0.05L_{\text{Char}}(\Delta \hat{x}_H, \Delta x_H)$. We observe clear improvements in Table 3 and Figure 4. As shown, the performance gain was orthogonal to existing enhancement techniques, e.g., test-time local converter (TLC) (Chu et al., 2022). From the experiments, we conclude that our $a$RGB representation indeed helps training image restoration models better than the RGB color representation in a variety of tasks, architectures, loss functions, and lead to synergic effect with a variety of other enhancement techniques, such as perceptual loss, adversarial training, edge loss, and test-time local converter. ### 4 Discussion In order to understand the representation learned by the $a$RGB autoencoder, we first explore the consequence of our two key design choices: the linear decoder and the mixture-of-experts encoder. Table 2: Results on real image denoising using NAFNet. \| Model \| Objective \| PSNR↑ \| SSIM↑ \| \|---------------------\|-----------------\|-------\|-------\| \| NAFNet-width32 \| $L_{PSNR}$ \| 39.9672 \| 0.9599 \| \| NAFNet-width32 \| $L_{PSNR,RGB}$ \| 39.9864 \| 0.9601 \| \| NAFNet-width32 \| $L_1,RGB$ \| 40.0106 \| 0.9602 \| \| NAFNet-width64 \| $L_{PSNR}$ \| 40.3045 \| 0.9614 \| \| NAFNet-width64 \| $L_1,RGB$ \| 40.3364 \| 0.9620 \| (a) Inverting orthogonal mixture of two aRGB embeddings. (b) Expert selection map of the MoE router $f_t$. (c) t-SNE plot of the aRGB embedding $\xi$ of pixels in image 5b. (d) Change of $L_2$ metrics in the aRGB space relative to the $L_2$ metrics in the RGB space. Figure 5: Understanding the learned aRGB representation. Figure 5a show a visual example of aRGB embedding inversion. Figure 5b and 5c reveal clear evidence that the experts of our aRGB encoder $f$ are specialized for a particular type of input structures, and that even the embedding vectors within a single patch are clustered in a complicated manner, justifying our usage of MoE architecture. Figure 5d shows how the distance metric changes in the aRGB space relative to the distance in the RGB space. Mean distances and their standard deviations are measured by MSE losses between an image and the same image with 100 AWGNs with the same standard deviation. Note that the aRGB space slightly exaggerates the distance more outside natural image domain, e.g., Gaussian noise, and the metric’s variance is negligibly small. Then, we quantify the effect of changing the representation space on the scale of metrics defined over the space and their gradients. We conclude our discussion with ablation studies. 4.1 NULLSPACE OF THE DECODER In addition to the design simplicity, our pixel-wise linear decoder enjoys an additional benefit: decomposability. Since our autoencoder is almost lossless as demonstrated in Table 8, we will consider that the RGB $x \in \mathbb{R}^3$ and the aRGB $\xi = f(x) \in \mathbb{R}^C$ representations of any given image equivalent. That is, $x' = g(\xi) = A\xi + b = x$. As a result of the linearity of our decoder $g$, the aRGB representation $\xi$ can be decomposed into the sum of two orthogonal components: $$\xi = \xi_\parallel + \xi_\perp, \quad \text{s.t.} \quad \xi_\parallel = A^\dagger A \xi =: f_\parallel(x) \quad \text{and} \quad \xi_\perp = (I - A^\dagger A)\xi =: f_\perp(x),$$ where $A^\dagger$ is the Moore-Penrose pseudoinverse of $A$. The parallel component $\xi_\parallel$ of the aRGB representation lies in the three-dimensional subspace of $\mathbb{R}^C$ that is projected onto the RGB colors by the decoder $g$, i.e., $A\xi_\parallel = AA^\dagger A \xi = A \xi$. The remaining perpendicular part $\xi_\perp$ can be regarded as the information the aRGB space encodes in addition to the RGB colors. The contribution of the two components can be visualized by inverting the encoder $f$ with respect to a mixed embedding: $$f^{-1}(\xi_{mix}) = \arg\min_z \\|f(z) - \xi_{mix}\\|^2_2, \quad \text{s.t.} \quad \xi_{mix} = \xi_\parallel + \xi_\perp = A^\dagger A f(x_1) + (I - A^\dagger A)f(x_2).$$ We use a SGD optimizer with a learning rate of 0.1 for 50 iterations. As shown in Figure 5a and Appendix E, the inversion of the mixed embedding inherits color information from the parallel embedding $\xi_\parallel$, while the perpendicular part $\xi_\perp$ contributes to the high-frequency edge information. 4.2 SPECIALIZATION OF THE EXPERTS AND LEARNED STRUCTURES Figure 5b visualizes how individual pixels of a natural image are distributed into $K = 20$ experts. Unlike in semantic segmentation, where segmentation maps are chunked into large blocks of semantically correlated pixels, our pixel-wise router $f_t$ generates fine-grained distributions of pixels. That is, multiple experts jointly involve in encoding the same texture such as the blue sky and the leafy trees. Another salient feature we can observe in the figure is that edges of different orientations are dealt with different experts, implying their specialization. Visualizing the aRGB embedding space using t-SNE (van der Maaten and Hinton, 2008) provides us with additional insights on the topology of Table 3: Results on motion blur deblurring using MPRNet. \| Model \| Objective \| PSNR↑ \| SSIM↑ \| PSNR↑ \| SSIM↑ \| \|----------------\|-----------------\|-------\|-------\|-------\|-------\| \| MPRNet \| $L_{Char} + 0.05L_{Edge}$ \| 32.6581 \| 0.9589 \| 30.9622 \| 0.9394 \| \| MPRNet \| $L_{Char,RGB} + 0.05L_{Edge}$ \| 32.7118 \| 0.9594 \| 31.0248 \| 0.9398 \| \| MPRNet-TLC \| $L_{Char} + 0.05L_{Edge}$ \| 33.3137 \| 0.9637 \| 31.1868 \| 0.9418 \| \| MPRNet-TLC \| $L_{Char,RGB} + 0.05L_{Edge}$ \| 33.3886 \| 0.9642 \| 31.2082 \| 0.9421 \| the space. Figure 5c reveals that the aRGB embeddings cluster into multiple disconnected groups in two different types: common groups where multiple experts are involved in encoding process and specialized groups where a single expert is exclusively allocated for the embeddings. These observations align well with our initial design principles in Section 2.1, where the feature embeddings occupy highly complicated, disconnected set, and an MoE architecture effectively deals with this structure by specializing each expert to a subset of the embedding space. ### 4.3 aRGB Metric Space and Produced Gradients The main purpose of our the aRGB space is to provide alternative supervision to the existing image restoration framework. This supervision is realized with a metric defined over the space and its gradients generated from pairs of images. To this end, we first visualize the correlation between $L_2$ distances defined in the RGB and our aRGB spaces in Figure 5d. We plotted additional figure with title $X - X_0X_1$ to show the deviation of the graph over the straight line, showing clear convexity of the graph. This implies that the metrics within aRGB spaces are inflated when the given two images are similar. Figure 6 shows the gradients from two per-pixel $L_1$ losses between a restored image and its high-quality counterpart defined over both spaces. Unlike RGB $L_1$ loss which exhibits a highly off-centered, discrete distribution, the $L_{1,aRGB}$ loss shows smooth and centered distribution of gradients. We believe that this allows for the stable training of the image restoration models despite its huge scale of the generated gradients from the $L_{1,aRGB}$ loss, which is more than a hundredfold as shown in the x axis of Figure 6b. In the RGB domain, the same scale of gradient is achievable only through increasing the learning rate, which leads to destabilization of the training. Overall, the analyses show how our aRGB encoder helps the training of image restoration models. ### 4.4 Ablation Study Lastly, we provide ablation studies to determine the best hyperparameters for our aRGB autoencoder. We compare the models by the results of training an RRDBNet (Wang et al., 2018b) only on DIV2K dataset. The results are summarized in Table 4. More information is elaborated in Appendix B. #### Number of experts. The first four rows of Table 4 show the effect of the number of experts of the aRGB encoder $f$ on its supervision quality. From the results, we choose to fix the number of experts to 20 throughout our experiments. #### Dataset dependence. As the second part of Table 4 presents, the training data for the aRGB autoencoder decides the quality of supervision the model gives. This implies that our aRGB autoencoder utilizes structural priors of its training data. Appendix 7 provides additional theoretical and empirical evidence that our aRGB autoencoder learns image structures to reconstruct given images. #### Regularizers. In the last row of Table 4, we observe that the regularizing noise $z$ added at the end of the encoder during training helps the aRGB encoder to produce stronger supervision for image restoration models. In practice, we observe more than tenfold reduction in the scale of produced gradients when the aRGB autoencoder trained without the regularizing noise is applied. This correlates to our discussion in Section 4.3, that our aRGB encoder helps training image restoration models by stably increasing the scale of gradients. --- Table 4: Ablation studies on the aRGB autoencoder. RRDBNets (Wang et al., 2018b) are trained with DIV2K (Agustsson and Timofte, 2017) for 300k iterations for $4 \times$ SISR tasks with only the $L_1$ loss between the aRGB embeddings. \| # experts \| Routing \| aRGB train set \| Reg. noise \| Set14 PSNR \| SSIM \| Urban100 PSNR \| SSIM \| DIV2K-Val PSNR \| SSIM \| \|-----------\|---------\|----------------\|------------\|-------------\|------\|---------------\|------\|----------------\|------\| \| 1 \| MoE \| DIV2K \| ✓ \| 26.87 \| 0.7467 \| 24.75 \| 0.7735 \| 29.08 \| 0.8222 \| \| 5 \| MoE \| DIV2K \| ✓ \| 26.87 \| 0.7477 \| 24.83 \| 0.7745 \| 29.12 \| 0.8231 \| \| 10 \| MoE \| DIV2K \| ✓ \| 26.89 \| 0.7474 \| 24.84 \| 0.7750 \| 29.11 \| 0.8231 \| \| 20 \| MoE \| DIV2K \| ✓ \| 26.91 \| 0.7471 \| 24.87 \| 0.7745 \| 29.14 \| 0.8227 \| \| 30 \| MoE \| DIV2K \| ✓ \| 26.89 \| 0.7476 \| 24.84 \| 0.7750 \| 29.11 \| 0.8231 \| \| 20 \| MoE \| GoPro \| ✓ \| 26.89 \| 0.7459 \| 24.83 \| 0.7728 \| 29.12 \| 0.8220 \| \| 20 \| MoE \| SIDD \| ✓ \| 26.86 \| 0.7420 \| 24.80 \| 0.7690 \| 29.06 \| 0.8186 \| \| 20 \| MoE \| None \| ✓ \| 26.63 \| 0.7441 \| 24.66 \| 0.7722 \| 28.86 \| 0.8212 \| \| 20 \| MoE \| DIV2K \| ✓ \| 26.91 \| 0.7469 \| 24.85 \| 0.7722 \| 29.13 \| 0.8223 \| 5 RELATED WORK Pairwise loss in image restoration. Training a deep neural network that translates low-quality images into high-quality estimates has undoubtedly become the standard way of solving image restoration. While most of the advancements have been made in the network architecture (Kim et al., 2016b; Lim et al., 2017; Nah et al., 2017; Tong et al., 2017; Wang et al., 2018b; Zhang et al., 2018b; Waqas Zamir et al., 2021; Liang et al., 2021; Waqas Zamir et al., 2022; Chen et al., 2022), the importance of loss functions is also widely acknowledged. Since SRCNN (Dong et al., 2016), the first pioneer, employed the MSE loss, the first image restoration models had been trained with the MSE loss (Kim et al., 2016a;b; Nah et al., 2017; Zhang et al., 2017). However, after EDSR (Lim et al., 2017) reported that better convergence can be achieved with $L_1$ loss, various pairwise loss functions are explored. LapSRN (Lai et al., 2017) rediscovers Charbonnier loss (Bruhn et al., 2005), a type of smooth $L_1$ loss, for image super-resolution, which is also employed in image deraining (Jiang et al., 2020) with a new edge loss, defined as a Charbonnier loss between Laplacians, which is then employed in general restoration by MPRNet (Waqas Zamir et al., 2021). NAFNet (Chen et al., 2022), on the other hand, uses the PSNR score directly as a loss function. In accordance with these approaches, we attempt a more general approach to design a representation space, over which those loss functions can be redefined. Structural prior of natural images. It is generally recognized that a convolutional neural network, either trained (Simonyan and Zisserman, 2015) or even untrained (Ulyanov et al., 2018), contains structural prior that resonates with the internal structure of natural images. This prior information permeates through the network into its output space. Attempts to exploit this information include the perceptual loss (Johnson et al., 2016) and various perceptual metrics (Zhang et al., 2018a; Ding et al., 2020). Those are pairwise distance metrics defined over the range space of pre-trained classifier networks (Krizhevsky et al., 2012; Simonyan and Zisserman, 2015). However, as mentioned in Section 1, such losses cannot be used alone when it is required to respect the strong correspondence between the generated and the desired images. Different from the strategies sought for the perceptual metrics, our aRGB encoder is designed to preserve the full information of its inputs by a scale-preserving architecture and a linear decoder to strictly constrain the representation. Mixture of Experts. Instead of relying on a single model to handle complex large-scaled data, a more effective approach is to distribute the workload among multiple workers. To achieve this, a routing strategy (Shazeer et al., 2017) can be employed to divide information between different models, each of which processes a subset of the training data. These individual models, referred to as experts, collectively form a Mixture of Experts (MoE) (Jacobs et al., 1991). Recent studies (Zhou et al., 2022; Fedus et al., 2022) have shown the advantages of MoE in deep learning. However, there are two main challenges when working with multiple experts: limited computational resources and training stability. The conventional routing strategy can lead to unstable training of the MoE unless appropriate regularization methods are applied. Moreover, without advanced techniques (Fedus et al., 2021; He et al., 2021), MoE experience longer processing times as the number of experts increases. In response to these challenges, we employ a balancing loss (Fedus et al., 2022) to ensure the stable training of expert networks and incorporate MoE exclusively during the training phase, leaving the testing phase unaffected. 6 CONCLUSION It is a well-known phenomenon (Ledig et al., 2017) that per-pixel pairwise loss functions, such as $L_1$ or $L_2$ distances, defined in the RGB color space have a strong tendency to guide the trained image restoration model to produce blurry, unrealistic textures. We hypothesize that such problem can be alleviated if we have a representation space that contains accurate color information as well as the local structural information of an image. Our augmented RGB (aRGB) representation is designed with a nonlinear mixture-of-experts encoder and a linear decoder to meet the requirements. From diversified set of experiments, we demonstrate the improved performance across a variety of image restoration tasks such as perceptual super-resolution, denoising, and deblurring could be achieved by only changing the representation space to our aRGB space. Given our results suggesting that the RGB color space may not be the optimal representation space for low-level computer vision tasks, we hope our work spurs more interests and exploration in this research direction. REFERENCES Abdelrahman Abdelhamed, Stephen Lin, and Michael S. Brown. A high-quality denoising dataset for smartphone cameras. In CVPR, 2018. Eirikur Agustsson and Radu Timofte. NTIRE 2017 challenge on single image super-resolution: Dataset and study. In CVPR Workshop, 2017. Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer normalization. arXiv preprints, 2016. Marco Bevilacqua, Roumy Aline, Guillemot Christine, and Morel Marie line Alberi. 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TOE6N8dp4w	The authors follow similar direction to the prior work (Bommasani et al., 2019; Yue et al., 2022; Putta et al., 2023; Mattern et al., 2022) and fine-tune a publicly pre-trained LM with DP and generate synthetic data samples. It seems to me that the difference from prior work (Putta et al., 2023) and (Mattern et al., 2022) is that the authors do not augment the training objective and the difference from prior work (Yue et al., 2022) is merely applying parameter efficient fine-tuning instead of full fine-tuning. Looking at the results of Table 1, the reviewer observes that there is 2-3% difference in performance between real and non-private (private) synthetic. Similar results were demonstrated in prior work as well (+ authors here use much larger 8B model compared to GPT2 series and prior work also consider multiclass classification). Therefore, can authors explain how they arrive to the statements such as	HARNESSING LARGE-LANGUAGE MODELS TO GENERATE PRIVATE SYNTHETIC TEXT Anonymous authors Paper under double-blind review ABSTRACT Differentially private training algorithms like DP-SGD protect sensitive training data by ensuring that trained models do not reveal private information. An alternative approach, which this paper studies, is to use a sensitive dataset to generate synthetic data that is differentially private with respect to the original data, and then non-privately training a model on the synthetic data. Doing so has several advantages: synthetic data can be reused for other tasks (including for hyper parameter tuning), retained indefinitely, and shared with third parties without sacrificing privacy. However, generating private synthetic data is much harder than training a private model. To improve performance on text data, recent work has utilized public data by starting with a pre-trained generative language model and privately fine-tuning it on sensitive data. This model can be used to sample a DP synthetic dataset. While this strategy seems straightforward, executing it has proven problematic. Previous approaches either show significant performance loss, or have, as we show, critical design flaws. In this paper we demonstrate that a proper training objective along with tuning fewer parameters results in excellent DP synthetic data quality. Our approach is competitive with direct DP-training of downstream classifiers in terms of performance on downstream tasks. Further, we demonstrate that our DP synthetic data is not only useful for downstream classifier training, but also to tune those same models. 1 INTRODUCTION Machine learning models can memorize their training data (Carlini et al., 2019) and it is possible to extract the training data from a model (Carlini et al., 2021). Training a model with differential privacy (DP) (Abadi et al., 2016) provably reduces the risk of memorization (Ponomareva et al., 2022), which is critical when ML models are trained on sensitive data. However, DP training only ensures that the model does not release private information, and just releasing the model or its predictions is not adequate for many applications. For example, other researchers might want to use the data for analysis, or to build a different predictive model. It would therefore be ideal to release the dataset itself while protecting the privacy of the users that contributed to it. Local differential privacy has been proposed as a method of preprocessing low-dimensional datasets before public release (Ponomareva et al., 2023). Local DP adds noise to individual data points in the training data. While protecting privacy, local DP generally leads to much lower utility, due to the large amount of noise that must be added compared to central differential privacy, where DP is applied to the model or statistical output (Wang et al., 2017; Bassily et al., 2017; Team, 2017). Generally there is an inherent tension between privacy and utility when releasing private datasets: we want to release a dataset that protects the privacy of the underlying data while at the same time we want the dataset to be as useful as the original data for any possible downstream task. Therefore, we focus on central DP and consider generating private synthetic data. Generating such synthetic data involves creating a generative model that learns the original data distribution. To protect the original data, either the generative model should be made private, via DP training, or privacy should be enforced at inference time (e.g., during the generation of synthetic data items, so-called private prediction). Private inference has been shown to be inferior to DP training when a large number of inferences is required (van der Maaten & Hannun, 2020). Since we seek to generate at least as much data as in the original dataset, DP training is the clear choice. Several works proposed using publicly pre-trained large language models (LLM) for private synthetic data generation (Bommasani et al., 2019; Yue et al., 2022; Putta et al., 2023; Mattern et al., 2022). This approach involves privately fine-tuning an LLM using class labels as prompts for the model and subsequently sampling from this model. However these attempts have had mixed success: they either reported poor utility even for non-private synthetic data, or had to augment standard NLP loss metrics to assist the LLM to correctly respond to prompts during the generation process. Additionally, none of the previous work considered privacy leakage from a pre-trained LLM itself. The privacy leakage happens because these papers used academic datasets (like IMDB (Maas et al., 2011)) as sensitive dataset and they utilized GPT-2 LLM (Radford et al., 2019) which was pre-trained on these datasets without any privacy guarantees. Although we follow a similar recipe conceptually, in that we use a DP-finetuned LLM model to generate private synthetic data, we highlight the following differences in our execution of this idea: 1. Privacy leakage mitigation. We draw attention to the need to account for the data that went into pre-training of the LLMs used for generation. Our de-duplication of the pre-training data ensures that no privacy leakage, possibly present in previous works, takes place. 2. Reporting: We use a long sequence of text (512 tokens, representing full reviews like IMDB or Yelp) as our privacy unit. Our privacy guarantees (Appendix A) are tight and transparent, and we tune the hyperparameters of the downstream classifier on private synthetic data only. 3. Method: We demonstrate that the standard approach to private fine-tuning does not yield the desired quality of generated data. Instead of augmenting the LLM’s objective or architecture for fine-tuning as in (Putta et al., 2023; Mattern et al., 2022), we identify a loss function, well known to the NLP community, that is particularly suitable for private fine-tuning. Additionally, we argue that parameter-efficient fine-tuning, especially LoRA tuning, is beneficial for synthetic data generation. Our contributions can be summarized as follows: 1. We demonstrate state-of-the-art results in terms of quality of synthetic data. Specifically, we show in multiple experiments that the quality of the model trained on private synthetic data is comparable to or even better than the quality of the downstream model trained on real data with DP. 2. We demonstrate that parameter efficient fine-tuning like prompt-tuning and LoRA-tuning is superior to full fine-tuning when the tuning is performed privately. In particular, LoRA-tuning results in up to 11 percentage points lift in downstream model performance. To the best of our knowledge, we are the first to demonstrate that parameter-efficient tuning performs better than full fine-tuning when each is combined with DP, whereas the opposite often holds for non-DP training (Shin et al., 2020; Brown et al., 2020; Zhong et al., 2021). 3. We show that generating more synthetic data than the size of the original dataset is helpful, especially for simpler downstream models. 4. We show that DP synthetic data can be used to tune the hyperparameters of the downstream classifiers. We achieve ranking correlation with the ordering of trials performed on real data of up to 87%, even for $\epsilon = 1$. ## 2 RELATED WORK Privacy-preserving synthetic data generation requires that the generated data is both high-fidelity (i.e., exhibits similar distributional characteristics as the original data) and anonymized to preserve the privacy of the users who contributed their data. For complex data like text, images, audio and video, most existing approaches build a generative model, for example a GAN-based model (Guan et al., 2018). However, in most previous work the data is anonymized using heuristic methods, without providing formal privacy guarantees. For example, Melamud & Shivade (2019) attempted to de-identify summaries of clinical discharge notes using heuristic rules for an LSTM model and only empirically demonstrated the privacy of the synthetic data. DP-fine tuning is a standard method for fine tuning LLMs that satisfies differential privacy guarantees and has been shown to perform well with appropriate hyperparameter tuning (Li et al., 2021; Yu et al., DP-fine tuning involves using a pre-trained model and a modification of a training algorithm like DP-SGD to fine tune the model on private data. For private synthetic text generation, Bommasani et al. (2019) suggested using a pre-trained GPT-2 model and then DP-fine tuning it on private data with word-level privacy, but did not implement or evaluate any method. In similar vein, Yue et al. (2022) DP-fine tuned pre-trained GPT models of various sizes. While they do obtain good results on some of the benchmarks, they also observe up to 25% drop of downstream model accuracy on synthetic data (even without DP) on other benchmarks. Putta et al. (2023) attempted a similar recipe on a pre-trained distilGPT2 model, but also reported a large performance drop of the classifier trained on synthetic data. Additionally, they proposed modifying the fine tuning process to also include a discriminator that attempts to distinguish between the labels to improve the separability of learned representations for two binary classes of the text data. Similarly, Mattern et al. (2022) proposed augmenting the training objective with an additional term penalizing the generation of sample with the wrong label. None of the prior work takes into account problem of data contamination between LLM pre-training dataset and dataset used in downstream task. As we show in Appendix D this problem is real. In particular some of both training and test samples examples from downstream datasets could be found in GPT-2 pre-training data, which is used by all prior work. This may potentially invalidate DP-guarantees and may result in overestimated accuracy on downstream tasks. Additionally none of the works on DP synthetic data mentioned above explored parameter-efficient fine tuning. To the best of our knowledge, we are the first to demonstrate that parameter-efficient finetuning like LoRA tuning can produce better quality synthetic DP data than full finetuning. 3 PRELIMINARIES Differential privacy Differential Privacy (DP) (Dwork et al., 2006b) is considered the gold standard for ensuring data anonymization. Throughout this work we employ a notion of DP called $(\epsilon, \delta)$-DP. Definition 1 ($(\epsilon, \delta)$-Differential Privacy, (Dwork et al., 2006a)). Consider neighbouring datasets to be datasets that differ only in addition or removal of one record only. Given non-negative $\epsilon$ and $\delta \leq 1$, a mechanism $A$ is $(\epsilon, \delta)$-DP if for any two neighbouring datasets $D$ and $D'$ and for any $S \subseteq Range(A)$, $$P[A(D) \in S] \leq \exp(\epsilon) \times P[A(D') \in S] + \delta.$$ (1) The $\epsilon$ and $\delta$ values determine the strength of the privacy guarantees, with smaller values corresponding to stronger guarantees. The post-processing property of a DP mechanism means that applying any data-independent transformation to its output will remain DP with the same guarantees. DP in context of ML models In context of ML, DP can be introduced either at the input level, during the training of a model (DP-Training), or during model serving (prediction) (Ponomareva et al., 2023). DP synthetic data falls into the first category and in general is a harder task than introducing DP during the training. This is because DP synthetic data ensures that any ML model trained on this data is DP with respect to the original training data. This is in contrast with DP-Training that only ensures that a particular ML model is DP. Therefore, it is expected that any model trained on DP synthetic data should perform at most as well as the downstream DP-Trained ML model on real data. However the idea of using a pre-trained generative LLM to aid generation of synthetic data means that we inject massive amount of public data, making the task of DP synthetic data generation less daunting. The most practical methods of DP-Training for non convex losses are gradient-noise injection methods like DP-SGD (Abadi et al., 2016), which work by clipping per example gradients to limit the sensitivity of the loss, and noising aggregated clipped gradients with Gaussian noise to make them private. The noise level is proportional to the clipping norm (the sensitivity) and the strength of $\epsilon$ guarantees. The same recipe can be adopted to adaptive optimizers like Adafactor (Shazeer & Stern, 2018), where the noised gradients are passed to the optimizer to figure out the optimal learning rate. LLMs Throughout the paper we will use the terms of pre-training and fine-tuning of LLMs: pre-training is the initial training of a LLM with a large public dataset, for example C4 (Raffel et al., Fine-tuning is an adaptation of a pre-trained model to perform some concrete task, for example question-answering, which involves running several epochs of an optimizer over the additional task training data. 4 METHODOLOGY As a motivational example, consider the task of medical data sharing for research purposes: a medical provider has a sensitive dataset with patients records and wants to accomplish some machine learning task. They may want to share the dataset with external researchers and academic institutions to get their help in solving the downstream task, while preserving the privacy of the original data. We assume that we have a sensitive dataset $D$ consisting of $(D_{train}, D_{valid}, D_{test})$, where the privacy of each record must be protected (see additional details on the unit of privacy in Appendix A). We want to accomplish some task on this dataset, such as training some downstream machine learning model. Additionally, we would like to allow a non-trusted third party to be able to perform the downstream task without violating privacy. To achieve this, we aim to create a synthetic dataset $D^{synth}$, which is DP with respect to the dataset $D$. Our dataset $D^{synth}$ will consist of synthetic training and validation splits. Figure 1 illustrates our methodology of data generation and evaluation: 1. Privately finetune (e.g., using DP-Training) a publicly pre-trained generative LLM $G$ on $D_{train}$, using $D_{valid}$ for hyperparameter tuning. To tune hyperparameters for DP-Training, we follow an algorithm outlined in (Ponomareva et al., 2023) (Section 5.4.1). 2. Independently sample $G$ to generate two new synthetic datasets $D^{synth}_{train}$ and $D^{synth}_{valid}$ which will serve as synthetic training and validation data. 3. Train a downstream model $M$ on $D^{synth}_{train}$ and use $D^{synth}_{valid}$ for hyperparameter tuning. 4. Evaluate the final performance of the model on real dataset $D_{test}$. 4.1 USING AN LLM FOR DATA SYNTHESIS Both encoder-decoder or decoder-only pretrained language models can generate synthetic data; we use decoder-only LLMs in our experiments. To finetune LLM for the synthetic data generation task, we use the next token prediction objective setup as follows. Given an example from the sensitive dataset with text $x$ and label $y$, we generate a prefix $p = "[TaskName] [LabelName_y] " $, where "[TaskName]" is the name of the task (for example "[imdb]"), and "[LabelName_y]" is "[negative]" when $y = 0$ or "[positive]" when $y = 1$. We finetune the model using the Prefix-LM objective (Raffel et al., 2020) using $p$ as a model input and $x$ as a target. Below we outline how the Prefix-LM way of splitting of the training example into input and target is advantageous for DP-training. Let’s consider some example from the dataset which is tokenized into input prefix $p = \{z_1, \ldots, z_k\}$ and target $x = \{z_{k+1}, \ldots, z_n\}$. Typically, weighted next token prediction cross entropy loss looks like the following: $L(\tilde{z}, \tilde{w}, \theta) = -\sum_{i=1}^{n} w_i z_i \log P(z\|z_{<i}, \theta)$ where $\theta$ - model parameters, $\tilde{z} = \{z_1, \ldots, z_n\}$ is tokenized training example (including input and target tokens), each $z_i$ as one hot encodings of token, $P(z\|z_{<i})$ is the probability of $i$-th token given values $z_{<i}$ of all previous tokens and $\tilde{w} = \{w_1, \ldots, w_n\}$ is weights vector for each token in the loss. Standard next-token prediction loss assigns weights $ w_i = 1 $ to all tokens, including those in the prefix $ p $. As a result, prefix tokens will be included in the gradient of the loss $ \frac{\partial L}{\partial \theta} $, thus essentially forcing the model to learn the distribution of tokens in the prefix as well. On the other hand, the Prefix-LM formulation assigns zero weights to the prefix $ i.e. \forall i \leq k : w_i = 0 $, so the total loss looks like the following: \[ L_{\text{PrefixLM}}(z, w, \theta) = -\sum_{i=k+1}^{n} z_i \log P(z\|z_{<i}, \theta) \] As a result, the LLM is not forced to learn distribution of input prefix $ p $ which we found to be beneficial for differentially-private training. DP-Training adds the noise to all the gradients; in a standard setup this will result in the gradients from the prefix portion being corrupted with the noise. This in turn means that prompting the DP-Trained LLM to generate synthetic data will not work as well as expected. We believe this is the same phenomenon that was observed in works by Putta et al. (2023) and Mattern et al. (2022), where authors had to add an adversarial head or augment the loss respectively, to aid the model in differentiating different types of prompts. Prefix-LM in turn is a standard loss well known to the community, and this comes with the benefits of knowing approximate hyperparameter values for its tuning. The aforementioned Prefix-LM setup allows to train one model for all the class labels and can be easily extended beyond the binary classification setup. ### 4.2 Parameter-Efficient Fine Tuning Full finetuning of large models is expensive, and empirically, tuning very large number of weights with DP-finetuning often results in substantial utility drop. Many techniques exist that update the pretraining model without resorting to full model weights update. In this work, we consider two popular ones - Prompt Tuning and LoRA. Prompt tuning (Lester et al., 2021) is a technique which prepends a small prompt tensor in front of the model’s input in the embedding space, freezes the rest of the model’s parameters and then finetunes only the prompt tensor weights. We found that combining prompt tuning with differentially private training allows us to achieve much higher utility of the trained generative model compared to full model fine-tuning. This could be explained by the fact that the prompt tensor is much smaller compared to the size of entire model (we used prompt tensor with 20480 parameters vs 8B weights in the full model) and smaller models tend to have smaller gap between private and non-private utility (Bassily et al., 2014; Bun et al., 2014), probably due to the total amount of noise injected during the training. It should be noted that prompt tuning as described in original paper (Lester et al., 2021) showed very poor utility when trained with differential privacy. We observed that even in the best runs LLM quality metrics (perplexity, next token prediction accuracy) fluctuated significantly. No amount of typical hyperparameter tuning could improve prompt tuning utility in DP-regime. Borrowing some ideas from (Mehta et al., 2022) and experimenting with various optimizers and ways to initialize prompt tensor proved to be the key to making prompt-tuning work. Eventually we found out that the main culprit of poor utility was prompt tensor initialization. (Lester et al., 2021) initializes prompt tensor by using embeddings of some real tokens from vocabulary. Changing prompt tensor initialization to random uniform with small range $[-0.01, 0.01]$ significantly improved utility. Additionally we observed that change of optimizer from Adafactor to Adam or Momentum helped to make training more stable, which simplified hyperparameter tuning (Appendix E). LoRA tuning (Hu et al., 2021) (Low-rank Adaptation) is a technique that freezes all the pre-trained model weights and introduces trainable low-rank decomposition matrices into each dense layer (MLP and Attention). This results in fewer trainable weights than full fine tuning but the number of trainable weights in LoRA is significantly larger than in Prompt tuning. For example, rank 8 LoRA updates 20M trainable parameters, as opposed to 41K prompt tuning vs 8B full fine tuning. Empirically we find (Section 5) that LoRA results in superior performance, surpassing that of both full finetuning and Prompt finetuning and that tuning both MLP layers and Attention blocks is preferred, see Appendices F and I.5 for more details. As a conclusion, we advocate for the use of parameter-efficient techniques when performing DP-training, with LoRA being the most promising so far. --- 1Original paper (Raffel et al., 2020) only describes bidirectional attention over prefix and omits the description of loss weights. Nevertheless zero weighting of the prefix is implemented in the T5 code. 4.3 Data sampling To generate one synthetic example we first randomly select example label $ y $, create a prefix $ p = "[\text{TaskName}] [\text{LabelName}_y]" $ (Section 4.1), feed prefix $ p $ as an input to the language model, and autoregressively sample the output. We repeat this process many times until we reach the desired amount of synthetic data. For each task we sampled at least the same amount of synthetic data as in original training dataset. We observed that generating more synthetic examples generally improves downstream task performance, but this benefit eventually diminishes and compute is typically the limiting factor (Appendix G). 5 Experiments Generative LLM In our experiments we used a model with architecture similar to Lamda 8B (Thop, pilan et al., 2022), which we pre-trained on The Pile dataset (Gao et al., 2020) using a standard next-token prediction loss. We stress that for our experimental results to be valid we must ensure that the pre-trained model was not itself trained on data that is considered private for the downstream task. For example, the GPT-2 model used in (Mattern et al., 2022) seemingly contained IMDB data in its pre-training dataset (Radford et al., 2019), but this model was subsequently used to generate a synthetic version of IMDB, see also appendix D for details. To prevent privacy leakage we modified the pre-training dataset by de-duplicating it against all sensitive datasets used in downstream tasks, following the recipe and scripts from (Lee et al., 2022). The outline of the de-duplication approach is as follows. First we tokenized and constructed a suffix array for each involved dataset (The Pile, IMDB, Yelp, AGNews). Then we used the suffix arrays to find common sequences of 50 or more tokens which appear in The Pile and any other dataset. Finally we cut all those common sequences from The Pile dataset. Note that this de-duplication is “stronger” than simply removing the datasets from the Pile. After cutting the sequences we de-tokenized the dataset back to strings and used it for pre-training. Refer to Appendix C for additional details. Datasets and classification problems We conducted our experiments on IMDB (Maas et al., 2011), Yelp (Zhang et al., 2015a) and AGNews (Zhang et al., 2015b) datasets. All these datasets only provide a training and test set, so in each case we use the first 90% of the training set for training and the remaining 10% for validation. For each dataset we formulated a binary classification problem (sentiment classification) as the downstream prediction task. 5.1 Downstream classifier performance We investigate the utility of using private synthetic data for a downstream task. For each dataset, we consider two types of models. First one is a (encoder-only) BERT model (Devlin et al., 2018a) with classification head. BERT is publicly pretrained and then fine tuned using either real data or our generated synthetic data. This model benefits from public pre-training data. We also consider a word-level CNN model (Johnson & Zhang, 2015) that does not utilize any public data. For each model, we report the performance on real data with no DP guarantees (an entry "Real" with $ \epsilon = \infty $ in Table 1). This serves as a upper bound of downstream classifier performance. We also report the performance of doing DP-Training on the downstream classifier directly (entries "Real" with $ \epsilon \in \{1, 3, 10\} $, referred to as "DP-on-real" in the text) and report the results on synthetic data generated from fine-tuned (Fine-tuned-SD), prompt tuned (Prompt-tuned-SD) and LoRA-Tuned (LoRA-tuned-SD) models. We would like to highlight however that in the case of using the real data directly for DP-Training, only the resulting downstream model is DP, and the real data can’t be shared freely or used for hyperparameter tuning (or such tuning should be accounted for in privacy guarantees). DP Synthetic data however can be shared freely and used for feature engineering, hyperparameter tuning, etc. Non-private synthetic data Firstly, our results in Table 1 indicate that obtaining good fidelity non-private synthetic data is possible, contrary to the results reported in (Yue et al., 2022) and Putta et al. (2023). Both Fine-tuned-SD and LoRA-tuned SD exhibits better performance than Prompt-tuned-SD, in line with current understanding that for a non-DP setting, tuning more model parameters is beneficial (Shin et al., 2020; Brown et al., 2020; Zhong et al., 2021). Interestingly, even for non DP setting, downstream models trained on LoRA synthetic data outperform those trained on fully fine-tuned synthetic data in 2 out of 3 datasets. Private synthetic data While there is a clear utility drop when going from non-private SD data to private SD, DP LoRA-tuned-SD is a clearly superior way of obtaining DP synthetic data. Prompt-tuned DP SD is better than fully fine tuned DP SD, however LoRA outperforms the Prompt-tuned DP synthetic data in majority of the cases. We hypothesize that it might be due to less total noise being added in DP LoRA models, due to fewer parameters being updated than with the full fine-tuning. Prompt tuning on the other hand updates the minimal number of parameters, however this minimum update hurts the utility of SD, suggesting that like with everything in ML, there is a “sweet spot” on the number of parameters trained with DP. The difference between the performance is significant, with LoRA-tuned-SD exhibiting of up to 10-11% lift on downstream BERT classifier tasks, compared to model trained on fine-tuned-SD. For CNN model that is more dependent on the quality of the data than BERT (that essentially reaps some additional benefits from Transfer Learning), the results are even more significant, with a boost from prompt-tuned-SD (vs fine-tuned-SD) reaching up to 22%. Private synthetic data vs DP-Training on real data To obtain a DP downstream ML model, we can either use DP synthetic training data or introduce DP directly during downstream model training (DP-on-real). As previously mentioned, the former is a harder setup. When comparing BERT models, we can see that private LoRA-tuned-SD achieves performance similar or even superior (e.g., for IMDB and Yelp datasets) to DP-on-real for all levels of privacy, but an additional benefit of such synthetic data is that it can be additionally shared freely and used for hyperparameter tuning and feature engineering. For CNN model, LoRA-tuned-SD (and even prompt-tuned SD) exhibits better performance than DP-on-real. This is due to the fact that private synthetic data benefits from massive amount of public data that was used for pretraining of the LLM (CNN model itself is trained from scratch, as opposed to BERT that is itself a pre-trained model, albeit with smaller amount of public data than the 8b Lamda model we used for SD generation). This indicates that for simpler models synthetic data can be a preferred way of injecting additional public knowledge. This is an interesting result since it is commonly assumed that for Transfer Learning to work, public data should come from a similar distribution as the target data. However in case of synthetic data, we inject public data from different distributions (crawl of the web) than that of the downstream task (e.g. Yelp reviews). \| IMDB \| $\epsilon$ \| Real \| Synthetic \| \| IMDB \| $\epsilon$ \| Real \| Synthetic \| \|------\|------------\|------\|-----------\|------\|------\|------\|------\|-----------\| \| \| $\infty$ \| 93.7 ± 0.1 \| 93.2 ± 0.2 \| 92.0 ± 0.1 \| 91.6 ± 0.2 \| 90.1 ± 0.1 \| 89.8 ± 0.1 \| 87.4 ± 0.1 \| 89.0 ± 0.1 \| \| \| 10 \| 90.6 ± 0.1 \| 84.0 ± 0.7 \| 90.7 ± 0.2 \| 91.3 ± 0.2 \| 78.2 ± 0.4 \| 80.0 ± 0.5 \| 86.9 ± 0.1 \| 87.7 ± 0.2 \| \| \| 3 \| 89.7 ± 0.2 \| 83.9 ± 0.6 \| 87.4 ± 0.2 \| 90.6 ± 0.2 \| 74.8 ± 0.6 \| 74.2 ± 0.1 \| 85.4 ± 0.5 \| 87.4 ± 0.3 \| \| \| 1 \| 88.6 ± 0.1 \| 79.1 ± 1.7 \| 88.1 ± 0.4 \| 90.0 ± 0.3 \| 69.3 ± 0.6 \| 64.7 ± 0.5 \| 85.4 ± 0.1 \| 87.6 ± 0.4 \| \| Yelp \| $\infty$ \| 97.6 ± 0.1 \| 95.9 ± 0.1 \| 93.9 ± 0.1 \| 96.4 ± 0.1 \| 95.6 ± 0.1 \| 89.3 ± 0.5 \| 91.6 ± 0.1 \| 93.7 ± 0.0 \| \| \| 10 \| 94.0 ± 0.1 \| 84.2 ± 0.7 \| 94.1 ± 0.1 \| 95.1 ± 0.1 \| 90.1 ± 0.1 \| 71.9 ± 0.6 \| 89.1 ± 0.4 \| 90.6 ± 0.1 \| \| \| 3 \| 94.6 ± 0.1 \| 84.0 ± 0.1 \| 93.5 ± 0.1 \| 95.6 ± 0.1 \| 90.9 ± 0.2 \| 67.9 ± 2.6 \| 80.5 ± 0.1 \| 93.6 ± 0.1 \| \| \| 1 \| 94.3 ± 0.1 \| 84.1 ± 0.3 \| 94.1 ± 0.1 \| 95.5 ± 0.1 \| 89.8 ± 0.1 \| 71.1 ± 0.4 \| 91.1 ± 0.3 \| 93.4 ± 0.1 \| \| AGNews \| $\infty$ \| 93.7 ± 0.1 \| 91.1 ± 0.1 \| 88.3 ± 0.3 \| 91.8 ± 0.2 \| 91.3 ± 0.1 \| 87.7 ± 0.1 \| 84.7 ± 0.1 \| 88.5 ± 0.2 \| \| \| 10 \| 90.9 ± 0.2 \| 65.1 ± 5.4 \| 86.9 ± 0.1 \| 90.0 ± 0.1 \| 85.2 ± 0.2 \| 45.2 ± 1.8 \| 83.5 ± 0.2 \| 88.9 ± 0.1 \| \| \| 3 \| 90.4 ± 0.2 \| 65.3 ± 2.9 \| 86.5 ± 0.2 \| 89.6 ± 0.3 \| 83.4 ± 0.1 \| 45.2 ± 1.8 \| 83.5 ± 0.2 \| 86.6 ± 0.2 \| \| \| 1 \| 89.8 ± 0.2 \| 65.7 ± 2.9 \| 84.9 ± 0.8 \| 88.4 ± 0.4 \| 79.9 ± 0.2 \| 46.8 ± 1.5 \| 80.4 ± 0.6 \| 85.8 ± 0.1 \| Amount of synthetic data vs downstream classifier performance We studied how much synthetic data we should generate relative to amount of real data. Table 2 demonstrates that generating more synthetic data can be beneficial, but has diminishing returns for BERT (0.8% lift going from 1x to 3x times the data), with benefits more pronounced for simple models like WordCNN (1.4% lift from increasing the amount of synthetic data 3x). \| Model \| 1x \| 2x \| 3x \| 4x \| 5x \| 6x \| \|-------\|----\|----\|----\|----\|----\|----\| \| BERT \| 87.2 ± 0.4 \| 87.9 ± 0.4 \| 88.0 ± 0.1 \| 88.1 ± 0.4 \| 88.4 ± 0.1 \| 88.7 ± 0.1 \| \| WordCNN \| 83.2 ± 0.2 \| 84.3 ± 0.4 \| 84.6 ± 0.1 \| 83.4 ± 0.1 \| 83.7 ± 0.3 \| 83.8 ± 0.2 \| One can also potentially combine the synthetic data with training with DP on real data, by pre-training the downstream model with DP synthetic data and then fine-tuning with DP on real data. This will however require spreading the privacy budget between DP synthetic data and DP-Training of the downstream classifier. We leave this for future work. Comparison with prior work While works below don’t provide sufficient (or any) information on their privacy unit (as we do in Appendix A), we assume that privacy unit that was used is one example (e.g. 1 full yelp or imdb review etc); we also assume central DP setting, that δ values are the same or comparable etc. Additionally, none of the works below take into account the fact that pre-training data might have contained the data they deem private (as we highlight in Appendix D), potentially invalidating their reported DP guarantees. Yue et al. (2022) used Yelp dataset for multiclass (rating) classification, so our results are not directly comparable. Putta et al. (2023) used AGNews dataset. Their approach is a combination of next token prediction (similar to our setup) and additional loss term from a new head that attempts to learn to distinguish between various classes directly (instead of simply relying on the prompts in the next token prediction head). Putta et al. (2023) reports 0.867 accuracy of downstream task for ε of 3, while we obtain 89.6 (the baseline performance of downstream classifier for our and their work is comparable, 0.938, suggesting that we are using comparable downstream classifiers). Mattern et al. (2022) suggested a modification of the loss (prompt-mismatch loss, to discourage the generation of text inconsistent with the prompt, like generating a negative review when positive prompt was given). They performed experiments on IMDB dataset. Their best IMDB experiments reporting worse accuracy on DP synthetic data (89.1% theirs vs 90.6% ours for ε = 3). They also seem to have worse performance on real data despite using the same model (BERT-classifier). ### 5.2 Tuning downstream model hyperparameters on synthetic data With the following experiments on IMDB data, we want to demonstrate that private synthetic data is useful for hyperparameter tuning of the downstream classifier. For all our experiments, when tuning the downstream classifier, we use validation accuracy on set-aside portion of synthetic data for hyperparameter selection. We tune weight decay and learning rate for both CNN and BERT models. For synthetic data, we create vectors of accuracy on validation (synthetic) data and performance on real test data for all possible combinations of hyperparameter values tried. We then report the ranking correlation between performance as indicated by validation accuracy (synthetic data) and test accuracy computed on real data. We also report the ranking correlation of accuracies on real validation and real test data, to provide an upper bound. Additionally, we report rank-biased overlap ranking metric (Webber et al., 2010), which is a weighted metric that gives more weight to the top of the ranking (we use parameters that give 85% of the weight to the first top 25% of ranking). Table 3 demonstrates excellent ranking correlation on synthetic data. Interestingly, prompt-tuned synthetic data metrics, in particular the mean and standard deviation of the top 25% of trials, suggest that BERT classifier performance is less sensitive to hyperparameters on better fidelity (prompt or LoRA tuning) data than on worse fidelity data (fine-tuning). ### 5.3 Estimating synthetic data quality It is useful to have an efficient method of evaluating the quality of a synthetic dataset without relying on specific downstream tasks. For one, a key use case for privacy preserving synthetic data is to enable data sharing without a definitive end use case. For another, training the generator LLM has multiple hyperparameters that can be tuned, and it can be prohibitive to evaluate candidate models using full data synthesis and downstream training (which itself might require tuning hyperparameters). Instead, lighter weight proxy metrics can be used. Commonly used proxy metrics are: perplexity, n-gram statistics, and MAUVE (Pillutla et al., 2021). We investigate the effectiveness of each of these metrics by comparing their correlation to downstream performance (Table 4). These metrics are used to compare datasets, and thus their absolute value is uninformative. For n-gram statistics we determine the frequency of unigrams, bigrams, and sample lengths in characters for both the original and synthetic datasets. We then compute the area under the divergence frontier between these two frequency distributions as is done by MAUVE. MAUVE works by computing the difference between two datasets by first embedding each example, then clustering the datasets, followed by comparing (via divergence frontiers) the histogram of cluster membership across the two datasets. It has recently been shown to be an effective metric for synthetic text datasets (Yue et al., 2022) (Mattern... Table 3: Ranking correlations (full list) and rank-biased overlap (RBO) \cite{webber2010} for top 25% of hyperparameter tuning trials. Real data metrics are calculated on the performance of a model as reported on real validation and real test. For synthetic data, metrics are calculated on synthetic validation and real test data. Mean 25% and STD 25% show mean and std of real test accuracy evaluated on top 25% trials (ordered by validation accuracy on synthetic data). \| Model \| $ c $ \| Method \| RBO 25% \| Spearman \| Kendall \| Mean 25% \| STD 25% \| \|-----------\|---------\|----------------\|---------\|----------\|---------\|----------\|---------\| \| BERT \| $ \infty $ \| Real data \| 0.56 \| 0.96 \| 0.93 \| 93.55 \| 0.50 \| \| \| 3 \| Fine-tuning \| 0.33 \| 0.94 \| 0.86 \| 79.27 \| 0.75 \| \| \| 10 \| Prompt-tuning \| 0.32 \| 0.73 \| 0.60 \| 88.00 \| 0.00 \| \| \| 3 \| LoRA-tuning \| 0.29 \| 0.86 \| 0.79 \| 90.00 \| 0.00 \| \| \| 10 \| \| 0.3 \| 0.78 \| 0.66 \| 91.18 \| 0.39 \| \| WordCNN \| $ \infty $ \| Real data \| 0.92 \| 0.92 \| 0.84 \| 90.00 \| 0.00 \| \| \| 3 \| Fine-tuning \| 0.63 \| 0.79 \| 0.65 \| 72.09 \| 2.37 \| \| \| 10 \| Prompt-tuning \| 0.64 \| 0.73 \| 0.59 \| 84.36 \| 1.49 \| \| \| 3 \| LoRA-tuning \| 0.69 \| 0.81 \| 0.67 \| 87.45 \| 0.66 \| et al.] \cite{kour2022} \cite{kour2022}, which our results support. We compute the MAUVE score as given in Pillutla et al. \cite{pillutla2021} using the suggested hyperparameters unless noted. We investigated modifying these hyperparameters and confirm they make little difference to the relative ranking, with the notable exception of the model used to embed examples. Unlike the original paper, we find larger models to be much more effective. In particular, embedding using Sentence-T5 \cite{ni2021} has much higher correlation to downstream performance than BERT or any other model we tried. For more details see appendix K. Our results match many of the results given in Kour et al. \cite{kour2022}. All metrics are at least somewhat noisy with standard test-set perplexity performing very well. Given its ease to compute while finetuning, perplexity is our recommended proxy metric when available. Table 4: The Spearman’s rank correlation for each metric compared against downstream classifier performance. Metrics are used to select candidate datasets, and thus their relative rank is what's most important for the metrics to reflect. \| Perplexity \| Unigram \| Bigram \| Length \| BERT \| MAUVE Sft5-base \| Sft5-3B \| \|------------\|---------\|--------\|--------\|------\|----------------\|--------\| \| 0.91 ± 0.02 \| 0.74 ± 0.11 \| 0.83 ± 0.09 \| 0.88 ± 0.26 \| 0.84 ± 0.62 \| 0.88 ± 0.04 \| 0.93 ± 0.10 \| 6 CONCLUSION We have shown that training downstream models on DP synthetic training data is an effective alternative to training such models with DP directly on real data for text classification tasks. We explored two methods for privately generating the synthetic training data, both of which involve modifying the weights of an existing LLM. One method privately fine-tuned all the layers of the LLM, while the other method used parameter efficient fine tuning (‘prompt-tuning’ and ‘LoRA-tuning’). Our experiments demonstrated that LoRA tuning is a superior way of obtaining DP-synthetic data, which provides performance on the downstream task that is comparable or even better than directly DP-Training on real data. We showed that the standard NLP Prefix-LM loss is well suited for DP-finetuning. Private synthetic data can be used freely for all purposes, such as feature engineering, hyperparameter tuning, debugging and monitoring, and sharing, but without any privacy-related concerns. We also showed that while Mauve is a good proxy metric for evaluating the quality of the synthetic data, simpler metrics like perplexity, when available, perform well. 7 ETHICS STATEMENT We expect that our proposed method of generating DP synthetic data will facilitate safer data sharing and that the societal impact will be positive, since entities who own private data but do not necessarily have the knowledge or resources to train predictive models can share private synthetic data with specialists for model creation, benefiting from their expertise without comprising the privacy of the users who contributed their data. The main limitation of our approach is that we only conducted experiments on English datasets, however we expect that methods should work on multilingual datasets as long as public multi-lingual data are available for LLM pre-training. 8 Reproducibility Statement All of our experiments are based on open sourced frameworks and public datasets, refer to Appendices H and M. We further provided necessary details to reproduce our experiments in Appendices C, E, F, G, and I. References Martin Abadi, Andy Chu, Ian Goodfellow, H Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC conference on computer and communications security. ACM, oct 2016. doi: 10.1145/2976749.2978318. URL https://doi.org/10.1145%2F2976749.2978318 Raef Bassily, Adam Smith, and Abhradeep Thakurta. Private empirical risk minimization: Efficient algorithms and tight error bounds. Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 464–473, 12 2014. doi: 10.1109/FOCS.2014.56. Raef Bassily, Kobbi Nissim, Uri Stemmer, and Abhradeep Guha Thakurta. Practical locally private heavy hitters. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017. URL https://proceedings.neurips.cc/paper_files/paper/2017/file/3d779cae2d46cf6a8a99a35ba4167977-Paper.pdf Sid Black, Stella Biderman, Eric Hallahan, Quentin Anthony, Leo Gao, Laurence Golding, Horace He, Connor Leahy, Kyle McDonell, Jason Phang, Michael Pieler, USVSN Sai Prashanth, Shivanshu Purohit, Laria Reynolds, Jonathan Tow, Ben Wang, and Samuel Weinbach. GPT-NeoX-20B: An open-source autoregressive language model. In Proceedings of the ACL Workshop on Challenges & Perspectives in Creating Large Language Models, 2022. URL https://arxiv.org/abs/2204.06745. Avrim Blum, Katrina Ligett, and Aaron Roth. A learning theory approach to non-interactive database privacy. CoRR, abs/1109.2229, 2011. URL http://arxiv.org/abs/1109.2229 Rishi Bommasani, Steven Wu, and Xanda Schofield. Towards private synthetic text generation. In NeurIPS 2019 Machine Learning with Guarantees Workshop, 2019. Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners. CoRR, abs/2005.14165, 2020. URL https://arxiv.org/abs/2005.14165 Mark Bun, Jonathan Ullman, and Salil Vadhan. Fingerprinting codes and the price of approximate differential privacy. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’14, pp. 1–10, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450327107. doi: 10.1145/2591796.2591877. URL https://doi.org/10.1145/2591796.2591877 Nicholas Carlini, Chang Liu, Úlfar Erlingsson, Jernej Kos, and Dawn Song. The secret sharer: Evaluating and testing unintended memorization in neural networks. In 28th USENIX Security Symposium (USENIX Security 19), pp. 267–284, 2019.
3zQo5oUvia	Moreover, there is a notable difference in the results reported for the TNC on the HAR dataset between the original TNC paper [2] and this manuscript. In the origianl paper, it was reported AUPRC 0.94, Accuracy 88 while in this manuscript, it is reported AUPRC 0.98 and Accuracy 94. More information about the potential factors leading to these discrepancies would be beneficial for the reader's comprehension.	REBAR: Retrieval-Based Reconstruction for Time-Series Contrastive Learning Maxwell A. Xu¹, Alexander Moreno¹, Hui Wei³, Benjamin M. Marlin³, James M. Rehg² ¹ Georgia Tech, ² UIUC, ³ UMass Amherst maxxu@gatech.edu, jrehg@illinois.edu Abstract The success of self-supervised contrastive learning hinges on identifying positive data pairs, such that when they are pushed together in embedding space, the space encodes useful information for subsequent downstream tasks. Constructing positive pairs is non-trivial as the pairing must be similar enough to reflect a shared semantic meaning, but different enough to capture within-class variation. Classical approaches in vision use augmentations to exploit well-established invariances to construct positive pairs, but invariances in the time-series domain are much less obvious. In our work, we propose a novel method of using a learned measure for identifying positive pairs. Our Retrieval-Based Reconstruction (REBAR) measure measures the similarity between two sequences as the reconstruction error that results from reconstructing one sequence with retrieved information from the other. Then, if the two sequences have high REBAR similarity, we label them as a positive pair. Through validation experiments, we show that the REBAR error is a predictor of mutual class membership. Once integrated into a contrastive learning framework, our REBAR method learns an embedding that achieves state-of-the-art performance on downstream tasks across various modalities. 1 Introduction Self-supervised learning uses the underlying structure within a dataset to learn rich and generalizable representations without labels, enabling fine-tuning on various downstream tasks. This reduces the need for large labeled datasets, which is attractive for many machine learning tasks, and particularly useful in the analysis of time series data for health applications. Due to advances in sensor technology, it is increasingly feasible to capture a large volume of health-related time-series data (Nasiri & Khosravani, 2020), but the cost of labeling this data remains high. For example, in mobile health applications, acquiring labels requires burdensome real-time annotation (Rehg et al., 2017) by participants. Additionally, in medical applications such as ECG analysis, annotation is costly as it requires specialized medical expertise. Contrastive learning is a powerful self-supervised approach to learning semantically-meaningful representations, which is based on constructing and embedding positive and negative pairs of unlabeled samples. In order to obtain useful representations, pairs should capture important structural properties of the data. In the vision applications that have driven this approach, augmentations are used to construct positive pairs by exploiting invariances of the imaging process (e.g. transformations such as flipping and rotating that change the data vector without changing its meaning). Unfortunately, general time-series do not possess a large and rich set of such invariances. Shifting, which addresses translation invariance, is widely-used, but other augmentations such as shuffling or scaling can destroy the signal semantics. For example, shuffling an ECG waveform destroys the temporal structure of the QRS complex, and scaling it can change the clinical diagnosis (Nault et al., 2009). Moreover, there is no consistent consensus of augmentations in the literature; methods such as TF-C (Zhang et al., 2022) incorporate jittering and scaling, while TS2Vec (Yue et al., 2022) finds that these augmentations impair downstream performance. In this work, we introduce a novel approach for identifying positive pairs for time-series contrastive learning. Our key idea is that instead of generating positive pairs via augmentation, we use a learned similarity measure to identify positive pairs that naturally occur in extended time-series recordings. Figure 1: This figure demonstrates the intuition of our Retrieval-Based Reconstruction (REBAR) approach. If we can successfully retrieve information from another subsequence to aid in reconstruction, then the two subsequences should form a positive pair in a contrastive learning framework. We first use the context-window, designated by the grey box, of Walk Subseq 1 to query for information in Walk Subseq 2 or in Sit Subseq. Upper b) shows that the context window in Walk Subseq 2 provides a good match with a similar double peak motif, leading to a good reconstruction. Lower b) shows that Sit Subseq has no matching motif, leading to a poor reconstruction. In our framework, we conceptualize a time-series as a composition of a sequence of subsequences, each of which has a class label. This framing describes many real-world physiological signals. For example, a daily record of an accelerometry signal from a wrist-worn smartwatch contains many repeated subsequences corresponding to frequent activities like walking or sitting. Each subsequence from the same activity class will in turn be comprised of brief temporal patterns or “motifs” such as a “swing up” hand motion during walking. Likewise, the deflections in an ECG signal due to the depolarization of the heart define motifs within the QRS complex (Bouaziz et al., 2014). If two subsequences contain similar motifs, then they are likely to share the same class label and are therefore a good candidate to form a positive pair. We operationalize the idea of matching motifs with our Retrieval-Based Reconstruction (REBAR)$^1$ approach. In order to avoid explicitly modeling and detecting motifs, we adopt a reconstruction-based approach in which masked samples in one subsequence are reconstructed directly from values retrieved from a second, candidate subsequence, as illustrated in Fig. 1. A context window is taken around each masked sample, and the REBAR cross-attention model learns to compare each context window to the windows in the candidate subsequence to be retrieved for reconstruction. When two subsequences have many motifs in common, high quality matches can be obtained that minimize reconstruction error. Therefore, the REBAR reconstruction error is a learned measure that captures motif similarity, and pairs with a lower error can then form positive examples in contrastive learning. Such pairs are likely to share semantic meaning, so that the resulting learned embedding space is class-discriminative. We are able to demonstrate this by showing that REBAR achieves state-of-the-art performance on a diverse set of time-series. The full REBAR approach can be seen in Fig. 2, and our public code repository can be found here: https://github.com/maxxu05/rebar. Our main contributions in this work are: 1. This is the first work to use a similarity measure to select positive and negative pairs in time-series contrastive learning. We do so with our REBAR measure, which captures motif-similarity between subsequences using a convolutional cross-attention architecture. 2. We demonstrate that our learned measure predicts mutual class membership in a nearest neighbor sense, which validates that our positive pairs are implicitly capturing the subtle invariances within time series signals, as required for contrastive learning. 3. Our REBAR contrastive learning approach achieves SOTA performance against a representative set of contrastive learning methods that encompass the different ways in which positive and negatives pairs can be generated. Note that our contrastive training method also beats a fully-supervised training approach. $^1$Note that we will interchangeably use “REBAR” to refer to the REBAR contrastive learning approach, the REBAR cross-attention, or the REBAR measure. The specific meaning will be evident from the context. 2 RELATED WORK Augmentation-based Contrastive Learning: Augmentation-based methods are the most studied type of contrastive learning method in time-series research, due to the success of augmentation-based strategies in computer vision (He et al., 2020; Chen et al., 2020; Chen & He, 2021; Caron et al., 2021). However, it is unclear which augmentation strategies are most effective for time-series, and the findings across different works are inconsistent. TS2Vec (Yue et al., 2022) uses cropping and masking to create positive examples, and their ablation study found that jittering, scaling, and shuffling augmentations led to performance drops. Conversely, TF-C (Zhang et al., 2022) included jittering and scaling, along with cropping, time-shifting, and frequency augmentations. TS-TCC (Eldele et al., 2023) augments the time-series with either jittering+scaling or jittering+shuffling. This is in spite of how shuffling breaks temporal dependencies, and scaling changes the semantic meaning of a bounded signal. Other augmentation works (Woo et al., 2022; Yang & Hong, 2022; Yang et al., 2022b; Ozyurt et al., 2022; Lee et al., 2022) also use some combination of scaling, shifting, jittering, or masking. Empirical performance was used to justify the augmentation choice, but differences in datasets, architectures, and training regimes make it difficult to draw a clear conclusion on what the best set of augmentations are. Our REBAR method instead uses a sampling-based approach to identify positive instances from a set of real subsequences, rather than generating a positive instance from an inconsistent set of augmentations. Sampling-based Contrastive Learning: After sampling an anchor subsequence, TLoss (Franceschi et al., 2019) creates the positive subsequence as a crop of the anchor and the negative as a crop from a different time-series. CLOCS (Kiyasseh et al., 2021) samples pairs of temporally-adjacent subsequences and pairs of subsequences across channels from the same time-series as positives. TNC (Tonekaboni et al., 2021) randomly samples a positive example from the anchor subsequence’s neighborhood region and an unlabeled example from outside. The neighborhood is found via a stationarity test, resulting in TNC’s run-time being 250x slower than TS2Vec (Yue et al., 2022), and it utilizes a hyperparameter to estimate the probability that the unlabeled example is a true negative. Our REBAR approach is sampling-based, but unlike previous work, our positive examples are not selected based on temporal proximity to the anchor. Instead, positive examples are selected on the basis of their similarity to the anchor, measured by retrieval-based reconstruction. Other Self-Supervised Learning Methods: CPC is a contrastive learning method learns to contrast future points against incorrect ones (Oord et al., 2018). There have also been contrastive learning methods designed for specific sensor modalities with expert knowledge, such as for EEG data (Zhang et al., 2021). Another method is the Masked Autoencoder, which involves masked reconstruction but is fundamentally different from REBAR. See Appendix A.1.5 for further discussion. Time-Series Motifs: A motif is a brief temporal shape that repeats itself approximately across the time-series and is potentially class-discriminative. Much work has been done in identifying motifs via works such as matrix profile (Yeh et al., 2016; 2018; Gharghabi et al., 2018), and there are many classical time-series approaches that use template-matching methods to classify motifs (Frank et al., 2012; Okawa, 2019; Niemattrakul et al., 2012). However, instead of decomposing our time-series into specific motifs as the classical literature does, our REBAR method uses cross-attention to retrieve motifs that are useful in the context of reconstruction. Then, we can utilize the reconstruction error to capture motif-similarity in a novel contrastive learning context for identifying positive pairs. 3 NOTATION The dataset is designated by $A \in \mathbb{R}^{N \times U \times D}$, with $N$ long time-series of $U$ temporal length and $D$ channels. $A^{(i)} \in \mathbb{R}^{U \times D}$ is the $i$th time-series in the dataset. $X^{(i)} \in \mathbb{R}^{T \times D}$ is a subsequence of the $A^{(i)}$ with length $T$, where $X^{(i)} = A^{(i)}[t : t + T]$, for some $t \in \mathbb{N}$ where $T \ll U$. $(i)$ will be omitted for brevity when not relevant. $x \in \mathbb{R}^D$ refers to a specific time-point’s data found in $X$. Throughout the paper, a subscript is used to describe a specific subsequence, $X_{\text{description}}$. Within cross-attention, $X_q$ and $X_k$ designate the subsequences that serve as the query or key, respectively. A bar in $\bar{X}$ designates that $X$ has been partially masked out. In contrastive learning, $X_{\text{anchor}}$ is the anchor, and $X_{\text{cand}}$ is a candidate. We then identify which of the candidates $X_{\text{cand}}$, should be labeled as positive, $X_{\text{pos}}$, or negative, $X_{\text{neg}}$. Figure 2: 1) First, our REBAR cross-attention is trained to retrieve information from the key to reconstruct a masked-out query. 2) Next, it is frozen and utilized to identify the positive instance. After sampling subsequences from the time-series, the subsequence that reconstructed the anchor with the lowest REBAR error is labeled as positive, and the others are labeled as within-time-series negatives. These negatives capture how time-series dynamics can change over time. Subsequences from other time-series within a data batch are labeled as between-time-series-negatives, and these negatives capture differences among patients. 3) We use the assigned labels to train an encoder. 4 REBAR APPROACH Self-supervised contrastive learning methods learn an embedding by constructing positive and negative instance pairs and then pushing the positive pairs together and negative pairs apart. In order to construct positive and negative pairs via retrieval, we designate one subsequence as the anchor, and use our REBAR measure to quantify the similarity between the anchor and other instances from the same time-series. The most similar instance forms a positive pair with the anchor, while the other instances, including instances from other time-series, form the negative pairs. Sec. 4.1 describes how we design our REBAR cross-attention module to produce the REBAR measure, and Sec. 4.2 explains how we apply the measure for sequence comparison in a contrastive learning framework. Sec. 4.3 tests the hypothesis that the REBAR measure can capture semantic relationships by demonstrating that REBAR predicts mutual class membership. REBAR($\mathbf{X}_{\text{anchor}}, \mathbf{X}_{\text{cand}}$) cross-attention reconstructs $\tilde{\mathbf{X}}_{\text{anchor}}$ by retrieving motifs in $\mathbf{X}_{\text{cand}}$ that match the context window. Then, REBAR error serves as a distance measure$^2$ between two sequences, shown in Eq. 1. We hypothesize that if $d(\mathbf{X}_{\text{anchor}}, \mathbf{X}_{\text{cand}})$ is small, then it predicts if $\mathbf{X}_{\text{cand}}$ is the same class as $\mathbf{X}_{\text{anchor}}$ (i.e. mutual class membership), allowing us to identify positive pairs. $$d(\mathbf{X}_{\text{anchor}}, \mathbf{X}_{\text{cand}}) := \\| \text{REBAR}(\tilde{\mathbf{X}}_{\text{anchor}}, \mathbf{X}_{\text{cand}}) - \mathbf{X}_{\text{anchor}} \\|_2^2$$ (1) 4.1 DESIGN OF THE REBAR CROSS-ATTENTION We would like to design our retrieval-based reconstruction error to be class-discriminative, such that pairs with better reconstruction and lower distance are more likely to share classes and thus be semantically related. As such, REBAR identifies the motifs from the candidate, $\mathbf{X}_{\text{cand}}$, that best match to the context window of the anchor, $\tilde{\mathbf{X}}_{\text{anchor}}$ and retrieves these motifs to reconstruct the anchor. For example, take the visualizations shown in Fig. 1. The error resulting from reconstructing Walk Subsequence 1 from the retrieved matching motif in Walk Subsequence 2 is lower than reconstructing from the retrieved motif in the Sit Subsequence. The reconstruction performance is dependent on how closely the motifs in the candidate are able to match with the anchor, which allows for the REBAR measure to be class-discriminative. Cross-attention learns to produce weighted averages of a transformation of the key time-series and is an attractive method for modeling this paradigm. This is because the retrieval function, $p(x_k \| x_q)$ as shown in Eq. 2, can be interpreted as identifying the $x_k$ that best matches $x_q$. Cross-attention is most commonly used with text acting as a query to retrieve relevant regions in an image (Lee et al., 2018a; Miech et al., 2021; Zheng et al., 2022), but it has been used occasionally for supervised $^2$We refer to this as a measure because it is not a valid distance metric, violating symmetry+triangle inequality. time-series tasks (Garg & Candan, 2021; Yang et al., 2022a). We describe cross-attention below for a given query time-point, $ x_q $ (biases, norms, scaling factor, and linear layer are omitted for brevity): \[ \text{CrossAttn}(x_q; X_k) = \sum_{x_k \in X_k} \frac{\exp(\langle x_q W_q, x_k W_k \rangle)}{\sum_{x_k' \in X_k} \exp(\langle x_q W_q, x_k' W_k \rangle)} (x_k W_v) \] After generalizing $ \langle \cdot, \cdot \rangle = \text{sim}(\cdot, \cdot) $ and $ xW = f(x) $ and reformulating for reconstruction with $ \bar{x} $, we have REBAR and its retrieval formulation in Eq. 3 and Eq. 4, respectively: \[ \text{REBAR}(\bar{x}_q; X_k) = \sum_{x_k \in X_k} \frac{\exp(\text{sim}(f_q(\bar{x}_q), f_k(x_k)))}{\sum_{x_k' \in X_k} \exp(\text{sim}(f_q(\bar{x}_q), f_k(x_k')))} f_v(x_k) \] The retrieval and reconstruction steps of our REBAR($ \bar{x}_q, X_k $) cross-attention is visualized in Fig. 3. We utilize stacks of dilated convolutions for REBAR’s $ f_{k/q/v} $, similar to WaveNet (van den Oord et al., 2016), instead of a linear layer. This allows for the retrieval function’s similarity function, $ \text{sim}(\cdot, \cdot) $, to compare the motifs from the context window around $ \bar{x}_q $ with those in the window around $ x_k $ (Xu et al., 2022). Vanilla cross-attention’s linear layer only compares individual time-points with each other. This comparison is illustrated in Fig. 4. The retrieval function, $ p(x_k\|\bar{x}_q) $, identifies regions surrounding an $ x_k \in X_k $ that are useful for reconstructing $ \bar{x}_q $, and then the $ f_v $ consolidates information from that region for reconstruction. See Appendix A.1.1 for further details. To restate: the objective of REBAR is to learn a class-discriminative measure that will reconstruct well when $ X_k $ has matching motifs with the $ X_q $, but will reconstruct poorly when it does not. Therefore, we would like to emphasize a good retrieval function, $ p(x_k\|\bar{x}_q) $ that effectively compares and retrieves motifs and avoid a complex model that may achieve an accurate reconstruction even when $ X_q $ and $ X_k $ are dissimilar. As such, we design our model so that the query is not directly used for reconstruction: it is only used to identify regions in the key subsequence to retrieve with $ p(x_k\|\bar{x}_q) $. In other words: for some function $ g : \{\Delta^T\}^T \times (T \times D) $, \[ \text{REBAR}(\bar{x}_q, X_k) = g(p(X_k\|\bar{x}_q), X_k) \] where $ \Delta^T $ is the $ T $-dimensional probability simplex: that is, the reconstruction only depends on the query through the probability weights in $ p(X_k\|\bar{x}_q) \in \{\Delta^T\}^T $. The model cannot simply borrow information from within the query to reconstruct itself. For example, if there are 3 time-points on an upwards line and the middle-time point is missing, the model is unable to directly learn a simple linear interpolation to reconstruct that point. Instead, the model is forced to identify a similar upwards line in the key and use this retrieved window for reconstruction. The model can only reconstruct the query from retrieved regions of the key with $ f_v(x_k) $. As such, reconstruction ability is directly dependent on how similar the motifs in the key are to the query. Figure 4: Comparison of different $ f_{q/k} $ within $ \text{sim}(f_q(\bar{x}_q), f_k(x_k)) $. The REBAR’s $ f := \text{Dilated Convolution} $ allows for semantically-meaningful motif comparison within the retrieval function, unlike in the vanilla’s $ f := xW $, in which single time-points are compared with another. Now, we note that training REBAR ($ \bar{X}_q, X_k $) to learn how to retrieve similar motifs for reconstruction is done in the pre-contrastive learning stage, and this should be done without labels so that we can later use REBAR to identify positive pairs for the self-supervised setting. However, given a random $ \bar{X}_q, X_k $ pair, we do not know if they share class labels, and thus we do not know if reconstruction error should be minimized to learn a motif-matching similarity function between them. Therefore, during training, we set $ X_q $ and $ X_k $ to be the same value, so that REBAR learns a motif similarity function that is able to retrieve the regions from the key $ X_k $ that match the missing region from the query, $ \bar{X}_q $, for reconstruction. Note that we use $ X_q $ and $ X_k $ to indicate two separate variables as inputs to cross-attention, and their values can be the same or different. As previously noted, they are the same during training, but during application, when we use REBAR to identify positive pairs for contrastive learning, $ X_q $ and $ X_k $ are different instances with different values, and REBAR uses its learned motif-retrieval function to reconstruct the query from the most salient motifs in the key. See Appendix A.1.2 for further details on masking methodology during training and application. ### 4.2 Applying our REBAR Measure in Contrastive Learning In contrastive learning, the anchor and positive instance are pulled together and the anchor and negative instances are pushed apart in the embedding space. Due to the REBAR measure’s aforementioned class-discriminative properties (which is further empirically validated in Sec. 4.3), we can use REBAR to label candidate instances as being positive or negative relative to the anchor. The trained REBAR cross-attention is used to attempt to reconstruct $ \bar{X}_{\text{anchor}} $ from $ X_{\text{cand}} $. Note that the anchor subsequence $ X_{\text{anchor}}^{(i)} $ and set of candidate subsequences, $ S_{\text{cand}}^{(i)} $, are randomly sampled from the time-series $ A^{(j)} $. Across all of our downstream experiments and datasets, we set $ \|S_{\text{cand}}\| = 20 $. Then, we label the candidates either to be the positive, $ X_{\text{pos}}^{(i)} $, or to be in the within-time-series negative set $ S_{\text{within-neg}}^{(i)} $ based on the reconstruction performance. These labels are then used in our within-time-series loss, $ L_w $, modeled by NT-Xent (Sohn, 2016), shown below. \[ X_{\text{pos}}^{(i)} := \argmin_{X_{\text{cand}}^{(i)} \in S_{\text{cand}}^{(i)}} d(X_{\text{anchor}}^{(i)}, X_{\text{cand}}^{(i)}) \] \[ S_{\text{within-neg}}^{(i)} := S_{\text{cand}}^{(i)} \setminus \{X_{\text{pos}}^{(i)}\} \] \[ L_w = - \log \frac{\exp(\cos(X_{\text{anchor}}^{(i)}, X_{\text{pos}}^{(i)})/\tau)}{\sum_{X_{\text{neg}}^{(i)} \in S_{\text{within-neg}}^{(i)}} \exp(\cos(X_{\text{anchor}}^{(i)}, X_{\text{neg}}^{(i)})/\tau) + \exp(\cos(X_{\text{anchor}}^{(i)}, X_{\text{pos}}^{(i)})/\tau)} \] (6) This within-time-series loss, $ L_w $, captures how a time-series can change class labels over time. It learns to pull the anchor and the subsequence that is most likely to be of the same class as the anchor, according to REBAR, together in the embedding, while pushing those that are less likely, apart. Next, in order to capture the relationships between-time-series, the other anchor subsequences from the other time-series in our batch, $ A^{(j)} $ with $ j \neq i $, are set to be the between-time-series negatives set, $ S_{\text{between-neg}} $. Then, along with our original $ X_{\text{pos}}^{(i)} $, we have our between-time-series loss, $ L_b $. \[ S_{\text{between-neg}}^{(i)} := \bigcup_{j \neq i} X_{\text{anchor}}^{(j)} \] \[ L_b = - \log \frac{\exp(\cos(X_{\text{anchor}}^{(i)}, X_{\text{pos}}^{(i)})/\tau)}{\sum_{X_{\text{neg}}^{(j)} \in S_{\text{between-neg}}^{(i)}} \exp(\cos(X_{\text{anchor}}^{(i)}, X_{\text{neg}}^{(j)})/\tau) + \exp(\cos(X_{\text{anchor}}^{(i)}, X_{\text{pos}}^{(i)})/\tau)} \] (7) This between-time-series loss, $ L_b $, captures differences in time-series, as well as differences in patients, because commonly, each time-series originates from a different patient. This approach is most similar to augmentation-based methods (e.g. SimCLR) that draw their negatives from the batch. We utilize a convex combination of these two losses from Eq. 6 and 7 to create our final loss function, $ L $, in Eq. 8, to give us the flexibility between emphasizing learning differences found within-ts or between-tS. For example, for an accelerometry signals, we could emphasize learning how activity changes over time for a given user by decreasing $\alpha$, and for an ECG signal, we could emphasize how a patient’s heart condition is different from other patients’ by increasing $\alpha$. See Appendix A.1.3 for further discussion on how $\alpha$ is chosen and its impact on downstream performance. $$L = \alpha L_b + (1 - \alpha)L_w \text{ with } 0 \leq \alpha \leq 1$$ (8) 4.3 REBAR Nearest Neighbor Validation Experiment Before using REBAR in contrastive learning, we assess whether the REBAR-identified positive pairs are meaningful, by evaluating whether the REBAR measure effectively predicts mutual class membership. This validation experiment is shown in Eq. 9 and is done by borrowing the class-labels that would typically only be used in downstream experiments. The labels are used in a nearest neighbor classification of an anchor, where distance is measured by REBAR. $$P(c_{\text{pred}} = c \| c_{\text{true}}) = \mathbb{E}_{A^{(i)} \sim D} \left[ \mathbb{E}_{X^{(i)} \sim A^{(i)}} \left[ \mathbb{I}_{c_{\text{true}}} \left( \argmin_{c \in \{1,\cdots,C\}} d(X^{(i)}_{\text{anchor}}, X^{(i)}_{\text{cand},c}) \right) \right] \right]$$ (9) This gives us a conditional probability of a predicted class being $c$, given that the anchor is of class $c_{\text{true}}$. One trial randomly segments an anchor subsequence and one candidate subsequence from each of the $C$ classes. This trial is repeated for the given time-series, $A^{(i)}$, and for all $A^{(i)}$ in our dataset to obtain each empirical expectation estimate. The specific algorithm details are in Appendix A.4.1. The confusion matrices in Fig. 5 help visualize REBAR’s strong results. ![Figure 5](image) Figure 5: There is a high concentration on the diagonals of the confusion matrices across all of our datasets. This shows that REBAR, although trained with a reconstruction task without class labels, is able to predict mutual class membership, validating our idea for using REBAR to identify positive pairs in contrastive learning. Our three datasets are further explained in the Sec. 5 and in Appendix A.2, but what is most important to note is that each of them represent distinctively different sensor modalities. Given this, across all three distinctive domains, each true label’s highest prediction label is still always itself, such that $c_{\text{true}} = \argmax_c P(c_{\text{pred}} = c \| c_{\text{true}})$. This implies that REBAR-identified positive pairs will match the anchor with its correct class more often than any of the other individual classes, and so using REBAR to train our contrastive learning framework will encourage mutual classes to be pushed together in the embedding space over time. This validates our usage of REBAR in the unsupervised setting. As previously noted in Sec. 4.2, we use REBAR to compare an anchor subsequence with a set of randomly sampled candidate subsequences in order to identify the positive candidate. 5 Downstream Experiments and Results In this section, we detail our experimental design for evaluating REBAR against other contrastive learning methods, and the results with an ablation study are shown in Sec 5.1. Benchmarks: We aim to assess how differing methods perform based upon their contrastive objective, so the encoder architecture (Yue et al., 2022) is kept constant across all benchmarks. Each benchmark represents a specific time-series contrastive learning paradigm, and they are listed below. Further implementation details are found in Appendix A.3. Figure 6: Qualitative Clusterability results with t-SNE Visualizations of each benchmark’s encodings on the HAR dataset. Most methods are able to encode “Lay” into its own cluster, and also “Sit” and “Stand” into nearby but generally distinct clusters. REBAR is the only method able to successfully separate out the “Walk”, “Walk Up”, and “Walk Down” labels into disjoint clusters. - TS2Vec is a strong augmentation-based method with time-stamp representations (Yue et al., 2022). - TNC is a sampling-based method that samples a nearby subsequence as a positive and utilizes a hyperparameter to estimate whether a distant sample is negative (Tonekaboni et al., 2021). - CPC contrasts based on future timepoint predictions (Oord et al., 2018). - SimCLR is a simple augmentation-based method (Chen et al., 2020), which we have adapted for time-series with the most common augmentations (i.e. shifting, scaling, jittering). - Sliding-MSE is a simplified REBAR that instead uses a sliding-MSE comparison as a measure. Data: We utilize 3 datasets from 3 different sensor domains with time-series that have their classification labels change over time: Human Activity Recognition (HAR) with accelerometer and gyroscopic sensors to measure activity (Reyes-Ortiz et al., 2015), PPG to measure stress (Schmidt et al., 2018), and ECG to measure heart condition (Moody, 1983). Each of these modalities have drastically different structures, and the class-specific temporal patterns vary differently within a modality. Please find further dataset descriptions and visualizations in Appendix A.2. Downstream Evaluation: To evaluate their class-discriminative strengths, we learn a linear probe (i.e. logistic regression) on each model’s frozen encoding for downstream classification and use the Accuracy, AUROC, and AUPRC metrics to quantify the results. Additionally, a fully supervised model composed of an encoder, identical to that used by the baselines, with a linear classification layer is benchmarked. This matches our baselines’ linear probe evaluation, only trained end-to-end. We then also assess cluster agreement for further corroboration. After a $k$-means clustering of the frozen encoding, with $k$ as the number of classes, we assess the similarity of these clusters with the true labels with the Adjusted Rand Index and Normalized Mutual Information metrics. ### 5.1 Results Linear Probe Classification: Tbl. 1 shows that the linear probe trained on our REBAR representation consistently achieved the strongest results, even beating the fully supervised model in PPG and HAR, achieving the same accuracies, but higher AUROC and AUPRC. For ECG, REBAR achieves better accuracy, but lower AUROC and AUPRC. This demonstrates our REBAR’s methods strength in learning a representation that is better at handling class imbalance than the fully-supervised model. \| Model \| HAR ↑ Accuracy \| HAR ↑ AUROC \| HAR ↑ AUPRC \| PPG ↑ Accuracy \| PPG ↑ AUROC \| PPG ↑ AUPRC \| ECG ↑ Accuracy \| ECG ↑ AUROC \| ECG ↑ AUPRC \| \|----------------\|---------------\|-------------\|-------------\|---------------\|-------------\|-------------\|---------------\|-------------\|-------------\| \| Fully Supervised \| 0.9535 \| 0.9835 \| 0.9531 \| 0.4138 \| 0.6241 \| 0.3689 \| 0.7814 \| 0.9329 \| 0.9260 \| \| TS2Vec \| 0.9324 \| 0.9931 \| 0.9766 \| 0.4023 \| 0.6428 \| 0.3959 \| 0.7612 \| 0.8656 \| 0.8516 \| \| TNC \| 0.9437 \| 0.9937 \| 0.9788 \| 0.2989 \| 0.6253 \| 0.3730 \| 0.7340 \| 0.8405 \| 0.8195 \| \| CPC \| 0.8662 \| 0.9867 \| 0.9438 \| 0.3448 \| 0.5843 \| 0.3642 \| 0.7775 \| 0.8377 \| 0.8223 \| \| SimCLR \| 0.9465 \| 0.9938 \| 0.9763 \| 0.3448 \| 0.6119 \| 0.3688 \| 0.9024 \| 0.8784 \| 0.8063 \| \| Sliding-MSE \| 0.9352 \| 0.9931 \| 0.9763 \| 0.3333 \| 0.6456 \| 0.3831 \| 0.7751 \| 0.8755 \| 0.8574 \| \| REBAR (ours) \| 0.9535 \| 0.9965 \| 0.9891 \| 0.4138 \| 0.6977 \| 0.4457 \| 0.8154 \| 0.9146 \| 0.8985 \| Table 1: Linear Probe Classification Results with Accuracy, AUROC, and AUPRC Our REBAR method demonstrates a much stronger performance than Sliding-MSE, showing the necessity of learning a retrieval-reconstruction distance rather than using a simple measure to identify positive pairs. Our improved performance compared to TNC highlights the value of identifying positives that are not necessarily near the anchor. SimCLR’s subpar results demonstrate that even if common time-series augmentations are used, this does not guarantee strong performance and the set of augmentations should be tuned. Although TS2Vec is a state-of-the-art method, it is unable to consistently achieve the strongest performance among the other benchmarks. We suspect that... this is because TS2Vec was evaluated on short class-labeled time-series rather than time-series with class-labeled subsequences. REBAR’s sampling-based approach successfully exploits this structure to sample positive pairs from subsequences across the time to achieve strong performance. Clusterability Evaluation: Tbl. 2 shows that when measuring the cluster agreement with the true class labels, REBAR continues to achieve the best ARI and NMI, corroborating the strong classification results. This is unlike other methods, such as TS2vec in PPG, that achieve strong linear probe results, but low cluster agreement. Additionally, the t-SNE visualizations shown in Fig. 6 for HAR and in Appendix A.4.2 for remaining datasets show that REBAR’s encodings push mutual classes together and distinct classes apart, even distinct classes that are semantically similar, such as "Walk", "Walk Down", and "Walk Up". These results support the idea that REBAR’s embedding space is particularly class-discriminative. \| Model \| HAR \| PPG \| ECG \| \|-------------\|--------------\|--------------\|--------------\| \| \| ↑ ARI ↑ NMI \| ↑ ARI ↑ NMI \| ↑ ARI ↑ NMI \| \| TS2Vec \| 0.4654 0.6115 \| -0.0353 0.1582 \| 0.2087 0.1701 \| \| TNC \| 0.4517 0.5872 \| 0.0958 0.1666 \| 0.2186 0.1753 \| \| CPC \| 0.1603 0.2217 \| 0.1110 0.1867 \| 0.0532 0.0724 \| \| SimCLR \| 0.5805 0.6801 \| 0.1535 0.3081 \| 0.2182 0.1751 \| \| Sliding-MSE \| 0.5985 0.7019 \| 0.1083 0.2141 \| -0.0081 0.0180 \| \| REBAR (ours)\| 0.6258 0.7721 \| 0.1830 0.3422 \| 0.2260 0.1796 \| Table 2: Clusterability Results with Adjusted Rand Index and Normalized Mutual Information REBAR approach analysis: Fig. 7 visualizes the positive pairing that was identified by our REBAR measure from a randomly sampled set of candidates for a given anchor for the HAR dataset. We see that even when there is no exact match of the anchor within the candidates, REBAR’s motif-comparison retrieval and reconstruction is able to identify a positive example that shares the same class as the anchor. Please find a large gallery of positive pairing visualizations in Appendix A.4.4. ![Figure 7: Positive pairings identified by REBAR from the candidates, for an anchor of each class.](image) The key component of our REBAR cross-attention design is the dilated convolutions used for motif comparison and retrieval. We find that removing and replacing these convolutions with the vanilla linear layer, results in performance drops of a 6.4% decrease in accuracy of the linear probe, which is worse than all but one of the benchmarks, and a 34.1% NMI decrease in the clusterability evaluation. Additionally, we find that REBAR is fairly robust against hyperparameter tuning. When we modify the size of the dilated conv’s receptive field, the size of masks, or REBAR cross-attention reconstruction training epochs, performance remains consistent. See Appendix A.1.4 for further details and additional specific model ablation results. ## 6 Conclusion In this paper we introduced REBAR, a novel approach to time-series contrastive learning. By using cross-attention to retrieve class-specific motifs in one subsequence to reconstruct another subsequence, we can predict mutual class membership. Then, if we use this REBAR measure to identify positive pairs, we are able to achieve state-of-the-art results in learning a class discriminative embedding space. Our REBAR method offers a new perspective in time-series self-supervised learning with our measure-focused approach, and we hope that this work will drive future research into how to best capture and encode semantic relationships between time-series. 7 ACKNOWLEDGEMENTS We would like to thank Catherine Liu for her help and support on this work. This work is supported in part by NIH P41-EB028242-01A1, NIH 1-R01-CA224537-01, and the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2039655. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. 8 ETHICS STATEMENT Our paper works on creating models for health-related signals, and it has the potential to improve health outcomes, but at the same time could lead to a loss of privacy and could possibly increase health-related disparities by allowing providers to characterize patients in more fine-grained ways. In the absence of effective legislation and regulation, patients may lack control over use of their data, leading to questions of whether autonomy, a key pillar of medical ethics, are being upheld. Overall though, we hope that our work leads to a net positive as it helps further the field towards creating personalized health recommendations, allowing patients to receive improved care and achieve better health outcomes, directly contributing to patient safety and overall well-being. 9 REPRODUCIBILITY STATEMENT Our Methods section in Section 4 details the way in which we set-up our method, and our Experiments section in Section 5 details our experimental design. Additionally, in the Appendix A.3, we itemize each of the hyperparameters we used to tune each of our benchmarks. Upon acceptance, we will release our GitHub code publicly, which will have the set seeds and exact code we used to run our experiments. We will also make our model checkpoints downloadable. The datasets used are publicly available, and we describe how we curate each of them for our task in Appendix A.2. Additionally, our code can be found at https://github.com/maxxu05/rebar. 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Timemae: Self-supervised representations of time series with decoupled masked autoencoders. arXiv preprint arXiv:2303.00320, 2023. Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018. Emadeldeen Eldele, Mohamed Ragab, Zhenghua Chen, Min Wu, Chee-Keong Kwoh, Xiaoli Li, and Cuntai Guan. Self-supervised contrastive representation learning for semi-supervised time-series classification. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2023. Jean-Yves Franceschi, Aymeric Dieuleveut, and Martin Jaggi. Unsupervised scalable representation learning for multivariate time series. Advances in neural information processing systems, 32, 2019.
rzF0R6GOd4	Since the mathematical derivation matters here, please state what assumptions go into the paragraph between equation 5 and 6, which talks about the continuity and differentiability of a time-dependent SDF. In theory, there is nothing preventing an object from appearing out of nowhere, an SDF does not inherently have any restrictions on its temporal evolution (while it does have restrictions in space, namely the Eikonal equation). In real-world cases, geometry noise (say, due to the sensor or an imperfect reconstruction) pops randomly into existence and vanishes randomly over time, which leads to discontinuities w.r.t. the time parameter. Unless point x is meant to be a Lagrangian particle rather than an Eulerian grid coordinate? Figure 2 looks Eulerian though. --- Please state the assumptions that go into that paragraph.	Neural SDF Flow for 3D Reconstruction of Dynamic Scenes Wei Mao Australian National University wei.mao@anu.edu.au Richard Hartley Australian National University & Google richard.hartley@anu.edu.au Mathieu Salzmann CVLab, EPFL & SDSC, Switzerland mathieu.salzmann@epfl.ch Miaomiao Liu Australian National University miaomiao.liu@anu.edu.au Abstract In this paper, we tackle the problem of 3D reconstruction of dynamic scenes from multi-view videos. Previous dynamic scene reconstruction works either attempt to model the motion of 3D points in space, which constrains them to handle a single articulated object or require depth maps as input. By contrast, we propose to directly estimate the change of Signed Distance Function (SDF), namely SDF flow, of the dynamic scene. We show that the SDF flow captures the evolution of the scene surface. We further derive the mathematical relation between the SDF flow and the scene flow, which allows us to calculate the scene flow from the SDF flow analytically by solving linear equations. Our experiments on real-world multi-view video datasets show that our reconstructions are better than those of the state-of-the-art methods. Our code is available at https://github.com/wei-mao-2019/SDFFlow.git. 1 Introduction The 3D reconstruction of a dynamic scene from multi-view videos is a very challenging research topic compared to its counterpart for static scenes. Yet, it has many important applications ranging from virtual/augmented reality to videos games, where it is required to model changes in the 3D environment, i.e., surface deformations. To handle such deformations, traditional non-rigid structure from motion methods [Bregler et al., 2000; Akhter et al., 2008] require 2D correspondences across time. While more recent works [Pumarola et al., 2021; Park et al., 2021a; Li et al., 2021] tackle this problem with neural rendering techniques, i.e., NeRF [Mildenhall et al., 2021], almost all those works [Pumarola et al., 2021; Park et al., 2021a; Li et al., 2021] directly model the movements of 3D points. Despite their great success, these methods mainly focus on synthesizing photo-realistic novel views and cannot obtain good 3D geometry due to the shape radiance ambiguity [Zhang et al., 2020]. To resolve such ambiguity, a commonly used strategy is to parameterize the density with Signed Distance Function (SDF) [Wang et al., 2021b; Yariv et al., 2021]. In this work, we aim for the reconstruction of an unconstrained dynamic scene and for recovering the 3D motion of the scene, i.e., scene flow, using NeRF [Mildenhall et al., 2021]. Previous works [Yang et al., 2022; Xu et al., 2021; Wang et al., 2022; Grassal et al., 2022; Hong et al., 2022; Guo et al., 2023] on dynamic object reconstruction are restricted to either a single articulated object such as a human [Yang et al., 2022; Xu et al., 2021; Wang et al., 2022; Guo et al., 2023] or a pre-defined template such as human head [Grassal et al., 2022; Hong et al., 2022]. To handle an unconstrained scene that may contain multiple (non-)rigid objects, existing NeRF-based methods [Cai et al., 2022; Shao et al., 2023] either require mapping the 3D points to a higher dimensional space to account for topology changes [Cai et al., 2022] or directly predict the SDF at each time step [Shao et al., 2023], thus preventing them to recover the scene flow. By contrast, we propose a novel representation, namely SDF flow, which naturally captures the topology changes and allows us to infer the scene flow. Based on the observation that the SDF of any given point in a dynamic scene is continuous and almost everywhere smooth with respect to time, our SDF flow is defined as the first-order derivative of the SDF with respect to time. Given such an SDF flow, the SDF at any point and at any given time is simply the integral of its SDF flow. We then develop a NeRF-based method that, instead of directly predicting the SDF, is trained to estimate the SDF flow, allowing us to extract the 3D scene motion. To obtain such a 3D scene motion, we derive the mathematical relationship between the SDF flow and the scene flow. Specifically, we show that the SDF flow of a 3D point can be expressed as a linear function of its location, surface normal, and the scene flow. We then demonstrate that, without any supervision, we can analytically compute the scene flow from the SDF flow. Although such scene flow can be noisy due to the reconstruction error, we showcase that the linear relationship provides a good regularization on both the scene flow and SDF flow resulting better scene flow estimation and 3D reconstruction. We believe that revealing this relationship will be valuable for future research. Our contributions can be summarized as follows: i) we propose the SDF flow as a novel 3D representation of unconstrained dynamic scenes; ii) we unify the SDF flow and the scene flow with a linear function; iii) With our SDF flow representation, we introduce a NeRF-based pipeline that can reconstruct the 3D geometry of a dynamic scene given multi-view videos. We evaluate our method on real world multi-view videos, and our model reconstructs more accurate surfaces than those of the state-of-the-art dynamic scene reconstruction methods. 2 RELATED WORK Neural radiance field for dynamic scenes. Given multi-view images, a neural radiance field (NeRF) (Mildenhall et al., 2021) optimizes a continuous function that maps any 3D location to its density and radiance. Such a function has been proven to be effective for novel view synthesis of static scenes. Recent works (Park et al., 2021a,b; Pumarola et al., 2021; Li et al., 2021; Tretschk et al., 2021; Du et al., 2021; Wang et al., 2021a; Song et al., 2022; Fang et al., 2022; Li et al., 2022, 2023) further extend it to dynamic scenes. Most of them propose to optimize additional functions that deform the observed points to a canonical space (Park et al., 2021a,b; Pumarola et al., 2021; Tretschk et al., 2021; Fang et al., 2022) or over time (Li et al., 2021; Wang et al., 2021a; Du et al., 2021; Li et al., 2023). Despite their good quality of novel view synthesis, these methods cannot reconstruct faithful 3D scene geometry due to the “shape-radiance ambiguity” (Zhang et al., 2020). 3D reconstruction of dynamic scenes. Traditional non-rigid structure from motion (NRSfM) methods (Bregler et al., 2000; Akhter et al., 2008) reconstruct deformable 3D shapes from a set of 2D correspondences. Such correspondences are sometimes hard to obtain, making these methods not suitable to complex real-world scenes. Although some works (Blanz et al., 2003; Cao et al., 2014; Ichim et al., 2015; Thies et al., 2016; Guo et al., 2018; Gafni et al., 2021; Yang et al., 2021a,b; Xu et al., 2021; Yang et al., 2022; Wang et al., 2022; Hong et al., 2022; Grassal et al., 2022; Guo et al., 2023) can reconstruct non-rigid objects without requiring 2D correspondences, they assume the reconstructed object to be either articulated or follow certain pre-defined templates such as, (Cao et al., 2013; Li et al., 2017; Blanz & Vetter, 2023). Such assumptions make these methods not suitable for unconstrained scenes where there may be multiple non-rigid moving objects. Other works (Newcombe et al., 2015; Immann et al., 2016; Slavcheva et al., 2017; Lin et al., 2022) that can handle unconstrained scenes, require depth map as inputs. Two existing NeRF-based methods can nonetheless handle unconstrained scenes without depth information: NDR (Cai et al., 2022) and Tensor4D (Shao et al., 2023). NDR (Cai et al., 2022) introduces a bijective function that maps the points in observation space to a canonical space. It requires to extend the 3D input to a higher dimensional space to account for the topology changes (Park et al., 2021b). Tensor4D (Shao et al., 2023) represents the dynamic scene with a 4D tensor and further decomposes the tensor into several 2D planes to speed up training and inference. However, since their method directly estimate the SDF at each time step, it cannot recover the scene flow. Our SDF flow naturally captures the smooth deformations of the surface and handles topology changes by design. Given the SDF flow, we can further obtain the scene flow. 3 OUR APPROACH In this section, we first briefly introduce the neural radiance field and the SDF-based parameterization of the density (Section 3.1). We then describe our SDF flow to capture the dynamic scenes (Section 3.2). Lastly, we derive the mathematical relationship between the SDF flow and the scene flow (Section 3.3). 3.1 Preliminaries Neural radiance field (NeRF). The main idea of NeRF (Mildenhall et al., 2021) is to represent a static scene as a 5D continuous function that maps a 3D location $ x \in \mathbb{R}^3 $ and a viewing direction $ d \in \mathbb{R}^2 $ to the RGB radiance $ c \in \mathbb{R}^3 $ and the density $ \sigma \in \mathbb{R} $, i.e., \[ c, \sigma = f_\Theta(x, d), \] where the function $ f $ is typically implemented as a Multi-Layer Perceptron (MLP) with $ \Theta $ as trainable parameters. Given such function, for each ray $ r(r) = o + rd $ shooting from the camera origin $ o \in \mathbb{R}^3 $ along direction $ d $, one can obtain the pixel intensity via the volume rendering function \[ C(r) = \int_{r_n}^{r_f} T(r)\sigma(r(r))c(r(r), d)dr, \] where $ r_n $ and $ r_f $ are the bound of the 3D scene; $ T(r) = e^{-\int_{r_n}^{r} \sigma(r(l))dl} $ is the accumulated opacity; $ C(r) \in \mathbb{R}^3 $ is the rendered color of this ray. The model can then be trained by minimizing the loss between the rendered color $ C(r) $ and the ground-truth $ \bar{C}(r) $, i.e., \[ L_{RGB} = \\|C(r) - \bar{C}(r)\\|_1. \] Volume rendering with SDF. It has been shown that NeRF may not recover the correct 3D geometry due to the shape radiance ambiguity (Zhang et al., 2020). To address this issue, a few works have proposed to regularize the density by parameterizing it as an SDF (Wang et al., 2021b; Yariv et al., 2021). Taking VolSDF (Yariv et al., 2021) as an example, the density $ \sigma $ is defined as \[ \sigma = \frac{1}{\beta}\Psi_\beta(s(x)), \] where $ \Psi_\beta $ is the Cumulative Distribution Function (CDF) of the Laplace distribution with zero mean and scale $ \beta $. Instead of directly estimating the density in Equation [1], the function $ f $ outputs the SDF $ s(x) $. To extend NeRF to dynamic scenes, the most commonly adopted strategy is to jointly optimize an additional function that models the deformation in 3D space such function either maps all observation spaces to a canonical one such as (Cai et al., 2022) or models the temporal motion of the scene such as (Li et al., 2021). In the next section, we propose a drastically different representation, i.e., SDF flow, which tries to directly model the change of the dynamic scenes over time. 3.2 SDF Flow Let $ \Omega \subset \mathbb{R}^3 $ represent the 3D space occupied by a scene and $ \partial \Omega $ be the scene surface. The Signed Distance Function (SDF) $ s(x) $ is defined as \[ s(x) = \begin{cases} -\min_{y \in \partial \Omega} \\|x - y\\|_2 & \text{if } x \in \Omega \\ \min_{y \in \partial \Omega} \\|x - y\\|_2 & \text{otherwise}. \end{cases} \] For any given 3D point $ x $ in a dynamic scene, we can treat its SDF $ s(x, t) $ as a function of time $ t $. As shown in Figure 2, such a function is always continuous and almost everywhere differentiable in the real world scenario. To model the continuous function, we propose to estimate its first-order derivative $ \frac{\partial s}{\partial t} $ (SDF flow) as $ \frac{\partial s(x,t)}{\partial t} = f(x,t) $, where $ f $ is the function to be optimized during training. The SDF of point $ x $ at time $ t $ is then the integral of its SDF flow, i.e., \[ s(x,t) = \int_{t_0}^{t} f(x,t) dt + s(x,t_0), \] (5) where $ s(x,t_0) $ is the SDF of $ x $ at the initial time $ t_0 $, which can be produced by another function as $ s(x,t_0) = f_0(x) $. We can also obtain its normal as \[ n(x,t) = \nabla_x s = \int_{t_0}^{t} \nabla_x f(x,t) dt + n(x,t_0), \] (6) where $ n(x,t_0) = \nabla_x s(x,t_0) = \nabla_x f_0(x) \in \mathbb{R}^3 $. Figure 1 provides an overview of our pipeline. Given the SDF $ s(x,t) $ of point $ x $ at time $ t $, we can compute its density $ \sigma(s(x,t)) $ using Equation 4. We follow the neural rendering pipeline to further optimize another function that produces the radiance $ c $. Given the density and the radiance, we use the volume rendering equation defined in Equation 2 to obtain the final rendered RGB color $ C(r,t) $. Our training loss consists of two parts: \[ L = L_{RGB} + \lambda L_{SDF}, \] (7) where $ L_{RGB} = \mathbb{E}_{r \in P, t \in T} \\|C(r,t) - \tilde{C}(r,t)\\|_1 $ is the average color loss over all sampled rays $ P $ across all times $ T $, and $ L_{SDF} = \mathbb{E}_{x \in X, t \in T} (\\|n(x,t)\\|_2 - 1)^2 $ is the eikonal constraint of the SDF for all sampled points $ X $ across all times. $ \lambda $ is a balancing weight. Since it is often beneficial to obtain the 3D correspondences described by the scene flow for many applications or downstream tasks, in the next section, we derive the mathematical relation between the proposed SDF flow and the scene flow. ### 3.3 Relation between SDF Flow and Scene Flow Given any point $ x $ on the surface $ \partial \Omega $, we first define its $ \epsilon $-neighbor as a local region on the surface that contains that point, i.e., $ N_\epsilon(x) = \{ y \| \\|y - x\\|_2 < \epsilon, y \in \partial \Omega, x \in \partial \Omega \} $. When considering the scenario where the surface is evolving within time $ \Delta t > 0 $, we make the following assumption and derive a theorem, which we will illustrate via a toy example at the end of the section. Assumption 1 As time period $ \Delta t $ approaches zero, with sufficiently small $ \epsilon $, the motion of a surface point $ x $'s $ \epsilon $-neighbor $ N_\epsilon(x) $ is rigid and can be represented as a rotation $ \Delta R \in SO(3) $ and a translation $ \Delta T \in \mathbb{R}^3 $. Thus, we can obtain the corresponding location $ x' $ of point $ x $ after $ \Delta t $ as \[ x' = \Delta R x + \Delta T. \] (8) Theorem 2 Given a 3D location $ x $ which is on a locally smooth surface deforming smoothly, the SDF change of $ x $ thus the first order derivative of its SDF with respect to time is the negative projection of its scene flow on to its normal. Specifically, \[ \frac{\partial s}{\partial t} = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} \] (9) \[ = -\frac{\partial x}{\partial t}^T n(x), \] (10) where $ \frac{\partial x}{\partial t} \in \mathbb{R}^3 $ is the scene flow and $ n(x) \in \mathbb{R}^3 $ is the surface normal at location $ x $. We provide the proof in Section A.1 and illustrate the proof in Figure 3. Combining the Assumption 1 and Theorem 2, we have \[ \frac{\partial s}{\partial t} = -(\omega \times x + v)^T n(x), \] (11) Figure 2: Considering an ellipse moving right at constant speed (top), the SDF of the point as a function of time is always differentiable (bottom). Figure 3: 2D example of the relation between the scene flow $(x' - x)$ and the SDF flow $\Delta s$. The solid curve is the surface around $x$. The dashed one is the deformed surface after a very short time period. Figure 4: (a) 2D toy example of an ellipse moving with angular velocity $\omega$ and velocity $v$. $o$ is the origin. (b) We plot the SDF flow computed from Equation 9 and 11. Here, we use the polar coordinate system to represent points on the ellipse. The $x$-axis represents the points on the ellipse of different polar angles. The SDF flow computed from the scene flow (Equation 11) well matches that from the definition (Equation 9) for any point on the ellipse. where $\omega = [\frac{\partial \theta_x}{\partial t}, \frac{\partial \theta_y}{\partial t}, \frac{\partial \theta_z}{\partial t}]^T$ is the angular velocity of the surface ($\theta_x, \theta_y, \theta_z$ are the 3 rotation angles); $v = \frac{\partial T}{\partial t} \in \mathbb{R}^3$ is the velocity. The angular velocity and velocity define the 3D surface motion and thus the scene flow. The detailed derivation is provided in Section A.2. As will be demonstrated in Section 4, we would also like to compute scene flow directly from SDF flow with the derived relation linking them. To this end, we first transform Equation 11 to \[ \frac{\partial s}{\partial t} = -a^T \begin{bmatrix} \omega \\ v \end{bmatrix}, \] where $[\omega; v] \in \mathbb{R}^6$, $a_{1:3} = x \times n(x)$ and $a_{4:6} = n(x)$. In principle, given the SDF flow of at least 6 points that are moving rigidly, one can solve for the scene flow.$^1$ In practice, to handle the noise, we select more than 6 points and obtain the optimal scene flow by minimizing the least-square error (details are in Section A.3). Toy example. In Figure 4(a), we provide a 2D toy example to verify Equation 11 where the initial position of an ellipse as well as its angular velocity $\omega \in \mathbb{R}$ and velocity $v \in \mathbb{R}^2$ are given. As shown in Figure 4(b), the SDF flow computed from the scene flow (Equation 11) closely matches that from the definition (Equation 9). 4 EXPERIMENTS 4.1 DATASETS We evaluate our method quantitatively on the CMU Panoptic dataset (Joo et al., 2017) and qualitatively on the Tensor4D dataset (Shao et al., 2023). The CMU Panoptic dataset (Joo et al., 2017) captures various kinds of scenes including multi-person activities and human object interactions using multiple RGB(-D) cameras. Each scene is $^1$Note that, there exists exceptions where the solution may not match the true scene motion. We discuss such exceptions in Section A.3. Table 1: Quantitative results on the CMU Panoptic dataset. We report the accuracy (top), completeness (middle), and overall (bottom) in millimeter. For each sequence, the accuracy and completeness are averaged across all 24 frames, and the “avg” column is the average over all 5 scenes. \| acc (mm) \| Ian3 \| Haggling_b2 \| Band1 \| Pizza1 \| Cello1 \| avg \| \|----------\|------\|-------------\|-------\|--------\|--------\|-----\| \| NDR (Cai et al., 2022) \| 21.8 \| 12.5 \| 15.9 \| 17.7 \| 23.1 \| 18.2 \| \| Tensor4D (Shao et al., 2023) \| 15.4 \| 13.7 \| 17.1 \| 18.3 \| 17.9 \| 16.5 \| \| Ours \| 14.1 \| 8.3 \| 13.0 \| 11.5 \| 12.3 \| 11.8 \| \| comp (mm) \| Ian3 \| Haggling_b2 \| Band1 \| Pizza1 \| Cello1 \| avg \| \|-----------\|------\|-------------\|-------\|--------\|--------\|-----\| \| NDR (Cai et al., 2022) \| 20.7 \| 22.8 \| 23.7 \| 25.0 \| 19.5 \| 22.3 \| \| Tensor4D (Shao et al., 2023) \| 22.8 \| 25.3 \| 29.2 \| 27.4 \| 23.5 \| 25.6 \| \| Ours \| 17.5 \| 18.6 \| 21.4 \| 20.6 \| 15.2 \| 18.7 \| \| overall (mm) \| Ian3 \| Haggling_b2 \| Band1 \| Pizza1 \| Cello1 \| avg \| \|--------------\|------\|-------------\|-------\|--------\|--------\|-----\| \| NDR (Cai et al., 2022) \| 21.3 \| 17.7 \| 19.8 \| 21.3 \| 21.3 \| 20.3 \| \| Tensor4D (Shao et al., 2023) \| 19.1 \| 19.5 \| 23.2 \| 22.9 \| 20.7 \| 21.1 \| \| Ours \| 15.8 \| 13.5 \| 17.2 \| 16.1 \| 13.7 \| 15.2 \| captured by 10 RGB-D and hundreds of RGB cameras. In this paper, we only use the images from 10 RGB-D cameras. We obtain the ground-truth point cloud at each time step by registering the depth maps taken from those cameras using the provided camera poses and intrinsics. We select 5 challenging clips: “Ian3”, “Haggling_b2”, “Band1”, “Pizza1”, and “Cello1”. Our selected sequences cover activities like multi-person socializing (“Haggling_b2”), a band with multiple persons and musical instruments (“Band1”) and a mother playing with a little child (“Ian3”). Each clip contains 24 frames from 10 camera view thus 240 images. The resolution of the image is $1920 \times 1080$. Since our goal is 3D reconstruction, we use all 10 camera views for training and only evaluate the meshes. The Tensor4D dataset (Shao et al., 2023) is captured by a sparse-view system with RGB cameras. It contains a single person performing different actions like thumbs-up and waving hands. We select the 3 sample sequences provided on their official github page: “Boxing_v12”, “Dance_v4”, and “Thumbsup_v4”. The “Boxing_v12” sequence is captured by 12 cameras in a circle surrounding the human. The “Dance_v4”, and “Thumbsup_v4” sequences are taken by 4 forward-facing cameras. For each sequence, we select a clip of 12 frames. The image resolution is $1024 \times 1024$. Since no ground-truth geometry is available, we only provide qualitative comparison on this dataset. ### 4.2 Metrics, Baselines & Implementation Metrics. We follow the standard evaluation protocol in the multi-view stereo literature (Yao et al., 2018) to evaluate our method with accuracy, completeness and overall distance. Specifically, given the ground-truth point cloud $\mathcal{P}$, and the predicted point cloud $\hat{\mathcal{P}}$, the accuracy and completeness are defined as $$\text{Acc} = \frac{1}{\|\mathcal{P}\|} \sum_{p \in \mathcal{P}} \min_{\hat{p} \in \hat{\mathcal{P}}} \\| p - \hat{p} \\|_2$$ $$\text{Comp} = \frac{1}{\|\mathcal{P}\|} \sum_{p \in \mathcal{P}} \min_{\hat{p} \in \hat{\mathcal{P}}} \\| p - \hat{p} \\|_2 .$$ The overall distance is the average of the accuracy and completeness. Baselines. We compare our method with two recent NeRF-based dynamic scene reconstruction methods: NDR (Cai et al., 2022) and Tensor4D (Shao et al., 2023). NDR (Cai et al., 2022) attempts to find a bijective mapping between the observation space and the canonical space. Tensor4D (Shao et al., 2023) decomposes the 4D space into several 2D planes to speed up the model training. For both methods, we use their official implementation. Figure 5: Qualitative results on the CMU Panoptic dataset. Best viewed on screen. Implementation details. We implement our method using Pytorch (Paszke et al., 2017) and use the Adam optimizer (Kingma & Ba, 2014) to train our model with a 0.0005 learning rate. The batch size is set to 1024. We use the second-order Runge-Kutta method to solve the integration in Equation 5. We train our model for 2000 epochs, which takes around 7 days on 2 NVIDIA 4090 GPUs for ten $1920 \times 1080$ videos of 24 frames. The rendering of one ray takes around 1.5 ms. The balancing weight $\lambda$ is set to 0.1. During testing, for all baselines and our method, we construct a 3D grid of resolution $512 \times 512 \times 512$ and query the SDF of each voxel in this grid from the trained model. We then use the marching cube algorithm (Lorensen & Cline, 1996) to obtain the mesh. We uniformly sample 10000 points from the 3D mesh and compare them to the ground-truth point clouds. 4.3 Results Quantitative results. We provide quantitative results on the CMU Panoptic dataset (Joo et al., 2017) in Table 1. Our method consistently outperforms the baselines for all scenes in all metrics. For each scene, the reported accuracy (top), completeness (middle), and overall (bottom) are averaged across all frames, and we also report the average distance over all scenes (last column). Qualitative comparisons on the CMU Panoptic dataset. We compare our results to those of the baselines on the CMU Panoptic dataset in Figure 5. Here, we show the reconstruction results of 3 different time steps for 2 scenes: “Ian3” and “Haggling_b2”. As highlighted by the red box, the baselines (second and third columns) sometimes reconstruct overly smooth surfaces or even produce non-existing geometry. By contrast, the reconstructed meshes from our method are sharper with more details. More qualitative results on this dataset are provided in Section A.4 and the supplementary video. Qualitative comparisons on the Tensor4D dataset. The results are shown in Figure 6. Although NDR (Cai et al., 2022) performs well on “Dance_v4” (middle), it sometimes fails on “Thumbsup_v4” (right) and its reconstructions of “Boxing_v12” (left) are over smoothed. As for Tensor4D (Shao et al., 2023), although it can sometimes reconstruct more details, such as the face in ‘Thumbsup_v4”, all of its results share similar artifacts which we conjecture are due to the tensor decomposition. Our method performs comparably to the baselines, with fewer artifacts. Table 2: Optical flow evaluation on the CMU Panoptic dataset. We report the average end-point-error (EPE) in pixel. \| EPE (pixel) \| Ian3 \| Haggling_b2 \| Band1 \| Pizza1 \| Cello1 \| avg \| \|------------\|------\|-------------\|-------\|--------\|--------\|-----\| \| NDR (Cai et al., 2022) \| 6.86 \| 4.43 \| 1.66 \| 3.11 \| 2.41 \| 3.69 \| \| Ours \| 3.49 \| 2.97 \| 1.18 \| 1.18 \| 1.34 \| 2.04 \| Scene flow from SDF flow. We further demonstrate that using our SDF flow lets us derive the scene flow, i.e., the angular velocity and velocity. We first show the derived scene flow of a toy example where two rigid objects are moving in the scene. For any surface point, we use the surrounding surface points to compute the scene flow. The results are shown in Figure 7. Note that, here, we do not have any prior knowledge of the number of rigid objects in the scene. As shown in the second and fourth column, the scene flow clearly distinguishes the two moving objects. We provide quantitative evaluation on the scene flow of this synthetic sequence in Section A.9. We also compare our scene flow to that of NDR (Cai et al., 2022) on the real-world sequences of the CMU Panoptic dataset. Since the ground-truth scene flow is not available, we evaluate the optical flow instead and use the optical flow estimated from RAFT (Teed & Deng, 2020) as pseudo-ground truth. Specifically, we project our scene flow to the image plane to obtain the corresponding optical flow. For NDR (Cai et al., 2022), we use the bijective mapping to obtain the scene flow. We report the average end-point-error of the optical flow. The evaluation results is shown in Table 2. As also evidenced in Figure 8, our optical flow better matches the ground truth. We also visualize the 3D scene flow in Section A.5 and the supplementary video. Our scene flow better reflects the real scene motion. 5 CONCLUSION In this paper, we have proposed to exploit the SDF flow to represent a dynamic scene with the first-order derivative of its SDF with respect to time. Our SDF flow naturally captures the deformations of the scene surface. We have designed a NeRF-based pipeline using our SDF flow to reconstruct a dynamic scene from multi-view videos. Our experiments show that our method yields state-of-the-art performance. We have further derived a mathematical relation between the SDF flow and the scene flow. Such a relation allows us to calculate the scene flow from the SDF flow analytically by solving linear equations. We have demonstrated that the resulting scene flow correctly reflects the real motion. In the future, we would like to explore the potential of applying our method to monocular videos. ACKNOWLEDGEMENTS This research was supported in part by the Australia Research Council DECRA Fellowship (DE180100628) and ARC Discovery Grant (DP200102274). The authors would like to thank NVIDIA for the donated GPU (Titan V). REFERENCES Ijaz Akhter, Yaser Sheikh, Sohaib Khan, and Takeo Kanade. Nonrigid structure from motion in trajectory space. Advances in neural information processing systems, 21, 2008. Volker Blanz and Thomas Vetter. A morphable model for the synthesis of 3d faces. In Seminal Graphics Papers: Pushing the Boundaries, Volume 2, pp. 157–164. 2023. 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C5u71ph75Q	“To ensure this, we add an auxiliary regularizing loss in the form of a mean absolute error over λ −1 ,” Does this not introduce a Laplace likelihood (or prior) into the posterior? It’d be great to explore that relationship and decision a bit more.	INTERNAL-COORDINATE DENSITY MODELLING OF PROTEIN STRUCTURE: COVARIANCE MATTERS Anonymous authors Paper under double-blind review ABSTRACT After the recent ground-breaking advances in protein structure prediction, one of the remaining challenges in protein machine learning is to reliably predict distributions of structural states. Parametric models of fluctuations are difficult to fit due to complex covariance structures between degrees of freedom in the protein chain, often causing models to either violate local or global structural constraints. In this paper, we present a new strategy for modelling protein densities in internal coordinates, which uses constraints in 3D space to induce covariance structure between the internal degrees of freedom. We illustrate the potential of the procedure by constructing a variational autoencoder with full covariance output induced by the constraints implied by the conditional mean in 3D, and demonstrate that our approach makes it possible to scale density models of internal coordinates to full protein backbones in two settings: 1) a unimodal setting for proteins exhibiting small fluctuations and limited amounts of available data, and 2) a multimodal setting for larger conformational changes in a high data regime. 1 INTRODUCTION Proteins are macro-molecules that are involved in nearly all cellular processes. Most proteins adopt a compact 3D structure, also referred to as the native state. This structure is a rich source of knowledge about the protein, since it provides information about how the protein can engage biochemically with other proteins to conduct its function. The machine learning community has made spectacular progress in recent years on the prediction of the native state from the amino acid sequence of a protein (Jumper et al., 2021; Senior et al., 2020; Wu et al., 2022b; Back et al., 2021; Wu et al., 2022a). However, the static picture of the structure of a protein is misleading: in reality a protein is continuously moving, experiencing both thermal fluctuations and larger conformational changes, both of which affect its function. One of the remaining challenges in machine learning for structural biology is to reliably predict these distributions of states, rather than just the most probable state. We discuss the state of the density modelling field in Section 5 (Related work). Modelling the probability density of protein structure is non-trivial, due to the strong constraints imposed by the molecular topology. The specific challenges depend on the chosen structural representation: if a structure is represented by the 3D coordinates of all its atoms, these atom positions cannot be sampled independently without violating the physical constraints of e.g. the bond lengths separating the atoms. In addition, an arbitrary decision must be made about how the structure is placed in a global coordinate system, which implies that operations done on this representation should preferably be invariant or equivariant to this choice. An alternative is to parameterize the structure using internal coordinates, i.e. in terms of bond lengths, bond angles and dihedrals (rotations around the bonds). The advantage of this representation is that internal degrees of freedom can be sampled independently without violating the local bond constraints of the molecule. It also makes it possible to reduce the number of degrees of freedom to be sampled – for instance fixing the bond lengths to ideal values, since they fluctuate much less than the torsion angles and bond angles. For the reasons given above, an internal coordinate representation would appear to be an attractive choice for density modelling. However, one important problem reduces the appeal: small fluctuations in internal coordinates will propagate down the chain, leading to large fluctuations remotely downstream in the protein. As a consequence, internal-coordinate density modelling necessitates careful modelling of the covariance structure between the degrees of freedom in order to ensure that small fluctuations in internal coordinates result in small perturbations of the 3D coordinates of the protein. Such covariance structures are typically highly complex, making direct estimation difficult. In this paper, we investigate whether density modelling of full-size protein backbones in internal coordinates is feasible. We empirically demonstrate the difficulty in estimating the covariance structure of internal coordinates from data, and instead propose a technique for inducing the covariance structure by imposing constraints on downstream atom movement using the Lagrange formalism. Rather than estimating the covariance structure from scratch, we can instead modulate the covariance structure by choosing appropriate values for allowed fluctuations of downstream atoms. We demonstrate the procedure in the context of a variational autoencoder (Fig. 1). Given a prior on the internal coordinate fluctuations and a predicted mean, we impose constraints on the atom fluctuations in 3D space to obtain a full covariance structure over the internal coordinates. We show that this allows us to generate valid structures in terms of both internal and Cartesian coordinates. Our method is validated in two regimes: a low data regime for proteins that exhibit small, unimodal fluctuations, and a high data regime for proteins that exhibit multimodal behavior. We anticipate that this method could serve as a building block applicable more generally for internal-coordinate density estimation, for instance internal-coordinate denoising diffusion models. Our main contributions are: - We formulate a procedure for inducing full protein backbone covariance structure in internal coordinates, based on constraints on atom fluctuations in 3D space. - Rather than predicting a full covariance matrix over internal coordinates, our proposed method only requires to predict one Lagrange multiplier for each atom, from which the full covariance matrix can be constructed. For $M$ atoms, this corresponds to a reduction from $(2 \times M - 5)^2$ to simply $M$ predicted values. - We design a variational autoencoder which models fluctuations for full-length protein backbones in internal coordinates. Even though constraints are formulated in Euclidean space, the model is not dependent on a global reference frame (i.e. it is rotationally invariant). - We demonstrate that our model provides meaningful density estimates on ensemble data for proteins obtained from experiment and simulation. Scope. The focus of this paper will be on modelling distributions of protein structure states in internal coordinates. We are thus concerned with thermodynamic ensembles, rather than the detailed dynamics that a molecule undergoes. Dynamics could potentially be modelled on top of our approach, for instance by fitting a discrete Markov model to describe transitions between states, and using our approach to model the thermal fluctuations within a state, but this is beyond the scope of the current work. Another perspective on our approach is that we wish to describe the aleatoric uncertainty associated with a static structure. 2 BACKGROUND 2.1 CARTESIAN VS INTERNAL COORDINATES As stated before, Cartesian coordinates and internal coordinates each have advantages and disadvantages. Assume we have a 3D protein structure in Euclidean space with atom positions $x$. Throughout this paper, we only consider backbone atoms N, C$_\alpha$ and C, which means that the total number of atoms $M = 3 \times L$, with $L$ the number of amino acids. The Euclidean setting thus results in $3 \times M$ coordinates. Even though in this setting each of the atoms can fluctuate without affecting other atoms in the backbone chain, there is no guarantee for chemical integrity, i.e. conservation of bond lengths. and respecting van der Waals forces. This can lead to backbone crossings and generally unphysical protein structures. One way to ensure chemical integrity is to parameterize protein structure in internal coordinate space using dihedrals $\kappa_1$, bond angles $\kappa_2$ and bond lengths $\kappa_3$. Here, dihedrals are torsional angles that twist the protein around the bond between two consecutive atoms, bond angles are angles within the plane that is formed by two consecutive bonds, and bond lengths are the distances between two consecutive backbone atoms. Since bond length distributions have very little variance, we choose to fix them, thereby reducing the number of variables over which we need to estimate the covariance. We will refer to the remaining two internal coordinates together as $\kappa$ to avoid notation clutter. As dihedrals are defined by four points (the dihedral is the angle between the plane defined by the first three points and the plane defined by the last three points) and bond angles are defined by three points, the resulting protein structure representation will have $(2 \times M) - 5$ coordinates. Not only does this result in less coordinates to determine a full covariance structure over, the coordinates are also automatically rotation and translation invariant, as opposed to Cartesian coordinates. The remaining problem is that small changes in one internal coordinate can have large consequences for the global structure of the protein, since all atoms downstream of the internal coordinate will move together, acting like a rigid body. It is therefore challenging to preserve global structure while altering internal coordinates, since they are mostly descriptive of local structure. ### 2.2 Standard Precision Estimators Do Not Capture Global Fluctuations Because of the limitations of internal coordinates mentioned in Section 2.1, it is a highly non-trivial task to capture a full covariance structure over $\kappa$ which also conforms to constraints in Euclidean space that are inherent to the protein. As an example, we use a standard estimator to get a precision matrix (i.e. the inverse of the covariance matrix) over $\kappa$ for a short molecular dynamics (MD) simulation on “1pga”, also known as “protein G” (Fig. 2). Details about the simulation can be found in Appendix A. We see that when we take samples from a multivariate Gaussian over $\kappa$ with the true mean (based on the dataset) and the estimated precision, the samples exhibit atom fluctuations that are much higher than the original simulation, and with a very different pattern. To overcome the limitations that regular covariance and precision estimators have, we incorporate constraints on atom fluctuations in Euclidean space. ![Figure 2](image.png) Figure 2: When a standard estimator is used to get the precision structure over internal coordinates, resulting atom fluctuations significantly deviate from MD simulations. Blue arrows and red helices represent secondary structural elements. The variance is calculated as the mean of the variances over the x, y and z axis, in Å². ### 3 Internal-Coordinate Density Modelling with Constraints #### 3.1 Setup We parameterize a 3D protein structure in terms of internal coordinates (i.e. dihedrals and bond angles, while bond lengths are kept fixed), which together will be referred to as $\kappa$. Our aim is to obtain a multivariate Gaussian distribution over the deviations from the mean $p(\Delta \kappa)$, centered at zero, with a full precision structure. This target distribution is subject to constraints over atom fluctuations, enforcing the preservation of global structure. We have a prior $q(\Delta \kappa)$, which we will call the $\kappa$-prior, over the internal coordinate distribution, where the mean is zero and the precision is a diagonal matrix with the diagonal filled by the inverse variance over all $\kappa$ values $\sigma_{\kappa, data}^{-2}$, estimated from our input data. The strength of the $\kappa$-prior can be tuned using hyperparameter $a$. The $\kappa$-prior is defined as $$q(\Delta \kappa) = \frac{1}{Z_q} \exp \left( -\frac{1}{2} \Delta \kappa^T \Sigma_{\kappa,\text{prior}}^{-1} \Delta \kappa \right),$$ where $Z_q = \sigma_{\kappa,\text{data}} \sqrt{2\pi a}$ is the normalization constant for the $\kappa$-prior distribution and $\Sigma_{\kappa,\text{prior}}^{-1} = a \cdot \text{diag}(\sigma_{\kappa,\text{data}}^2)$. Our approach will be to construct a new distribution $p$ which is as close as possible to $q$, but which fulfills a constraint that prohibits the downstream 3D coordinates from fluctuating too much. We thus wish to minimize the Kullback-Leibler divergence between the objective distribution and $\kappa$-prior: $$D_{\text{KL}}(p\|\|q) = \int p(\Delta \kappa) \ln \frac{p(\Delta \kappa)}{q(\Delta \kappa)} d\Delta \kappa,$$ adding constraints on the expected value over squared atom displacements of each downstream atom: $$E_{\Delta \kappa \sim p(\Delta \kappa)} [\Delta x_m^2] = C_m$$ where $E_{\Delta \kappa \sim p(\Delta \kappa)} [\Delta x_m^2]$ is the expected value for the squared displacement of atom $m$, and $C_m$ is a constant equivalent to the variance of the atom position $\sigma_{x_m}^2$ assuming equal variance in all directions (isotropic Gaussian). Since every $\Delta x_m$ is a function of $\Delta \kappa$ with probability density function $p(\Delta \kappa)$, we can use the law of the unconscious statistician to reformulate the constraints as follows: $$E_{\Delta \kappa \sim p(\Delta \kappa)} [\Delta x_m^2] = \int \Delta x_m^2 p(\Delta \kappa) d\Delta \kappa = C_m$$ ### 3.2 Lagrange Formalism to Incorporate Constraints Employing Jaynes’ maximum entropy principle [Jaynes, 1957], we use the Lagrange formalism to incorporate $M$ of these constraints, with $M$ the number of atoms, under the conditions that our probability density $p(\Delta \kappa)$ has zero mean and sums to one.\footnote{Even though throughout this derivation we have included the normalization constant for rigor, in practice we work with unnormalized densities and normalize post hoc, since we recognize the final result to be Gaussian.} This leads to Lagrangian $$\tilde{D}(p,q) = \int p(\Delta \kappa) \ln \frac{p(\Delta \kappa)}{q(\Delta \kappa)} d\Delta \kappa + \lambda_0 \left( \int p(\Delta \kappa) d\Delta \kappa - 1 \right) + \sum_{m=1}^{M} \lambda_m \left( \int \Delta x_m^2 p(\Delta \kappa) d\Delta \kappa - C_m \right).$$ Next, we take the functional derivative and set it to zero: $\frac{\partial \tilde{D}(p,q)}{\partial p(\Delta \kappa)} = 0$, leading to the following well-established result [Jaynes, 1957; Kesavan & Kapur, 1989]:\footnote{Note that $\frac{\partial}{\partial y} \ln y/q + \lambda_0(y - 1) + \sum_{m=1}^{M} \lambda_m (\Delta x_m^2 y - C_m) = \ln \frac{y}{q} + y \cdot \frac{1}{q} + \lambda_0 + \sum_{m=1}^{M} \lambda_m \Delta x_m^2$} $$0 = \ln \frac{p(\Delta \kappa)}{q(\Delta \kappa)} + 1 + \lambda_0 + \sum_{m=1}^{M} \lambda_m \Delta x_m^2 \Rightarrow p(\Delta \kappa) = \frac{1}{Z_p} q(\Delta \kappa) \exp \left( -\sum_{m=1}^{M} \lambda_m \Delta x_m^2 \right)$$ with $Z_p = \exp(-1 - \lambda_0)$ the normalization constant of the target distribution. Note that $\frac{\partial^2 \tilde{D}(p,q)}{\partial p(\Delta \kappa)^2}$ is positive, therefore we know our solution will indeed be a minimum. ### 3.3 First Order Approximation for Atom Fluctuations In order to use Eq. (6) to satisfy the imposed constraints, we need to express $\Delta x^2$ in terms of $\Delta \kappa$. To first order, we can express the displacement vectors $\Delta x_m$ of each atom as a regular small angle approximation (first order Taylor expansion): $$\Delta x_m \approx \sum_i \frac{\partial x_m}{\partial \kappa_i} \Delta \kappa_i$$ where $ x_m $ is the position of the $ m $-th atom, under the condition that the atom is post-rotational (Bottaro et al., 2012), i.e., the location of atom $ m $ is downstream of the $ i $-th internal coordinate. From Eq. (7) it follows that the squared distance can be approximated by \[ \Delta x_m^2 \approx \left( \sum_i \frac{\partial x_m}{\partial \kappa_i} \Delta \kappa_i \right)^2 = \sum_{ij} \left( \frac{\partial x_m}{\partial \kappa_i} \Delta \kappa_i \cdot \frac{\partial x_m}{\partial \kappa_j} \Delta \kappa_j \right) = \Delta \kappa^T G_m \Delta \kappa , \] where $ G_{m,j}^{i} = \frac{\partial x_m}{\partial \kappa_i} \cdot \frac{\partial x_m}{\partial \kappa_j} $ is a symmetric and positive-definite matrix. Substituting Eq. (8) and our $ \kappa $-prior expression from Eq. (1) into our target distribution from Eq. (6) gives a new Gaussian distribution: \[ p(\Delta \kappa) \approx \frac{1}{Z} \exp \left( -\frac{1}{2} \Delta \kappa^T \left( \Sigma^{-1}_{\kappa,\text{prior}} + \Sigma^{-1}_{\kappa,\text{constr}} \right) \Delta \kappa \right) = \mathcal{N}(0, \hat{\Sigma}_\kappa) , \] where $ Z $ is the new normalization constant, $ \Sigma^{-1}_{\kappa,\text{constr}} = 2 \sum_{m=1}^{M} \lambda_m G_m $ and the covariance matrix of the new Gaussian distribution $ \hat{\Sigma}_\kappa = \left( \Sigma^{-1}_{\kappa,\text{prior}} + \Sigma^{-1}_{\kappa,\text{constr}} \right)^{-1} $. ### 3.4 Satisfying the Constraints The final step in the constrained optimization is to establish the values for the Lagrange multipliers. A closed form solution for this is not readily available, but using the findings from Section 3.3 we can now rewrite the constraints from Eq. (4) as \[ C_m = \mathbb{E}_{\Delta \kappa \sim \mathcal{N}(0, \hat{\Sigma}_\kappa)} \left[ \Delta \kappa^T G_m \Delta \kappa \right] = \text{tr}(\hat{\Sigma}_\kappa G_m) \] where the last simplification step comes from standard expectation calculus on a quadratic form $ (\Delta \kappa^T G_m \Delta \kappa) $, where $ \Delta \kappa $ has zero mean (Eq. 378 in Petersen et al., 2008). Although it is nontrivial to express Lagrange multipliers $ \lambda $ in terms of atom fluctuations $ C $, we thus see that it is possible to evaluate $ C $ given a set of Lagrange multipliers $ \lambda $. In the following, we will therefore construct our models such that our networks predict $ \lambda $, directly. ### 3.5 VAE Pipeline VAE model architecture. To demonstrate how our method works within a modelling context, we use a variational autoencoder (VAE), as illustrated in Fig. 3. The VAE has a simple linear encoder that takes internal coordinates $ \kappa $ (dihedrals and bond angles, bond lengths are kept fixed) as input and maps to latent space $ z $, where we have a standard Gaussian as a prior on the latent space, which we call $ z $-prior to avoid confusion with the $ \kappa $-prior. The decoder outputs the mean over $ \kappa $, which is converted into Cartesian coordinates using pNeRF (AlQuraishi, 2018). This mean structure in 3D coordinates is used for two purposes. First, using the structure we can evaluate the partial derivatives of atom positions with respect to the individual $ \kappa $ as in Eq. (7). Second, the predicted mean over $ \kappa $ is used to get a pairwise distance matrix $ d $ that serves as the input to a U-Net (Ronneberger et al., 2015), from which we estimate values for the Lagrange multipliers for each constraint. This allows the variational autoencoder, conditioned on the latent state $ z $, to modulate the allowed fluctuations. Implementation-wise, the U-net is concluded with an average pooling operation that for each row-column combination computes one Lagrange multiplier $ \lambda $. Together with our fixed-variance $ \kappa $-prior over $ \kappa $ and hyperparameter $ a $ determining the strength of this $ \kappa $-prior, a new precision matrix is formed according to Eq. (9). The model can generate new structures through simple ancestral sampling: first generating $ z $ from the standard normal $ z $-prior, and subsequently sampling from a multivariate Gaussian distribution with the decoded mean and the constructed precision matrix. For specific model settings see Appendix A. Loss. We customarily optimize the evidence lower bound (ELBO) using the Gaussian likelihood on the internal degrees of freedom as constructed above. This likelihood does not ensure that the predicted Lagrange multipliers are within the range within which our first order approximation of the fluctuations is valid. To ensure this, we add an auxiliary regularizing loss in the form of a mean absolute error over $ \lambda^{-1} $, which prevents the $ \kappa $-prior from dominating. By tuning the weight $ w_{\text{aux}} $ on the auxiliary loss, we can influence the strength of the constraints. Figure 3: Model overview. The encoder (left) embeds internal coordinates into the latent space. The decoder (right) predicts a mean, from which constraints are extracted to obtain a precision matrix. Together with the $\kappa$-prior over the precision matrix based on the input data, a new precision matrix is formed which can be used to sample from a multivariate Gaussian. 4 EXPERIMENTS 4.1 TEST CASES Unimodal setting in low data regime. We consider three test proteins for small fluctuations in a low data regime: 1unc, 1fsd, and 1pga. 1unc corresponds to the solution structure of the human villin C-terminal headpiece subdomain. This protein contains 36 residues, corresponding to 108 backbone (N, C$_\alpha$ and C) atoms. This solution nuclear magnetic resonance (NMR) dataset is freely available from the Protein Data Bank and contains 25 conformers. 1fsd, a beta beta alpha (BBA) motif, is also a freely available NMR dataset containing 41 structures. This system has 28 residues with 84 backbone atoms. 1pga, corresponding to B1 immunoglobulin-binding domain protein G, is a 56 amino acid long protein with 168 backbone atoms. We have a short in-house molecular dynamics (MD) simulation, which is 20ns long and structures were saved at a 50ps interval, resulting in 400 structures for this protein. See Appendix A for more details about the simulation. Multimodal setting in high data regime. We also include two test cases for larger fluctuations following a multimodal distribution in a high data regime. Both are known as “fast-folders” and the MD datasets were obtained from Lindorff-Larsen et al. (2011). We refer the reader to this work for detailed descriptions of the simulations. Chignolin (cln025) is a peptide with a hairpin structure, containing 10 residues and thus 30 backbone atoms. The simulation is 106 µs long, saved at a 200 ps interval, resulting in 534,743 data points. The second test case, 2f4k, is the chicken villin headpiece, with 35 residues and 105 backbone atoms. The simulated trajectory is 125 µs and also saved every 200 ps, yielding 629,907 structures. 4.2 METRICS For the unimodal setting, we choose two simple measures of local and global structure, respectively. To evaluate local structure fluctuations, we show Ramachandran plots, a well-known visualization tool in the context of protein structures, where $\phi$ and $\psi$ dihedrals, which are the torsional angles around the $N-C_\alpha$ and $C_\alpha-C$ bonds, are plotted against each other. As a global measure, we report the variance over atom positions, averaged over three dimensions, across superposed (i.e. structurally aligned) samples to evaluate global structure fluctuations. For the multimodal setting, we report free energy landscapes, parameterized by the first two components of time-lagged independent component analysis (TICA) (Molgedey & Schuster 1994). Similar to e.g. PCA, TICA fits a linear model to map a high-dimensional input to a lower-dimensional output, but TICA also incorporates the time axis. The resulting components are ranked according to their capacity to explain the slowest modes of motion. Taking the first two components corresponds to selecting reaction coordinates that underlie the slowest protein conformational changes, which is highly correlated with (un)folding behavior. We fit the TICA model on the time-ordered MD data, and pass samples from the VAE and baselines through the fitted model to create free energy landscapes. Baselines. Apart from comparing the generated samples from our model to reference distributions from MD or NMR, we include four baselines. The first baseline, named “$\kappa$-prior (fixed)” is a VAE trained to predict $\mu_\kappa$ given a fixed covariance matrix that is equal to $\Sigma_{\kappa,\text{prior}}^{-1}$. In other words, this is the same as our full VAE setup, but omitting the imposed 3D constraints. The second baseline is “$\kappa$-prior (learned)”, which corresponds to a more standard VAE-setting where the decoder directly outputs a mean and a variance (i.e. a diagonal covariance matrix). The third baseline does not involve a VAE, but samples structures from a multivariate Gaussian with a mean based on the dataset and a precision matrix computed by a standard estimator. This is an empirical estimator for MD datasets, and an Oracle Approximating Shrinkage (OAS) estimator \cite{Chen2010} for NMR datasets, since empirical estimators do not converge for such low amounts of samples. Finally, we include the flow-based model from \cite{Kohler2023} as a fourth baseline, which, to the best of our knowledge, is the current state of the art in density modelling for internal coordinate representations. 4.3 INTERNAL-COORDINATE DENSITY MODELLING RESULTS Unimodal setting in low data regime. The unimodal, low data regime test cases exhibit small fluctuations around the native protein structure, where the largest fluctuations correspond to the loops connecting different secondary structure elements. Fig. 4 demonstrates that 1unc, 1fsd and 1pga structures sampled from the VAE conform to local and global constraints, with valid Ramachandran distributions when compared to the reference as well as improved atom position variance along the chain compared to the baselines (see quantitative results in Table A2). Even in the extremely low data regime of 25 and 41 data points for 1unc and 1fsd, respectively (top two rows in Fig. 4), the VAE is able to estimate a full covariance matrix that approximates the distribution better than the baselines, especially in loop regions. This can lead to unphysical structures with backbone crossings, even though the local structure is preserved. These effects can also be observed in the 3D visualization of the sampled ensembles in Fig. A2. ![Figure 4](image) Figure 4: Modelling fluctuations in the unimodal setting for 1pga, 1fsd, and 1unc. Left: structure visualization, with $\alpha$-helices in red and $\beta$-sheets as blue arrows. Middle: Ramachandran plots for the MD reference and VAE samples. Right: variance along the atom chain for VAE samples, MD reference, and baselines. Secondary structure elements are indicated along the x-axis. The third test case, 1pga (bottom row in Fig. 4), has a more complex structure with two $\beta$-strands at the N-terminus forming a sheet together with two $\beta$-strands from the C-terminus. These global constraints are not captured well by the baselines in this low data regime, resulting in very high fluctuations in loop regions which violate the native structure (additional visualizations can be found in Fig. A2). For our VAE, we see the benefits of imposing global constraints, resulting in much better density estimation compared to the baselines. Moreover, we can control the interplay between local and global constraints by adjusting the hyperparameters of our model, as exemplified in Appendix D.1. However, the complexity of this protein prevents perfect density estimation in a low data regime. Interestingly, we show in Appendix C.1 that the variance of the atom positions highly correlates to the imposed constraints $C$ calculated from a set of predicted Lagrange multipliers using Eq. (9). Multimodal setting in high data regime. Here, we explore the use of our approach for modelling more complex behavior in a high data regime. Fig. 5 shows the free energy landscape in terms of the two first TICA components for cln025 and 2f4k. When comparing the VAE and the baselines to the MD reference (see also quantitative results in Table A3), it is clear that the learned prior and standard estimator do not capture all modes in the free energy landscape. The flow-based model performs best, suggesting that in this multimodal setting with plenty of available data, our proof-of-concept VAE is not as expressive as this state-of-the-art model. Moreover, the benefit of imposing 3D constraints on top of the fixed $\kappa$-prior (see baseline) seems beneficial for cln025, but the effect is not as strong for the $\alpha$-helical 2f4k, where local constraints might dominate (a similar effect can be seen when comparing the VAE to the fixed $\kappa$-prior baseline for lunc in the unimodal setting). However, our simple VAE setup is evidently able to model large conformational changes through its latent space, demonstrating how our general-purpose method for modeling fluctuations using 3D constraints can be incorporated into more expressive models to model complex behavior. Similar to the unimodal case, there is a tradeoff between local and global constraints which we can modulate using hyperparameters, as demonstrated in Appendix D.2. In addition, we show in Appendix C.3 how distinct regions of the VAE latent space map to different clusters in the TICA free energy landscape, and visualize the corresponding structures. ![Figure 5: Modelling (un)folding behavior in the multimodal setting for cln025 and 2f4k. Left: structure visualization. Right: TICA free energy landscapes for MD reference, VAE, and baselines.](image) ### 5 RELATED WORK There is a large body of work on models for analyzing trajectories of molecular dynamics simulations, either through Markov state models (Chodera & Noé [2014], Singhal & Pande [2005], Sarich et al. [2013], Schütte et al. [1999], Prinz et al. [2011]), or more complex modelling strategies (Mardt et al. [2018], Hernández et al. [2018], Sultan et al. [2018], Mardt et al. [2020], Xie et al. [2019]). Typically, these focused on dimensionality reduced representations of the molecular structures, and are therefore not density models from which samples can be drawn. To our knowledge, the first generative density model of full protein coordinates was the Boltzmann generator (Noé et al. [2019]), a normalizing flow over the Cartesian coordinates of protein ensembles. An extension of this approach was later used to estimate $C_\alpha$-only coarse-grained force fields for molecular dynamics simulations (Köhler et al. [2023]). This method, which uses internal coordinate inputs, demonstrated the ability of augmented flows to sample structural ensembles for proteins up to 35 amino acids. Other approaches involve latent variable models. One example is the IG-VAE, which generates structures in 3D coordinates but expresses the loss in terms of distances and internal coordinates to maintain SE(3) invariance. Similar approaches have been used to analyze cryo-EM data, where the task is to generate ensembles of structures given the observed cryo-EM image data. Since cryo-EM data provides information at slightly lower resolution than the full-atomic detail we discuss here, the output of these approaches are often density maps in 3D space (Zhong et al. [2021], Punjani & Fleet [2021]). One example of atomic-level modelling in this space is Rosenbaum et al. (2021), which decodes deterministically into 3D coordinates, but describes the variance in image space. Finally, diffusion models have recently provided a promising new approach to density modelling, with impressive examples of density modelling at the scale of full-size proteins (Watson et al., 2022; Ingraham et al., 2022; Anand & Achim, 2022). The primary objective in our paper is to investigate whether internal-coordinate density modelling with a full covariance structure is feasible using a simple, parsimonious setup. Internal-coordinate probabilistic models of proteins have traditionally focused on protein local structure, i.e. correct modelling of angular distributions of the secondary structure elements in proteins. Early work was based on hidden Markov models of small fragments (Camproux et al., 1999, 2004; de Brevern et al., 2000; Benros et al., 2006). The discrete nature of the fragments meant that these models did not constitute a complete probabilistic model of the protein structure. Later models solved this issue by modelling local structure in internal coordinates, using different sequential models and angular distributions (Edgoose et al., 1998; Bystroff et al., 2000; Hamelryck et al., 2006; Boomsma et al., 2008, 2014; Thygesen et al., 2021). Due to the downstream effects of small internal-coordinate fluctuations, these models are not by themselves capable of modelling the distribution of entire protein structures, but they are useful as proposal distributions in Markov chain Monte Carlo (MCMC) simulations of proteins (Irback & Mohanty, 2006; Boomsma et al., 2013). Using deep learning architectures to model the sequential dependencies in the protein chain, recent work has pushed the maximum length of fragments that can be reliably modelled to length 15 (Thygesen et al., 2021), where the fragment size is limited due to the challenges in estimating the necessary covariance structure. Our work was inspired by methods used for constrained Gaussian updates in MCMC simulation, first introduced by Favrin et al. (2001), and later extended by Bottaro et al. (2012). Our method generalizes the approach to global updates of proteins, derives the relationship between the Lagrange multipliers and corresponding fluctuations in Euclidean space, and uses neural networks to govern the level of fluctuations in order to modulate the induced covariance structures. Recent work has demonstrated that internal-coordinate modelling can also be done using diffusion models (Jing et al., 2022). So far this method has been demonstrated only on small molecules. We believe the method we introduce in this paper might help scale these diffusion approaches to full proteins. In Cartesian space, the Chroma model (Ingraham et al., 2022) demonstrated the benefits of correlated diffusion arising from simple constraints between atoms. Our method can be viewed as an extension of this idea to richer covariance structures. 6 DISCUSSION Although protein structure prediction is now considered a solved problem, fitting the density of structural ensembles remains an active field of research. Many recent activities in the field focus on diffusion models in the Cartesian coordinate representation of a protein. In this paper, we take a different approach, and investigate how we can describe small-scale fluctuations in terms of a distribution over the internal degrees of freedom of a protein. The main challenge in this context is the complex covariance between different parts of the chain. Failing to model this properly results in models that produce disruptive changes to the global structure even for fairly minor fluctuations in the internal coordinates. Instead of estimating the covariance matrix from data, we show that it can be induced by imposing constraints on the Cartesian fluctuations. In a sense, this represents a natural compromise between internal and Cartesian coordinates: we obtain samples that are guaranteed to fulfill the physical constraints of the local protein topology (e.g. bond lengths, and bond angles), while at the same time producing meaningful fluctuations globally. We implement the idea in the decoder of a variational autoencoder on two protein systems. This is primarily a proof of concept, and this implementation has several limitations. First of all, the standard deviations in the $\kappa$-prior of the internal degrees of freedom are currently set as a hyperparameter. These could be estimated from data, either directly within the current VAE setup, or using a preexisting model of protein local structure. Another limitation is the current model is that the produced fluctuations are generally too small to fully cover the individual modes of the target densities. This could be solved by constructing a hierarchical VAE, where samples are constructed as a multi-step process, similar to the generation process in diffusion models. In fact, we believe that our fundamental approach of induced covariance matrices could be a fruitful way to make diffusion models in internal coordinates scale to larger systems, by allowing for larger non-disruptive steps. We leave these extensions for future work. 7 CODE AND DATA AVAILABILITY Code and in-house generated data will be made available upon acceptance. REFERENCES Mohammed AlQuraishi. pnerf: Parallelized conversion from internal to cartesian coordinates. bioRxiv, pp. 385450, 2018. Namrata Anand and Tudor Achim. Protein structure and sequence generation with equivariant denoising diffusion probabilistic models. arXiv preprint arXiv:2205.15019, 2022. Minkyung Baek, Frank DiMaio, Ivan Anishchenko, Justas Dauparas, Sergey Ovchinnikov, Gyu Rie Lee, Jue Wang, Qian Cong, Lisa N Kinch, R Dustin Schaeffer, et al. 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wtJS8YDQBc	In the end of the first paragraph, it was mentioned minor delays markedly amplifies the risk in self-driving. Is there any reference? Following this question, are there more examples (better with data) of practical scenarios that suffers heavily from delays?	DEER: A Delay-Resilient Framework for Reinforcement Learning with Variable Delays Anonymous authors Paper under double-blind review Abstract Classic reinforcement learning (RL) frequently confronts challenges when handling tasks involving delays. These delays introduce a mismatch between the received observations and the subsequent actions to be executed, evidently deviating from the Markov property. Existing approaches usually tackle this issue with end-to-end solutions using state augmentation, often by augmenting the state space with a predefined maximum dimension to accommodate random delays. However, this black-box approach, characterized by incomprehensible intermediate processes and redundant information in augmented states, can result in instability and even undermine the overall performance. To alleviate the delay challenges in RL, we propose DEER (Delay-resilient Encoder-Enhanced RL), a framework that can effectively enhance the interpretability and address the random delay issues. DEER employs a pretrained encoder to encode delayed states along with their variable-length past action sequences due to different delays. Specifically, we leverage delay-free environment datasets to train the encoder and convert delayed states and their corresponding action sequences into hidden states, which serve as novel delay-free states for further policy training. In a variety of delayed scenarios, the trained encoder can smoothly integrate with standard RL algorithms without extra modifications and enhance the delay-solving capability by simply adapting the input dimension of the original algorithms. We evaluate DEER through extensive experiments on Gym and Mujoco, which confirm that DEER is superior to state-of-the-art RL algorithms in both constant and random delay environments. 1 Introduction Deep reinforcement learning has made substantial development in games (Mnih et al., 2013; Silver et al., 2016) and large language models (Ouyang et al., 2022; Carta et al., 2023), where most works are based on the assumption that action execution and state observation occur instantaneously. However, delays are inevitable in real-world tasks such as robotics (Duan et al., 2016; Hwangbo et al., 2017), remote control (Lampe et al., 2014) and distributed communication (Moon et al., 1999). Prior research (Gu & Niculescu, 2003; Dugard & Verriest, 1998) has revealed the substantial impact of delays on an agent’s decision process, which not only leads to performance degradation but also holds the potential to induce instability in dynamic systems, posing severe risks in real-world applications. Notably, in self-driving scenarios, even minor delays in the observation and execution modules can markedly amplify the risk of accidents. Despite the ubiquity of delay as a practical challenge, related research in the domain of RL remains scarce. Existing methods largely fall into two categories: model-free and model-based approaches. Most model-free approaches (Katsikopoulos & Engelbrecht, 2003a; Nath et al., 2021a; Ramstedt & Pal, 2019; Xiao et al., 2020; Schuitema et al., 2010; Agarwal & Aggarwal, 2021; Bouteiller et al., 2021) rely on state augmentation to transform delayed MDPs into equivalent undelayed ones. Though being successful to some extent, their effectiveness is limited by the augmented state space’s dimension. On the one hand, fixed input dimension methods are tailored for environments with constant delays, making them unsuitable for new tasks with different or random delays. On the other hand, the dimension of the augmented state space grows linearly with the length of delay, leading to exponential computational requirements and suboptimal policy learning by the agent. By contrast, model-based methods (Walsh et al., 2007; Fester & Stone, 2013; Chen et al., 2021; Firou et al., 2018; Derman et al., 2021) aim to predict the current state using the agent’s recently received delayed state and action sequence. While being effective in static contexts, their robustness in dynamic environments requires further enhancement. For instance, Frou et al. (2018) proposed a predictive model using unrolled Gated Recurrent Unit (GRU) (Chung et al., 2014) modules to iteratively generate a single action, and Derman et al. (2021) introduced the Delayed-Q algorithm for making decisions based on iterative forward dynamic predictions. However, both methods suffer from issues including inference time, model precision, and cumulative errors, that can significantly impact their overall performance. Considering the outlined challenges, we propose Delay-resilient Encoder-Enhanced RL (DEER) that leverages an encoder pretrained on offline datasets to enhance online learning in delayed environments. Instead of making direct decisions using augmented states, we initially map these states into a hidden space known as the context representation space. The actions are subsequently inferred based on these context representations. The overview of DEER is shown in Fig. 1. Specifically, we employ an undelayed offline dataset mainly consisting of random trajectories, complemented by a small number of expert trajectories, for the pretraining of an encoder-decoder model. The model’s encoder module is designed to generate a context representation that presents a semantic embedding of the delayed state and its corresponding action sequence. This embedding encapsulates the implicit information about both the current state and historical states, effectively serving as a high-dimensional state representation without delay, and can be directly used by standard RL algorithms to generate the current action. This process features three key advantages: (1) The trained encoder can easily generalize to diverse delay environments, as it has been trained across various delay settings. Even when facing an unknown delay in a new environment, the pretrained encoder combined with standard RL algorithms can still work effectively; (2) The proposed approach is versatile in addressing both constant and random delay environments. Since the encoder transforms the augmented state into a fixed-length vector, there is no need to modify the agent’s structure for different delay scenarios; (3) This method explicitly breaks down the end-to-end decision process into two distinct stages: encoding the augmented state and making decisions based on the embedding, which significantly improves the interpretability of the entire process. Furthermore, DEER can be seamlessly integrated with any standard RL algorithm. In this paper, we employ Soft Actor-Critic (SAC) as the decision module, and comprehensive experiments on Gym and Mujoco confirm that our approach is superior to state-of-the-art methods in both constant and random delay environments. The main contributions of this paper are summarized as follows: - DEER innovatively leverages offline datasets from delay-free environment tasks to aid in handling tasks occurring within delayed environments. - A versatile framework DEER is introduced to enhance agent performance in delayed environments, which can be smoothly integrated with standard RL algorithms without any additional modifications. - With SAC as the decision module, extensive experiments on Gym and Mujoco demonstrate that DEER achieves competitive or superior learning efficiency and performance compared with previous state-of-the-art methods. 2 RELATED WORKS Offline assisted Online RL A large number of works have been done to improve an agent’s online performance with the aid of offline RL techniques and they can be categorized as follows: (1) Combining offline data with online learning. Several early works made an attempt to initialize a replay buffer by the demonstration data (Vecerik et al., 2017; Hester et al., 2018), while other works (Lee et al., 2022; Mao et al., 2022; Ball et al., 2023; Nar et al., 2018; Hansen et al., 2022) designed new prioritized sampling schemes to improve learning efficiency and control distribution shift in the online learning stage. (2) Pretraining in representation or policy. The former (Yang & Nachum, 2021) adopted standard contrastive learning methods to extract the features from a variety of offline datasets, which can be applied to downstream tasks including online RL, imitation learning and offline policy optimization. The latter (Rajeswaran et al., 2017; Nar et al., 2020; Zhao et al., 2022; Rudner et al., 2021; Uchendu et al., 2022) called offline-to-online RL has been more prevalent in recent years and commonly executes offline RL algorithms followed by online fine-tuning including parameter transferring, policy regularization, etc. Our method shares the same key concept with the influential work by Yang & Nachum (2021), yet features significant distinctions in data source, loss function, and working principle. Precisely, we develop an encoder-decoder model to map augmented states, composed of the delayed state and subsequent action sequence, into a common hidden space. This model is trained on a dataset primarily containing random data, with a minor portion of expert data from undelayed environments. Encoders in RL Encoders have gained widespread usage in reinforcement learning for extracting representations as input to the policy. The RL4Rec framework (Chen et al., 2019; Zhao et al., 2018) employs a state encoder to compress users’ historical interactions into a dense representation, capturing user preferences for further inference. Liu et al. (2020) evaluated diverse state encoders and claimed that an attention-based variant can produce the optimal recommendation performance. Generally, encoders in RL4Rec are trained in an end-to-end manner with RL algorithms, distinguishing them from our approach. In visual RL, pretrained encoders are employed to efficiently extract visual features and reduce image input dimensions. Studies such as Shah & Kumar (2021) and Parisi et al. (2022) demonstrated that pretrained ResNet representations can achieve performance comparable to state-based inputs with the aid of expert demonstrations. Additionally, Yuan et al. (2022) investigated the efficacy of the image encoder to enable agents to generalize to unseen visual scenarios with a substantial distributional shift in a zero-shot manner. Moreover, Ge et al. (2021) employed a multi-view state encoder to process input states from multiple perspectives, enhancing generalization abilities via adaptive traffic signal control transfer learning. Nonetheless, the exploitation of pretrained models in delay scenarios remains limited in the current literature. 3 PRELIMINARY 3.1 Markov Decision Property (MDP) The sequential decision-making problem is typically formulated as a discounted Markov Decision Process (MDP), denoted by a tuple $(S, A, \rho, p, r, \gamma)$. Here, $S$ and $A$ are state and action spaces, respectively; $\rho$ is the initial state distribution; $p : S \times A \rightarrow S$ is the transition function; $r : S \times A \rightarrow \mathbb{R}$ gives the reward to any transitions and $\gamma \in [0, 1)$ is a discount factor. During the interaction between the agent and the environment, the agent follows a policy $\pi : S \rightarrow A$, resulting in a sequence of transitions or an entire trajectory $\tau = (s_t, a_t, r_t)_{t \geq 0}$. The cumulative return is calculated as $R(\tau) = \sum_{t=0}^{\infty} \gamma^t r_t$ and the primary objective in RL is to identify a return-maximizing policy $\pi^* = \text{argmax}_\pi \mathbb{E}[R(\tau)]$. 3.2 Random Dropping Delayed Markov Decision Process (RDDMDP) In real-world scenarios, especially in tasks such as remote control and distributed communication, delays resulting from long-distance transmission or heavy data transfers have a significant impact on agent performance, which are denoted by an intrinsic delay parameter $d_I$ in our work. Moreover, during the process of information transmission and interaction, states may be dropped due to obstacles or network malfunctions. Consequently, apart from the first state that is always observable, the instances of state dropout in subsequent steps follow a Bernoulli distribution with parameter $\mu$, representing the probability of dropout. Furthermore, the maximum number of extra dropping steps based on $d_I$, labeled as $d_M$, is defined to ensure that the overall delay is within the agent’s capacity limit. Therefore, at each time step $t$, the agent is expected to receive a state $s_{t-d_I}$ and a corresponding reward $r_{t-d_I}$. However, each state dropout follows a Bernoulli distribution: $\omega_t \sim \text{Bern}(\mu)$, which implies that when $\omega_t$ equals 0, the agent receives complete information including the state and reward, and when $\omega_t$ is 1, it receives nothing. As a result, the agent works in an environment with inherent random delays, deviating from the concept discussed in Katsikopoulos & Engelbrecht (2003b) and Nath et al. (2021b). A detailed elaboration on the discrepancies is available in Appendix A.1. The Random Dropping Delayed Markov Decision Process (RDDMDP) is proposed as follows: Definition 1 The RDDMDP can be defined as a 9-tuple $(d_I, d_M, I_z, A, \rho, p, r, \gamma, \mu)$: (1) Intrinsic delay value: $d_I \in \mathbb{Z}^+$, which is caused by long distance transmission or heavy data transfers; (2) Maximum number of extra dropping steps: $d_M \in \mathbb{Z}^+$, which is defined to ensure that $d_I + d_M$ remains within the agent’s capacity; (3) Information state space: $I_z = S \times A^z$, where $z$ denotes the random delay value with $d_I \leq z \leq d_I + d_M$, $S$ and $A$ are the same as the definition in MDP ; (4) Action space: $A = A$; (5) Initial information state distribution: $\rho(i_0) = \rho(s_0, a_0, ..., a_{d_I-1}) = \rho(s_0) \prod_{i=0}^{d_I-1} \delta(a_i - c_i)$, where $\rho$ is the initial state distribution in MDP and $\{c_i\}_{i=0}^{d_I-1}$ are actions selected randomly at the initial of trajectories when states are not observed, and $\delta$ is the Dirac delta function; (6) Transition distribution: $p(i_{t+1}\|i_t, a_t)$, where $a_t \in A$ and the information state $i_t \in I_z$ is described in detail below; (7) Reward function: $r_t = r_{t-z_t}$, where $z_t$ denotes the random delay value at time $t$; (8) Discount factor: $\gamma \in [0, 1)$; (9) Dropping probability: $\mu \in [0, 1)$, and when $\mu = 0$, the RDDMDP is reduced to the constant delayed MDP (CDMDP) and the details are provided in Appendix A.2. Figure 2: Process of model pretraining. Firstly, the information state dataset is created based on the original undelayed dataset. All state sequences are standardized to a uniform length $D$, where $D$ represents the maximum delay in the environment. Next, these datasets are fed into the seq2seq model and trained in a supervised manner. At each time $t$, there is a chance of $\mu$ that the agent does not receive the delayed state $s_{t-d_I}$, leading to a potential dropout of state. Thus, the random delay value $z_t$ is defined in the following manner: $$ z_t = \begin{cases} d_I, & \text{if } z_{t-1} = d_I + d_M \text{ or with probability } 1 - \mu, \\ z_{t-1} + 1, & \text{others}. \end{cases} $$ The information state is defined correspondingly as $\hat{i}_t$: $$ \hat{i}_t = (s_{t-z_t}, (a^{(t)}_{t-n})_{n=z_t:1}) = \begin{cases} (s_{t-d_I}, (a^{(t)}_{t-n})_{n=d_I:1}), & \text{if } z_{t-1} = d_I + d_M \text{ or with probability } 1 - \mu, \\ \text{concatenate}(\hat{i}_{t-1}, a_{t-1}), & \text{others}. \end{cases} $$ Accordingly, the reward function is expressed as: $$ r_t = r_{t-z_t} = \begin{cases} r_{t-d_I}, & \text{if } z_{t-1} = d_I + d_M \text{ or with probability } 1 - \mu, \\ r_{t-1}, & \text{others}. \end{cases} $$ After finishing the aforementioned delay modeling, the agent will continue to take actions based on the current information state $\hat{i}_t$, akin to its behavior in a delay-free environment. 4 METHOD In this section, we present Delay-resilient Encoder-Enhanced RL (DEER), a concise and effective framework designed to address delays in RL, which capitalizes on the encoder pretrained on undelayed datasets to extract informative features and can properly handle both constant and random delays. The algorithmic framework of DEER is provided in Algorithm 1. 4.1 PRETRAINED ENCODER DEER explicitly utilizes pretrained models as feature extractors, requiring no alteration of the RL algorithm. The pretrained encoder projects information states into embeddings of equal length, helping the agent handle delay challenges without the prior knowledge of environment delays. During the policy learning across training tasks, the encoder’s parameters remain fixed to acquire universal information representations. To acquire a competent encoder, the training of the encoder-decoder model is conducted on the datasets composed of trajectories generated by a random policy along with a few expert trajectories collected by a well-trained SAC agent, all from undelayed environments. The input and output of 1The superscript of $a^{(t_2)}_{t_1}$ shows that the action is an element of the information state $\hat{i}_{t_2}$ and the subscript indicates that the action is taken at timestep $t_1$. the model are referred to as the information state $I_t = (s_t, a_t, ..., a_{t+d-1})$ and the state sequence $(s_{t+1}, ..., s_{t+d})$, respectively, and the encoder-decoder model is employed as a regression model for state predication. In view of the capabilities of the encoder-decoder model, the hidden features extracted by the encoder are expected to contain valuable information about delays, enabling the agent to make proper decisions. Moreover, to make the encoder smoothly generalize across various constant delays and effectively handle random delays, the training dataset consists of information states with diverse action sequence lengths, while maintaining a fixed dimension for the hidden features, so that the encoder can acquire features that can be directly employed by the agent, irrespective of the specific delay conditions. We use a Seq2Seq (Sutskever et al., 2014) model as the encoder-decoder framework, a simple yet effective choice for handling the delay problem. Multi-Layer Perceptrons (MLPs) are firstly applied to encode each element in the information state, including a state and a series of actions, to generate corresponding embeddings. Subsequently, these embeddings are fed into a GRU module to produce the hidden feature vector whose dimension is a hyperparameter. The Seq2Seq model is optimized based on the MSE loss to improve the accuracy of state sequence predictions, consequently refining the hidden feature’s representation of the information state. The complete process of model pretraining is shown in Fig.2, and the detailed network structure and parameter configurations are presented in Appendix B.1. ### 4.2 Encoder-enhanced Policy Learning The pretrained encoder plays a crucial role in the policy learning phase: extracting essential representations of delayed information and enabling standard RL algorithms to learn effectively regardless of environment delays. The encoder provides the context representation based on delayed information, offering distinct advantages in both constant and random delay environments. In constant delay settings, its strength lies in the generalization to different types of delays based on the universal training data. This enables direct transformation of information states with unknown lengths into fixed-length representations, avoiding policy input dimension adjustments. In random delay environments, original information states of varying lengths are encoded into hidden features of constant lengths, facilitating the adoption of standard RL algorithms that depend on fixed-length inputs. The entire process of the encoder-enhanced policy learning is shown in Fig.1. ## 5 Experimental Results In this section, we thoroughly evaluate the effectiveness of our approach by comparing DEER with state-of-the-art RL algorithms in both constant and random delay environments. We investigate various aspects of the context representation’s performance within the same scenario, analyzing the impact of decision space dimensions on the final performance. Additionally, we conduct an ablation study to highlight the efficacy of the context representation generated by the pretrained encoder, which is distinct from the predicted state produced by the same model. Moreover, we consider and discuss more factors that influence the experimental outcomes, further elucidating the efficacy of DEER in addressing tasks with delays. We use SAC (Haarnoja et al., 2018) for decision making, a popular choice for continuous control tasks due to its integration of the actor-critic architecture and the maximum entropy principle. When the context representation is produced by the pretrained encoder, the agent takes the action based on the new state and updates its policy, similar to its behavior in undelayed environments. All experiments are conducted under the MuJoCo environments from the gym library, including Ant, HalfCheetah, Hopper, Swimmer, Walker2d, and Reacher. Furthermore, each algorithm is executed with 5 different seeds in each environment. The details regarding the number of trajectories used in the pretraining phase are provided in Appendix B.3. ### 5.1 Evaluation The following algorithms are used in comparative studies to illustrate the effectiveness of our proposed method: • Reinforcement Learning with Random Delays (RLRD; Bouteiller et al., 2021). RLRD introduces a technique where past actions are relabeled using the current policy. This relabeling procedure generates on-policy sub-trajectories, providing an off-policy and planning-free approach applicable to environments with constant or random delays. • Delay-Aware Trajectory Sampling (DATS; Chen et al., 2020). The effectiveness of DATS can be attributed to the synergistic combination of its unique dynamics model, which incorporates both the known part resulting from delays and the unknown part inherited from the original MDP, and its effective planning method, PETS. • Soft Actor-Critic with Augmented States (SACAS). The implementation of SACAS aligns with the principles described in Katsikopoulos & Engelbrecht (2003a). Considering the differences in reward settings between DATS and other methods, we normalize the cumulative rewards by $ \frac{\text{Return} - \text{min\_return}}{\text{Expert\_return} - \text{min\_return}} $. The parameters remain consistent within each algorithm but may vary across different algorithms. Return represents the cumulative rewards obtained in each episode; min_return corresponds to the minimum return observed throughout all experiments; Expert_return indicates the level of expertise achieved in undelayed environments. Constant Delays. The initial experiments focus on environments with constant delays. Four algorithms are compared in environments where delay values are set to 1, 2, 4, 6 and 8, respectively. As shown in Figure 3, it is clear that: 1) As delay increases, the performance of all compared algorithms diminishes; 2) In Ant, Swimmer, Walker2d, and Reacher, DEER outperforms other algorithms, evident from their respective performance curves, while in HalfCheetah and Hopper, DEER’s performance is similar to that of other algorithms or slightly lower with certain delay values (e.g., Hopper with delay = 8); 3) DEER consistently outperforms the expert in Swimmer across various delays, further highlighting the effectiveness of the context representation in making informed decisions. Random Delays. Randomly delayed environments present a tougher challenge compared with constant delays due to the increased risk of information dropout. We evaluate the four aforementioned algorithms with $ d_I = 2, d_M = 4 $, and dropping probabilities $ \mu = 0.2, 0.4, $ and $ 0.6 $, respectively. Figures 5 to 7 show the performance comparison of different algorithms under random delays with different $ \mu $. To better analyze the results, we take $ \mu = 0.2 $ and $ \mu = 0.4 $, and summarize the final results of different algorithms on different tasks in Table 2. Evidently, the loss of information accounts for a notable performance decline across all algorithms, even if they are capable of achieving satisfactory results without random delays. Nevertheless, DEER consistently outperforms its counterparts. In summary, the context representation generated by DEER’s pretrained encoder can effectively extract valuable information from delayed states, readily applicable across varying delay settings. Table 1: Comparison of algorithms with dropping probability 0.2 and 0.4. \| Drop \| 0.2 \| 0.4 \| \|------\|-----\|-----\| \| Algorithm \| DEER \| RLRD \| DATS \| SACAS \| DEER \| RLRD \| DATS \| SACAS \| \| Ant \| 0.47 \| 0.16 \| 0.23 \| 0.08 \| 0.38 \| 0.02 \| 0.17 \| 0.03 \| \| HalfCheetah \| 0.32 \| 0.28 \| 0.29 \| 0.3 \| 0.26 \| 0.08 \| 0.08 \| 0.01 \| \| Hopper \| 0.82 \| 0.48 \| 0.64 \| 0.64 \| 0.51 \| 0.31 \| 0.5 \| 0.31 \| \| Swimmer \| 1.79 \| 0.83 \| 0.94 \| 0.83 \| 1.16 \| 0.67 \| 0.82 \| 0.68 \| \| Walker2d \| 0.62 \| 0.49 \| 0.44 \| 0.26 \| 0.32 \| 0.34 \| 0.35 \| 0.158 \| \| Reacher \| 0.91 \| 0.72 \| 0.61 \| 0.8 \| 0.88 \| 0.64 \| 0.75 \| 0.88 \| 5.2 Influence of Key Parameter The context representation plays a crucial role as the input to the decision model and within the encoder-decoder architecture. Next, we investigate the impact of dimension of the context representation on the agent’s performance in delayed environments. The results of DEER for dimensions with 128, 256, and 512 across various delays are presented in Figures 8 - 15. Similarly, to better analyze the impact of context representation dimensions on the performance of the agent under different conditions, we have summarized the results in Tables 2 and 6. From these tables, it can be observed that the performance of DEER in Reacher appears less sensitive to dimension changes and is primarily sensitive to the delay factor. Moreover, in tasks such as HalfCheetah and Swimmer, higher dimensions correspond to improved performance, while Hopper and Walker2d present an opposite trend. This observation suggests that the final performance of the agent depends on the representation capability of the pre-trained encoder, when the training strategy is kept the same. Only when the context representation can well represent the delay information, that is, when the pre-trained encoder can well represent the information state, is it beneficial to the agent’s decision-making. Therefore, the dimension of the representation information is not necessarily related to the final performance. In summary, taking into account factors including computational complexity and overall performance, opting for a 256-dimensional context representation is generally recommended. Table 2: Comparison of DEERs performance in various dimensions with delay values of 4, 6 and 8. \| Delay \| 4 \| 6 \| 8 \| \|-------\|---\|---\|---\| \| Dimension \| 128 \| 256 \| 512 \| 128 \| 256 \| 512 \| 128 \| 256 \| 512 \| \| Ant \| 1344 \| 2574 \| 2415 \| 889 \| 1653 \| 1932 \| 617 \| 1072 \| 1381 \| \| HalfCheetah \| 5236 \| 5780 \| 5337 \| 3320 \| 3853 \| 4234 \| 1874 \| 2924 \| 3111 \| \| Hopper \| 2713 \| 2918 \| 2198 \| 2197 \| 2565 \| 1737 \| 1908 \| 2462 \| 1891 \| \| Swimmer \| 46 \| 78 \| 118 \| 50 \| 83 \| 105 \| 44 \| 48 \| 86 \| \| Walker2d \| 3600 \| 4119 \| 2098 \| 2712 \| 3546 \| 677 \| 1011 \| 3074 \| 239 \| \| Reacher \| -7.9 \| -7.9 \| -8 \| -9.9 \| -9.8 \| -9.9 \| -11.7 \| -11.5 \| -11.5 \| 5.3 Ablation Study The ablation study aims to demonstrate the importance of the context representation compared with the predicted state, termed "Decision on Last Predicted State" (DOLPS). DOLPS can be inferred from the decoder module trained with the encoder in DEER at the same time. Experimental results in Figures 16, 17 and Table 3 consistently confirm DEER’s advantage over DOLPS across various environments and delays, with DOLPS showing limited effectiveness in Ant, Hopper, and Walker2d. It is clear that the context representation effectively mitigates prediction errors, which is crucial for decisions in the original decision space. Furthermore, it captures historical information embedded in delayed states and action sequences, which is shown to be advantageous for decision-making in the context of delayed scenarios. Table 3: Comparison of DEER and DOLPS under delay values of 4 and 6. \| Algorithm \| Delay 4 \| Delay 6 \| \|-------------\|---------\|---------\| \| \| DEER \| DOLPS \| DEER \| DOLPS \| \| Ant \| 2574 \| -10.8 \| 1653 \| -16 \| \| HalfCheetah \| 5780 \| 2975 \| 3853 \| 1574 \| \| Hopper \| 2918 \| 663 \| 2565 \| 652 \| \| Swimmer \| 78 \| 42 \| 83 \| 45 \| \| Walker2d \| 4119 \| 238 \| 3546 \| 264 \| \| Reacher \| -7.9 \| -17 \| -9.8 \| -21 \| 5.4 More analysis on DEER In this section, we’ll delve deeper into DEER from six aspects related to its design, execution, and outcomes, aiming to highlight its effectiveness in handling tasks with delays. These aspects encompass various comparisons and discussions: a time performance contrast between DEER and other algorithms, a comparison between online and offline DEER, an analysis against state-of-the-art algorithms in Offline to Online RL, a discussion on the impact of different context representation dimensions on agent performance, a showcase of the effects of three distinct offline datasets on resolving delayed tasks, and a fresh comparison in an alternative scenario of random delays. All of those are provided in Appendix C.4. 5.5 Limitation The experimental results confirm DEER’s efficacy in addressing delay problems, especially highlighting the significant performance gains achieved by well-pretrained encoders. However, pretrained encoders show certain level of sensitivity to the quantity and state distribution of trajectories used for training. During the process of model pretraining, we carefully selected the number and the type of trajectories based on the specific task to train a better encoder. In-depth discussion and analysis are provided in Appendix C.5. 6 Conclusion and Future Work In this paper, we introduce DEER, a concise framework designed to effectively tackle delay issues in RL, including both constant delays and random delays, and enhance the interpretability of the entire process. In DEER, an encoder is pretrained using trajectories collected from delay-free environments to map augmented states containing the delayed information into hidden features called context representation, which is subsequently used by the agent to derive new actions. Experiments on DEER combined with SAC demonstrate that our method achieves competitive or superior learning efficiency and performance in comparison with state-of-the-art methods, which validate the effectiveness and efficacy our approach in addressing delay-related challenges. Future work will focus on extending DEER to visual reinforcement learning, where agents receive and process visual information as states. Additionally, efforts will be made to deploy our approach to real-world systems, such as remote control systems or physical robots, further assessing its performance and applicability in practical scenarios. References Mridul Agarwal and Vaneet Aggarwal. Blind decision making: Reinforcement learning with delayed observations. Pattern Recognition Letters, 150:176–182, 2021. Philip J Ball, Laura Smith, Ilya Kostrikov, and Sergey Levine. Efficient online reinforcement learning with offline data. arXiv preprint arXiv:2302.02948, 2023. 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lBUUNj0Fnz	Could the authors provide clarification regarding the memory consumption associated with the algorithm, particularly as the volume of unlabeled images increases? Understanding how the algorithm's memory usage scales in response to larger datasets would be crucial for assessing its practical applicability and efficiency.	Active Learning for Image Segmentation with Binary User Feedback Anonymous authors Paper under double-blind review Abstract Deep learning algorithms have depicted commendable performance in a variety of computer vision applications. However, training a robust deep neural network necessitates a large amount of labeled training data, which is time-consuming and labor-intensive to acquire. This problem is even more serious for an application like image segmentation, as the human oracle has to hand-annotate each and every pixel in a given training image, which is extremely laborious. Active learning algorithms automatically identify the salient and exemplar samples from large amounts of unlabeled data, and tremendously reduce human annotation effort in inducing a machine learning model. In this paper, we propose a novel active learning algorithm for image segmentation, with the goal of further reducing the labeling burden on the human oracles. Our framework identifies a batch of informative images, together with a list of semantic classes for each, and the human annotator merely needs to answer whether a given semantic class is present or absent in a given image. To the best of our knowledge, this is the first research effort to develop an active learning framework for image segmentation, which poses only binary (yes/no) queries to the users. We pose the image and class selection as a constrained optimization problem and derive a linear programming relaxation to select a batch of (image-class) pairs, which are maximally informative to the underlying deep neural network. Our extensive empirical studies on three challenging datasets corroborate the potential of our method in substantially reducing human annotation effort for real-world image segmentation applications. 1 Introduction Semantic segmentation (labeling every pixel in an image to the category it belongs to) is one of the core tasks of visual recognition and is extensively used in a variety of applications, including autonomous driving, medical imaging and video surveillance among others (Ghosh et al., 2020). With the advent and popularity of deep learning, several deep architectures have been studied for image segmentation, which have depicted state-of-the-art results (Zhu et al., 2019; Yuan et al., 2020; Liu et al., 2021). However, for these models to work reliably, a large amount of training data (in the form of pixel-level annotated images) is required, which requires significant time and human labor. Thus, an algorithm to reduce human annotation effort is critically important to train deep learning models for image segmentation applications. Active Learning (AL) algorithms identify the most informative samples from vast amounts of unlabeled data (Settles, 2010). This tremendously reduces the human annotation effort in training a machine learning model, as only the samples that are selected by the algorithm need to be labeled manually. Further, since the model gets trained on the exemplar samples from the data, it typically depicts better generalization performance than a passive learner, where the training data is sampled at random. AL has been successfully used in a variety of applications, including computer vision (Yoo & Kweon, 2019), text analysis (Tong & Koller, 2001), bioinformatics (Osmanbeyoglu et al., 2010) and medical diagnosis (Gorritz et al., 2017) among others. The growing popularity of deep learning has motivated research in the field of deep active learning, to efficiently train the data-hungry deep learning models (Ren et al., 2021). The paucity of human labor and the need to use it more efficiently is even more pronounced for an application like image segmentation, due to the enormous time and effort associated with labeling. every pixel in an image. This necessitates specialized query and annotation mechanisms for the AL algorithms to be feasible in a real-world setting. In this paper, we propose a novel AL algorithm to address this challenging problem, in an effort to alleviate the labeling burden on human oracles while inducing a deep learning model for image segmentation. Our algorithm queries a batch of (image-class) pairs and for each pair, poses the question: “Does the image $ i $ contain the semantic class $ j $?”$^2$ The human annotator merely has to provide a binary “yes / no” feedback for each query. This is depicted in Figure 1. Providing such feedback is extremely easy and less prone to annotation errors; it is also significantly less time-consuming and burdensome than providing pixel-level annotations. Our contributions in this paper can be summarized as follows: - We present a novel AL framework for image segmentation, which poses only binary (“yes / no”) queries regarding the presence / absence of a semantic class in a given image. To our knowledge, this is the first active learning framework for semantic image segmentation which poses only binary queries to the human annotators. - We pose the image and class selection as a constrained optimization problem, and derive a linear programming relaxation to select a batch of (image-class) pairs, which are maximally informative to the underlying deep neural network. - We conduct user studies to estimate the time and human effort required to annotate an image at the pixel-level, region-level and binary-level (our method). This can provide valuable insights and enable us to study the trade-off between the human annotation effort and the generalization capability of the trained deep neural network, for different categories of annotation strategies. - We conduct extensive empirical studies on three benchmark datasets to study the performance of our framework against competing baselines. Figure 1: Figure showing the conventional active learning query (left) and the proposed binary query mechanism (right). Best viewed in color. 2 RELATED WORK Active Learning: AL is a well-researched problem in the machine learning community (Settles 2010; Zhan et al. 2022). Uncertainty sampling is the most common strategy for active learning, where unlabeled samples with the highest prediction uncertainties are queried for annotation. Several strategies have been explored to quantify uncertainty, such as Shannon’s entropy (Li & Guo 2013; Joshi et al. 2010), disagreement among a committee of classifiers regarding the label of a sample (Freund et al. 1997), the Fisher information matrix (Hoi et al. 2006), mutual information between the labeled and unlabeled samples (Guo & Greiner 2007) among others. The growing success and popularity of deep learning have motivated researchers to explore the problem of deep active learning (DAL), where the goal is to select the informative unlabeled samples to train a deep neural network (Ren et al. 2021). Common DAL techniques include incorporating a loss prediction module to predict the loss value of an unlabeled sample and querying samples accordingly (Yoo & Kweon 2019), selecting informative unlabeled samples for AL and simultaneously, searching for the best neural architectures on-the-fly (Geifman & El-Yaniv 2019), a sampling technique based on diverse gradient embeddings (BADGE) (Ash et al. 2020), a technique which captures the information balance between the uncertainty of underlying softmax probability and the label variable and queries samples accordingly (Woo 2023) and a technique to select a coreset of samples, such that the model learned over the selected subset is competitive for the remaining data points (Sener & Savarese 2018). Techniques based on adversarial learning have depicted particularly impressive --- 1the terms user, annotator, oracle and labeler are used interchangeably in this paper 2the term class is used to mean semantic class in this paper performance in this context (Sinha et al., 2019; Mayer & Timofte, 2020; Zhang et al., 2020). A segment of AL research has focused on weak / noisy labels, where annotators can provide noisy annotations or can provide annotations at different levels of precision (Olmin et al., 2023; Wu et al., 2017; Younestian et al., 2021; Lu et al., 2017). Beyond the conventional label query, a body of research in AL has focused on the development of novel query and annotation mechanisms to further reduce the labeling burden on human users. Binary feedback mechanism has been studied, where the active learner queries a pair of images, and the human annotator has to specify whether or not the two images belong to the same category (Joshi et al., 2010; Fu et al., 2014). In another variant, the learner queries an unlabeled image together with a class label, and the human annotator has to specify whether the selected image belongs to that class (Hu et al., 2019; Bhattacharya & Chakraborty, 2019). Along similar lines, AL has been exploited in clustering, where a pair of samples is queried and the oracles need to specify whether or not the samples in a pair correspond to the same cluster (Biswas & Jacobs, 2012). Although the query mechanism is binary, these methods query the label of an image as a whole, and not the presence of a semantic class within an image, and hence are not directly applicable to the problem of image segmentation. Active Learning for Image Segmentation: Providing pixel-level annotations to train an image segmentation model is a time-consuming and expensive process. To address this challenge, weakly supervised semantic segmentation techniques have been developed, such as providing the presence or absence of classes in an image during training (Xu et al., 2014; Pinheiro & Collobert, 2015), pointing to an object of interest (Bearman et al., 2016), bounding box annotations (Papandreou et al., 2015), free-form squiggles (Lin et al., 2016) and noisy web tags (Ahmed et al., 2014). However, these methods utilized the weak supervision only during model training (as a term in the training loss function) and did not use active learning to identify the informative images or the semantic classes within an image. As in conventional AL, uncertainty and diversity based metrics have been exploited for AL in the context of semantic segmentation (Yang et al., 2017). Metrics like view-point entropy have been studied for multi-view datasets (Siddiqui et al., 2020). Xie et al. proposed DEAL, a difficulty aware AL algorithm for image segmentation, which focused on the difficulty of different semantic areas in selecting samples for annotation (Xie et al., 2020). A body of research has focused on identifying the informative regions in an image and getting them annotated by the human labelers, rather than the entire image. Various strategies have been explored to identify the informative regions, such as deep reinforcement learning (Casanova et al., 2020), uncertainty quantification using superpixel entropies (Kasarla et al., 2019), informativeness, combined with annotation cost and the spatial coherency of an image (Mackowiak et al., 2018), margin-based sampling combined with diversity (Shin et al., 2021) and self-consistency under equivariant transformations (Golestaneh & Kitani, 2020). Although annotating image regions is less strenuous than providing pixel-level annotations, it still requires the human oracles to meticulously label all the pixels in the queried regions, which can be quite time-consuming, particularly if the queried region involves multiple semantic classes. In contrast, our framework requests only binary feedback regarding the presence / absence of specific classes in an image, which requires much lesser annotation effort and facilitates an easier mode of interaction between the user and the system. We now describe our framework. ### 3 Proposed Framework #### 3.1 Problem Formulation Consider an active image segmentation problem where we are given a labeled training set $L$ and an unlabeled set $U$. Let $N$ denote the number of unlabeled images, $N = \|U\|$. Images in $L$ are provided with pixel-level annotations. Let $w$ be the deep neural network trained on $L$, and $C$ be the number of semantic classes in the dataset. We are given a query budget $B$ and a parameter $C_{max}$ which denotes the maximum number of classes that can be queried per image (to ensure that the queries are distributed across a large number of images). Our objective is to select a batch of images, together with a list of classes for each image for binary user query, such that the total number of queries does not exceed the budget $B$, and the user response about the presence/absence of the semantic classes augments maximal information to the deep learning model. In order to identify the optimal set of images and semantic classes to be queried, we need a metric to quantify the utility score of a batch of (image-class) pairs. We used a criterion based on class presence uncertainty and image redundancy for this purpose. The first criterion ensures that we query those (image-class) pairs where there is maximal uncertainty regarding the presence of the given class in the given image; the redundancy criterion ensures that we query a diverse set of images in our batch and avoid duplicate image queries. These are detailed below. Computing Class Presence Uncertainty: Let $ p_{ij} $ denote the probability that image $ i $ contains the semantic class $ j $ (computed using the current deep neural network $ w $, as the average probability of pixels belonging to the semantic class $ j $ within image $ i $). We used Shannon’s entropy to compute the prediction uncertainty of the presence of semantic class $ j $ in image $ i $: \[ H_{ij} = -p_{ij} \log p_{ij} - (1 - p_{ij}) \log(1 - p_{ij}) \] Using this, we computed a confidence matrix $ G \in \mathbb{R}^{C \times N} $, where $ G(j, i) $ denotes the confidence of the deep model in predicting the presence of class $ j $ in image $ i $ (high entropy corresponds to low confidence and vice versa): \[ G(j, i) = \frac{\alpha}{H_{ij}} \quad i = 1, \ldots, N, \quad j = 1, \ldots, C \] where $ \alpha $ is a constant. Computing Image Redundancy: We computed a redundancy matrix $ R \in \mathbb{R}^{N \times N} $, where $ R(i, j) $ denotes the redundancy between images $ x_i $ and $ x_j $ in the unlabeled set. The cosine similarity was used to quantify the redundancy between a pair of samples; negative values were replaced with 0, so that $ R $ contains only non-negative entries: \[ R(i, j) = \max(0, \cos(\mathcal{F}(x_i), \mathcal{F}(x_j))) \] where $ \cos(\mathcal{F}(x_i), \mathcal{F}(x_j)) = \frac{\mathcal{F}(x_i)^T \mathcal{F}(x_j)}{\|\|\mathcal{F}(x_i)\|\| \|\|\mathcal{F}(x_j)\|\|} $ and $ \mathcal{F}(x) $ denotes the deep feature representation of image $ x $. A low value of $ R(i, j) $ implies that images $ x_i $ and $ x_j $ have low redundancy between them. Cosine similarity has been previously used to compute similarity in AL research, with promising results (Coleman et al., 2022). Depending on the application, other metrics can be used to compute the uncertainty and redundancy terms. ### 3.2 Active Sampling Framework Given $ G $ and $ R $, our objective is to query a batch of (image-class) pairs such that in each pair, the deep model has low confidence in predicting the presence of the given class in the given image, and the queried images have minimal redundancy among them. We define a binary matrix $ M \in \{0, 1\}^{N \times C} $, where each row corresponds to an unlabeled image and each column corresponds to a semantic class. A value of 1 in a row denotes that the image should be selected for annotation, and the position(s) of 1 in a particular row of $ M $ denote the semantic class(es) that should be used to pose the binary queries for this image. We also define a binary vector $ v \in \{0, 1\}^{N \times 1} $ where $ v_i = 1 $ denotes that image $ x_i $ is selected for annotation, and $ v_i = 0 $ denotes that it is not selected. The active selection of (image-class) pairs can thus be posed as the following optimization problem: \[ \begin{align} \min_{M, v} & \quad \text{Tr}(MG) + \lambda v^T Rv \\ \text{s.t.} & \quad \langle M, E \rangle = B \\ & \quad (M.e)_i \leq C_{\text{max}}, \forall i \\ & \quad v_i = \min(1, (M.e)_i), \forall i \\ & \quad v_i, M_{ij} \in \{0, 1\}, \forall i, j \end{align} \] where $ \lambda > 0 $ is a weight parameter governing the relative importance of the two terms, $ E $ is a matrix of size $ N \times C $ (same size as $ M $) with all entries 1, $ e $ is a vector of size $ C \times 1 $ with all entries 1, $ B $ is the labeling budget, $ \langle , \rangle $ denotes the inner product operator and $ \text{Tr} $ denotes the trace of a matrix. The first term in the objective function denotes that the deep model has low confidence in predicting the presence of the selected semantic classes in the corresponding selected images; the second term ensures that the selected images have minimal redundancy among them. The first constraint denotes the total number of queries posed by $ M $ is equal to the specified budget; the second constraint ensures that the number of 1s in each row of $ M $ is less than or equal to $ C_{\text{max}} $, that is, the number of queries posed for each image is less than or equal to the pre-specified limit $ C_{\text{max}} $; the third constraint denotes that $ v_i $ is equal to 1 if there is at least one entry with value 1 in row $ i $ of $ M $ (image $ x_i $ is selected for annotation), and $ v_i $ is equal to 0 if all the entries in row $ i $ of $ M $ have value 0 (image $ x_i $ is not selected); the fourth constraint denotes that $ v $ is a binary vector and $ M $ is a binary matrix. We now present a theorem to solve this optimization problem. Theorem 1. The optimization problem defined in Equation (7) can be expressed as an equivalent linear programming (LP) problem. Please refer to Section A.1 of the Appendix for the proof of this theorem. We relax the integer constraints into continuous constraints and solve the problem using an off-the-shelf LP solver. After obtaining the continuous solution, we recover the integer solution using a rounding approach where the $ B $ highest entries in $ M $ are reconstructed as 1 and the other entries as 0, observing the constraints. The pseudo-code of our algorithm is depicted in Algorithm 1 (for one active learning iteration). Algorithm 1 The Proposed Active Learning Algorithm with Binary User Feedback Require: Labeled training set $ L $, unlabeled set $ U $, query budget $ B $, parameters $ \alpha, C_{\text{max}} $ and $ \lambda $, a deep neural network architecture for image segmentation 1: Train the deep model on the training set $ L $ 2: Compute the confidence matrix $ G $ using the probabilities of the trained deep model (Equation (2)) 3: Compute the redundancy matrix $ R $ (Equation (3)) 4: Solve the LP problem in Equation (8) in the Appendix after relaxing the integer constraints 5: Round the solution to derive the matrix $ \tilde{M} $ 6: Select the unlabeled images and the corresponding semantic classes to pose the binary queries based on the entries in $ \tilde{M} $ 7: Update the deep model with the user response to the binary queries (detailed in Section E.1 in the Appendix) ### 4 EXPERIMENTS AND RESULTS #### 4.1 DATASETS We used three challenging datasets to study the performance of our framework: (i) Flickr-Landscapes [Park et al., 2019]; (ii) Cityscapes [Cordts et al., 2016]; and (iii) PASCAL VOC12 [Hariharan et al., 2011]. All these are benchmark datasets commonly used to validate the performance of image segmentation algorithms. #### 4.2 COMPARISON BASELINES We used a total of five methods as comparison baselines that annotate images at the pixel-level, region-level and binary-level. These are detailed below. Pixel-level annotation: In this category, a batch of unlabeled images were queried and all the pixels of all the queried images were annotated. We used two AL algorithms to query a batch of unlabeled images: Entropy [Settles, 2010], a commonly used AL method which selects samples with the highest degree of uncertainty as computed by entropy (the entropy of an image in our image segmentation application, was computed as the average entropy of every pixel in the image, obtained from the softmax probabilities furnished by the deep network); and Coreset [Sener & Savarese, 2018], a widely used AL technique which queries a batch of images such that a model trained on the queried subset is competitive for the remaining data samples. Region-level annotation: Here, a batch of regions were queried from the unlabeled images and all the pixels in the queried regions were annotated. We used the region-based active learning (RAL) method proposed by Kasarla et al. [Kasarla et al., 2019] where the SLIC algorithm was used to compute the superpixel of an image, and the regions with the highest uncertainties (defined by the superpixels) were queried for annotation. Binary-level annotation: In this category, binary queries were posed regarding the presence / absence of specific semantic classes in the unlabeled images (similar to our method). This is the first AL framework with binary-level annotation for image segmentation; we hence used the following methods as comparison baselines: Random-Random (RR), which randomly selects a subset of images and randomly queries $B$ semantic classes from the selected images; and Entropy-Entropy (EE), where a batch of images were selected based on the entropy of the underlying model and the semantic classes producing the highest prediction entropy values were queried from each. We used the DeepLabV3+ model with the ResNet101 backbone (pre-trained on ImageNet) as our base model due to its promising performance in image segmentation applications ([Chen et al., 2018](#)). The same architecture was used for all the baseline methods, for fair comparison. Evaluation Metrics: The mean intersection-over-union (mIoU) was used as the evaluation metric, as commonly done in image segmentation research ([Chen et al., 2018](#)). Since our comparison baselines span different categories of annotation, we also used the annotation time as an evaluation metric. ### 4.3 Experimental Setup Each dataset was divided into three parts: (i) an initial training set $L$; (ii) an unlabeled set $U$; and (iii) a test set. The number of images in the initial training, unlabeled and test sets were 1,500, 1,200 and 1,000 respectively for all three datasets. All the images in $L$ were provided with pixel-level annotations. A query budget $B$ (taken as 200 for Cityscapes and PASCAL and 400 for Flickr) was imposed in each AL iteration, and the experiments were conducted for 25 AL iterations. The query budget denotes the number of binary queries that can be posed (for the binary-level annotation methods, RR, EE and our method) or the number of image regions that can be queried (for the region-level annotation method, RAL). However, since we had 1200 images in our unlabeled set, using a query budget of 200 for the pixel-level annotation baselines would have exhausted the unlabeled pool after 6 AL iterations. We hence set the query budget to 48 (= 1200/25) in each AL iteration for the pixel-level baselines, so that the unlabeled pool is completely exhausted after 25 AL iterations. Also, since each queried image was annotated at the pixel level for Entropy and Coreset, these baselines represent an upper bound on the AL performance among the methods studied. After each AL iteration, the selected samples were annotated and appended to the training set; the deep neural network was retrained and tested on the test set. The objective was to study the improvement in performance on the test set with increasing number of label queries. The value of $\alpha$ in Equation (2) was set as 1, the parameter $C_{max}$ in Equation (4) was taken as 5, and the weight parameter $\lambda$ in Equation (4) was taken as 1 for all the datasets. All the results were averaged over 3 runs (with different training, unlabeled and test sets) to rule out the effects of randomness. ### 4.4 Implementation Details Please refer to Section F of the Appendix for details on implementation and model parameters. Please refer to Section F.1 of the Appendix for details on updating the deep neural network with binary user feedback. We also provide a few visual illustrations showing the performance of our binary query AL framework (in Section F.2 of the Appendix). ### 4.5 User Study to Estimate Annotation Time To accurately estimate the human annotation time (and hence, effort) required to annotate an image at the pixel-level, region-level and binary-level, we conducted a user study. 10 images were selected at random from each of the three datasets. For each image, the following tasks were posed: (i) Annotators were asked to segment each image at the pixel level with the different categories of objects and mark each category with a different color (pixel-level annotation) (ii) Annotators were asked to annotate all the pixels within a given region (super-pixel) of an image with the different categories of objects and mark each category with a different color (region-level annotation) (iii) Annotators were asked a question regarding the presence of an object in each image and had to provide a binary response: “YES / NO” (binary-level annotation). Annotators were provided with the LabelMe annotation tool (Russell et al., 2007) to segment the images. The time taken for each annotation task was noted. The annotators were also asked to provide a rating, denoting the ease of annotation for each task, on a scale of 1 to 10, 1 being VERY DIFFICULT and 10 being VERY EASY. Each image was annotated (at the pixel, region, and binary levels) by 3 human annotators independently. \| Annotation Task \| Flickr \| Cityscapes \| PASCAL VOC12 \| \|-----------------\|--------\|------------\|--------------\| \| \| Time \| Ease \| Time \| Ease \| Time \| Ease \| \| Pixel-level \| 7.8±2.9 mins \| 5.5±1.2 \| 37.5±6.3 mins \| 3.6±1.6 \| 18.2±4.3 mins \| 5.2±1.7 \| \| Region-level \| 1.6±1.2 mins \| 7.3±2.7 \| 3.6±0.7 mins \| 5.5±1.8 \| 2.7±1.1 mins \| 6.7±2.3 \| \| Binary-level \| 2±0.3 secs \| 10±0.0 \| 4±0.8 secs \| 10±0.0 \| 3±1.4 secs \| 10±0.0 \| Table 1: User study results. The table reports the average time (and ease of annotation) to annotate one complete image at the pixel-level, one region within an image at the pixel-level, and to answer one binary query posed for a given image, for the three datasets. The results were averaged across all images for a given dataset and all annotators. The user-study results are reported in Table 1, which depicts the average time (and ease of annotation) across all images and annotators, for the three datasets. The resolution of each image was $513 \times 513$ for Flickr, $768 \times 768$ for Cityscapes and $513 \times 513$ for PASCAL VOC. As evident from the table, pixel-level annotation entailed the maximum amount of time (and hence, human labor). Annotating a given region within an image took considerably less amount of time. As expected, binary-level annotations were the most efficient in terms of time and took only a few seconds for each image. We also note that the pixel-level annotations were the most difficult to provide, followed by region-level annotations. All the annotators reported that binary annotations were the easiest and the most convenient to provide and it consistently received the highest rating of 10. This user study demonstrates the tremendous savings in human annotation effort that can be achieved by the proposed binary-level annotation technique for image segmentation applications. Note that the user study was conducted to estimate the annotation time for the three annotation tasks, which will be used in our empirical analysis (detailed below). To train the DeepLabV3+ model in our experiments, we used the ground truth annotations that are provided for each dataset, since it will be extremely time-consuming to obtain human annotations for all the training images used in our study. ### 4.6 Active Learning Performance The active learning performance results are shown in Figure 2. In each graph, the $x$-axis denotes the iteration number and the $y$-axis denotes the mean IoU on the test set. From the results, we conclude the following: ![Active Learning Performance Comparison](image) Figure 2: Active Learning performance comparison. The $x$-axis denotes the iteration number and the $y$-axis denotes the mean IoU on the test set. Query budget = 200 for Cityscapes and PASCAL and 400 for Flickr in each AL iteration. Here, B denotes binary-level annotation, R denotes region-level annotation and P denotes pixel-level annotation. Best viewed in color. The proposed method comprehensively outperforms the two other AL techniques that utilize binary user feedback: RR and EE. In almost all the iterations across all three datasets, our framework depicts \| Dataset \| RR(B) \| EE(B) \| RAL(R) \| Entropy(P) \| Coreset(P) \| Proposed(B) \| \|-----------\|-----------\|-----------\|-----------\|------------\|------------\|-------------\| \| Flickr \| 71.9 ± 0.44 \| 74.95 ± 0.21 \| 76.9 ± 0.41 \| 76.8 ± 0.58 \| 76.8 ± 0.57 \| 75.7 ± 0.13 \| \| Cityscapes\| 75.56 ± 0.37 \| 76.76 ± 0.25 \| 79.30 ± 1.48 \| 79.4 ± 0.07 \| 79.4 ± 0.52 \| 78.5 ± 0.47 \| \| PASCAL \| 74.9 ± 0.17 \| 75.46 ± 0.35 \| 76.1 ± 0.23 \| 76.4 ± 0.17 \| 76.4 ± 0.29 \| 75.96 ± 0.06 \| Table 2: Final mIoU achieved by all the methods after 25 AL iterations. Here, B denotes binary-level annotation, R denotes region-level annotation and P denotes pixel-level annotation. \| Dataset \| Binary-Level \| Region-Level \| Pixel-Level \| \|-----------\|--------------\|--------------\|-------------\| \| Flickr \| 5.56 \| 266.67 \| 156 \| \| Cityscapes\| 5.56 \| 300 \| 750 \| \| PASCAL \| 4.16 \| 225 \| 364 \| Table 3: Approximate total time (in hours) to be expended for annotation (for the binary-level, region-level and pixel-level methods) over 25 AL iterations for all the three datasets. Query budget = 200 for Cityscapes and PASCAL and 400 for Flickr in each AL iteration. Query budget denotes the number of binary queries answered for binary-level annotation methods, and number of image regions annotated for the region-level annotation methods. Pixel-level annotation methods annotate all the 1,200 unlabeled images at the pixel-level (48 images in each AL iteration for all the datasets). A better mIoU value compared to these two baselines. The final mIoU achieved by our method after 25 AL iterations is also higher than RR and EE, for all three datasets. This shows that our algorithm can successfully identify the exemplar (image-class) pairs which augment maximal information to the deep learning model, thereby enabling it to attain much better generalization capabilities. The RAL method (which requires human users to annotate pixels within given image regions) as well as the Entropy and Coreset methods (which requires users to annotate all the pixels in a given image) marginally outperform the proposed algorithm (for the Flickr and Cityscapes datasets). Coreset depicts the best performance for Cityscapes and Flickr while Entropy depicts the best performance for PASCAL VOC. Table 2 shows the final mIoU attained by all the methods after 25 AL iterations. We note that RAL, Entropy and Coreset all achieve a marginally higher mIoU than our method. However, these methods also entail a significantly higher human annotation effort than our binary query framework. Table 3 depicts an estimate of the total annotation time (in hours) that has to be expended over the 25 AL iterations, for all the methods studied. These figures were obtained by multiplying the values in Table 1 by the number of annotations performed in each AL iteration and the total number of AL iterations. For instance, for the Cityscapes dataset, the time for pixel-level annotation was computed as: 37.5mins (time taken to annotate one image at the pixel-level) × 48 (no. of images annotated in each AL iteration) × 25 (no. of AL iterations); similarly, the time for region-level annotation was computed as: 3.6mins (time taken to annotate the pixels in one region in an image) × 200 (number of regions annotated in each AL iteration) × 25 (no. of AL iterations); and the time for the proposed binary annotation was computed as: 4secs (time taken to answer one binary query) × 200 (number of binary queries answered in each AL iteration) × 25 (no. of AL iterations). From Figure 2 and Table 3, it is evident that our method requires substantially less annotation time and effort, while producing mIoU values that are comparable to RAL, Entropy and Coreset. For the PASCAL VOC dataset for instance, the final mIoU achieved by our binary query framework is 75.96, and the difference is less than 0.5% compared to the values achieved by RAL, Entropy and Coreset (Table 2). However, the total annotation time required by the region-level (RAL) and pixel-level annotation (Entropy and Coreset) methods are 54.08 times and 87.5 times greater than our method respectively (Table 3). These results corroborate the promise and potential of our binary query and annotation technique to substantially reduce human annotation effort, with a marginal loss in performance, in an application like image segmentation, where annotating a single data instance is extremely time-consuming and laborious. From Table 3, we also note that region-level annotation can sometimes take more time than pixel-level annotation, depending on the number of regions annotated and the resolution of the images. 4.7 Study of Backbone Network Architecture In this experiment, we studied the effect of the backbone network architecture used in the DeepLabV3+ model (we used ResNet-101 as the default backbone architecture). The results on the Cityscapes dataset (with query budget 200) using the XceptionNet (Chollet, 2017) and ResNet50 backbones are shown in Figure 3. Our framework once again outperforms the binary-level annotation baselines RR and EE and depicts comparable performance to the region-level (RAL) and pixel-level (Entropy and Coreset) annotation baselines. Table 4 depicts the final mIoU values attained by all the methods after 25 AL iterations; since we have only changed the backbone network architecture (and not the query budget), the total annotation time computed in Table 3 for the Cityscapes dataset is also applicable for this experiment. From Table 4, we note that, for the Xception backbone, our binary query framework depicts the highest mIoU after 25 AL iterations; for the ResNet-50 backbone, our algorithm’s final mIoU is marginally less than that of RAL, Entropy and Coreset. However, as evident from Table 3, the total annotation time required by the region-level (RAL) and pixel-level annotation (Entropy and Coreset) methods are 53.95 times and 134.89 times greater than our method respectively. Our framework thus depicts comparable (and sometimes, marginally better) performance than the region-level and pixel-level annotation baselines, and is significantly more efficient in terms of the total annotation time required for the entire experiment. This shows the robustness of our framework to the backbone network architecture. ![Figure 3](image.png) (a) XceptionNet Backbone (b) ResNet50 Backbone Figure 3: Study of backbone network architecture on the Cityscapes dataset. Query budget = 200. Best viewed in color. \| Backbone \| RR(B) \| EE(B) \| RAL(R) \| Entropy(P) \| Coreset(P) \| Proposed(B) \| \|----------\|-------------\|-------------\|-------------\|--------------\|-------------\|-------------\| \| Xception \| 72.9 ± 0.18 \| 71.25 ± 0.63\| 72.4 ± 0.31 \| 72.8 ± 0.38 \| 72.8 ± 0.43 \| 73.2 ± 0.11 \| \| ResNet-50\| 67.2 ± 0.91 \| 67.5 ± 0.07 \| 68.2 ± 0.26 \| 68.2 ± 0.67 \| 68.2 ± 0.29 \| 67.95 ± 0.36\| Table 4: Final mIoU achieved by all the methods after 25 AL iterations (as shown in Figure 3) are depicted in the table. Here, B denotes binary-level annotation, R denotes region-level annotation and P denotes pixel-level annotation. We also conducted the following experiments, which are reported in the Appendix, due to space constraints: study of query budget (Section B); ablation study (Section C); analysis of the computation time of all the methods (Section D); study of the parameter $C_{max}$ (Section E); study of the initial training set size (Section G); and comparison against the fully supervised baseline (Section H). ## 5 Conclusion and Future Work In this paper, we proposed a novel active learning framework for semantic image segmentation, which poses only binary queries regarding the presence / absence of a semantic class in a given image. To the best of our knowledge, this is the first research effort to develop such an active query mechanism in the context of image segmentation. We posed the image and class selection as a constrained optimization problem and derived an LP relaxation to identify a batch of (image-class) pairs for active query. 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Xd46Q82QEO	if the assumption is that the neighbours of a single point in two representation spaces matter in the construction of useful similarity scores, then I think an easy and effective approach would be Jaccard distance, and its variants that take distances into consideration. I wonder how the proposed approach compares to Jaccard distance.	EXPLORING POINTWISE SIMILARITY OF REPRESENTATIONS Anonymous authors Paper under double-blind review ABSTRACT Representation similarity measures have emerged as a popular tool for examining learned representations. Many existing studies have focused on analyzing aggregate estimates of similarity at a global level, i.e. over a set of representations for $ N $ input examples. In this work, we shed light on the importance of investigating similarity of representations at a local level, i.e. representations of a single input example. We show that peering through the lens of similarity of individual data points can reveal previously overlooked phenomena in deep learning. Specifically, we investigate the similarity in learned representation of inputs by architecturally identical models that only differ in random initialization. We find that while standard models represent (most) inputs similarly only when they are drawn from training data distribution, adversarially trained models represent a wide variety of out-of-distribution inputs similarly, thus indicating that these models learn more “stable” representations. We design an instantiation of such a pointwise measure, named Pointwise Normalized Kernel Alignment (PNKA), that provides a way to quantify the similarity of an individual point across distinct representation spaces. Using PNKA, we additionally show how we can further understand the effects of data (e.g. corruptions) and model (e.g. fairness constraints) interventions on the model’s representations. 1 INTRODUCTION The success of deep neural network (DNN) models can be attributed to their ability to learn powerful representations of data that enable them to be effective across a diverse set of applications. However, the impressive performance of these models is often overshadowed by a variety of reliability concerns that arise when they are deployed in real-world scenarios [Geirhos et al., 2018; Hendrycks & Dietterich, 2019; Taori et al., 2020; Szegedy et al., 2013; Papernot et al., 2016; Athalye et al., 2018; Moosavi-Dezfooli et al., 2017; Angwin et al., 2016; O’neill, 2017]. These concerns have led to a surge in interest in better understanding the internal representations of these models before deploying them [Alain & Bengio, 2016; Davari et al., 2022; Kriegeskorte et al., 2008]. One promising line of research that offers a deeper understanding of model representations is representation similarity [Kornblith et al., 2019; Laakso & Cottrell, 2000; Raghu et al., 2017; Morcos et al., 2018]. At their core, representation similarity measures provide an overall score that quantifies how a set of points are positioned relative to each other within the representation spaces of two models. While aggregate measures have proved to be a useful tool to better understand many properties of deep learning [Nguyen et al., 2021a,b; Nanda et al., 2022; Raghu et al., 2021; Moschella et al., 2022], in this work we show that many other intriguing phenomena in deep learning can be understood by measuring the similarity of representations at the level of individual data points. Consider the well-studied case of two architecturally identical DNNs that only differ in random initialization. Prior works have independently concluded that two such models learn “similar” representations (indicated by a high aggregate representation similarity score on the test set) [Kornblith et al., 2019; Raghu et al., 2017]. However, when analyzing similarity at the level of individual points, we find that not all points are represented similarly across these two models. Instead, we observe a few points whose representations obtain lower similarity scores. We refer to these as unstable points. We find that such unstable points hold some properties that can have implications for the models’ performances on these points, i.e. models are more likely to disagree on the predictions for unstable points. We further show how the use of a pointwise representation measure enables a deeper and better understanding of the connections between a model’s representations and several other aspects, including its behavior and the impact of interventions on the acquired representations, both on the data employed (e.g., how changing the data distribution affects the representations of individual points) as well on the model itself (e.g., how training with fairness constraints changes representations of individual points). To this end, we design an instantiation of such a pointwise representation similarity measure, which we call Pointwise Normalized Kernel Alignment (PNKA), that builds on the well-studied and broadly used Centered Kernel Alignment (CKA) (Kornblith et al., 2019) and assigns similarity scores to each point being evaluated across two distinct representations. Intuitively, for PNKA to assign a high similarity score to a point across two representation spaces, that point should be positioned similarly relative to the other points in both representations. Analogous to CKA, how to define the relative position of a point for PNKA can be changed flexibly by using the appropriate kernel function. PNKA can be seen as a local decomposition of global representation similarity measures, by providing a distribution of similarity scores that when aggregated, provides an overall similarity estimation that is related to the aggregate measures broadly used today. Our key contributions are summarized as follows: - We highlight the importance of analyzing representation similarity at the granularity of individual data points. To this end, we design an instantiation of a measure, PNKA, that can provide similarities at a pointwise granularity. - While the widely used aggregate representation similarity measures assign a high overall similarity score to the penultimate layer representations of two models that differ solely due to stochastic factors, e.g., in their random initialization, we show that not all individual inputs score equally highly. We call the points with lower representation similarity as unstable. - Through a pointwise representation similarity measure (PNKA) we are able to investigate the properties that these points hold under different scenarios of data distribution shifts. We find that models are more likely to disagree on the predictions of unstable points. We also show that while non-robust models represent (most) points similarly only under an in-distribution context, adversarially trained models represent a wide variety of out-of-distribution samples similarly, thus indicating that these models learn more “stable” representations. - Finally, using PNKA, we analyze how interventions to a model modify the representations of individual points. Applying this approach to the context of learning fair representations, we show that debiasing approaches for word embeddings do not modify the targeted group of words as expected, an insight overlooked by current evaluation metrics. 1.1 Related Work Representation Similarity Measures. Recently, approaches that compare the representational spaces of two models by measuring representation similarity have gained popularity (Laakso & Cottrell, 2000; Li et al., 2015; Wang et al., 2018; Raghu et al., 2017; Morcos et al., 2018; Kornblith et al., 2019). Raghu et al. (2017) introduced SVCCA, a metric based on canonical correlation analysis (CCA) (Hotelling, 1992), which measures similarity as the correlation of representations mapped into an aligned space. Morcos et al. (2018) build on this work by introducing PWCCA, another CCA-based measure that is less sensitive to noise. More recently, CKA (Kornblith et al., 2019) has gained popularity and has now been extensively used to study DNN representations (Nguyen et al., 2021a; Ramasesh et al., 2020; Raghu et al., 2019; 2021). CKA is based on the idea of first choosing a kernel and then measuring similarity as the alignment between these two kernel matrices. We take inspiration from this insight to propose PNKA. We refer readers to (Klabunde et al., 2023) for a comprehensive overview of similarity measures. Understanding Representations of Individual Data Points using Neighbourhoods The broad idea of comparing nearest neighbors of instances in the representation space has been introduced in prior works, albeit for different motives, e.g., changes in linguistic styles (Hamilton et al., 2016), analyzing node embeddings (Wang et al., 2020), and for robust prediction (Papernot & McDaniel, 2018). While 1Similar to the CKA paper we use a linear kernel for all our experiments. our method is inspired by the higher-level idea of comparing neighborhoods across representations, we differ significantly from these works since we offer a concrete measure of pointwise similarity that is general-purpose and can be broadly applied to understand many phenomena in deep learning, across different data modalities. Recent work by Shah et al. (2023) proposes a method to estimate the contribution of individual points to learning algorithms. However, their work is mainly focused on understanding what features of inputs are encoded in the representations and does not evaluate the similarity of representations. Instead, in our work, we focus on showing the importance of analyzing whether two models represent individual inputs similarly. Work by Moschella et al. (2022) also relates to ours as their proposed model stitching method resembles our proposed measure (PNKA). However, we note here that the goal, contributions, and assumptions made in their paper differ drastically from ours. More importantly, their method assumes that the angles between elements of the latent space are kept the same for all elements, which we show in Section 4 is not being the case. 2 Why study representation similarity at finer granularity Previous studies have primarily focused on inspecting the aggregate-level representation similarity of DNNs. As a consequence, these studies do not provide insights into the distribution of similarity scores at the granularity of individual data points. We illustrate this in Figure 1a where we compare representation spaces of two models, namely $Y$ and $Z$. The majority of points (in black) are positioned highly similarly relative to the other points, in both representations, while a minority of points (in red) are positioned highly dissimilarly. We need fine-grained pointwise similarity scores to enable us to distinguish between these stably represented (black) and unstably represented (red) points. In Figure 1b, we demonstrate the need for such a fine-grained measure with a concrete example. Figure 1b shows the distribution of pointwise similarity scores, on the CIFAR-10 test set, for two ResNet-18 models that only differ on their random initialization, but are otherwise trained using the same procedure on CIFAR-10. We also illustrate some data points sampled at different points in the distribution. We see that most of the points exhibit high similarity scores, which also aligns with the high CKA score obtained. However, there exist some (unstable) points in the tail of the distribution with lower representation similarity scores. Identifying unstable points whose representational stability is impacted solely by stochastic factors (i.e., randomness) within the training process is not --- 2 More information on training details as well as test set accuracies can be found in Appendix A. 3 We expand this analysis to other architectures and datasets in Appendix B. only valuable but also crucial. As we show later in Section 4, these points are not only more likely to originate from out-of-distribution sources but are also prone to higher misclassification rates. Finally, we note that some prior studies using aggregate similarity measures implicitly assume that representational instability arising from randomness in the training procedure would be limited to very few points. The original CKA paper (Kornblith et al., 2019) claimed that two models that differ only in their random initialization would learn highly similar representations at the penultimate layer, without qualifying that the observation holds true only for inputs drawn from the training data distribution. This observation has since been even proposed as a sanity check to audit different representation measures, e.g., for Ding et al. (2021), a reliable similarity measure must assign high similarity to representations from models that only differ in random initialization. As we show in Section 4, the stability of learned representations for models with different random initializations is strongly influenced by other factors such as whether models use robust or standard learning procedures. Thus, the validity of conducting such a sanity check becomes questionable. 3 MEASURING POINTWISE REPRESENTATION SIMILARITY In order to analyze representation similarity at a local level, we design an instantiation of a pointwise representation similarity measure, named Pointwise Normalized Kernel Alignment (PNKA), which builds on the well-studied and broadly used Centered Kernel Alignment (CKA). Notation. We denote by $ Y \in \mathbb{R}^{N \times d_1}, Z \in \mathbb{R}^{N \times d_2} $ two sets of $ d_1 $ and $ d_2 $ dimensional representations for a set of $ N $ inputs, respectively. We assume that $ Y $ and $ Z $ are centered column-wise, i.e., along each dimension. We aim to measure how similarly the $ i $-th point is represented in $ Y $ and $ Z $. We denote a pointwise similarity measure between representations $ Y $ and $ Z $ for point $ i $ by $ s(Y, Z, i) $. Formally defining PNKA. To design PNKA, we leverage the simple, but powerful insight from prior works, which states that while we cannot directly compare similarity across representations, we can do so within the same representation (Kornblith et al., 2019; Kriegeskorte et al., 2008). Therefore, to determine whether the representations $ Y_i $ and $ Z_i $ of point $ i $ are similar, we can first compare how similarly $ i $ is positioned relative to all the other points within each representation. We then compare the relative position of $ i $ across both representations. More formally, given a set of representations $ Y $ and a kernel $ k $, we can define a pairwise similarity matrix between all $ N $ points in $ Y $ as $ K(Y) $ with $ K(Y)_{i,j} = k(Y_i, Y_j) $. In our work, we use linear kernels, i.e., $ k(Y_i, Y_j) = Y_i \cdot Y_j^\top $, but other kernels, e.g., RBF (Kornblith et al., 2019) could be used as well. We leave the exploration of other types of kernels for future work. Given two similarity matrices $ K(Y) $ and $ K(Z) $, we measure how similarly point $ i $ is represented in $ Y $ and $ Z $ by comparing its position relative to the other points. To this end, we define \[ \text{PNKA}(Y, Z, i) = \cos(K(Y)_i, K(Z)_i) = \frac{K(Y)_i^\top K(Z)_i}{\|\|K(Y)_i\|\| \|\|K(Z)_i\|\|}, \] where $ K(Y)_i $ and $ K(Z)_i $ denote how similar point $ i $ is to all other points in $ Y $ and $ Z $, respectively. We use cosine similarity to compare the relative positions across representations for two reasons. First, cosine similarity provides us with normalized similarity scores for each point. Second, by normalizing by the length of the similarity vectors $ K(Y)_i $ and $ K(Z)_j $, we compare the relative instead of the absolute similarity of points, i.e., how similar point $ i $ is represented relative to points $ j $ and $ j' $. PNKA can also be extended into an aggregate version, that has empirically shown to be correlate with CKA (Kornblith et al., 2019) (Appendix C.1), by computing \[ \overline{\text{PNKA}}(Y, Z) = \frac{1}{N} \sum_{i=1}^{N} \text{PNKA}(Y, Z, i), \] Computing PNKA with stable reference points. As PNKA works by comparing how a point is positioned relative to other reference points across two representation spaces, one may wonder if the reference points themselves should be required to have stable representations. For instance, in Figure 1a, computing PNKA scores using unstable (red) points as reference points might yield low similarity scores for all points. To this end, one can construct a particular case of PNKA, restricting the set of $ N $ reference points to $ L $ stable points. We establish that reference points in this context must adhere to two essential properties: (1) stability: points should remain stably positioned relative to each other, i.e. have high PNKA amongst themselves, and (2) spatial diversity: points should be well-distributed in the representation space, i.e. points should not be collapsed. We show in Appendix C.2 that these two properties hold for our choice of reference points. The reference points can come from the training set or as a subset of the test set distribution $ L \subseteq N $, where $ L = N $ is the general case previously presented). In the experiments of the following section, we draw $ L = 1,000 $ reference points from the training set, i.e. we compute the relative position of the $ N $ test set points with respect to a subset of $ L $ stable and spatially diverse points from the training set. Formally, given the representations of points $ A \in \mathbb{R}^{N \times d_1}, C \in \mathbb{R}^{N \times d_2} $, and respective reference points $ B \in \mathbb{R}^{L \times d_1}, D \in \mathbb{R}^{L \times d_2} $, from two models with dimensions $ d_1 $ and $ d_2 $, respectively, we define a pairwise similarity matrix as $ K(A, B) $ with $ K(A_i, B_j) = k(A_i, B_j) $. Thus, in this specific case PNKA is defined as \[ \text{PNKA}(Y, Z, i) = \cos(K(A, B)_i, K(C, D)_i) = \frac{K(A, B)_i^\top K(C, D)_i}{\|\|K(A, B)_i\|\| \|\|K(C, D)_i\|\|}, \] where $ K(A, B)_i $ and $ K(C, D)_i $ denote how similar point $ i $ is to the $ L $ reference points in each of the models. Properties. We empirically show that PNKA holds important properties (Kornblith et al., 2019), such as invariance to both orthogonal transformations and isotropic scaling (Appendix C.3). We also empirically show that PNKA captures the overlap of neighbors across two representations and that if the PNKA score of point $ i $ is higher than that of $ j $, then there is a higher chance that $ i $'s nearest neighbors overlap more across representations $ Y $ and $ Z $ than those of $ j $ (Appendix C.4). 4 USING POINTWISE ANALYSIS TO UNDERSTAND DATA INTERVENTIONS In this section, we use PNKA to investigate the properties of unstable points, i.e. points represented less similarly, between models that differ solely due to their random initialization, and analyze if these points possess some distinct properties. We deliberately chose to focus on comparing representations of models that differ on random initialization as in this scenario, unstable points represent inputs whose representations are heavily influenced by random chance, and using such unstable representations for downstream tasks can be worrisome. We analyze the (in)stability of representations under three scenarios: (1) in-distribution data points (Section 4.1), e.g. the test set, which exemplifies a usual scenario where the model will be used for the same downstream task that it has been previously trained for; (2) subset of data points is out-of-distribution (Section 4.2), which might illustrate a practical scenario in which individuals seek to evaluate models on “in-the-wild” data while already possessing a set of trusted (in-distribution) data points; (3) all data points are OOD (Section 4.3), which portrays a scenario where the features of the models might be used for a different task than the model was previously trained for, e.g., transfer learning. In the remainder of this section, we report an average PNKA score over 3 runs of two models trained on CIFAR-10 (ResNet-18 (He et al., 2016)) differing only in their random weight initialization. 4.1 MODELS MORE LIKELY TO DISAGREE ON UNSTABLE POINTS We first examine unstable points for inputs that fall within the training distribution, i.e. CIFAR-10 test set. We also expand this analysis to CIFAR-10.1 (Recht et al., 2018), which attempts to construct another CIFAR-10 test set, closely following the methodology of the original dataset, but which has been shown to cause a significant drop in accuracy (4 – 10%). Given that unstable points exhibit greater dissimilarity across models trained with different initializations, a reasonable hypothesis is that these models will be more prone to disagreeing on the predictions for these unstable points. In Figure 2, we show the percentage of instance predictions on which the models agree, relative to their ranked similarity score. The points were first sorted according to their similarity scores, with the leftmost end (0) representing the group with the lowest scores and the rightmost end (9) representing the group with the highest scores, and then grouped into deciles, with each bar representing 10% of the total points in the test set. The vertical dotted line shows the aggregate scores (PNKA) for each group. We can see that the fraction of points whose predictions the models disagree on are mainly at the tail of the distribution, i.e. being less similarly represented, for both CIFAR-10 and CIFAR-10.1 test sets. In Appendix D.1, we show the same pattern for other choices of architecture and dataset. We $^{4}$10% of the total amount of test set points of CIFAR-10 and CIFAR-100 datasets. Figure 2: Percentage of instance predictions on which the models agree, relative to their ranked similarity score, for both CIFAR-10 (a) and CIFAR10.1 (b) test sets. The x-axis represents groups of points sorted based on their pointwise representation similarity according to PNKA, with each group (bar) containing 10% of the total amount of instances. The y-axis represents the fraction of those points on which models agree (blue) or disagree (red). The vertical dotted line shows the aggregate scores (PNKA) for that group. Results are averaged over 3 runs, each one containing two models trained on CIFAR-10 with different random initialization. The more unstable a point is, i.e., lower its representation similarity, the more likely models are to disagree on its prediction. also show in Appendix D.1.2 that these points are not only classified in different ways but that most of them are misclassified as well. Lower accuracy is to be expected since if two models disagree on a prediction, at most one of them can be correct. This finding also aligns with previous work on calibration [Baek et al., 2022; Jiang et al., 2021; Garg et al., 2022] which uses a model’s outputs to detect which instances are more likely to be misclassified. Therefore, unstable points are those for which models exhibit the greatest prediction disagreement and incorrect predictions. 4.2 Out-of-distribution points more likely to have unstable representations Next, we examine the case where some points do not come from the training distribution. To inspect that, we perturbed p% of the test set points with naturally occurring perturbations, e.g., blurring, color jitter, and elastic transformation. We then compute PNKA on the test set with p% perturbed and 1 − p% non-perturbed (i.e., originally from the test set) points for models that differ in their random initialization. We hypothesize that the representations of models are similar for points that have a high likelihood under the models’ training distributions, but that the representations of models will be dissimilar on OOD points. In Figure 3, we show the percentage of perturbed instances, relative to their ranked similarity scores. As previously, points were sorted according to their similarity score and then grouped into deciles. We use p = 10% and show that, for different types of perturbations, perturbed points are more likely to obtain lower similarity scores compared to non-perturbed (i.e., in-distribution) points. Thus, under this scenario, unstable points are more likely to be OOD than points with higher representation similarity. We expand this analysis for other choices of p, architectures, and datasets in Appendix D.2. 4.3 Robust models are less influenced by stochastic factors Finally, we investigate the extreme scenario of data distribution shift, where all the samples are out-of-distribution, i.e., p = 100%. Prior work [Ding et al., 2021; Davari et al., 2023; Nguyen et al., 2021a,b; McCoy et al., 2019] has employed global measures of representation similarity to examine models’ representations when exposed to out-of-distribution (OOD) data. It has been observed that these models exhibit dissimilar representations, even when the sole difference lies in their random initialization. Under this scenario, we also study unstable points for adversarially trained (i.e., robust) models as they are trained to be more resilient to adversarial examples, i.e., samples that are slightly perturbed to alter the model’s behavior. For both types of models, we again compute the pointwise representation similarity between models that differ only in random initialization. Figure 4 shows the similarity scores distribution for both robust and non-robust models trained on CIFAR-10 and evaluated under their original distribution, CIFAR-10 test set (Figure 4a), as well as two different distribution shifts: CIFAR-100 (Figure 4b) and images with complete random noise (Figure 4c). We Figure 3: Percentage of perturbed instances, relative to their ranked similarity score. The $x$-axis represents groups of points sorted based on their pointwise representation similarity according to PNKA, with each group containing 10% of the total amount of instances. The $y$-axis represents the fraction of the points that are perturbed (red) or not perturbed (green). The vertical dotted line shows the aggregate scores (PNKA) for that group. We consider three possible perturbations: (a) blurring, (b) color jitter, and (c) elastic transformation. Results are over CIFAR-10 test set instances, averaged over 3 runs, each one containing two models trained on CIFAR-10 with different random initializations. Note that the more unstable a point is, i.e., lower the representation similarity, the more likely a point is to be OOD. Figure 4: Distribution of similarity scores for standard (non-robust) models (blue) and adversarially trained (robust) models (red). Results are averaged over 3 runs, each one containing two models trained on CIFAR-10 with different random initialization. The pointwise similarity scores are shown for (a) CIFAR-10 test set (in-distribution), as well as (b) CIFAR-100 and (c) complete random noise. While standard models represent (most) inputs similarly only when they are drawn from training data distribution (left-most figure), adversarially trained models represent a wide variety of out-of-distribution inputs similarly, thus indicating that these models learn more “stable” representations. We can see that under a similar training distribution (Figure 4b), both robust and non-robust models have similar PNKA distributions. However, as we use points further away from the distribution, the robust models seem to obtain more stable representations than the non-robust model. Even for complete random noise, the robust model represents several points similarly, i.e., PNKA score > 0.9. This suggests that robust models learn more “stable” representations across a wide variety OOD data. We expand this analysis to other types of OOD data, as well as models trained on other datasets in Appendix D.3. 5 USING POINTWISE ANALYSIS TO UNDERSTAND MODEL INTERVENTIONS Pointwise representation similarity can also be a useful tool to better understand the effects of interventions on a model. We can use PNKA to compute pointwise similarity scores between the representations of the original and the modified (i.e., intervened) models and analyze the inputs that are most affected by the intervention. We showcase the use of PNKA in the context of interventions to learn fair ML models. An important goal of the fair ML literature is non-discrimination, where we attempt to mitigate biases that affect protected groups in negative ways (Angwin et al., 2016; O’Neil, 2017). A popular approach to achieve non-discrimination is through learning debiased or fair representations (Zemel et al., 2013; Creager et al., 2019; Louizos et al., 2015). These approaches transform or train model representations in a way that minimizes the information they contain about the group membership of inputs. However, today, we often overlook how the interventions targeting (macro-)group-level fairness affect representations at the (micro-)individual-level and whether the changes in individual point representations are desirable or as intended. By applying PNKA to the original and the debiased representations, we can understand the effects of the debiasing intervention at the level of individual inputs, and analyze the inputs whose representations underwent the biggest change. We demonstrate how this ability can be leveraged in the context of natural language word embeddings to investigate whether the debiasing approaches indeed work as intended. Approaches to debias word embeddings: Many word embedding approaches have been found to produce biased representations with stereotypical associations (Bolukbasi et al., 2016; Gonen & Goldberg, 2019; Zhao et al., 2018), and several methods have been proposed with the goal of reducing these stereotypical biases (Bolukbasi et al., 2016; Gonen & Goldberg, 2019; Zhao et al., 2018; Kaneko & Bollegala, 2019). In this work, we choose two approaches with the goal of using PNKA to analyze whether debiasing successfully decreases stereotypical associations. Both debiasing techniques are based on the original GloVe (Kaneko & Bollegala, 2019): (1) Gender Neutral (GN-)GloVe (Zhao et al., 2018) focuses on disentangling and isolating all the gender information into certain specific dimension(s) of the word vectors; (2) Gender Preserving (GP-)GloVe (Kaneko & Bollegala, 2019) targets preserving non-discriminative gender-related information while removing stereotypical discriminative gender biases from pre-trained word embeddings. The latter method can also be used to finetune GN-GloVe embeddings, generating another model namely, GP-GN-GloVe. Evaluation of debiased word embeddings: In order to evaluate the impact of the debiasing methods, both GP- and GN-GloVe use the SemBias dataset (Zhao et al., 2018). Each instance in SemBias consists of four word pairs: a gender-definition word pair (e.g. “waiter - waitress”), a gender-stereotype word pair (e.g. “doctor - nurse”), and two other word-pairs that have similar meanings but no gender relation, named gender-neutral (e.g. “dog - cat”). The goal is to evaluate whether the debiasing methods have successfully removed stereotypical gender information from the word embeddings, while simultaneously preserving non-stereotypical gender information. To this end, GP- and GN-GloVe evaluated how well the embeddings can be used to predict stereotypical word pairs in each instance of the SemBias dataset. The details and results of this prediction task are in Appendix E.1. The evaluation shows that GP-GloVe embeddings offer only a marginal improvement, while GN- and GP-GN-GloVe embeddings offer substantial improvement at the prediction task. Using PNKA to understand debiased word embeddings: We applied PNKA to the original and the debiased GloVe embeddings to examine whether the methods are indeed reducing bias as claimed. Figure 5 shows the distribution of PNKA scores for words in SemBias dataset grouped by their category (i.e., gender defining, gender neutral, and gender stereotype). We highlight two observations. First, GP-GloVe representations are very similar to GloVe (Figure 5a) for almost all of the words, whereas GN-GloVe (Figure 5b) and GP-GN-GloVe (Figure 5c) considerably change the representations for a subset set of the words. This observation aligns well with results of prior evaluation which found that GP-GloVe achieves similar results to GloVe, while GN-GloVe and GP-GN-GloVe achieve better debiasing results. Second, Figure 5 also shows that across all three debiasing methods, the --- (a) GloVe × GP-GloVe. (b) GloVe × GN-GloVe. (c) GloVe × GP-GN-GloVe. Figure 5: Distribution of PNKA scores per group of words for SemBias dataset (Zhao et al., 2018). We compare the baseline (GloVe) model and its debiased versions. Words with the lowest similarity scores are the ones that change the most from the baseline to its debiased version. Surprisingly, across all debiased embeddings, the words whose embeddings change the most are the gender-definition words. Figure 6: Relationship between PNKA scores and percentage change in magnitude of the projection onto the gender vector from the baseline GloVe. A positive change indicates an increase in magnitude along the canonical gender direction. Word embeddings that change their gender information are the ones that obtain low PNKA scores. Words whose embeddings change the most are the gender-definition words. Note that this observation is in complete contradiction to the expectation that with debiasing, the embeddings that would change the most are the gender-stereotypical ones, while the embeddings that would be preserved and not change are the gender-definitional ones. Put differently, the pointwise similarity scores suggest a very different explanation for why GN-GloVe and GP-GN-GloVe achieve better debiasing evaluation results over SemBias dataset: rather than remove gender information from gender-stereotypical word pairs, they are amplifying the gender information in gender-definition word pairs, resulting in better performance in distinguishing gender-stereotypical and gender-definition word pairs. We confirm our alternate explanation by measuring for each word how much its embedding changed in terms of gender information, when compared to the original GloVe embedding, by projecting it onto the canonical gender vector $\vec{he} - \vec{she}$ (more in Appendix E.2), generating the percentage difference in magnitude. Figure 6 shows that the GN-GloVe and GP-GN-GloVe debiasing methods primarily amplify the gender information in gender-definition words, rather than reduce it for gender-stereotype words. In fact, the words that change their gender information the most are the low-similarity ones. This analysis illustrates how pointwise similarity scores can offer new insights, trigger new investigations, and lead to a better understanding of the effects of model training interventions. 6 DISCUSSION In this work, we demonstrate the power of investigating representations at the level of individual data points. First, we show that not all data points obtain a high similarity score, even for models that differ solely due to differences in random weight initialization. Under this context, we define the lower similarity points as unstable. We then investigate some of the characteristics of unstable points, including a higher likelihood of model prediction disagreements and the possibility that these points might be out-of-distribution. We then show that while standard (i.e., non-robust) models represent (most) inputs similarly only when they are drawn from the training data distribution, adversarially trained (i.e., robust) models exhibit higher representation similarity for a broader range of out-of-distribution points. This finding suggests that robust models learn more “stable” representations. Finally, we use the context of fairness to show that pointwise similarity measures can be a useful tool for understanding which individuals are most affected by model interventions, thus shedding light on the internal characteristics of such modifications. A limitation of our work lies in the restricted consideration of only a few model variations. Other applications of pointwise representation similarity analysis. Employing pointwise representation similarity measures unveils several intriguing directions for exploration. For instance, one could examine differences in different architectures through the lens of similarity of individual points. An initial exploration in this direction is presented in Appendix E. Another promising line of work could potentially analyze how points are represented (dis)similarly across different layers of a neural network. We offer an initial analysis in this direction in Appendix G. Finally, one can use PNKA to delve deeper into the understanding of individual neurons within a neural network layer. We provide an initial analysis of the influence of single neuron units on pointwise representation similarity in Appendix H. Reproducibility. We run all our experiments using publicly available, open-source frameworks, architectures and datasets. Thus, all our results can be seamlessly reproduced. We also attach our code to aid reproducibility. To ensure correctness, we also report all our results over 3 random seeds. All other details about pre-processing, learning rate, epochs, model architectures, and more information can be found in Appendix A and are also included in our attached code. REFERENCES Guillaume Alain and Yoshua Bengio. Understanding intermediate layers using linear classifier probes. arXiv preprint arXiv:1610.01644, 2016. Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner. Machine bias. In Ethics of data and analytics, pp. 254–264. Auerbach Publications, 2016. Anish Athalye, Logan Engstrom, Andrew Ilyas, and Kevin Kwok. Synthesizing robust adversarial examples. In International conference on machine learning, pp. 284–293. PMLR, 2018. Christina Baek, Yiding Jiang, Aditi Raghunathan, and J Zico Kolter. Agreement-on-the-line: Predicting the performance of neural networks under distribution shift. Advances in Neural Information Processing Systems, 35:19274–19289, 2022. David Bau, Bolei Zhou, Aditya Khosla, Aude Oliva, and Antonio Torralba. Network dissection: Quantifying interpretability of deep visual representations. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 6541–6549, 2017. Tolga Bolukbasi, Kai-Wei Chang, James Y Zou, Venkatesh Saligrama, and Adam T Kalai. Man is to computer programmer as woman is to homemaker? debiasing word embeddings. Advances in neural information processing systems, 29, 2016. Nick Cammarata, Gabriel Goh, Shan Carter, Ludwig Schubert, Michael Petrov, and Chris Olah. Curve detectors. Distill, 5(6):e00024–003, 2020. Elliot Creager, David Madras, Jörn-Henrik Jacobsen, Marissa Weis, Kevin Swersky, Toniann Pitassi, and Richard Zemel. Flexibly fair representation learning by disentanglement. In International conference on machine learning, pp. 1436–1445. PMLR, 2019. MohammadReza Davari, Nader Asadi, Sudhir Mudur, Rahaf Aljundi, and Eugene Belilovsky. Probing representation forgetting in supervised and unsupervised continual learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 16712–16721, 2022. MohammadReza Davari, Stefan Horoi, Amine Natik, Guillaume Lajoie, Guy Wolf, and Eugene Belilovsky. Reliability of cka as a similarity measure in deep learning. In International Conference on Learning Representations, 2023. Frances Ding, Jean-Stanislas Denain, and Jacob Steinhardt. Grounding representation similarity through statistical testing. Advances in Neural Information Processing Systems, 34:1556–1568, 2021. Logan Engstrom, Andrew Ilyas, Hadi Salman, Shibani Santurkar, and Dimitris Tsipras. Robustness (python library), 2019. URL https://github.com/MadryLab/robustness. Saurabh Garg, Sivaraman Balakrishnan, Zachary C Lipton, Behnam Neyshabur, and Hanie Sedghi. Leveraging unlabeled data to predict out-of-distribution performance. arXiv preprint arXiv:2201.04234, 2022. Robert Geirhos, Carlos R. M. Temme, Jonas Rauber, Heiko H. Schütt, Matthias Bethge, and Felix A. Wichmann. Generalisation in humans and deep neural networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL https://proceedings.neurips.cc/paper/2018/file/0937fb5864ed06fffb59ae5f9b5ed67a9-Paper.pdf.
dqWobzlAGb	In some cases, the proposed method is never better than the previous state-of-the-art, e.g., on celegans with 3-dim (0 out of 30) and on mouse3 with 3-dim (0 out of 30). Is there any analysis of the failure cases?	MODELLING BRAIN CONNECTOMES: SOLV IS A WORTHY COMPETITOR TO HYPERBOLIC GEOMETRY! Anonymous authors Paper under double-blind review ABSTRACT Finding suitable embeddings for connectomes (spatially embedded complex networks that map neural connections in the brain) is crucial for analyzing and understanding cognitive processes. Recent studies have found two-dimensional hyperbolic embeddings superior to Euclidean embeddings in modelling connectomes across species, especially human connectomes. However, those studies had some limitations: geometries other than Euclidean, hyperbolic or spherical were not taken into account. Following the suggestion of William Thurston that the networks of neurons in the brain could be successfully represented in Solv geometry, we study goodness-of-fit of the embeddings for 21 connectome networks (8 species). To this end, we suggest an embedding algorithm based on Simulating Annealing that allows us embed connectomes to Euclidean, Spherical, Hyperbolic, Solv, Nil, and also product geometries. Our algorithm tends to find better embeddings than the state of the art, even in the hyperbolic case. Our findings suggest that while in many cases, three-dimensional hyperbolic embeddings yield the best results, Solv embeddings perform reasonably well. 1 INTRODUCTION Connectomes are comprehensive maps of neural connections in the brain. Understanding the interactions shaped by them is a key to understanding cognitive processes. As connectomes are spatially embedded complex networks with the structure shaped by physical constraints and communication needs, they seem to exhibit traits inherent to non-Euclidean geometries. That is why a vast amount of research interest has been recently devoted to finding the suitable embeddings for connectome networks. Recent studies (e.g., Whi et al. (2022); Allard & Serrano (2020)) have found two-dimensional hyperbolic embeddings superior to Euclidean embeddings in modelling connectomes across species, especially human connectomes. However, those studies had some limitations: geometries other than Euclidean, hyperbolic or spherical were not taken into account. Our study broadens the perspectives for the suitable embeddings. We analyze the goodness-of-fit (measured with widely-used quality measures) of the embeddings for 21 connectome networks (8 species) to 15 unique tessellations (Euclidean, Spherical, Hyperbolic, Solv, Nil, and also product geometries). We include both two-dimensional manifolds and three-dimensional ones. Following the suggestion of William Thurston that the networks of neurons in the brain could be successfully represented in Solv geometry (one of eight so-called Thurston geometries), we stipulate that this using this geometry would outperform using hyperbolic geometry. Against this background, our contribution in this paper can be summarized as follows: • We provide a new embedding method based on Simulated Annealing (SA). Our experiments show that our algorithm tends to find better embeddings than the state of the art, even in the hyperbolic case, measured using the standard measures from the literature (mAP, MeanRank, greedy routing success and stretch). • To our best knowledge, we are the first to compare embeddings of connectomes to all Thurston geometries. Thus, we introduce new possibilities in modelling of connectomes. • We find that while in many cases three-dimensional hyperbolic geometry yields the best results, there are other geometries worth consideration, e.g., Solv. As our results are based on an extensive simulation scheme, they are more robust in comparison to previous work. Figure 1: Tessellations of the hyperbolic plane. From left to right: (a) bitruncated order-3 heptagonal tiling (\{7, 3\}), (b) infinite-order triangular tiling (\{3, ∞\}), (c) binary tiling. 2 HYPERBOLIC EMBEDDINGS The $n$-dimensional sphere is $S^n = \{x \in \mathbb{R}^{n+1} : g(x,x) = 1\}$, where $g$ is the Euclidean inner product, $g(x,y) = x_1y_1 + x_2y_2 + \ldots + x_{n+1}y_{n+1}$. The distance between two points $a, b$ on the sphere is the length of the arc connecting $a$ and $b$, which can be computed as $d(a,b) = \text{acos } g(a,b)$. Similarly, $n$ dimensional hyperbolic geometry can be defined using the Minkowski hyperboloid model. In this model, $H^n = \{x \in \mathbb{R}^{d+1} : x_{d+1} > 0, g^-(x,x) = -1\}$, where $g^-$ is the Minkowski inner product, $g^-(x,y) = x_1y_1 + x_2y_2 + \ldots + x_ny_n$. The distance is $d(a,b) = \text{acosh } g^-(a,b)$. Figure 1 depicts three tessellations of the hyperbolic plane $H^2$ in the Poincaré disk model—a projection of $H^2$ to the Euclidean plane that distorts the distances. In each of these tessellations, all the shapes (of the same color) are actually of the same hyperbolic size, even though ones closer to the boundary look smaller in the projection. Figure 1 shows the tree-like structure of hyperbolic geometry. This tree-likeness has found application in the visualization of hierarchical structures (Lamping et al., 1995; Munzner, 1998), and then in the modelling of complex networks. The hyperbolic random graph model (Boguña et al., 2010) is parameterized by parameters $N, R, T, \alpha$. Each node $i \in \{1, \ldots, n\}$ is assigned a point $m(i)$ in the hyperbolic disk of radius $R$; the parameter $\alpha$ controls the distribution. Then, every pair of points $a, b \in \{1, \ldots, n\}$ is connected with probability $1/(1 + \exp((d - R)/T))$, where $d$ is the hyperbolic distance between $a$ and $b$. Graphs generated according to this model have properties typical to scale-free networks, such as high clustering coefficient and power law degree distribution (Papadopoulos et al., 2012; Boguña et al., 2010). 3 THURSTON GEOMETRIES By the uniformization theorem, every closed two-dimensional topological surface can be given spherical ($S^2$), Euclidean ($E^2$), or hyperbolic ($H^2$) geometry, that is, there exists a Riemannian manifold with the same topology as $M$ and locally isometric to a sphere, Euclidean plane, or hyperbolic plane. William Thurston conjectured (Thurston, 1982) that three-dimensional topological manifolds can be similarly decomposed into fragments, each of which can be given one of eight Thurston geometries, which are homogeneous Riemannian manifolds. The eight Thurston geometries include: - isotropic geometries: spherical ($S^3$), Euclidean ($E^3$), and hyperbolic ($H^3$). - product geometries: $S^2 \times \mathbb{R}$ and $H^2 \times \mathbb{R}$. In geometry $A \times B$, the distance $d_{A \times B}$ between $(a_1, b_1), (a_2, b_2) \in A \times B$ is defined using the Pythagorean formula: \[d_{A \times B}((a_1, b_1), (a_2, b_2)) = \sqrt{d_A(a_1, a_2)^2 + d_B(b_1, b_2)^2}.\] Intuitively, the third dimension is added to $S^2$ or $H^2$ in the Euclidean way. - Twisted product geometries: twisted $E^2 \times \mathbb{R}$, also known as Nil, and twisted $H^2 \times \mathbb{R}$, referred to as Twist in this paper, also known as the universal cover of $SL(2, \mathbb{R})$. - Solv geometry, also known as Solve or Sol, which is fully anisotropic. In low-dimensional topology, three-dimensional geometry is especially challenging, in particular, the Poincaré conjecture was the most challenging in three dimensions. On the other hand, our inter- est in two-dimensional and three-dimensional geometries is based on their visualization possibilities (Kopczyński & Celińska-Kopczyńska, 2020; Coulon et al., 2020) and potential application to geometric embeddings. The original research into geometric embedding of networks used $\mathbb{H}^2$; more recently, higher-dimensional hyperbolic spaces are also studied (Jankowski et al., 2023; Whi et al., 2022). Similar embeddings are also used in machine learning, in particular, in (Gu et al., 2019) product geometries are studied. Up to our knowledge, twisted product or Solv geometry have not been studied in this context. We are especially interested in the intriguing suggestion of William Thurston from 1997 that the architecture of brain might be based on Solv geometry (Schwartz, 2020). The more exotic Thurston geometries have been successfully visualized only very recently (Kopczyński & Celińska-Kopczyńska, 2020; Coulon et al., 2020), and thus are much less known than isotropic geometries. We refer to these papers and explanatory videos (Rogue, 2023; 2022) and demos (Coulon et al., 2022) for detailed explanations of Solv and Nil geometries. In the rest of this section, we include a brief explanation of Solv and an intuitive explanation of twisted product geometries. We also discuss how their properties might prove beneficial for modeling networks. To explain Solv, we should start first with the horocyclic coordinate system of $\mathbb{H}^2$. Horocycles are represented in the Poincaré disk model as circles tangent to the boundary; these can be seen as hyperbolic analogs of circles with infinite radius and circumference, centered in an ideal point (point on the boundary of the Poincaré disk). Concentric horocycles are seen in Figure 1; the distance between two adjacent horocycles in this picture is $\log(2)$, and if two points $A$ and $B$ on given horocycle are in distance $x$, then the distance between their projections on the next (outer) horocycle is $2x$. For a point $P \in \mathbb{H}^2$, we project $P$ orthogonally to $Q$ on the horocycle going through the center $C$ of the Poincaré model. The $x$ coordinate is the (signed) length of the horocyclic arc $CQ$, and $y$ is the (signed) length of the segment $PQ$. (This is similar to the upper half-plane model (Cannon et al., 1997), except that we take the logarithm of the $y$ coordinate.) In this coordinate system, the length of the curve $((x(t), y(t)) : t \in [a, b])$ is defined as $$\int_a^b \sqrt{(x'(t) \exp y(t))^2 + y'(t)^2} dt.$$ A similar coordinate system for $\mathbb{H}^3$ defines the length of the curve $((x(t), y(t), z(t)) : t \in [a, b])$ as $$\int_a^b \sqrt{(x'(t) \exp z(t))^2 + (y'(t) \exp z(t))^2 + z'(t)^2} dt.$$ The surfaces of constant $z$ are called horo-spheres; the geometry on a horosphere is Euclidean. Solv geometry is obtained by switching the sign in this formula. That is, each point also has three coordinates $(x, y, z)$, but the length of a curve is now defined as $$\int_a^b \sqrt{(x'(t) \exp z(t))^2 + (y'(t) \exp -z(t))^2 + z'(t)^2} dt.$$ The distance between two points in the length of the shortest curve connecting them; this length is difficult to compute (Coulon et al., 2020; Kopczyński & Celińska-Kopczyńska, 2022). Intuitively, the Solv geometry is based on two hierarchies (the hyperbolic plane $y = \text{const}$ and the hyperbolic plane $x = \text{const}$), which are opposed to each other, due to the opposite sign used with $z$ in the distance formula. This gives us hope that Solv geometry can be used to represent hierarchies in three-dimensions which cannot be represented using other two- or three-dimensional geometries exhibiting simpler hierarchical structure ($\mathbb{H}^2$, $\mathbb{H}^3$, $\mathbb{H}^2 \times \mathbb{R}$). A similar effect of two opposing hierarchies could be also obtained in $\mathbb{H}^2 \times \mathbb{H}^2$, however, that is a four-dimensional geometry, and thus less suitable for visualization. In Nil, we have well-defined directions in every point, which we can intuitively call North, East, South, West, Up and Down. However, while in Euclidean geometry, after moving 1 unit to the North, East, South, then West we return to the starting point, in Nil such a loop results in a move by 1 unit in the Up direction. In general, the vertical movement is equal to the signed area of the projection of the loop on the horizontal plane. Twist is based on the same idea, but the horizontal plane is now hyperbolic. An interesting property of Nil geometry is that it is a three-dimensional geometry where volume of a ball of radius $R$ has $\Theta(R^4)$ growth, which suggests better embedding possibilities than $\mathbb{E}^3$, but worse than the exponentially-expanding geometries. ### 4 Our Embedding Algorithm Our goal is to find good quality embeddings of a connectome $(V, E)$ into some geometry $G$, that is, a map $m : V \rightarrow G$. As in the hyperbolic random graph model, we assume that our embedding has two parameters $R$ and $T$. The probability that an edge exists between $i$ and $j$ is again $p_1(d) = 1/(1 + \exp((d - R)/T))$, where $d$ is the distance between $m(i)$ and $m(j)$. We use MLE method to find the embedding, that is, we aim to maximize the likelihood $\prod_{1 \leq i < j \leq N} p(i, j)$, where $p(i, j) =$ $p_1(d_G(m(i), m(j)))$ in case if the edge between $i$ and $j$ exists, and $p(i, j) = 1 - p_1(d_G(m(i), m(j)))$ otherwise. Equivalently, we maximize the loglikelihood $\sum_{1 \leq i < j \leq N} \log p(i, j)$. Prior algorithms learning embeddings may be specifically tailored to the specific geometry. Furthermore, prior algorithms assume that $d_G$ is easy to compute, which is not the case for Solv. Therefore, a new embedding algorithm is necessary. As in Celinska-Kopczynska & Kopczynski (2022), our algorithm is based on a uniform grid in geometry $G$. Natural grids exist in all Thurston geometries of interest. While in the HRG model the network is mapped to a disk of radius $R$, here we map the network to the set $D$ of all grid points in $G$ which are in distance at most $d_R$ from some fixed origin. We choose $d_R$ so that the number of points inside $D$ is fixed; in most experiments we pick $M = 20000$ points (actually, there may be slightly more points due to ties). We compute the distance $d_G$ for every pair of points in $D$, thus obtaining a $\|D\| \times \|D\|$ array that can be used to find the distance between pairs of points quickly. In case of Solv, it turns out that the method to compute the Solv distances from Kopczynski & Celinska-Kopczynska (2020), while applicable to visualization, is not applicable to computing this table of distances due to long ranges. Therefore, for longer distances, we approximate by $d(a, b)$ as the smallest possible $d(a, a_1) + d(a_1, a_2) + \ldots + d(a_k, b)$, where intermediate points are also in $D$, and each pair of consecutive points is within the range of the exact method. Dijkstra’s algorithm is used to find the path $(a_i)$. Now, we use the Simulated Annealing (SA) method to learn the embedding. We start with an arbitrary embedding $m : V \rightarrow D$. Then, we perform the following for $i = 1, \ldots, N_S$. First, introduce a small change $m'$ to the current embedding $m$. Then, compute $L$, the loglikelihood of $m$, and $L'$, the loglikelihood of $m'$. If $L' > L$, always replace $m$ with $m'$. Otherwise, replace $m$ with $m'$ with probability $\exp((L' - L)/\exp(T))$, where the parameter $T$ depends on the iteration index. In SA, we start with very high temperature $T$ (to accept all changes and thus explore the full space of possible embeddings without getting stuck on local maxima) and then we proceed to lower and lower temperatures (not accepting changes which yield much worse embeddings, but still experimenting with crossing lower valleys), eventually accepting only the changes which improve the embedding. In our experiments, $T$ decreases linearly from 10 to -15. We consider local changes of two possible forms: move $m'(i)$ for a random $i$ to a random point in $D$, and move $m'(i)$ for a random $i$ to a random point in $D$ that is close (neighbor) to $m(i)$. We start with some initial values of $R$ and $T$. Occasionally during SA we find the values of $R$ and $T$ that best fit the current embedding, and we use the new values for the remaining iterations. Since finding the correct values takes time, we do it relatively rarely (every $\|V\|$ iterations with successful moves) and only once SA rejects most changes. In our experiments, we repeat this setup 30 times; in the following iterations, we start with the values of $R$ and $T$ of the best embedding found so far. 5 DATA, TESSELLATIONS, AND THE SETUP OF THE SIMULATION Our implementation uses the tessellations implemented in RogueViz (Kopczyński & Celinska-Kopczyńska, 2023) and is based on the existing implementation of SA for finding hyperbolic visualizations (Celinska & Kopczyński, 2017). For our experiments, we use the same set of publicly available connectomes as Allard & Serrano (2020). See Table 1. We run 30 iterations of SA to try to find the best $R$ and $T$, with $N_S = 10000 \cdot \|V\|$. In the literature, the quality of embeddings is usually evaluated using the greedy routing measures (in the network science community, Boguña et al., 2010) and MeanRank/mAP measures (in the machine learning community, Nickel & Kiela, 2017). Thus, we evaluate the quality of embeddings using the following five measures, from 0 (worst) to 1 (perfect). SC Greedy routing success rate. This is the standard measure used in the literature on network embeddings (Boguña et al., 2010). SC is the probability that, for random pair of vertices $(x, y) \in V^2$, the greedy routing algorithm starting at $x$ eventually successfully reaches the target $y$. This routing algorithm moves in the first step from $x$ to $x_1$, the neighbor of $x$ which is the closest to $y$ (that is, $d_G(m(x_1), m(y))$ is the smallest). If $x_1 \neq y$, we continue to $x_2$, the neighbor of $x_1$ which is the closest to $y$, and so on. URL: https://github.com/networkgeometry/navigable_brain_maps_data/ \| name \| node \| zone \| $\|V\|$ \| $\|E\|$ \| source \| \|--------------\|--------\|-----------------------\|-------\|-------\|------------------------------\| \| CEllegans \| cell \| nervous system \| 279 \| 2290 \| Varshey et al. (2011) \| \| Cat1 \| area \| cortex \| 65 \| 730 \| Scannell et al. (1995) \| \| Cat2 \| area \| cortex and thalamus \| 95 \| 1170 \| Scannell et al. (1999) \| \| Cat3 \| area \| cortex \| 52 \| 515 \| Seannell et al. (1999) \| \| Drosophila1 \| cell \| optic medulla \| 350 \| 2886 \| Shinomiy a et al. (2022) \| \| Drosophila2 \| cell \| optic medulla \| 1770 \| 8904 \| Shinomiy a et al. (2022) \| \| Macaque1 \| area \| cortex \| 94 \| 1515 \| Kaiser & Hilgetag (2006) \| \| Macaque2 \| area \| cortex \| 71 \| 438 \| Young (1993) \| \| Macaque3 \| area \| cortex \| 242 \| 3054 \| Harriger et al. (2012) \| \| Macaque4 \| area \| cortex \| 29 \| 322 \| Markov et al. (2013) \| \| Mouse2 \| cell \| retina \| 916 \| 77584 \| Helmstaedter et al. (2013) \| \| Mouse3 \| cell \| retina \| 1076 \| 90810 \| Helmstaedter et al. (2013) \| \| Human1 \| area \| cortex \| 493 \| 7773 \| Hagmann et al. (2008) \| \| Human2 \| area \| cortex \| 496 \| 8037 \| Hagmann et al. (2008) \| \| Human6 \| area \| whole brain \| 116 \| 1164 \| Gray Roncal et al. (2013) \| \| Human7 \| area \| whole brain \| 110 \| 965 \| Gray Roncal et al. (2013) \| \| Human8 \| area \| whole brain \| 246 \| 11060 \| Gray Roncal et al. (2013) \| \| Rat1 \| area \| nervous system \| 503 \| 23029 \| Bota & Swanson (2007) \| \| Rat2 \| area \| nervous system \| 502 \| 24655 \| Bota & Swanson (2007) \| \| Rat3 \| area \| nervous system \| 493 \| 25978 \| Bota & Swanson (2007) \| \| ZebraFinch2 \| cell \| basal-ganglia (Area X)\| 610 \| 15342 \| Dorkenwald et al. (2017) \| Table 1: Connectomes in our experiments. Based on Allard & Serrano (2020) IST Greedy routing stretch. Stretch is the expected ratio of the length of the route found in the greedy routing procedure, to the length of the shortest route, under the condition that greedy routing was successful. IST is the reciprocal of stretch. IMR For an edge $(x,y) \in E$, $\text{rank}(x,y)$ is 1 plus the number of vertices which are closer to $x$ than $y$, but not connected with an edge. MeanRank is the expected value of $(x,y)$ over all edges. We use IMR=1/MeanRank. MAP For an edge $(x,y) \in E$, $P(x,y)$ is the ratio of vertices in distance of at most $d_G(m(x),m(y))$ to $x$ which are connected with $x$. $AP(x)$ is the average of $P(x,y)$ for all $y$ connected with $x$, and MAP is the average of $AP(X)$ over all $X$ ($MAP \in [0,1]$). NLL Last but not least, loglikelihood (LL), which is directly maximized by us, as well as in many other embedding algorithms. For a given connectome $(V,E)$, the best theoretically possible loglikelihood is obtained when an edge between $x$ and $y$ occurs if and only iff the distance $d_G(m(x),m(y))$ is below some threshold value and thus edges can be predicted with full certainty based on the distance (loglikelihood = 0), and the worst possible is obtained when the distance gives no information on edges, and thus the probability of each edge is predicted as $\|E\|/(\|V\|^2)$ (loglikelihood = H). Normalized loglikelihood, NLL, is defined as 1-LL/H, and is again from 0 to 1. The computations of SC, STR, MR and MAP care on the order of nodes $y \in V$ by distance from $x \in V$. However, since we are using a discrete set $D$, it is possible that $d_G(m(x),m(y)) = d_G(m(x),m(z))$ for $y \neq z$. In the case of tie, we assume a random order of the tied nodes. During the statistical testing, where necessary, we apply Bonferroni correction for multiple testing. In our main experiment, we work with the 15 unique tessellations listed in Table 2. Most of our tessellations are hyperbolic. Subdivided(d) means that each cube of the honeycomb has been subdivided into $d \times d \times d$ subcubes, and the point $D$ consists of the vertices and centers of these subcubes, approximating the set of centers of cells of the Euclidean bitruncated cubic honeycomb. In case of Nil and Solv, we do not get actual cubes, so this construction is approximate. For technical reasons, distances are rounded to the nearest integer multiple of 1/20 absolute unit, except sphere, where the unit is 1/200 of absolute unit. Thus, diameter 316 for a continuous tessellation is 15.8 absolute units, and sphere has diameter (i.e., half the circumference) $\pi$. \| name \| dim \| geometry \| closed \| nodes \| diameter \| description of the set $D$ \| \|----------\|-----\|------------\|--------\|-------\|----------\|----------------------------------------------------------------\| \| $\mathbb{H}^2$ \| 2 \| hyperbolic \| F \| 20007 \| 304 \| bitruncated $\{7,3\}$ (Figure 1a) \| \| $\mathbb{H}^2\&$ \| 2 \| hyperbolic \| T \| 17980 \| 157 \| closed hyperbolic manifold \| \| tree \| 2 \| tree \| F \| 20002 \| 396 \| $\{3,\infty\}$ (Figure 1b) \| \| $\mathbb{E}^3$ \| 3 \| euclid \| F \| 20107 \| 1070 \| bitruncated cubic honeycomb \| \| $\mathbb{E}^3\&$ \| 3 \| euclid \| T \| 19683 \| 450 \| torus subdivided into $27 \times 27 \times 27$ cells \| \| $\mathbb{H}^3$ \| 3 \| hyperbolic \| F \| 21365 \| 201 \| $\{4,3,5\}$ hyperbolic honeycomb \| \| $\mathbb{H}^3\ast$ \| 3 \| hyperbolic \| F \| 20039 \| 146 \| $\{4,3,5\}$ subdivided(2) \| \| $\mathbb{H}^3\&$ \| 3 \| hyperbolic \| T \| 9620 \| 102 \| subdivided(2) closed hyperbolic manifold \| \| Nil \| 3 \| nil \| F \| 20009 \| 1000 \| $\mathbb{Z}^3$ grid \| \| Nil* \| 3 \| nil \| F \| 20208 \| 290 \| $\mathbb{Z}^3$ grid, subdivided(2) \| \| Twist \| 3 \| twist \| F \| 20138 \| 152 \| twisted $\{5,4\} \times \mathbb{Z}$ \| \| $\mathbb{H}^2 \times \mathbb{R}$ \| 3 \| product \| F \| 20049 \| 29 \| bitruncated $\{7,3\} \times \mathbb{Z}$ \| \| Solv \| 3 \| solv \| F \| 20017 \| 246 \| analog of Figure 1c \| \| Solv* \| 3 \| solv \| F \| 20000 \| 143 \| analog of Figure 1c, subdivided(2) \| \| $S^3$ \| 3 \| sphere \| T \| 21384 \| 628 \| 8-cell, each cell subdivided(11) \| Table 2: Details on tessellations used in our study; * denotes finer grids. ## 6 COMPARISON AT MAXIMUM PERFORMANCES We start with a naive comparison among the tessellations based on the best results that were obtained for each tessellation for each connectome. Due to space limitations, we have moved the ranking figures and descriptive statistics to Appendix D. \| connectome \| NLL \| MAP \| IMR \| SC \| IST \| \|--------------\|------\|------\|------\|------\|------\| \| Cat1 \| 5.47 \| 1.29 \| 10.28\| 0.40 \| 0.65 \| \| Cat2 \| 4.84 \| 3.75 \| 8.94 \| 1.94 \| 1.63 \| \| Cat3 \| 6.22 \| 1.35 \| 11.04\| 0.09 \| 0.66 \| \| CElegans \| 7.46 \| 6.05 \| 8.38 \| 8.89 \| 6.30 \| \| Drosophila1 \| 5.46 \| 10.15\| 8.34 \| 12.19\| 9.47 \| \| Drosophila2 \| 12.52\| 32.87\| 11.48\| 27.32\| 25.87\| \| Human1 \| 9.13 \| 5.95 \| 29.08\| 11.94\| 7.06 \| \| Human2 \| 9.19 \| 6.20 \| 28.38\| 11.62\| 7.00 \| \| Human6 \| 7.69 \| 3.52 \| 26.79\| 7.29 \| 4.53 \| \| Human7 \| 8.13 \| 3.45 \| 25.58\| 7.23 \| 4.34 \| \| Human8 \| 6.38 \| 1.72 \| 17.92\| 0.23 \| 0.74 \| \| Macaque1 \| 3.95 \| 3.93 \| 10.21\| 2.87 \| 2.21 \| \| Macaque2 \| 7.22 \| 3.02 \| 16.74\| 6.11 \| 3.30 \| \| Macaque3 \| 4.99 \| 7.52 \| 9.05 \| 6.88 \| 5.84 \| \| Macaque4 \| 9.44 \| 0.27 \| 4.51 \| 0.00 \| 0.00 \| \| Mouse2 \| 9.68 \| 7.54 \| 10.86\| 3.78 \| 4.94 \| \| Mouse3 \| 10.85\| 8.84 \| 10.98\| 3.58 \| 5.14 \| \| Rat1 \| 44.60\| 32.51\| 66.25\| 10.25\| 8.18 \| \| Rat2 \| 44.32\| 31.33\| 68.97\| 10.02\| 8.13 \| \| Rat3 \| 40.76\| 27.42\| 62.36\| 9.85 \| 7.96 \| \| ZebraFinch2 \| 14.83\| 19.70\| 7.06 \| 16.29\| 12.50\| Table 3: Coefficients of variations (CV, in %) for the max performance of the geometries According to Table 4, we notice that the assessment of the performance of the geometry may vary with respect to the quality measure; there are also differences across species. E.g., in general, trees perform poorly in terms of measures other than greedy success rate, and no matter the measure, they are always the best choice for Rat’s connectomes (nervous system). Results for Rat’s and Drosophila2’s connectomes are also characterized by the relatively high variation among species (Table 3). For other species, the best performances are actually similar with respect to a quality measure: the differences in best performance among geometries measured with MAP, greedy rate success, and stretch are small (in most of the cases values of CVs are under 10%); especially for Cat’s connectomes they tend to be negligible (values of CVs even under 1%). \| geometry \| NLL \| MAP \| IMR \| SC \| IST \| \|----------\|-----\|-----\|-----\|----\|-----\| \| $\mathbb{H}^2$ \| 19.05 \| 23.81 \| 14.29 \| 80.95 \| 33.33 \| \| $\mathbb{H}^2\&$ \| 0.00 \| 0.00 \| 0.00 \| 0.00 \| 95.24 \| \| tree \| 23.81 \| 23.81 \| 14.29 \| 80.95 \| 47.62 \| \| $E^3$ \| 19.05 \| 23.81 \| 23.81 \| 9.52 \| 14.29 \| \| $E^3\&$ \| 19.05 \| 28.57 \| 47.62 \| 0.00 \| 4.76 \| \| $\mathbb{H}^3$ \| 66.67 \| 61.90 \| 33.33 \| 52.38 \| 66.67 \| \| $\mathbb{H}^3\ast$ \| 66.67 \| 76.19 \| 38.10 \| 61.90 \| 76.19 \| \| $\mathbb{H}^3\&$ \| 9.52 \| 19.05 \| 28.57 \| 0.00 \| 4.76 \| \| Nil \| 19.05 \| 9.52 \| 33.33 \| 4.76 \| 0.00 \| \| Nil* \| 38.10 \| 38.10 \| 57.14 \| 0.00 \| 19.05 \| \| Twist \| 61.90 \| 57.14 \| 38.10 \| 57.14 \| 71.43 \| \| $\mathbb{H}^2 \times \mathbb{R}$ \| 66.67 \| 52.38 \| 52.38 \| 42.86 \| 71.43 \| \| Solv \| 52.38 \| 47.62 \| 33.33 \| 47.62 \| 42.86 \| \| Solv* \| 38.10 \| 28.57 \| 61.90 \| 9.52 \| 23.81 \| \| $S^3$ \| 0.00 \| 9.52 \| 23.81 \| 0.00 \| 0.00 \| Table 4: Percentages: how often occurred within top or bottom five ranks (at the max performance) The results suggest that $\mathbb{H}^2\&$ and $S^3$ seem to be inefficient choices: the first one never enters the top five ranks; both often occur within the bottom five ranks, at their best performance being even the worst choices no matter the quality measure. In contrast, $\mathbb{H}^3$ and $\mathbb{H}^2 \times \mathbb{R}$ perform very well – they rarely occur within bottom five ranks. Twist and Solv or Solv* never happen to be the worst choices, all of them perform relatively well. Interestingly, the usage of finer grids may not increase the chance of obtaining the best performance, no matter the quality measure: while for $\mathbb{H}^3\ast$ vs $\mathbb{H}^3$ and Solv* vs Solv we notice that it reduces the chance of occurring within the bottom five ranks, the best performances of non-fine grids still outperform them when it comes to the occurrences within the five top ranks. Contrary, finer grid for Nil significantly increases percentage of occurrences among five best ranks. When it comes to Euclidean geometry, the results are inconsistent. The best performances of $E^3$ and $E^3\&$ often occur among the bottom five ranks of the geometries. However, there are cases in which those geometries perform excellently, e.g., for Human connectomes. 7 COMPARISON OF PERFORMANCES BASED ON DISTRIBUTIONS Comparison at the maximum performance from the previous section gives us intuition about the optimistic scenarios – what the limits for our embeddings are. However, due to the nature of SA, the maximum values we obtained are still realizations of random variables; that is why a closer inspection including information about the distributions of the simulation results is needed. To this end, we will compare geometries using voting rules, in particular, we will be interested in finding Condorcet winners and losers. As Condorcet winner may not exist in the presence of ties, we will refer to its simple modification: Copeland rule [Maskin & Dasgupta (2004)]. We say “geometry A wins against geometry B” if the probability that (for a given quality measure) a randomly chosen simulation result obtained by geometry A is greater than a randomly chosen simulation result obtained by geometry B is greater than 0.5. If that probability is equal to 0.5, we say that “there is a tie”, and otherwise, “geometry A loses”. To compute the score for a given geometry, we add 1 for every winning scenario, 0 for every tie, and -1 for every losing scenario. The geometries with the highest and lowest scores become Copeland winners and losers, respectively (we allow for more than just one candidate in both cases). Condorcet winners (as well as the winners based on the Copeland method) have interpretations – those are the candidates that beat the most of other candidates in pairwise contests. In our case, we could perceive them as the best options for embeddings. Based on the data in Table 5, we cannot name one universal winner. While it seems that $\mathbb{H}^3$ is a sound choice, we also notice that Solv and Twist are worthy attention. Interestingly, for Human connectomes, $E^3$ outperforms other geometries. See Appendix C for weighted directed networks constructed upon the voting rules. \| connectome \| NLL \| MAP \| IMR \| SC \| IST \| NLL \| MAP \| IMR \| SC \| IST \| \|-------------\|--------------\|--------------\|--------------\|--------------\|--------------\|--------------\|--------------\|--------------\|--------------\|--------------\| \| Cat1 \| Solv* \| H^3* \| Solv* \| H^3* \| Solv* \| H^2& \| tree \| tree \| H^3& \| tree \| \| Cat2 \| H^3* \| H^3* \| H^2 × R \| Twist \| H^2 × R \| H^2& \| S^3 \| tree \| H^3& \| tree \| \| Cat3 \| Solv* \| Solv* \| H^3& \| Nil* \| H^3& \| H^2& \| tree \| tree \| H^2& \| tree \| \| CElegans \| H^3* \| Nil \| Nil \| Nil \| H^2& \| H^2& \| tree \| tree \| H^2& \| tree \| \| Drosophila1 \| Twist \| H^3 \| H^3& \| H^3 \| H^3& \| H^2& \| S^3 \| tree \| H^2& \| tree \| \| Drosophila2 \| H^3 \| H^3 \| H^3 \| H^3 \| H^3 \| H^2& \| S^3 \| S^3 \| H^3& \| S^3 \| \| Human1 \| E^3 \| S^3 \| S^3 \| H^3 \| S^3 \| tree \| tree \| tree \| H^2& \| tree \| \| Human2 \| E^3 \| S^3 \| S^3 \| H^3 \| S^3 \| tree \| tree \| tree \| H^2& \| tree \| \| Human6 \| E^3 \| E^3 \| E^3 \| E^3 \| E^3 \| tree \| tree \| tree \| H^2& \| tree \| \| Human7 \| E^3 \| E^3 \| Solv \| E^3 \| tree \| tree \| tree \| tree \| H^2& \| tree \| \| Human8 \| H^3* \| H^3 \| H^2 \| E^3 \| tree \| tree \| tree \| tree \| H^2& \| tree \| \| Macaque1 \| Solv \| Solv \| Solv \| H^3* \| Solv \| S^3 \| S^3 \| tree \| H^3& \| tree \| \| Macaque2 \| Nil \| Nil \| Nil* \| H^2 \| Nil* \| tree \| tree \| tree \| H^2& \| tree \| \| Macaque3 \| H^3* \| H^3 \| H^2 × R \| H^2 \| H^2 × R \| H^2& \| S^3 \| tree \| H^2& \| tree \| \| Macaque4 \| E^3& \| E^3& \| Twist \| E^3& \| tree \| tree \| tree \| tree \| E^3 \| tree \| \| Mouse2 \| Twist \| H^3 \| H^2 × R \| H^2 \| H^2 × R \| S^3 \| S^3 \| H^2& \| S^3 \| H^2& \| \| Mouse3 \| Twist \| H^3 \| H^2 × R \| H^2 \| H^2 × R \| S^3 \| S^3 \| S^3 \| H^2& \| S^3 \| \| Rat1 \| tree \| tree \| H^3 \| tree \| H^3 \| S^3 \| S^3 \| S^3 \| S^3 \| S^3 \| \| Rat2 \| tree \| tree \| H^3 \| tree \| H^3 \| S^3 \| S^3 \| S^3 \| S^3 \| S^3 \| \| Rat3 \| tree \| tree \| H^3 \| tree \| H^3 \| S^3 \| S^3 \| S^3 \| S^3 \| S^3 \| \| ZebraFinch2 \| Solv \| H^3 \| Solv \| H^3 \| Solv \| S^3 \| S^3 \| Solv \| S^3 \| S^3 \| Table 5: Voting rules: Copeland winners and losers. 8 ROBUSTNESS CHECKS AND THREATS TO VALIDITY Ideally, there exists optimal embedding of $(V, E)$ into the whole geometry $G$, where $m_{\text{opt}} : V \rightarrow G$, and some values of $R$ and $T$ are used. Unfortunately, the embedding $m$ found by SA might be worse than $m_{\text{opt}}$ due to the following issues. See Appendix B for a detailed analysis. - The radius $d_R$ is too small, making $m_{\text{opt}}$ simply not fit, - The grid used is too coarse, hence the necessity of making $m(i)$ the grid point to closest to $m_{\text{opt}}(i)$, and thus reducing the loglikelihood, - The number of iterations of SA, $N_S$, is too small – while SA is theoretically guaranteed to find the optimal embedding for given $R$ and $T$ with high probability as $N_S$ tends to infinity, in practice, we are constrained by time limits, - The values of the parameters $R$ and $T$ have not been chosen correctly. Our results vs previous approaches To see how good is SA at obtaining good embeddings, we can compare it against the previously existing embedders. While we are the first to study Nil and Solv embeddings, there is a vast number of prior works on $H^2$ and $H^3$ embeddings. We have compared our results on the CElegans, Drosophila1, Human1 and Mouse3 connectomes. We use the results of comparison in Anonymous (2023). For $H^2$, we have compared against the BFKL embedder (Blasius et al., 2016), Mercator (García-Pérez et al., 2019) (fast and full version), 2D Poincaré embeddings (Nickel & Kiela, 2017) and 2D Lorentz embeddings (Nickel & Kiela, 2018). Each of the competing algorithms has been run five times, found the best result of these 25 runs, and compared to our results. We have also performed a similar analysis for $H^3$, against 3D Poincaré embeddings (BFKL and Mercator work only in $H^2$). Table 6 list our results for mAP and success rate (see Appendix E for other measures). In most cases, our result turned out to give better result in all 30 runs, and in almost all cases, we have received better results in most of the runs. We have not managed to beat Poincaré 3D embeddings on greedy success ratio and greedy stretch measures for Mouse3 and CElegans. Furthermore, our embeddings use smaller radius (7.7 for $H^2$, 3.7 for $H^3$), and use less time than Lorentz or Poincaré embeddings (about 220 seconds per run on Mouse3 in $H^3$). Smaller radius means that our em- \| connectome \| dim \| mAP \| method \| rad \| time \| ours \| better \| \|------------\|-----\|-----\|-----------------\|-----\|------\|------\|--------\| \| celegans \| 2 \| 0.500 \| Poincaré \| 7.2 \| 278 \| 0.540 \| 30 \| \| celegans \| 3 \| 0.583 \| Poincaré \| 10.1\| 274 \| 0.584 \| 21 \| \| drosophila1\| 2 \| 0.425 \| Mercator (full)\| 23.6\| 14 \| 0.483 \| 30 \| \| drosophila1\| 3 \| 0.488 \| Poincaré \| 11.4\| 365 \| 0.512 \| 30 \| \| human1 \| 2 \| 0.651 \| Lorentz \| 10.8\| 1085 \| 0.675 \| 30 \| \| human1 \| 3 \| 0.722 \| Poincaré \| 9.4 \| 827 \| 0.799 \| 30 \| \| mouse3 \| 2 \| 0.585 \| Mercator (full)\| 29.9\| 117 \| 0.612 \| 30 \| \| mouse3 \| 3 \| 0.654 \| Poincaré \| 12.2\| 9207 \| 0.655 \| 18 \| \| connectome \| dim \| success \| method \| rad \| time \| ours \| better \| \|------------\|-----\|---------\|-----------------\|-----\|------\|------\|--------\| \| celegans \| 2 \| 0.903 \| Poincaré \| 7.2 \| 267 \| 0.931\| 27 \| \| celegans \| 3 \| 0.958 \| Poincaré \| 10.1\| 274 \| 0.930\| 0 \| \| drosophila1\| 2 \| 0.769 \| Mercator (full)\| 23.6\| 14 \| 0.847\| 30 \| \| drosophila1\| 3 \| 0.844 \| Poincaré \| 11.4\| 365 \| 0.843\| 13 \| \| human1 \| 2 \| 0.889 \| Poincaré \| 12.2\| 1185 \| 0.929\| 21 \| \| human1 \| 3 \| 0.926 \| Poincaré \| 9.5 \| 835 \| 0.958\| 24 \| \| mouse3 \| 2 \| 0.962 \| Mercator (full)\| 34.5\| 74 \| 0.967\| 30 \| \| mouse3 \| 3 \| 0.971 \| Poincaré \| 12.2\| 8679 \| 0.952\| 0 \| Table 6: Our embeddings versus state-of-the-art. For each connectome and dimension, we list the best prior method and its result, the radius of the embedding, time elapsed in seconds, the best result of our method, and how many times (out of 30) our result was better. Beddings avoid numerical precision issues that tend to be a serious issue in hyperbolic embeddings (Blasius et al., 2018; Sala et al., 2018; Celinska-Kopczynska & Kopczynski, 2022), are better able to fully use both the large-scale (tree-like) and smaller-scale (Euclidean-like) nature of hyperbolic geometry (while large radius embeddings tend to be tree-like), and making them more applicable for visualization (in large-radius visualizations, less nodes are visible). 9 Conclusions In this paper, we presented an experimental analysis of embeddings of 21 connectomes to various geometries (both three- and two-dimensional). To our best knowledge, we are the first to compare embeddings to all Thurston geometries. We provided a new embedding method based on Simulated Annealing (SA) that outperforms previous methods. Although earlier studies suggested one universal winner geometry (usually pointing at $\mathbb{H}^2$), our results showed that if we allow for the third dimension, the universal winner ceases to exist. Especially, $\mathbb{H}^2$ embeddings tend to be worse than (non-Euclidean) 3D geometries, even if our $\mathbb{H}^2$ embeddings are actually good – better than Blasius et al. (2016); García-Pérez et al. (2019); Nickel & Kiela (2017, 2018). If we were to suggest a set of geometries that are worth attention while modelling connectomes, we would name $\mathbb{H}^3$, Solv, Twist, and $\mathbb{H}^2 \times \mathbb{R}$. Surprisingly, for Human connectomes, $\mathbb{P}^3$ is a suitable choice. There might be a correlation between the zone of the connectome and the best choice for the embedding, e.g., trees model nervous systems well. 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B4nhr6OJWI	The subnetwork can only be transferred without varying the architecture, right? E.g. there cannot be a subnetwork which is only made of two layers, and I plug these two layers into my model. Is this right? Or, if I understood wrong, how is the subnetwork transferred when the architecture changes?	Instilling Inductive Biases with Subnetworks Anonymous authors Paper under double-blind review Abstract Despite the recent success of artificial neural networks on a variety of tasks, we have little knowledge or control over the exact solutions these models implement. Instilling inductive biases—preferences for some solutions over others—into these models is one promising path toward understanding and controlling their behavior. Much work has been done to study the inherent inductive biases of models and instill different inductive biases through hand-designed architectures or carefully curated training regimens. In this work, we explore a more mechanistic approach: Subtask Induction. Our method discovers a functional subnetwork that implements a particular subtask within a trained model and uses it to instill inductive biases towards solutions utilizing that subtask. Subtask Induction is flexible and efficient, and we demonstrate its effectiveness with two experiments. First, we show that Subtask Induction significantly reduces the amount of training data required for a model to adopt a specific, generalizable solution to a modular arithmetic task. Second, we demonstrate that Subtask Induction successfully induces a human-like shape bias while increasing data efficiency for convolutional and transformer-based image classification models. Our code is available at the following anonymous repository link. 1 Introduction Neural networks have come to dominate most fields of machine learning (He et al., 2015a; Brown et al., 2020; Radford et al., 2022; Mildenhall et al., 2020), but we have little control over the algorithms these models learn during training. To address this problem, much work has been done to instill inductive biases — preferences for some solutions over others — into neural networks. Studying inductive biases is interesting for at least two reasons: (1) From a practical standpoint, inductive biases could be used to discourage models from adopting solutions that leverage incorrect or biased information to make decisions (e.g. sorting job candidates on the basis of protected characteristics, or exploiting heuristics that do not generalize to a larger domain). (2) From a theoretical standpoint, human learning is thought to be mediated by a variety of inductive biases, which enable better sample efficiency and better generalization capabilities (Lake et al., 2017). Contemporary deep learning systems demonstrate weaknesses related to both of the above: they require massive datasets and computing power to train (Touvron et al., 2023; Radford et al., 2021; Dosovitskiy et al., 2020) and can often be sensitive to small perturbations of inputs (Szegedy et al., 2014; Geirhos et al., 2019; Hermann & Kornblith, 2019). Thus, a better understanding of inductive biases and how to induce them could pave the way toward improving such systems. Current approaches to instilling inductive biases in models require either (1) limiting model expressivity through handcrafted architectural constraints, (2) metalearning over a large dataset (Griffiths et al., 2019), or (3) training or fine-tuning on augmented datasets, which may (Andreas, 2020) or may not (Huang et al., 2020; Khashabi et al., 2020) work. In contrast, we propose Subtask Induction, a method of instilling inductive biases by (1) localizing a subnetwork within a trained neural network that performs a specific subtask within an overall model, and (2) initializing another network with only these subnetwork weights, leaving the remaining weights randomly initialized. This instills a specific computation into a model from the outset, which provides a soft inductive bias towards solutions that leverage that subtask. We demonstrate that Subtask Induction is effective on a range of tasks and model architectures. While our results are an early proof of concept, they open a door for future research on more mechanistic approaches to instilling inductive biases. This approach is more flexible than architectural design, simpler and cheaper to train than metalearning-based approaches, and more reliable than data augmentation based approaches. Figure 1: Subtask Induction localizes a subnetwork that implements a certain subtask in a trained neural network and transfers it to a randomly initialized model, thereby instilling an inductive bias towards solutions utilizing the specific subtask. The figure above illustrates the 3 stages of Subtask Induction in our experiments: we first train for a binary weight-level mask representing the subnetwork for a specific subtask through subnetwork discovery, then perform subnetwork transfer by copying the subnetwork weights to a newly initialized model and keep it frozen while optimizing the re-initialized weights. We demonstrate through two experiments that transferring subnetworks effectively and reliably instills desired inductive biases. Our contributions are as follows: 1. We introduce Subtask Induction, a novel method that leverages recent advancements in interpretability to instill inductive biases. 2. We demonstrate the effectiveness of Subtask Induction on an arithmetic task, showing that Subtask Induction provides a preference for learning a particular solution with minimal training signal and significantly reduces the amount of data required for generalization. 3. We generate and release Mean-pooled ImageNet, a variant of the ImageNet dataset (Rusakovsky et al., 2015) where the pixel values of each image are mean-pooled within semantic segments of the image, effectively erasing local texture while retaining global shape. 4. We apply Subtask Induction to image classification on both ResNet18 and ViT models, instilling a human-like inductive bias towards classifying based on shape information, rather than texture information. 2 RELATED WORK Inductive Bias from Architectural Constraints Imposing architectural constraints is the standard approach for instilling inductive biases in artificial neural networks. For example, convolutional neural networks (LeCun et al., 1989) and recurrent neural networks (Hochreiter & Schmidhuber, 1997; Cho et al., 2014) are both designed to exploit useful properties of their input data (i.e. shift invariance and sequential structure). Neurosymbolic approaches give even stronger inductive biases by integrating neural networks with human-designed computations, thereby limiting the kinds of solutions a model can learn (Andreas et al., 2016; Feinman & Lake, 2020; Ruis & Lake, 2022). These approaches typically perform very well in the domain that they were crafted for, but require extensive knowledge about the domain. Inductive Bias from Data Augmentation and Meta-learning Data augmentation procedures have also been proposed to provide inductive biases. This approach has been validated in both vision (Geirhos et al., 2019; Hermann & Kornblith, 2019) and language (Andreas, 2020). However, the reliability of data augmentation for instilling inductive biases has been called into question (Jha et al., 2020; Huang et al., 2020; Khashabi et al., 2020). Relatedly, some work has explored a meta-learning approach toward instilling inductive biases (Griffiths et al., 2019; McCoy et al., 2019; Kumar et al., 2022; Lake, 2019). However, this approach requires meta-learning on a large dataset comprised of multiple related tasks, and the resulting model is still not guaranteed to adopt the desired inductive bias (Kumar et al., 2020). Mechanistic Interpretability Our work is inspired by recent advances in mechanistic interpretability – a burgeoning field whose goal is to reverse engineer the algorithms that neural networks learn. Several recent works have succeeded at this goal for both toy models (Olsson et al., 2022; Nanda et al., 2023; Chughtai et al., 2023) and more realistic models (Wang et al., 2022; Hanna et al., 2023; Merullo et al., 2023). Most closely related to the present article is recent work analyzing neural networks through the lens of subnetworks (Csordás et al., 2021; Lepori et al., 2023; Casper et al., 2022; Voss et al., 2021; Hamblin et al., 2022). This line of research has shown that trained neural networks are often composed of modular subnetworks, each of which implements specific subtasks. 3 LOCALIZING AND TRANSFERRING SUBNETWORKS Subtask Induction builds upon recent work in neural network interpretability and investigates the hypothesis that one can transfer subtasks from one model to another by transferring a subnetwork encoding that information, thereby instilling an inductive bias. If this hypothesis is true, transferring a certain subtask should bias a model towards learning solutions that use that subtask. In addition, we would also expect a greater sample efficiency and faster convergence if the inductive bias turns out to be helpful to the task. This section formalizes Subtask Induction as a two-stage process. We first localize a subnetwork within a trained model through subnetwork discovery (Section 3.1), which seeks to isolate a functional subtask captured by the original model. We then transfer the subnetwork (Section 3.2) to randomly initialized neural networks and train with a different objective to test if the transferred subtask provides significant inductive biases for solutions that rely on that subtask over those that do not. We provide a graphical illustration of our method in Figure 1. Our implementation is integrated with the Python package NeuroSurgeon (Lepori et al., 2023). 3.1 LOCALIZING SUBNETWORKS Given a trained neural network $M_\theta$ with parameters $\theta$, we define a subnetwork as a model where a binary mask $\gamma \in \{0, 1\}^{\|\theta\|}$ is applied over the original model parameters, such that $\theta_{\text{sub}} = \theta \odot \gamma$. In other words, a subnetwork is a variant of the original neural network where a subset of the parameters is kept the same, and the rest are set to zero. We say that a subnetwork implements a subtask if $M_{\theta_{\text{sub}}}$ produces the expected outcomes of a more basic task that potentially contributes to solving the original task. E.g. a subtask for an image classification model could be a curve detector, and a subtask in a language model could be a syntax parser. If we successfully find a subnetwork that achieves a subtask, we say that such a subtask is implemented within a model. Optimizing for a binary mask is practically intractable due to the $2^{\|\theta\|}$ possible combinations. We thus apply continuous sparsification (Savarese et al., 2020) to train a continuous approximate of the binary mask that is discretized at test time. Continuous sparsification re-parameterizes a binary mask with element-wise sigmoid functions and schedules a scale coefficient $\beta$ that increases through training to “anneal” a soft mask to a hard one. Our implementation of this algorithm is described in more details in Appendix A. In order to find a subnetwork for a particular subtask, we train the mask by defining a new training objective that captures the subtask and perform gradient descent to localize a set of parameters that minimizes loss on the subtask. We name this process subnetwork discovery. 3.2 TRANSFERRING SUBNETWORKS After obtaining a subnetwork with mask $\gamma_{\text{sub}}$, we initiate a subnetwork transfer by transferring the parameters within the subnetwork (i.e. parameters where $\gamma_{\text{sub}} = 1$) to a randomly initialized copy of the model. We then train the network on the new training objective. During training, we only optimize the randomly initialized parameters and keep the subnetwork frozen. Let $L_{\text{new}}$ denote the optimization objective of the new task, $\theta_{\text{original}}$ denote pretrained parameters, and $\theta_{\text{new}}$ denote the re-initialized parameters. The training objective then becomes $$\arg\min_{\theta_{\text{new}} \in \mathbb{R}^{\|\theta\|}} \left( L_{\text{new}} \left( M_{\gamma_{\text{sub}} \odot \theta_{\text{original}}} + (1 - \gamma_{\text{sub}}) \odot \theta_{\text{new}} \right) \right).$$ (1) Figure 2: Graphical illustration of our experimental setup. Tasks $T_1$ and $T_2$ are setup to be combinations of three subtasks, $S_1$, $S_2$, and $S_3$, where $S_1$ is shared between the two. We train a model on $T_1$, then perform Subtask Induction by localizing and transferring the shared subtask $S_1$ to instill inductive biases towards a new model trained on $T_2$. We find that transferring the subnetwork improves the model’s ability to learn $T_2$ significantly. 4 Arithmetic Experiments To verify the effectiveness of Subtask Induction, we train neural networks on an arithmetic dataset, where subtasks can be easily defined and tested. For this, we use tasks in the form of those studied by Power et al. (2022). In Power et al.’s experiments, an overparameterized neural network is trained on a synthetic dataset of some computation $a \circ b = c$, where $a$, $b$, and $c$ are discrete symbols and $\circ$ denotes an arithmetic operation with two arguments (for example, $a + b$ or $a^2 + ab$). We isolate a subnetwork implementing some particular subtask of the original training task. We then transfer this subnetwork to a new task that should benefit from having access to this subtask. 4.1 Dataset We algorithmically generate datasets by defining a computation $\circ$ and sampling two integers $a$ and $b$ from a chosen range $[0, \text{max}]$. We then formulate the expression into a sequence of four tokens $<a> <b> <sep> <c>$ where each element in a pair of brackets indicates a token. Here “sep” represents the special separator token, and $c$ is the expected output of the computation $a \circ b$. This formulation allows us to train a decoder-only transformer on the sequence with a standard next-token prediction objective. In all of the following experiments, we fix $\text{max} = 1000$. We tokenize each number into a discrete symbolic token, rather than an integer or floating point representation, and each token embedding is learned individually. Since each number is represented by a discrete token, we constrain the dataset such that each of the possible tokens must appear at least once in the training set. Following prior work (Power et al., 2022; Nanda et al., 2023), we take modulo of the output by a prime number $p$ to restrict the output space (i.e. the operation is always in the form “$a \star b \ (\text{mod } p)$”, where the modulo is taken after the two-place operator “$\star$”). In all our experiments we fix $p = 7$. 4.2 Experimental Setup We generate training data for two tasks, $T_1 := a + ab \ (\text{mod } p)$ and $T_2 := a^2 + ab \ (\text{mod } p)$. Note that the two tasks can be described as the combination of results from subtasks $S_1 := ab \ (\text{mod } p)$, $S_2 := a \ (\text{mod } p)$, $S_3 := a^2 \ (\text{mod } p)$, and $T_1$ and $T_2$ share the computation node $S_1$. We perform Subtask Induction from $T_1$ to $T_2$ by transferring $S_1$. Figure 2 demonstrates this procedure graphically. The experiment follows three steps: 1. Train a neural network on $T_1$, where it is expected to solve an arithmetic task. 2. Performing subnetwork discovery to localize a subnetwork that solves $S_1$. 3. Transferring the subnetwork to $T_2$ and test for inductive bias towards solutions utilizing $S_1$. In step 1, we generate training data for the computation $T_1$ by randomly sampling 20% of the total $1000^2$ combinations, which gives us 200,000 rows of training data. We use another independently generated set of 20,000 samples for test data. We train a decoder-only transformer on this dataset with a standard next token prediction objective, and report accuracy/loss on the last token, as the last token represents the solution to the problem. This task $a + ab \pmod{p}$ can be intuitively broken down into constituent subroutines: computing $a \pmod{p}$, computing $ab \pmod{p}$, and combining the results into the final output. We hypothesize that models also implicitly decompose the task in this manner. To probe for a subroutine responsible for the computation $ab \pmod{p}$, we generate 50,000 samples of the computation $ab \pmod{p}$, and perform subnetwork discovery. This step gives us a binary mask $\gamma_{\text{sub}}$, and the subnetwork $M\theta \circ \gamma_{\text{sub}}$ should perform the computation $ab \pmod{p}$ instead of the original training objective $a + ab \pmod{p}$. We then investigate if the subnetwork provides an inductive bias toward a solution utilizing the subtask. We intentionally make the training objective $T_2$ appear ambiguous by supplying the model a minimal dataset of 1000 samples of the format $X^{n=1000} = <i>, <i>, <sep>, <i \circ i>$, where the two inputs are identical. This ensures that each discrete token has appeared at least once while leaving the training task ambiguous. Concretely, the objective would be ambiguous between computations $2a^2 \pmod{p}$, $2b^2 \pmod{p}$, and $a^2 + b^2 \pmod{p}$. In addition to the minimal dataset above, we manipulate the number of disambiguation samples present in the training set, i.e., training examples in which the two inputs are no longer constrained to be identical. These are randomly sampled from the input space of $\{0, 1, 2, ..., 999\}^2$, and provides information to disambiguate the correct computation $T_2$ from other possible computations. We vary the number of disambiguation samples to quantify the inductive bias of neural networks. With a strong inductive bias towards the correct rule, a small number of disambiguating examples would be enough to disambiguate the task.\footnote{Ideally, with a sufficiently strong inductive bias, no unambiguous examples would be required, though in practice we do not obtain such a strong inductive bias.} If Subtask Induction is effective, it should enable the model to achieve higher accuracies with fewer disambiguating examples. The evaluation set and the test set always contain 1000 data points, each of which is generated independently from a random sample over all possible combinations. We experiment with several GPT2 configurations, varying the number of layers from 2 to 12. We vary the number of disambiguation samples from 10 to $10^4$ (0.001% to 1% of total possible combinations, respectively) with constant intervals for a total of 16 different sample sizes on each model. After transferring subnetwork weights, we train each model for 100 epochs and save the model with best accuracy on the evaluation set, and then report the accuracy achieved on the test set (See Appendix B.1 for model configuration and training details). ### 4.3 Results If Subtask Induction successfully instills an inductive bias, we would expect our model to achieve higher test accuracy with less training data, relative to a randomly initialized model. We find this to be the case: as shown in Figure 3, models initialized with subnetworks with as few as 3.2% of total parameters (see Table 2) representing subtask $S_1$ gain significant inductive bias towards the solution utilizing $S_1$. This is evidenced by the significantly higher sample efficiency: all model configurations trained with Subtask Induction achieve near-perfect accuracy with as few as 1000 disambiguation training samples (0.1% of total possible combinations). As a comparison, models trained from scratch only average 50.6% test accuracy when trained on the same data and never reach perfect generalization accuracy within the range of training samples tested (0 to $10^4$). We set up the following controls to validate the effectiveness of Subtask Induction: 1. Comparison with full model transfer: Since the subnetwork captures $S_1$, the only shared computation between $T_1$ and $T_2$, we hypothesize that it carries all the “helpful” information a neural network trained on $T_1$ could provide, and thus expect Subtask Induction to have comparable performance as transferring the entire model trained on $T_1$. This turns out to be the case: Across sample sizes and model configurations, transferring subnetworks of around 3% to 7% parameters achieves at least as good generalization accuracy and sample efficiency as transferring the entire model. Figure 3: Test accuracy vs number of disambiguation training samples. Left: average over all model configurations (GPT-2, 2 to 12 layers), right: One configuration (GPT-2, 12 layers) with standard deviation across 5 runs. The horizontal axis is in log scale. Trials shown in Figure include Subtask Induction compared against 3 controls: randomly initialized model, transferring randomly sampled subnetworks, and transferring the entire model trained on $T_1$. Despite transferring less than 10% of all parameters, Subtask Induction yields comparable and often higher accuracy compared to transferring the entire model and boosts data efficiency significantly compared to random controls. 2. Comparison with randomly sampled subnetwork: Intuitively, transferring a subset of parameters from a model trained on $T_1$ could provide benefits for training on $T_2$ purely due to the similarity of the two tasks. We control for this by sampling a random subnetwork containing the same number of parameters as a subnetwork localized through subnetwork discovery and transferring the sampled subnetwork. This gives uniformly worse results: while still better than random initialization, a randomly sampled subnetwork requires on average around 6 times as much data in order to reach perfect generalization accuracy. In addition to the results in Figure 3, all of the patterns reported above hold in each of the individual model configurations as well. We also experiment with a range of different arithmetic tasks (e.g., $a^3 + ab$) and subnetworks. We report these extended results and additional analysis in Appendix B. 5 VISION EXPERIMENTS In this section we apply Subtask Induction on image classification tasks, a highly complex domain for which no complete algorithmic solutions are known. While contemporary deep neural networks are able to meet or even exceed human-level accuracy on image classification (He et al., 2015b; Dosovitskiy et al., 2020), they often rely on a very different set of cues than humans do, thereby limiting their robustness and generalization capabilities (Dodge & Karam, 2017). Prominently, while human learners overwhelmingly rely on shape information (Landau et al., 1988), convolutional neural networks are primarily reliant on local texture (Gerhos et al., 2019). We show that by localizing and transferring subnetworks within pretrained models, it is possible to instill a more human-like bias towards shape information. 5.1 DATASET: MEAN-POOLED IMAGENET In order to quantify the shape and texture biases of image classification models, we introduce Mean-pooled ImageNet, a variant of ImageNet where local, high-frequency texture information of images is removed while maintaining global shape information. We use Segment Anything (Kirillov et al., 2023). To ensure as fair a comparison as possible, the randomly sampled subnetwork is sampled over the same layers as the subnetwork (i.e., all the attention layers and feed-forward MLPs, but not the embedding layers), and the number of parameters sampled at each individual layer is controlled to be the same as the trained subnetwork on the respective layer. This eliminates possibilities that simply sampling the right number of parameters per layer gives equivalent results. Figure 4: Qualitative evaluation of Mean-pooled ImageNet. Semantic segmentation followed by mean pooling retains most shape information in a naturalistic way while erasing local texture. to partition the image into semantic segments. After obtaining an image embedding, we query each image with a $16 \times 16$ grid of points to obtain semantic segments corresponding to each segment. To ensure that small but semantically relevant patches are not missed by the initial sampling, we further query on a $2 \times 2$ crop of the image and collect masks returned by the query. We then filter out masks that are smaller than 100 pixels and combine all masks for a non overlapping set of segments covering the entire image. Lastly, we replace each pixel value in the image by the mean pixel value of the segment it belongs to. We provide a few samples of Mean-pooled ImageNet for qualitative evaluation in Figure 4 and invite the reader to guess their corresponding classes. Mean-pooled ImageNet employs a naturalistic augmentation strategy as it does not shift the overall color scheme of images or intentionally occlude any information apart from local texture. For humans, this augmentation is unlikely to dramatically raise the difficulty of the task or impact a classification decision. However, we find this dataset to be challenging for image classification models. While ResNet18 reaches 95.4% accuracy when fine-tuned on 16-class ImageNet, its accuracy on the mean-pooled counterpart is only 36.8%. ViT performs much better on this dataset, but still only achieves 57.3% accuracy. 5.2 Experimental Setup Similar to the experiments on arithmetic tasks, we instill different inductive biases into image classification models by localizing a subnetwork within a pretrained image classification model using Mean-pooled ImageNet, and then transferring the subnetwork into a re-initialized model. We perform all our experiments on 16-class ImageNet (Gerhos et al., 2019) and its mean-pooled counterpart. Each class label is aggregated from one or multiple ImageNet classes. The dataset contains a total of 213k images from 16 common classes in the train split of ImageNet. As the dataset is unbalanced between classes, we additionally create two smaller but class-balanced subsets: a total of 13.9k randomly downsampled mean-pooled images are used to discover the subnetwork within a pretrained model, and an additional 1.54k images are used for evaluation and model selection. We de-duplicate our evaluation dataset with our training datasets and report accuracy on the validation split of ImageNet, which is not used for either training or model selection. We experiment with two model architectures: ResNet18 (He et al., 2015a) and ViT-base (Dosovitskiy et al., 2020). We perform subnetwork discovery on both models to localize a subnetwork that maximizes accuracy on mean-pooled images. Lastly, we transfer the subnetwork weights and re-train the model on 16-class ImageNet. As baselines, we compare against pretrained models that are finetuned on a data mixture of 213K 16-class ImageNet images and 15.4K mean-pooled images. This approach mimics the data augmentation approach to instilling inductive biases that have been explored in prior work (Andreas, 2020). We also compare against training these models from scratch and fine-tuning only the classification head of base models, which quantifies inherent inductive bias of the architecture and the performance base models, respectively. 5.3 Results Pretrained Models Capture Shape Subtasks For both ResNet18 and ViT, we are able to discover subnetworks achieving significantly higher accuracy on mean-pooled images than the original model, suggesting that shape-reliant subtasks exist within the original model. Within ResNet18, we find a subnetwork with 14.9% of the parameters achieving 73.8% classification accuracies on Figure 5: Training dynamics Comparison. Subtask Induction and training from scratch for ResNet18 and ViT. Left: evaluation accuracy on original ImageNet images, right: evaluation accuracy on Mean-Pooled Imagenet. Models initialized with Subtask Induction reach higher accuracies with fewer optimization steps and retain a much higher accuracy on Mean-pooled ImageNet. Table 1: Test accuracy of Subtask Induction compared with other training strategies. We note that: (1) Subtask Induction instills a strong shape bias (18.8% performance increase on Mean-pooled ImageNet for ResNet18, 8.7% for ViT) despite the re-initialized network never being directly trained on mean-pooled images, while data augmentation does not provide such bias, (2) Subtask Induction increases sample efficiency, as both ResNet and ViT reach much higher accuracy compared to from-scratch models when trained on 16-class ImageNet, (3) Subtask Induction gives much more robust models as seen on the Cue Conflict results, where our ResNet18 outperforms pretrained ResNet18 and reaches levels comparable to pretrained ViT. While ViT trained with Subtask Induction is not as strong, it still performs significantly better than data mixture and from-scratch baselines and has the best performance on mean-pooled images. \| Model \| Train Set Size \| ImageNet Original \| Pooled \| Cue Conflict Accuracy \| Robustness \| \|------------------------------\|----------------\|-------------------\|--------\|-----------------------\|------------\| \| RN18 + Subtask Induction \| 213k \| 80.7% \| 55.6% \| 27.1% \| 77.4% \| \| RN18 from scratch \| 213k \| 68.9% \| 24.7 % \| 15.9% \| 75.3% \| \| RN18 + Data Mixture \| 1.28M + 15.4k¹ \| 91.9% \| 38.3% \| 18.9% \| 55.3% \| \| RN18 Pretrained \| 1.28M \| 95.4% \| 36.8% \| 18.9% \| 56.0% \| \| ViT + Subtask Induction \| 213k \| 83.4% \| 66.0% \| 20.0% \| 72.1% \| \| ViT from scratch \| 213k \| 58.4% \| 23.4% \| 12.1% \| 70.3% \| \| ViT + Data Mixture \| 14.2M + 15.4k¹ \| 84.3% \| 35.1% \| 15.0% \| 64.7% \| \| ViT Pretrained \| 14.2M \| 97.1% \| 57.3% \| 28.5% \| 73.8% \| ¹ Data Mixture refers to the fine-tuning of a pretrained model with a mixture of original images and additional mean-pooled images (the same 15.4k used for subnetwork discovery) in order to instill a bias towards shape-based classification mean-pooled images. In ViT, we were able to localize a 14.6% parameter subnetwork achieving 76.1% accuracy on mean-pooled ImageNet. Both achieve a significant accuracy boost compared to pretrained models. Subtask Induction Increases Sample Efficiency In Figure 5, we show the training dynamics of ResNet and ViT trained with Subtask Induction compared against those trained from random initialization. We see that models initialized from subnetworks are much more data and computation efficient: on ResNet18, we observe that it achieves 11.8% better accuracy when trained on the same dataset; ViT proves to be much more data hungry as it fails to achieve competitive accuracies when trained on the 213k images of 16-class ImageNet. We also observe that the performance on mean-pooled images is maintained throughout training, suggesting that solutions learned by both models rely on the transferred subtask. In comparison, models trained from scratch with our small dataset do not generalize to mean-pooled images. Transferring Subnetworks Instills Stronger Shape Bias We present results of Subtask Induction compared against various baselines in Table 1. When the subnetworks are transferred and re-trained on 16-class ImageNet, we find that they achieve competitive accuracies on the original images and significantly better accuracies on mean-pooled images, suggesting a much stronger shape bias. In comparison, fine-tuning pretrained models and training from scratch with the mean-pooled data augmentation both fail to generalize to the held-out mean-pooled images. Notably, we show that Subtask Induction successfully instills a shape bias into ResNet, allowing it to achieve an accuracy comparable to pretrained ViT and 18.8% better than pretrained ResNet18, all while being trained on a much smaller dataset (17% and 1.5% the size of ResNet and ViT training set, respectively). While Subtask Induction gives weaker performance boosts to ViT, it still increases accuracy on mean-pooled images by 8.7% and performs much better in every benchmark compared to training a model from scratch on the same dataset. In addition, we also observe that fine-tuning the model with data augmentation achieves uniformly worse overall accuracy compared to using the pretrained model and only adapting the classifier layer, suggesting that the small mean-pooled dataset used for subnetwork discovery does not give model a shape bias when used for fine-tuning. This resonates with the finding in Pha et al. (2020): when a model is finetuned on a small out-of-domain dataset, data augmentation often hurts especially if the useful information in augmented data is hard to extract. 5.4 Analysis: Cue Conflict Next, we evaluate all of our models on the cue-conflict dataset introduced in Geirhos et al. (2019), a dataset consisting of images in which texture and shape cues are dissociated from one another. For example, this dataset contains images of dogs with the texture of an elephant overlaid on them. Cue-conflict images attempt to exploit a model’s texture bias to change their prediction. For each model, we report two metrics: (1) accuracy is the proportion of cue-conflict images that are classified correctly according to shape cues, (2) robustness is the proportion of image that are not classified according to misleading texture cues. Ideally we would want a model to achieve high performance on both accuracy and robustness. From the Cue Conflict columns of Table 1, we see that Subtask Induction consistently yields more accurate and robust models than fine-tuning with data augmentation. Consistent with our ImageNet results, we find that pretrained ViT already has a strong shape bias. However, it was also trained on orders of magnitude more data (14.2M vs 213k) than our ViT with Subtask Induction, which achieves comparable robustness on the cue-conflict data. Importantly, we also find that ResNet-18 trained with Subtask Induction achieves similar level of accuracy and robustness as pre-trained ViT, despite the small amount of training data and the inherent texture inductive bias of the ResNet architecture. 6 Discussion Inductive biases are crucial for understanding and controlling the solutions neural networks learn. We present a new technique, Subtask Induction, that leverages recent advances in our mechanistic understanding of trained models to instill such biases in neural networks. Across a range of experimental settings and model architectures, we demonstrated that Subtask Induction consistently confers the inductive bias that we expect, yielding increased sample efficiency and robustness to out of distribution stimuli. Furthermore, we demonstrated that our method has higher sample efficiency and outperforms data augmentation approaches to instilling inductive biases. Future Work Subtask Induction can be applied in wider contexts to instill specific inductive biases, either to encourage a model to learn particular solutions under limited data settings or to combat existing model heuristics. Though Subtask Induction is promising, we also note several limitations and avenues for future work. First, subtask induction requires supervised training of a binary mask to perform subnetwork discovery, which requires constructing custom-designed datasets. Future work might relax this constraint by decomposing a trained model in an unsupervised fashion, and transferring subnetworks that are discovered by this decomposition. Furthermore, Subtask Induction directly transfers subnetworks, which is only possible between models of identical architecture. Future work might seek to address this, perhaps by combining Subtask Induction with methods for re-scaling models, such as the Linear Growth Operator (Wang et al., 2023). 7 ETHICS STATEMENT We believe that the present work is in compliance with the ICLR code of ethics. Subtask induction can be used to influence the solutions that neural networks learn. This may have future implications for bias, fairness, and safety of neural network models. However, we emphasize that the current iteration of subtask induction is a proof of concept, and cannot and should not be used to render models free from social biases in real-world systems. 8 REPRODUCIBILITY STATEMENT To facilitate reproducibility, we provide detailed description of the models and training details in both the main text and the appendix. Specifically, the experimental setup section in both the arithmetic experiments (Section 4.2) and the vision experiments (Section 5.2) describes the configurations of our model and the baselines. 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NSyacfXOyX	Are there considerations or plans for deploying PatchSynth in real-world software development environments? How would the integration look like, and what kind of support or infrastructure would be required to ensure the model operates efficiently and securely in a production setting?	PATCHSYNTH: A PATCH-TEXT PRE-TRAINED MODEL Anonymous authors Paper under double-blind review ABSTRACT In recent years, patch representation learning has emerged as a necessary research direction for exploiting the capabilities of machine learning in software generation. These representations have driven significant performance enhancements across a variety of tasks involving code changes. While the progress is undeniable, a common limitation among existing models is their specialization: they predominantly excel in either predictive tasks, such as security patch classification, or in generative tasks such as patch description generation. This dichotomy is further exacerbated by a prevalent dependency on potentially noisy data sources. Specifically, many models utilize patches integrated with Abstract Syntax Trees (AST) that, unfortunately, may contain parsing inaccuracies, thus acting as a suboptimal source of supervision. In response to these challenges, we introduce PATCHSYNTH, a novel pre-training framework for patches and natural language text. PATCHSYNTH deploys a triple-loss training strategy for 1) patch-description contrastive learning, which enables to separate patches and descriptions in the embedding space, 2) patch-description matching, which ensures that each patch is associated to its description in the embedding space, and 3) patch-description generation, which ensures that the patch embedding is effective for generation. These losses are implemented for joint learning to achieve good performance in both predictive and generative tasks involving patches. Empirical evaluations focusing on patch description generation, demonstrate that PATCHSYNTH sets new state of the art performance, consistently outperforming the state-of-the-art in metrics like BLEU, ROUGE-L, METEOR, and Recall. 1 INTRODUCTION Patches are critical artefacts in software evolution. They bring the code modifications that are necessary for fixing bugs, including security vulnerabilities and performance issues, or enhancing features. As such, their accurate representation has a potent impact on various automation tasks of software engineering, notably towards assisting collaborative development, systematic documentation, and rapid code review processes. The research community has already engaged in various works towards developing techniques that can address the challenges of accurate patch representation. Most recently, pre-training approaches that build on programming language and natural language data have shown great promises [Feng et al., 2020]. However, learning to explicitly associate code-like data with text will lead to the emergence of Patch-Text Pre-training (PTP) paradigm, which will be a valuable asset in addressing various challenges such as generating the description of a patch [Xu et al., 2019], predicting whether a patch is solving a bug report [Tian et al., 2022]. The domain of PTP has witnessed a profusion of research endeavors, each attempting to bridge the gap between code modifications and textual descriptions. Early approaches primarily focused on deterministic models, extracting predefined patterns and attributes from patches to generate textual explanations [Allamanis et al., 2018]. However, the advent of deep learning ushered in a new era of possibilities [Elnaggar et al., 2021]. Advanced models, leveraging the capabilities of neural networks, sought to capture the nuanced semantics of patches and generate rich, context-aware descriptions [Hoang et al., 2020]. Yet, despite their sophistication, these models grapple with inherent challenges. A recurring concern is their pronounced specialization, wherein architectures exhibit 1In practice, developers submit code changes in the form of commits to conform to the version control system requirements. A commit includes the set of changes, i.e., the patch, and a text, i.e., the commit message, which is a natural language description of the changes. prowess either in patch understanding or in generation tasks, seldom both. Additionally, the reliability and accuracy of many PTP models are often compromised due to their reliance on data sources fraught with inconsistencies, particularly those integrated with Abstract Syntax Trees (AST) (Lin et al., 2022). Addressing the limitations in existing solutions, we introduce \textsc{PatchSynth}. Distinct from contemporary models, \textsc{PatchSynth} is underpinned by a harmonious synthesis of patch understanding and generation. To steer clear of the pitfalls of excessive specialization, our model is designed to effortlessly switch between these two essential tasks. The core part of \textsc{PatchSynth} lies a state-of-the-art synthetic description generator, purpose-built to extract and elucidate the multifaceted semantics embedded within patches. This robust core is complemented by a suite of advanced algorithms and methodologies, ensuring the generated narratives are not only accurate but also contextually rich and relevant. This paper embarks on a comprehensive exploration of \textsc{PatchSynth}, detailing its foundational principles, architectural nuances, and design philosophy. Through a structured exposition, we demystify the intricacies of our model, elucidating the rationale behind each design choice and its implications on performance. We also delve deep into the synergies between the various components of \textsc{PatchSynth}, highlighting how they collectively contribute to its superior capabilities. The presented narrative weaves together theoretical foundations with practical considerations, offering readers an all-encompassing understanding of our work. Our claims regarding \textsc{PatchSynth}'s capabilities are not merely theoretical postulates. Through rigorous empirical evaluations across diverse Patch-Text tasks, we substantiate the efficacy of our approach. Benchmarking \textsc{PatchSynth} against common metrics such as BLEU, ROUGE-L, METEOR, and Recall, our experiments consistently spotlight its dominance over the state-of-the-art technique proposed by Liu et al. (2023). Specifically, compared against with the state-of-the-art, \textsc{PatchSynth} achieves 10.76%, 11.62%, and 4.6% improvement for BLEU, ROUGE-L, and METEOR, respectively, on the patch description generation. Our manuscript makes the following contributions: - To the best of our knowledge, we are the first to propose a patch-text pre-trained framework with joint learning, capable of adapting to predictive and generative tasks. - Innovative Synthetic Description Generator: At the heart of \textsc{PatchSynth} lies a state-of-the-art synthetic description generator, carefully engineered to capture intricate semantics within patches. This component not only ensures contextually rich and accurate descriptions but also mitigates the challenges posed by inconsistent data sources. - Empirical Validation Against Benchmarks: Through comprehensive empirical evaluation, we validate the superior capabilities of \textsc{PatchSynth} on the task of patch description generation. Our results, benchmarked against revered metrics such as BLEU, ROUGE-L, METEOR, and Recall, outshining existing state-of-the-art systems. ## 2 RELATED WORK In light of the strides made in Patch-Text Pre-training (PTP), this section presents a detailed review of pertinent studies in the domain of patch representation and applications, while positioning our innovative approach, \textsc{PatchSynth}, within the broader landscape. ### 2.1 CODE-LIKE TEXT REPRESENTATION PARADIGMS Over the years, several approaches have been devised to represent code-like texts, from traditional source code mappings (Feng et al., 2020; Elnaggar et al., 2021) to specific patch representation strategies (Hoang et al., 2020). The comprehensive survey by Allamanis et al. (2018) offers deep insights into this realm. From graph-centric techniques, exemplified by control-flow graph representations (DeFreez et al., 2018), to the modern-day deep learning models (Elnaggar et al., 2021; Feng et al., 2020; Hoang et al., 2020), the trajectory of progress is evident. While earlier methods like those by Henkel et al. (2018) targeted symbolic trace generation for code embeddings, more recent architectures such as CC2Vec (Hoang et al., 2020) and CoDiSum (Xu et al., 2019) leverage deep learning for robust patch representation. CCRep by Liu et al. (2023) and the CACHE method proposed by Lin et al. (2022) are other notable mentions. Our approach transcends conventional methods, placing special emphasis on code change context and introducing an avant-garde graph intention embedding. ### 2.2 Utility Spectrum of Patch Representations Narrative Synthesis for Patches: Previous studies (Dyer et al., 2013; Dong et al., 2022) highlight the gaping void of descriptive commit messages in many projects, underlining the importance of auto-generating patch descriptions. Existing methods span the gamut from template-driven techniques (Buse & Weimer, 2010; Cortés-Coy et al., 2014), retrieval-centric solutions (Hoang et al., 2020; Liu et al., 2018; Huang et al., 2020), to generative models (Dong et al., 2022; Xu et al., 2019; Liu et al., 2020; Nie et al., 2021). PATCHSYNTH stands out with its bimodal approach, leveraging both the sequential and architectural nuances of patches via SeqIntention and GraphIntention integration. ### 2.3 Gaps and Constraints in Contemporary Approaches While recent Patch-Text pre-training solutions have showcased commendable efficacy, they’re not without challenges. A dominant concern remains their reliance on potentially error-prone data, especially those intertwined with Abstract Syntax Trees (AST). Such errors can inadvertently introduce inaccuracies, proving detrimental to model reliability. Additionally, the prevailing trend of model specialization — with a focus either on patch understanding or generative tasks — curtails the broader utility of these architectures. PATCHSYNTH seeks to bridge these gaps, presenting a versatile and reliable solution powered by an innovative synthetic description generator by building a triple-loss framework. ### 3 METHODOLOGY Our proposed approach, named Patch-Text Pretraining (PATCHSYNTH), employs a unified model to address the understanding and generation of patches alongside text. Below, we detail the architecture, functionalities, and the joint triple-loss pretraining scheme, as illustrated in Figure 1. #### 3.1 Architecture Our model is founded on the CodeBERT transformer architecture to act as our primary patch encoder. Given the significance of transformers in capturing relationships and dependencies in data, the chosen architecture promises to deliver effective encoding and decoding of both patch and textual information. Input Representation: Given a patch $p$ and its associated text $t$, the model processes these as sequences. For the patch, it is tokenized into a sequence of tokens $P = \{p_1, p_2, ..., p_n\}$ where $n$ is the length of the tokenized patch. Similarly, the text is tokenized as $T = \{t_1, t_2, ..., t_m\}$, with $m$ being the text length. Patch Encoder: The patch encoder ingests the tokenized patch $P$ and converts it into a dense representation using the CodeBERT architecture. The outcome is a sequence of embeddings $E_P = \{e_{p1}, e_{p2}, ..., e_{pn}\}$ corresponding to the tokenized patch. Textual Information Processing: For the textual data, a similar transformation process is adopted. The tokenized text $T$ is fed into a transformer encoder, yielding a sequence of embeddings $E_T = \{e_{t1}, e_{t2}, ..., e_{tm}\}$. Additionally, special tokens such as [CLS], [Encode], and [Decode], as described in the previous sections, are prepended or appended as necessary, influencing the subsequent encoding or decoding processes. #### 3.2 Functionalities of PATCHSYNTH The PATCHSYNTH system functions in three primary capacities: - Unimodal Encoder: Independently encodes patch and text. In the text encoder, akin to BERT (Devlin et al., 2018), a [CLS] token is prepended to the text input for sentence summarization. - Patch Description Encoder: Infuses patch details by introducing an extra cross-attention (CA) layer amidst the self-attention (SA) layer and the feed-forward network (FFN) within each transformer block of the text encoder. An [Encode] token is annexed to the text, with the [Encode] token’s output embedding serving as the multimodal representation for the patch-description pair. Patch Description Decoder: Shares parameters with the patch description encoder, using a [Decode] token to signal sequence commencement and an end-of-sequence token to denote its conclusion. ### 3.3 Joint Triple-Loss Pretraining The PATCHSYNTH’s efficacy derives from its ability to jointly optimize three distinct objectives during the pretraining phase: 1. The combined loss function $L_{\text{Joint}}$ incorporates the Patch-Description Contrastive Loss $L_{\text{PDC}}$ which enables to separate patches and descriptions in the embedding space, 2. The Patch-Description Matching Loss $L_{\text{PDM}}$ which ensures that each patch is associated to its description in the embedding space, and 3. The Patch Description Generation Loss $L_{\text{PDG}}$ which ensures that the patch embedding is effective for generation. Given the weights $\lambda_1$, $\lambda_2$, and $\lambda_3$ that balance the importance of each loss (in our work: $\lambda_1=\lambda_2=\lambda_3=1$), the joint objective is: $$L_{\text{Joint}} = \lambda_1 \cdot L_{\text{PDC}} + \lambda_2 \cdot L_{\text{PDM}} + \lambda_3 \cdot L_{\text{PDG}}$$ (1) This combined loss ensures that the model learns the individual objectives while also harmonizing their combined effect, producing a well-rounded representation and generation capability. To describe each loss in detail: Patch-Description Contrastive Loss (PDC): Using the unimodal encoder, the PDC loss focuses on creating a harmonized feature space between patch and text transformers. For a given positive patch-text pair $(p, t^+)$ and a negative text sample $t^-$, the PDC loss can be formulated as: $$L_{\text{PDC}}(p, t^+, t^-) = - \log \frac{\exp(f(p) \cdot f(t^+)/\tau)}{\exp(f(p) \cdot f(t^+)/\tau) + \exp(f(p) \cdot f(t^-)/\tau)}$$ (2) where $f$ denotes the encoder function, and $\tau$ is a temperature scaling parameter. This loss encourages the positive pairs to have representations closer in the embedding space compared to the negative pairs. Patch-Description Matching Loss (PDM): Activated by the patch description encoder, the PDM loss focuses on learning a combined representation of the patch and text. Given a patch $ p $ and its description $ t $, the binary classification loss can be represented as: \[ L_{PDM}(p, t) = -y \log(\sigma(g(p, t))) - (1 - y) \log(1 - \sigma(g(p, t))) \] where $ y $ is the ground truth label (1 for matched pairs and 0 for unmatched), $ g $ is the combined representation function, and $ \sigma $ denotes the sigmoid function. Patch Description Generation Loss (PDG): The PDG loss, facilitated by the patch description decoder, targets autoregressive text generation. For a given patch $ p $ and its corresponding textual description sequence $ T = \{t_1, t_2, ..., t_m\} $, the loss is: \[ L_{PDG}(p, T) = - \sum_{i=1}^{m} \log P(t_i \| t_1, ..., t_{i-1}, p) \] This cross-entropy loss encourages the model to maximize the likelihood of the correct next token in the sequence, based on the context of the previous tokens and the patch. To maximize pre-training efficiency with the joint triple-loss scheme, most parameters are shared between the text encoder and decoder, with the exception of those in the SA layers. This parameter-sharing strategy promotes training efficiency and leverages the advantages of triple-loss training. The balancing weights $ \lambda_1, \lambda_2, $ and $ \lambda_3 $ can be set based on the importance or sensitivity of each loss to the overall training objective. These weights also ensure that no individual loss dominates the training, preserving the multi-objective nature of the pretraining. Lastly, to ensure efficient pre-training while utilizing this joint triple-loss training approach, parameters are shared between the text encoder and decoder, except for the SA layers. Such sharing of parameters promotes training efficiency, benefiting from the joint training regimen. ### 3.4 Parameter Sharing Considerations The PATCHSYNTH strategically shares parameters between the text encoder and decoder due to the intrinsic overlap in their operations. This subsection will mathematically describe the parameter sharing and its implications. Shared Embeddings: The first point of parameter sharing is the embedding layer. Given an input token $ x $ from the vocabulary $ V $, the embedding layer transformation can be represented as: $ e(x) = W_e x $ where $ W_e $ is the shared embedding weight matrix. Shared Cross-Attention (CA) Layers: For each token in the patch sequence, a CA mechanism computes its attention over the textual sequence. Mathematically, for a patch token $ p $ and text token $ t $: \[ a(p, t) = \frac{\exp(\text{Score}(p, t))}{\sum_{i \in T} \exp(\text{Score}(p, i))} \] where $ \text{Score} $ is a function computing the alignment score, often a dot product between the two tokens. Shared Feed-Forward Network (FFN): Both the encoder and decoder leverage an FFN layer, defined by: \[ FFN(x) = W_2 \sigma(W_1 x + b_1) + b_2 \] where $ W_1, W_2 $ are weight matrices, $ b_1, b_2 $ are biases, and $ \sigma $ is an activation function, like ReLU. Exclusion of SA layers: The SA layers, despite their architectural similarities, encapsulate distinct nuances between encoding and decoding processes. For token $ x $: \[ SA(x) = \sum_{\hat{x} \in X} \frac{\exp(x \cdot \hat{x})}{\sum_{\hat{x} \in X} \exp(x \cdot \hat{x})} \hat{x} \] (7) The weights and biases in the SA layers remain unshared due to the layer’s distinct role in sequence self-alignment. By sharing parameters, especially in layers with similar functionalities, the PATCHSYNTH ensures consistent processing across the encoder and decoder. This design choice not only economizes on the number of parameters, leading to faster training but also imposes a form of regularization, promoting the synthesis of the joint triple-loss training scheme. 4 EXPERIMENTAL DESIGN This section elucidates our systematic experimental design, encompassing implementation specifics, research questions that drive our investigation, the comparative baseline models, the datasets employed, and the evaluation metrics harnessed. A cohesive understanding of these elements is pivotal for replicability and comprehension of the subsequent results. 4.1 IMPLEMENTATION DETAILS We have implemented our models using the PyTorch framework (Paszke et al., 2019), capitalizing on its flexibility and efficiency. The model is pre-trained on a robust hardware configuration of 4 A100 GPUs. In terms of initialization, the code transformer is derived from CodeBERT (Feng et al., 2020), while the text transformer owes its genesis to the BERT base model (Devlin et al., 2018). The pre-training regimen spans 50 epochs with batch sizes set at 32. Optimization is facilitated by the Adam optimizer (Kingma & Ba, 2014), with a learning rate initialized at 0.001. Parameter initialization adheres to the Xavier algorithm (Glorot & Bengio, 2010) for ensuring optimal weight values. The learning rate undergoes a warm-up to $ e^{-4} $ and subsequently experiences a linear decay at a rate of 0.85. Model dimensions are meticulously calibrated, with the hidden layer output dimension fixed at 512, and a conservative dropout rate of 0.1 ensures regularization. 4.2 GUIDING RESEARCH QUESTIONS Our empirical investigation is orchestrated around a triad of pivotal research questions: RQ-1: How does PATCHSYNTH perform concerning patch description generation compared to prevailing methods? RQ-2: Which architectural and design choices significantly influence the performance of PATCHSYNTH? 4.3 COMPARATIVE BASELINES To furnish a comprehensive perspective on PATCHSYNTH’s performance, we juxtapose it against a curated ensemble of state-of-the-art (SOTA) models. These include models specifically architected for patch representation learning and generic models previously adapted for patch-oriented tasks. A brief synopsis of each baseline is as follows: - CoDiSum (Xu et al., 2019): Leveraging an encoder-decoder paradigm, this model employs a multi-layer bidirectional GRU supplemented by a copying mechanism. - Coregen (Nie et al., 2021): A pure Transformer architecture targeting the nuanced task of commit message generation. - ATOM (Liu et al., 2020) is a commit message generation techniques, which builds on abstract syntax tree and hybrid ranking. - FIRA (Dong et al., 2022) is a graph-based code change representation learning approach for commit message generation. • CCRep (Liu et al., 2023) is an innovative approach that uses pre-trained models to encode code changes into feature vectors, enhancing performance in tasks like commit message generation, etc. ### 4.4 DATA CURATION The veracity of our results is contingent on the quality and comprehensiveness of our datasets. We have employed the following: - Patch Description Generation (PDG): Capitalizing on benchmarks from seminal works (Dyer et al., 2013; Hoang et al., 2020), our dataset, primarily focused on Java samples, includes 90,661 patches with their attendant descriptions. - Patch Description Matching (PDM): Inside the training batch, we generate PDM data points by paring patch and unoriginal paired description. All data is from PDG task. - Patch Description Contrastive Learning (PDC): We do contrastive learning between paired patch and description. ### 4.5 EVALUATION METRICS Quantitative evaluations are anchored in a suite of established metrics: - ROUGE (ROUGE, 2004): Primarily gauging text generation quality by contrasting generated content against human-produced references, with emphasis on the ROUGE-L metric. - BLEU (Papineni et al., 2002): A venerable metric in machine translation, BLEU ascertains the alignment of generated text sequences with reference sequences. - METEOR (Banerjee & Lavie, 2005): METEOR amalgamates precision and recall, producing an F-Score-oriented evaluation, particularly suited for text-generation models. - Recall Metrics (Tian et al., 2022): Specifically devised for patch correctness assessment, these metrics gauge the model’s proficiency in correctly predicting and filtering patches. ## 5 RESULTS FROM THE EXPERIMENTS ### 5.1 [RQ-1]: EVALUATING PATCHSYNTH’S PERFORMANCE IN GENERATING PATCH DESCRIPTIONS [Objective of the Experiment]: Our aim is to gauge the efficiency of the embeddings produced by PATCHSYNTH in a prevalent software engineering task: generating patch descriptions. We position PATCHSYNTH in comparison with the current state-of-the-art (SOTA) methodologies. [Design of the Experiment (RQ-1)]: For our experiment, we utilized the dataset sourced from FIRA. Since Dong et al. (2022) had already evaluated FIRA and other foundational methods using this dataset, we directly cite the performance results of these baselines from Table IV of the FIRA publication. This dataset comprises 75K, 8K, and 7.6K commit-message pairs for training, validation, and testing, respectively. Our assessment criteria for the patch descriptions generated in the test set are based on the BLEU, ROUGE-L, and METEOR metrics. Table 1 offers a comparative analysis of various methodologies employed in patch description generation. Each row represents a distinct approach, with references to their respective studies. The columns, on the other hand, showcase the performance metrics—Rouge-L, BLEU, and METEOR—expressed as percentages. These metrics are standard evaluation measures in the realm of natural language processing and provide insights into the quality of \| Approach \| Rouge-L (%) \| BLEU (%) \| METEOR (%) \| \|----------\|-------------\|----------\|------------\| \| Codsum \| Xu et al. (2019) \| 19.73 \| 16.55 \| 12.83 \| \| CoreGen \| Nie et al. (2021) \| 18.22 \| 14.15 \| 12.90 \| \| ATOM \| Ferencz (2020) \| 10.17 \| 8.35 \| 8.73 \| \| FIRA \| (Ferrara et al., 2020) \| 24.58 \| 17.67 \| 14.93 \| \| CCRep \| (Liu et al., 2023) \| 23.41 \| 19.70 \| 15.84 \| \| PATCHSYNTH \| \| 26.13 \| 21.82 \| 16.57 \| generated descriptions. Notably, \textsc{PatchSynth}, the focal point of our study, demonstrates commendable performance, achieving scores of 26.13% in Rouge-L, 21.82% in BLEU, and 16.57% in METEOR. When juxtaposed with other methods, this table underscores the efficacy of \textsc{PatchSynth} in the context of patch description generation, setting new benchmarks for future endeavors in this domain. [Outcomes of the Experiment (RQ-1)]: As depicted in Table 1, the average metric scores for descriptions generated by \textsc{PatchSynth} and its counterparts are presented. \textsc{PatchSynth} surpasses all other methods across the board in terms of performance metrics, with the sole exception being FIRA’s score in the ROUGE-L metric. Here, we take an example to indicate the performance of a different patch description generator. As illustrated in the example, the “add” line closely resembles the preceding “if” statement, facilitating their fusion during the generation of the patch description. This similarity can pose challenges during the generation of accurate and contextually rich patch descriptions. [Example]: ```java @@-,+@public class LogFormatterimplementsExchangeFormatter { SINGLE Exception exception = exchange.getException ( ); boolean caught = false ; if (showCaughtException && exception == null){ + if ((showAll \|\| showCaughtException) exception==null){ SINGLE exception=exchange.getProperty (Exchange.EXCEPTION_CAUGHT, Exception.class) ; caught = true ; ``` Ground Truth: Added missing showAll for caught exception. Codisum: Fix bug in showcaughtexception. Coregen: Add exception limitations. ATOM: Add showall. FIRA: Add showall in if condition. CCRep: Add showall and showcaughtexception. PatchSynth: Added absent showall for caught exception. From the above illustration, it is manifest that while many models capture the essence of the change (‘showall’ addition), they slightly deviate in capturing the exact contextual nuance. \textsc{PatchSynth}, on the other hand, aligns closely with the ground truth, demonstrating its fitness in generating precise patch descriptions. Its ability to discern and articulate the absence of ‘showAll’ in context with the caught exception underscores its efficiency and the advancements it brings to the table. [Performance cross different patch attention]: As introduced in Dong et al. (2022), the dataset contains three patch attention categories: fix, add, and remove. Figure 2 visually represents the distribution of different metrics—Precision, Recall, and F1-Score across these categories. Each category has three bars representing these metrics, with blue bars indicating Precision, orange for Recall, and green for F1-Score. In the ‘Fix’ category, the metrics are reported as follows: a Precision of 24.13, a Recall of 18.27, and an F1-Score of 13.58. Similarly, in the ‘Add’ category, the values are 26.89 for Precision, 23.14 for Recall, and 17.63 for F1-Score. Lastly, the ‘Remove’ category exhibits a Precision of 27.47, a Recall of 24.05, and an F1-Score of 18.5. These metrics are distinctly color-coded and patterned for clear differentiation. Alongside, a line graph indicating the count of each category in the dataset is plotted against a secondary y-axis, reporting counts of 14,600 for ‘Fix’, 68,765 for ‘Add’, and 7,296 for ‘Remove’. For the ‘Fix’ category, the Precision, Recall, and F1-Score are 24, 20, and 15 respectively. Similarly, for the ‘Add’ category, the values are 27, 23, and 17, and for the ‘Remove’ category, they are 25, 21, and 16. Alongside, a red line plot indicates the count of occurrences for each category, with counts being 14600 for ‘Fix’, 68765 for ‘Add’, and 7296 for ‘Remove’, providing a comparative insight into their respective volumes. The x-axis labels the categories, while the left y-axis denotes the metrics’ values, and the right y-axis denotes the count of occurrences. The metrics values for each category are also annotated at the top of each bar for clearer reference. Above the figure, a legend neatly organizes the information, indicating the color and pattern representation for each category’s metrics and the count line plot. This visual representation facilitates a comprehensive understanding of the dataset’s patch attention distribution and the relative volume of occurrences for each category, aiding in the analytical interpretation of the data. 5.2 [RQ-2]: ANALYSIS OF TRIPLE LOSS TRAINING [Experiment Goal]: We perform an ablation study to investigate the effectiveness of each loss in PATCHSYNTH. [Experiment Design]: We investigate the related contribution of $L_{PDC}$, $L_{PDM}$, and $L_{PDG}$ by building three variants of PATCHSYNTH where we remove either $L_{PDC}$ (i.e., denoted as PATCHSYNTH$_{PDC-}$), or $L_{PDM}$ (i.e., denoted as PATCHSYNTH$_{PDM-}$), or $L_{PDG}$ (i.e., denoted as PATCHSYNTH$_{PDG-}$). We evaluate the performance of these variants on the task of patch description generation. [Experiment Results (RQ-2)]: Table 2 presents an ablation study conducted to scrutinize the effect of different losses on the performance of the proposed model PATCHSYNTH. The PATCHSYNTH model, highlighted with cell coloring, serves as the baseline approach exhibiting a Rouge-L score of 26.13%, a BLEU score of 21.82%, and a METEOR score of 16.57%. Subsequent rows in the table delineate the performance metrics of PATCHSYNTH under different configurations, specifically PATCHSYNTH$_{PDC-}$, PATCHSYNTH$_{PDM-}$, and PATCHSYNTH$_{PDG-}$, which likely involve variations in the loss functions employed during training. The slight decrement in performance metrics from PATCHSYNTH$_{PDG-}$ to PATCHSYNTH$_{PDG-}$ illuminates the pivotal role the loss components play in optimizing the model for higher accuracy in patch description generation. Particularly, PATCHSYNTH$_{PDC-}$ registers a marginal decrease in performance compared to the baseline, with a Rouge-L score of 25.87%, a BLEU score of 21.76%, and a METEOR score of 16.53%. However, a more pronounced decline is observed in PATCHSYNTH$_{PDM-}$ and PATCHSYNTH$_{PDG-}$, especially for PATCHSYNTH$_{PDG-}$, indicating that generating task training is more important compared to the other two losses. [Conclusion (RQ-2)]: The ablation study manifests the critical role of different loss configurations on PATCHSYNTH’s performance. The more pronounced decline in metrics for PATCHSYNTH$_{PDG-}$ accentuates the essence of generating task training compared to the other loss configurations. 6 CONCLUSION We have designed a novel approach to overcome the limitations of existing approaches in representing patches for effectively automating relevant software engineering tasks. The pre-trained model, PATCHSYNTH, employs a triple loss training, which ensures that the rich information about patches and their associated descriptions are well captured, enabling it to achieve state of the art results. Notably, our evaluation in patch description generation show that PATCHSYNTH improves over the CCRep [Liu et al. (2023)] state of the art by 10.76%, 11.62%, and 4.6 % for BLEU, ROUGE-L, and METEOR, respectively. Open science. We provide a package to reproduce our experiments which is available at the following address: https://anonymous.4open.science/status/PatchSynth-8284 REFERENCES Miltiadis Allamanis, Earl T Barr, Premkumar Devanbu, and Charles Sutton. A survey of machine learning for big code and naturalness. ACM Computing Surveys (CSUR), 51(4):1–37, 2018. Satanjeev Banerjee and Alon Lavie. Meteor: An automatic metric for mt evaluation with improved correlation with human judgments. 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8rhHI6C8iC	Storing all the clients' plug-ins may also be a privacy risk, as there is not any aggregation that protects the clients' privacy from a malicious server. The threat model is not discussed by the paper.	ALL FOR ONE AND ONE FOR ALL: A COLLABORATIVE FL FRAMEWORK FOR GENERIC FEDERATED LEARNING WITH PERSONALIZED PLUG-INS Anonymous authors Paper under double-blind review ABSTRACT Personalized federated learning (PFL) mitigates the notorious data heterogeneity issue in generic federated learning (GFL) by assuming that client models only need to fit on local datasets individually. However, real-world FL clients may meet with test data from other distributions. To endow clients with the ability to handle other datasets, we theoretically formulate a new problem named as Selective FL (SFL), bridging the GFL and PFL together. To practically solve SFL, we design a general effective framework named as Hot-Pluggable Federated Learning (HPFL). In HPFL, clients firstly learn a global shared feature extractor. Next, with the frozen feature extractor, multiple personalized plug-in modules are individually learned based on the local data and saved in a modular store on the server. In inference stage, an accurate selection algorithm allows clients to choose and download suitable plug-in modules from the modular store to achieve the high generalization performance on target data distribution. We conduct comprehensive experiments and ablation studies following common FL settings including four datasets and three neural networks, showing that HPFL significantly outperforms advanced FL algorithms. Additionally, we empirically show the remarkable potential of HPFL to resolve other practical FL problems like continual federated learning and discuss its possible applications in one-shot FL, anarchic FL and an FL plug-in market. 1 INTRODUCTION Federated Learning (FL) is an effective framework that lets multiple users or organizations to collaboratively train a machine learning model with data privacy protection. The generic FL (Brendan McMahan et al., 2016) (GFL) was first proposed to obtain a global model (GM) performing well on test data from all clients. However, the performance of the classic FL algorithm FedAvg (Brendan McMahan et al., 2016) suffers from the client drift caused by the data heterogeneity (Kairouz et al., 2019), i.e. different data distributions on clients. To tackle the data heterogeneity problem, personalized FL (Collins et al., 2021; Chen & Chao, 2021) (PFL) is proposed with assuming that clients only need to perform well on its local test data. Usually, the distribution of local test data is similar to that of its local training data. Thus, PFL usually distinguishes its local models from the GM, and personalizes local models to better adapt to its training data while absorbing knowledge from the global training data. On the local test data, personalized models (PMs) in PFL (PFL-PM) significantly outperform the GM learned by GFL (GFL-GM) (Chen & Chao, 2021). However, in real-world scenarios, FL users may meet with test data which has different distribution from local training data (Liu et al., 2020; Luo et al., 2019; Hsu et al., 2020), instead similar one ever appeared in other clients. For example, when one is having a trip traveling abroad, weather app on their phone may collect entirely different temperatures from what it used to. Though prediction for the future temperature may be difficult solely with the forecast model trained before, there are users whose model trained on the local temperature which happens to possess similar pattern (if not identical) with the temperature the traveler is trying to predict. We offer more examples showing the Table 1: Test accuracy of PMs on generic FL (GFL-PM, G-P) and personalized FL (PFL-PM, P-P) test settings, with ResNet-18 trained on CIFAR-10 dataset. \| Algorithm \| FedAvg \| FedPer \| FedRep \| FedRoD \| \|-----------\|--------\|--------\|--------\|--------\| \| Test Settings \| G-P \| P-P \| G-P \| P-P \| G-P \| P-P \| G-P \| P-P \| \| Accuracy \| 81.5 \| 92.5 \| 74.1 \| 95.8 \| 85.1 \| 95.6 \| 85.3 \| 94.3 \| real scenes of our setting named GFL-PM, in Appendix E.1. In GFL-PM setting, the test set every client encounters comes from local test data of other clients, whose distribution is the same as that of clients’ local training data. In these realistic cases, classic PFL algorithms may not be suitable anymore, as the personalized client models cannot generalize well on other test data. We conduct an experiment to illustrate this. We train PMs with advanced PFL algorithms FedPer (Arivazhagan et al., 2019b), FedRep (Collins et al., 2021) and FedRoD (Chen & Chao, 2021), and test their performances on global and local test data. As Table 1 shows, PMs performs well when they are only required to deal with local test data (PFL-PM), but their performances significantly collapse when meeting with global test data (GFL-PM), i.e. clients equally meet with the test data from all clients. This performance degradation of PMs in GFL scenario leads to a practical and fundamental questions: Is it possible for FL clients to achieve the generalization performance in GFL as high as PFL? To answer this question, we theoretically formulate a new problem called Selective FL (SFL), bridging the GFL and PFL together. Both GFL and PFL can be seen as the special case of the SFL. Its core idea is to let clients select and inference with suitable personalized models (PMs) according to incoming test data. Thus, we give an affirmative answer to the above question. However, the naive solution to SFL faces privacy concerns and large system overheads. To this end, we propose a general effective framework named Hot-Pluggable Federated Learning (HPFL) to solve SFL practically. As shown in Figure 1, HPFL splits the model into two parts: a backbone module (also called feature extractor) and a “plug-in” module. The training process consists of two stages: backbone and plug-in training. When training the backbone, clients exploit GFL algorithms to help them learn a general representation of all datasets. Then, each client individually trains a “plug-in” based on the outputs from the shared backbone with PFL algorithms. All trained “plug-ins” will be uploaded and saved in a “plug-in” store on the server. During inference, clients could download a suitable “plug-in” from the server with respect to the test data, then “plug” it on the backbone to complete the inference. We summarize our contributions as follows: (1) We identify a substantial gap between GFL and PFL. Then we formulate a new problem SFL to bridge them together to address this performance gap (Section 3); (2) We propose a general efficient and effective framework, HPFL, which practically solves the SFL problem (Section 4); (3) We conduct comprehensive experiments and ablation studies on four datasets and three neural networks to show the effectiveness of HPFL (Section 5); (4) we show the remarkable potential of HPFL in federated continual learning (Section 5.4) and discuss possible applications of HPFL in one-shot FL, anarchic FL and an FL plug-in market (Section 7). 2 RELATED WORKS Generic Federated Learning. The convergence problem of FL with high non-IID data distribution has always been a vital problem in improving the performance of models trained with FL. To resolve this problem, FedProx (Li et al., 2020b) and MOON (Li et al., 2021b) propose to add regularization... terms to mitigate the negative effect caused by data heterogeneity. Some methods modify uploaded gradients to alleviate the dissimilarity (Wang et al., 2020; Karimireddy et al., 2019). Some works share intermediate features (Jeong et al., 2018; Hao et al., 2021) or extra data (Tang et al., 2022) to reduce client drift. Different from these works, we attempt to enhance the GFL performance with personalized models. Personalized Federated Learning. PFL exploits personalizing client models to better suit local heterogeneous training data. Meta-learning (Fallah et al., 2020), knowledge distillation (Yu et al., 2020b; Li & Wang, 2019), adaptive regularization and model mixtures (Hanzely & Richtárik, 2020; Dinh et al., 2020; Deng et al., 2020) are used to enhance personal knowledge learning of models. Some works like LG-FEDAVG (Liang et al., 2020) and LotteryFL (Li et al., 2021a) allow clients to learn different PM structures. FedRep (Collins et al., 2021) and FedRoD (Chen & Chao, 2021) propose to learn a global feature extractor and personalized classifiers. All of these works only consider PMs in PFL settings, i.e. in test time, local PMs only meet test data distribution similar to training distribution. Instead, we excavate the potential of PMs to solve problems in GFL. With divergent purposes, HPFL and those methods train and use these personalized models in quite different way. Unlike those PFL methods, clients can still perform well when meeting unseen test data distribution in HPFL. Test-time adaptation & domain adaptation methods in FL. There exist some works (Peng et al., 2019; Liu et al., 2021) that generalize a federated model trained on multiple source domains to unseen target domains. FedTHE (Jiang & Lin, 2023) discussed test-time distribution shift of PMs, which is similar to our problem setting. These methods enhance federated models by better training schemes. Different from them, HPFL is the first FL framework that selects flexible PMs to achieve this goal, which is orthogonal to existing works. Due to the limited space, we leave a more detailed discussion of the literature review in Appendix A. ### 3 Selective FL: Implementing Generic FL from Personalized FL #### 3.1 Generic FL The GFL aims to make $M$ clients collaboratively learn a global model parameterized as $\theta$. Each client has its local data distribution $D_m$. Thus, the local objective function $L_m(\theta)$ on client $m$ is also different. The global optimization object of GFL is defined as: $$\min_{\theta \in \mathbb{R}^d} L_G(\theta) := \sum_{m=1}^{M} p_m L_m(\theta) := \sum_{m=1}^{M} p_m \mathbb{E}_{\xi_m \sim D_m} \ell(f(\theta, \xi_m), \xi_m),$$ where $\xi_m \sim D_m$ is the data sampled from $D_m$, $f(\theta, \xi_m)$ is the prediction, $d$ is the number of model parameters, $p_m > 0$ and $\sum_{m=1}^{M} p_m = 1$. Usually, $p_m = \frac{n_m}{N}$, where $n_m$ denotes the number of client $m$’s samples and $N = \sum_{m=1}^{M} n_m$. $GM$ refers to the model obtained from optimizing GFL. #### 3.2 Personalized FL Different from the object function of GFL, the PFL aims to learn multiple personalized models which fit well on different datasets individually: (Li & Wang, 2019; Chen & Chao, 2021; Li et al., 2021c): $$\min_{\Omega, \theta_1, ..., \theta_M} L_P(\Omega, \theta_1, ..., \theta_M) := \sum_{m=1}^{M} p_m \mathbb{E}_{\xi_m \sim D_m} \ell(f(\theta_m, \xi_m), \xi_m) + R(\Omega, \theta_1, ..., \theta_M),$$ where $R$ is a regularizer (Chen & Chao, 2021) that varies with different algorithms, $\Omega$ is used to collaborate clients. We call each obtained locally personalized model $\theta_m$ as PM. #### 3.3 When PM Meets GFL In practice, PMs of clients may meet test data from other clients. Therefore, the learned PMs $\theta_1, ..., \theta_M$ need to perform well on all local data $D_1, ..., D_M$. We formulate the corresponding optimization goal with PMs in GFL scenario (GFL-PM) is: $$\min_{\Omega, \theta_1, ..., \theta_M} L_{P-G}(\Omega, \theta_1, ..., \theta_M) = \frac{1}{M} \sum_{i=1}^{M} \sum_{m=1}^{M} p_m \mathbb{E}_{\xi_m \sim D_m} \ell(f(\theta_i, \xi_m), \xi_m) + R(\Omega, \theta_1, ..., \theta_M),$$ which can be seen as a combination of GFL (Eq. 1) and PFL (Eq. 2): each PM is optimized to minimize the $\ell$ on all $D_m$, $m \in 1, ..., M$. When not personalize $\theta_i$ on $D_i$, Eq. 3 is reduced to GFL. And if each client’s PM only needs to perform well on its local data, Eq. 3 turns into PFL. One may think that there is no need to endow PMs with global generalization performance because one can optimize GFL to obtain a GM that generalizes well on all local datasets $\{D_m, m \in \{1, ..., M\}\}$. However, theoretically and empirically, optimization of GM is difficult (Karimireddy et al., 2019; Woodworth et al., 2020) under communication cost and data heterogeneity constraints. Additionally, PMs’ performance on local test data (PM on PFL) is usually significantly better than that of GM on global test data (GM on GFL) (Chen & Chao, 2021; Collins et al., 2021). However, PMs after PFL usually cannot achieve better performance on unseen data distributions than GM in GFL (Chen & Chao, 2021). FedRoD (Chen & Chao, 2021) simultaneously optimizes $L_G$ and $L_P$, aiming to learn models that perform well both in GFL and PFL. This shares a similar spirit of optimizing GFL-PM problem (Eq. 3). However, PMs obtained from FedRoD remain a trade-off between minimizers of PFL and GFL. It is challenging to obtain model parameters that are both minimizers of GFL and PFL simultaneously. Next, we show that GFL-PM can be naturally transformed into a Selective FL (SFL) problem (Eq. 5), which involves optimizing PFL and a model selection problem (Eq. 6 in section 3.4). And the solution of SFL could serve as the minimizer of both GFL and PFL. ### 3.4 Selective FL Successful personalization on client $i$ means the following equation (Chen & Chao, 2021; Kairouz et al., 2019; Tan et al., 2022a): \[ E_{\xi_m \sim D_m} \ell(f(\theta_i, \xi_m), \xi_m) \geq E_{\xi_m \sim D_m} \ell(f(\theta_j, \xi_m), \xi_m), i \neq m, \] which means that for any client $i$, its PM outperforms than all PMs of other clients (Chen & Chao, 2021). Now, we are ready to state the following theorem (proof in Appendix B.1). Theorem 3.1. With Equation 4 and the PMs obtained from optimizing Equation 2 as: \[ \Omega^{pfl}, \theta_1^{pfl}, ..., \theta_M^{pfl} = \arg \min_{\Omega, \theta_1, ..., \theta_M} L_P(\Omega, \theta_1, ..., \theta_M), \] we have \[ L_{P-G}(\Omega, \theta_1, ..., \theta_M) \geq L_P(\Omega^{pfl}, \theta_1^{pfl}, ..., \theta_M^{pfl}). \] Remark 3.1. Theorem 3.1 implies that $L_{P-G}$ is lower bounded by the minimum of $L_P$. Theorem 3.1 inspires us to think about a question: Is it possible to exploit PMs to improve the generalization performance on the global dataset? Based on Equation 4, the intuitive solution is to design a new forward function $\hat{f}$ to make client $i$ generate the same outputs of $f(\theta_m^{pfl}, \xi_m)$ when meeting data $\xi_m \sim D_m$. Thus, we propose the Selective FL (SFL) problem as the following: \[ \begin{align} \min_{H} & \quad L_S(\Theta, H) := \sum_{m=1}^{M} p_m E_{\xi_m \sim D_m} \ell(\hat{f}(\Theta, \xi_m, H), \xi_m) \\ \text{s.t.} & \quad \hat{f}(\Theta, \xi_m, H) = f(\theta_m^{pfl}, \xi_m), s = S(\Theta, \xi_m, H) \end{align} \] where $\Theta = \{\Omega^{pfl}, \theta_1^{pfl}, ..., \theta_M^{pfl}\} = \arg \min_{\Omega, \theta_1, ..., \theta_M} L_P(\Omega, \theta_1, ..., \theta_M)$, $S$ is called selection function that outputs the model index to select (or say “generate”) a model from the PMs based on the input $\xi_m$ and the auxiliary information $H$ (We will illustrate what can be the auxiliary information in Section 4). Now, we can state the following theorem to illustrate that we can solve problem 3 by SFL (proof in Appendix B.2): Theorem 3.2. With equation 4, $\Omega^{pfl}, \theta_1^{pfl}, ..., \theta_M^{pfl} = \arg \min_{\Omega, \theta_1, ..., \theta_M} L_P(\Omega, \theta_1, ..., \theta_M)$ and the $H^$ that guarantees $\theta_m^{pfl} = s(\Theta, \xi_m, H)$, we have \[ L_{P-G}(\Omega, \theta_1, ..., \theta_M) \geq L_P(\Theta) = L_S(\Theta, H^). \] Remark 3.2. Theorem 3.2 shows that if we can accurately select $\theta_m^{pfl}$ out from all PMs when meeting data samples $\xi_m \sim D_m$, the solution of SFL is also the lower bound of GFL-PM (Eq. 3). Therefore, solving SFL means that clients can achieve a generalization performance in GFL as high as PFL. ### 4 HPFL: A General Effective Framework to Solve Selective FL In this section, we will first illustrate that directly selecting PM faces some fatal obstacles, including the large system overheads and privacy concerns in Section 4.1. Then, we introduce the design of HPFL in Section 4.2 with the Algorithm 1. Lastly, the selection method is introduced in Section 4.3. #### 4.1 Problems of Directly Selecting PM With PMs $\Theta = \{\Omega^{pfl}, \theta_1^{pfl}, ..., \theta_M^{pfl}\} = \arg\min_{\Omega, \theta_1, ..., \theta_M} L_P(\Omega, \theta_1, ..., \theta_M)$, an intuitive idea is to choose PM $i$ based on the similarity between its local data $D_i$ and the input data $\xi_m \sim D_m$, thus the selection function [6] is implemented as: $s = S_\xi(\Theta, \xi_m, H) = \arg\min_{i \in M} d(D_i, \xi_m)$, where $d(\cdot, \cdot)$ is any distance measure, then do inference as $f(\theta_i^{pfl}, \xi_m)$. However, accessing data of other clients will cause privacy concerns. Moreover, communicating the whole model parameter $\theta_m$ is impractical due to large system overhead, especially for large language models and many clients. ### 4.2 Design of HPFL Training the complete model $\theta$. First, with any GFL algorithm, HPFL obtains a model $\theta$ that performs well (not as good as PMs in PFL) on all client datasets. Thus, the model $\theta$ owns a backbone $g$ that can extract general features from all client datasets. Due to the limited space, we chose the classic GFL algorithm FedAvg [McMahan et al., 2017] in our experiments. Future works can explore other advanced GFL algorithms to learn a better $\theta$. Training personalized plug-in module $\theta^p_m$. Usually, after training, early layers of a model learn more general features than late layers [Yosinski et al., 2014; Asano et al., 2020], which means that early layers can extract useful features from more datasets than late layers, but late layers are more specific to some particular datasets. Inspired by this, HPFL decomposes the model as $f_m = \rho \circ g$ for each client $m$. As shown in Figure 1, $g$ is a feature extractor, and $\rho$ is a model head that outputs the final model prediction. Clients can design a new personal plug-in module $\rho_m$ (or say model head) different from the original head $\rho$, based on different computation characteristics. Then, with the frozen general feature extractor $g$, each client individually trains personalized $\rho_m$ on local datasets $D_m$ by optimizing: $$\min_{\theta^p_m} L_P(\theta^p_m) := \mathbb{E}_{\xi_m \sim D_m} \ell(\rho_m \circ g(\xi_m), \xi_m).$$ Now, each client obtains a PM $f_m = \rho_m \circ g$, which enhances the generalization performance of $\rho_m \circ g$ on $D_m$, which is usually better than original GM $f = \rho \circ g$ due to the personalization. Thus, the $\theta^p_m$ in SFL problem [5] can be constructed by $\theta^p_m$, inference becomes as $f(\theta^p_m, \xi_m) = \rho_m \circ g(\xi_m)$. Inference and selecting plug-in module. In HPFL, we define some auxiliary information $H_m$ that will be exploited to select plug-in module and propose specific forms of it in Section 4.3. When training $\theta^p_m$, $H_m$ are collected by clients and uploaded to the server. Note that as a general framework, HPFL does not limit the specific form of $H_m$, which depends on the selection method. In this paper, We introduce a distance-based selection method in Section 4.3. We discuss and analyze the potential privacy risk of sharing the plug-ins in Appendix E.1. ### 4.3 Selection Methods Decomposing the DL model also helps to avoid accessing the raw data $\xi_m \sim D_m$. With the help of the shared feature extractor $g$, we can select the $\rho_m$ based on the intermediate features $h_m = g(\xi_m)$ rather than $\xi_m$ itself. There have been some works that exploit sharing intermediate features to improve FL [He et al., 2020a; Lin et al., 2020; Luo et al., 2021; Liang et al., 2020]. Distance based methods. Intuitively, now that each $\rho_m$ is trained based on local features $h_m$, we only need to compare the similarity between $h_m$ and $h_{test} = g(\xi_{test})$, where $\xi_{test}$ is the data that... needs testing. Now, the select problem turns from equation (6) into: $$S_{dist}(d, h_{test}, \hat{h}_1, ..., \hat{h}_M) = \arg\min_{m \in M} d(\hat{h}_m, h_{test}),$$ (8) in which $\hat{h}_m = (h_m + \kappa * \epsilon)/(1 + \kappa)$, where $\epsilon \sim N(\mu_m, \sigma_m)$ is the noise to enhance privacy protection. The $\mu_m$ and $\sigma_m$ are mean and variance of features $h_m$. $\kappa$ is the coefficient controlling the relative magnitude between Gaussian noise and the features. Clients receive the noised features $\hat{h}_m$ for plug-in selection. In this selection method, the $H_m = \hat{h}_m$. We discuss the potential privacy risk of the selection method in Appendix E.2 and testify sharing the noised feature stays safe from model inversion attack. MMD measures the Hilbert-Schmidt norm between kernel mean embedding of empirical joint distributions of source and target data (Long et al., 2017). In HPFL, we utilize it to measure the distance between features that plug-in modules train on and features of test data. Note that we can also choose other distance measures. Due to page limitation, results of HPFL based on SVCCA (Raghu et al., 2017) and CKA (Kornblith et al., 2019) are shown in Appendix D. We also provide an out-of-distribution confidence based selection method and its results in Appendix D.2. 5 EXPERIMENTS 5.1 EXPERIMENT SETUP Federated Datasets and Models. We conduct experiments on four commonly used image classification datasets in FL, including CIFAR-10 (Krizhevsky et al., 2009), CIFAR-100 (Krizhevsky et al., 2009), Fashion-MNIST (Xiao et al., 2017), and Tiny-ImageNet (Le & Yang, 2015), with Latent Dirichlet Sampling (Dir) partition method ($\alpha = 0.1$ and 0.05.) to simulate the data heterogeneity following (He et al., 2020b; Li et al., 2021b; Luo et al., 2021). We also evaluate the scalability of our proposed methods with different number of clients ($M = 10$ and 100). We train ResNet-18 (He et al., 2016), MobileNet and a simple-CNN on all datasets. We run all for 1000 communication rounds, with 1 local epoch in each round. Hyper-parameters and more details are explained in Appendix C. Baselines and Metrics. We compare HPFL with classic GFL algorithm FedAvg (McMahan et al., 2017), advanced PFL algorithms including FedPer (Arivazhagan et al., 2019a), FedRep (Collins et al., 2021), PerFedMask (Setayesh et al., 2023), FedRoD (Chen & Chao, 2021) which is for both GFL and PFL, and a test-time adaption method FedTHE (Jiang & Lin, 2023). For all algorithms, we validate the learned global model (GM) on the global test dataset (GFL), and the personalized models (PM) on the personalized dataset (PFL), also PMs on GFL. Note that PFL only focuses on individually testing on local datasets instead of all datasets. More details about metrics are stated in Appendix C. 5.2 EXPERIMENT RESULTS HPFL consistently outperforms baselines in PM on GFL while comparable with classic PFL methods in classic personalized setting. As shown in Table 2, in GFL-PM setting, HPFL performs the best in all of methods and most by a large margin, even surpasses accuracies in GFL-GM in most cases, while baselines perform poorly due to a lack of adaption to the test data. We attribute the significant performance gain to adaptation to test data implemented with precise plug-in selection, which we are going to discuss in Section 5.3. It is worth noting that FedTHE also attempts to adapt its model using test data, but only with the ensemble of its locally personalized and global classifier, thus not fully utilizes the knowledge of other clients and performs worse than HPFL. In terms of GFL-GM accuracy, HPFL actually shares the same GM with GFL backbone training method (in our case, i.e. FedAvg), so its GFL-GM accuracy is exactly the same as that of FedAvg and outperforms the classic PFL algorithms only focusing on PFL performance like FedPer (Arivazhagan et al., 2019a). As for PFL-PM accuracy, our proposed method HPFL reports comparable results to the PFL baselines. HPFL maintains fairly excellent robustness against non-IID degree. As shown in Table 2, the accuracy of HPFL is not only highest in GFL-PM, but also increases when the heterogeneity increases from Dir(0.1) to Dir(0.05) in a similar way as in PFL-PM in some cases. From this phenomenon, we infer that HPFL exploits local information from clients to ensemble a model in the form of plug-ins. The server holds these local information in the form of plug-ins instead of fusing these local knowledge in a single model, thus prevents the original local information from being corrupted in model aggregation as it occurs in highly heterogeneous data, and maintains a robustness against non-IID, which is a common issue in Federated Learning. Table 2: Experiment results. Noisy coefficient $\kappa=1$. §: we focus more on GFL setting. Numbers in ForestGreen highlight highest values in GFL setting. : FedAvg fine-tunes the whole model instead of partial model as in HPFL. Plug-in selection is implemented with MMD. $E_p$ denotes the epoch of fine-tuning. \| Clients \| 10 (sample 50% each round) \| 100 (5% each round) \| \|---------\|-----------------------------\|---------------------\| \| Non-IID \| Dir(0.1) \| Dir(0.05) \| \| Test Set\| GFL§ \| PFL \| \| Method/Model \| GM PM \| PM \| GM PM \| PM \| GM PM \| PM \| GM PM \| PM \| \| CIFAR-10 \| \| \| \| \| \| \| \| \| \| FedAvg $E_p = 1^$ \| 81.5 - \| 92.5 \| 62.4 - \| 96.1 \| 73.6 - \| 90.9 \| 47.9 - \| 91.5 \| \| FedAvg $E_p = 10^$ \| 81.5 - \| 92.8 \| 62.4 - \| 92.7 \| 73.6 - \| 91.6 \| 47.9 - \| 93.4 \| \| FedPer \| 74.1 \| 40.9 \| 95.8 \| 58.7 \| 27.3 \| 96.4 \| 44.5 \| 20.6 \| 89.7 \| 24.0 \| 14.3 \| 89.9 \| \| FedRoD \| 85.3 \| 41.6 \| 94.3 \| 67.6 \| 26.8 \| 96.9 \| 74.0 \| 20.1 \| 87.4 \| 66.7 \| 15.6 \| 91.2 \| \| FedRep \| 85.1 \| 51.3 \| 95.6 \| 73.2 \| 30.2 \| 85.3 \| 66.5 \| 27.4 \| 89.3 \| 59.2 \| 20.4 \| 89.1 \| \| PerFedMask $E_p = 5$ \| 57.8 \| 23.4 \| 83.1 \| 31.8 \| 15.1 \| 83.1 \| 53.8 \| 15.6 \| 82.1 \| 35.0 \| 12.5 \| 87.6 \| \| FedTHE \| 86.4 \| 51.6 \| 90.6 \| 68.0 \| 32.6 \| 89.2 \| 74.0 \| 41.5 \| 88.3 \| 66.7 \| 43.3 \| 87.9 \| \| HPFL $E_p = 1$ \| 81.5 \| 95.4 \| 95.4 \| 62.4 \| 96.0 \| 96.0 \| 73.6 \| 88.6 \| 94.9 \| 47.9 \| 82.2 \| 93.9 \| \| HPFL $E_p = 10$ \| 81.5 \| 95.7 \| 95.7 \| 62.4 \| 96.3 \| 96.3 \| 73.6 \| 85.7 \| 95.7 \| 47.9 \| 81.8 \| 95.3 \| \| FMNIST \| \| \| \| \| \| \| \| \| \| FedAvg $E_p = 1^$ \| 86.0 - \| 98.0 \| 76.1 - \| 99.1 \| 90.2 - \| 97.2 \| 86.1 - \| 97.9 \| \| FedAvg $E_p = 10^$ \| 86.0 - \| 98.2 \| 76.1 - \| 99.1 \| 90.2 - \| 97.8 \| 86.1 - \| 98.4 \| \| FedPer \| 73.5 \| 39.0 \| 87.5 \| 64.1 \| 27.5 \| 99.1 \| 69.0 \| 29.1 \| 95.9 \| 44.8 \| 22.6 \| 96.8 \| \| FedRoD \| 87.4 \| 44.1 \| 98.1 \| 72.5 \| 29.3 \| 98.9 \| 88.9 \| 47.0 \| 98.5 \| 84.8 \| 35.3 \| 98.2 \| \| FedRep \| 87.0 \| 43.0 \| 97.5 \| 74.7 \| 39.5 \| 98.0 \| 88.2 \| 72.4 \| 97.9 \| 84.4 \| 59.6 \| 98.3 \| \| PerFedMask $E_p = 5$ \| 80.1 \| 30.8 \| 95.8 \| 47.6 \| 27.1 \| 96.9 \| 89.3 \| 23.0 \| 93.5 \| 91.9 \| 21.3 \| 96.5 \| \| FedTHE \| 87.3 \| 64.8 \| 94.6 \| 73.6 \| 39.0 \| 97.7 \| 88.6 \| 17.1 \| 93.4 \| 84.8 \| 74.7 \| 95.7 \| \| HPFL(MMD) $E_p = 1$ \| 86.0 \| 98.3 \| 98.3 \| 76.1 \| 99.0 \| 99.1 \| 90.2 \| 97.6 \| 97.9 \| 86.1 \| 81.4 \| 98.1 \| \| HPFL(MMD) $E_p = 10$ \| 86.0 \| 98.4 \| 98.4 \| 76.1 \| 99.1 \| 99.2 \| 90.2 \| 97.9 \| 98.8 \| 86.1 \| 74.1 \| 98.7 \| \| CIFAR-100 \| \| \| \| \| \| \| \| \| \| FedAvg $E_p = 1^$ \| 69.1 - \| 79.5 \| 65.3 - \| 77.4 \| 59.7 - \| 60.0 \| 47.9 - \| 69.2 \| \| FedAvg $E_p = 10^$ \| 69.1 - \| 72.3 \| 65.3 - \| 80.9 \| 59.7 - \| 66.7 \| 47.9 - \| 75.1 \| \| FedPer \| 38.6 \| 22.5 \| 74.6 \| 33.9 \| 17.8 \| 82.8 \| 13.2 \| 7.0 \| 49.1 \| 4.1 \| 2.7 \| 46.7 \| \| FedRoD \| 69.4 \| 32.5 \| 77.2 \| 67.0 \| 23.6 \| 78.5 \| 52.8 \| 11.2 \| 55.4 \| 48.4 \| 7.3 \| 66.3 \| \| FedRep \| 68.4 \| 42.6 \| 72.4 \| 65.0 \| 37.3 \| 81.2 \| 47.9 \| 18.6 \| 56.5 \| 43.3 \| 14.1 \| 65.3 \| \| PerFedMask $E_p = 5$ \| 47.3 \| 7.0 \| 40.0 \| 49.4 \| 7.0 \| 39.7 \| 41.7 \| 3.8 \| 35.8 \| 42.1 \| 3.6 \| 35.2 \| \| FedTHE \| 69.8 \| 20.5 \| 69.0 \| 66.9 \| 14.2 \| 73.2 \| 53.7 \| 7.9 \| 51.9 \| 48.4 \| 3.6 \| 60.9 \| \| HPFL(MMD) $E_p = 1$ \| 68.6 \| 74.8 \| 83.3 \| 65.3 \| 75.8 \| 87.4 \| 59.7 \| 63.8 \| 81.2 \| 47.9 \| 72.3 \| 84.1 \| \| HPFL(MMD) $E_p = 10$ \| 68.6 \| 72.2 \| 85.7 \| 65.3 \| 73.9 \| 88.8 \| 59.7 \| 55.7 \| 84.1 \| 47.9 \| 70.9 \| 86.4 \| \| Tiny-ImageNet-200 \| \| \| \| \| \| \| \| \| \| FedAvg $E_p = 1^$ \| 56.5 - \| 69.5 \| 54.9 - \| 75.3 \| 47.2 - \| 53.3 \| 42.1 - \| 58.0 \| \| FedAvg $E_p = 10^$ \| 56.5 - \| 66.8 \| 54.9 - \| 73.6 \| 47.2 - \| 67.5 \| 42.1 - \| 68.9 \| \| FedPer \| 16.3 \| 0.5 \| 13.4 \| 0.5 \| 0.5 \| 2.4 \| 1.8 \| 23.5 \| 1.3 \| 25.1 \| 1.0 \| \| FedRoD \| 57.5 \| 26.1 \| 68.5 \| 55.3 \| 12.9 \| 52.9 \| 48.6 \| 49.3 \| 9.6 \| 43.7 \| 5.9 \| 53.7 \| \| FedRep \| 56.1 \| 28.7 \| 55.4 \| 54.5 \| 31.8 \| 69.6 \| 46.4 \| 18.6 \| 52.5 \| 40.3 \| 12.8 \| 58.6 \| \| PerFedMask $E_p = 5$ \| 26.9 \| 6.6 \| 35.9 \| 23.2 \| 4.2 \| 31.3 \| 29.9 \| 1.9 \| 23.5 \| 18.7 \| 1.6 \| 32.6 \| \| FedTHE \| 57.5 \| 15.6 \| 60.4 \| 55.3 \| 14.1 \| 71.2 \| 48.6 \| 15.8 \| 55.9 \| 43.7 \| 10.3 \| 56.9 \| \| HPFL(MMD) $E_p = 1$ \| 56.5 \| 51.9 \| 70.8 \| 54.9 \| 58.5 \| 74.7 \| 47.2 \| 50.7 \| 71.3 \| 42.1 \| 47.1 \| 74.7 \| \| HPFL(MMD) $E_p = 10$ \| 56.5 \| 50.9 \| 73.7 \| 54.9 \| 58.8 \| 77.0 \| 47.2 \| 48.0 \| 73.2 \| 42.1 \| 43.9 \| 76.5 \| HPFL has excellent scalability in terms of performance in accuracy. HPFL adopts a one-client-one-plug method to better modify final inference models according to the data distribution of clients’ local data. In this way, HPFL has inherent ability to allow more clients to come and go freely in the FL system. From Table 2, we observe that other PFL methods met with extreme problems when dealing with the situation that the number of clients was larger ($M=100$), with most of the accuracies lower than 30% on CIFAR-10, 20% on CIFAR-100. However, though with a little decay in accuracy, HPFL is still applicable in the situation where the system included larger number of clients. Table 3: Results with different architectures. \| Architecture \| Mobilenet \| Simple-CNN \| \|--------------\|-----------\|------------\| \| Method/Model \| GM PM \| PM \| GM PM \| PM \| \| FedAvg \| 55.7 - \| 92.3 \| 64.6 \| 85.4 \| \| FedPer \| 53.7 \| 10.0 \| 10.0 \| 44.1 \| 27.6 \| 85.5 \| \| FedRoD \| 76.3 \| 36.1 \| 92.3 \| 67.1 \| 28.8 \| 83.5 \| \| FedRep \| 74.1 \| 35.8 \| 85.0 \| 54.6 \| 10.0 \| 10.0 \| \| PerFedMask \| 13.0 \| 19.0 \| 76.4 \| 31.0 \| 10.0 \| 50.5 \| \| FedTHE \| 76.3 \| 36.1 \| 92.3 \| 67.1 \| 28.8 \| 83.5 \| \| HPFL \| 55.7 \| 92.8 \| 92.8 \| 64.6 \| 87.8 \| 87.8 \| A generalized framework applicable to different model architecture. As a general FL framework, HPFL can be seamlessly applied to model architectures where parameter decoupling is available. We deploy it on three different model architectures (ResNet-18, MobileNet (Howard et al., 2017), and a simple-CNN structure whose architecture is the same as simple-CNN in (Tang et al., 2022)), and HPFL outperforms baselines we use in the main experiment with all of the architectures, showing that HPFL can be extensively employed in different FL systems and improve their performance of GFL and adaptation ability to new clients. Results are in Table 3. Moreover, HPFL can exploit backbones trained with all kinds of GFL algorithms. An ablation study on GFL methods used to learn feature extractor of HPFL is demonstrated in Appendix D.6. A win-win deal: Efforts to protect privacy is not contradictory to the performance of HPFL. In HPFL, clients share auxiliary information with the server, which may raise privacy concern. To protect clients from the risk of data breaches during communication or improper storage on the server, we add noise to the auxiliary information. However, we surprisingly found that noise will not damage the performance of HPFL as shown in Table 4. We attribute the robustness toward noise to robust selection method of HPFL, which we study later in Section 5.3. Results of the model inversion attack against HPFL are shown in Appendix E. The more flexible the models are, the better? As shown in Figure 2, the accuracy of HPFL continuously decreases with the increasing number of plug-in layers, we propose two possible reasons leading to the phenomenon: (1) local clients’ samples are not sufficient for training big-scale plugs, resulting severe overfitting issue, and (2) The selection methods may not be suitable for middle features. However, according to Table 2, we believe that fine-tuning larger plug-ins does not lead to such a performance degradation, because FedAvg fine-tunes on the whole model without significant performance loss. Therefore, it is natural to give attention to the potential trouble big plug-ins may cause in plug-in selection. In Section 5.3, we conduct experiments to testify the speculation that the performance loss when increasing the plug-in layer is mainly due to the degradation of plug-in selection. Due to the page limit, we aim to provide an intuitive explanation in Appendix D.1. 5.3 Selection Accuracy Plug-in selection plays an important role in HPFL, so here we study how it is affected by the magnitude of noise added on features and the number of plug-in layers. Experiments in this section are carried out with $\alpha=0.1$, $M=10$ on CIFAR-10 dataset, we include the results of additional configurations in Appendix D.3. We observed the expected phenomenon conforming to our conjecture in Section 5.2 that it is harder for selection methods to correctly select plug-ins with more layers. With the increasing number of plug-in layers, the score map gradually begins to change. However, until it actually start to influence the result of selection, the performance of HPFL gets unaffected. Observed from Figure 3, despite the slight variation in the heatmaps of MMD score with the noise coefficient, selecting plug-in with the lowest MMD score instead of combining plug-ins with MMD score adds robustness towards noise to HPFL. The accuracy shows in Table 4. 5.4 Federated Continual Learning Federated continual learning (FCL) (Yoon et al., 2021) is a new problem where clients join FL training after initial training. The trained model must retain previous dataset knowledge and perform well on data from newly arrived clients. HPFL can address the forgetting problem of FCL by preserving previous training knowledge in a personalized plug-in and providing it for client inference as shown in Table 5. It is an application of HPFL on the temporal scale, where clients collaboratively learn models that generalize well over time. We present more details about the experiment and discussion in Appendix D.5. \| Test data \| GFL \| \|-----------\|-----\| \| Method/Model \| GM PM \| \| Naive FCL \| 69.5 \| 58.4 \| \| FCL under HPFL \| 62.2 \| 80.9 \| Table 5: Results of FCL Figure 3: Selection score maps with different noise coefficient. Blocks with green anchor mean the corresponding client selects the plug-in and download it. Blocks with green anchor lying in diagonal indicate that clients choose plug-ins of themselves when met with their own test data, which conforms to the aim of selection methods. Figure 4: Selection score maps with different number of plug-in layers 6 LIMITATIONS Accurate Plug-in Selection. As an initial trial, our proposed plug-in selection methods select sub-optimal plug-ins in some circumstances as shown in Figure 4, 3, 12 and 9 etc.. Future works may consider to design more accurately and robust selection methods. Training The Feature Extractor. In this work, we only consider using the classic GFL algorithm FedAvg to train the feature extractor while achieving superior performance. Designing methods to obtain a better feature extractor will be an important direction to enhance the practicality of HPFL. 7 BROADER IMPACT Federated continual learning. As shown in Section 5.4, HPFL can effectively tackle the forgetting problem in FCL, benefit from its ability to losslessly maintain the knowledge learned in a dataset and recover it when in need. The superiority of HPFL meets the need of FCL: FCL can be regarded as a distribution shift problem at Federated Learning within the temporal scale since the distribution of training data shifts as participants of FL change with time. One-shot FL. Once an average backbone is accessible like a pre-trained model, HPFL is able to directly train plug-ins in a single communication round and go straight into the inference stage. The same procedure also applies to the situation where a new client takes part in the FL system. Anarchic FL. In anarchic FL (Yang et al., 2022), clients can decide to join or quit training at any time, which severely harms FL convergence. To this end, HPFL naturally allows this kind of working paradigm. Like one-shot FL, once the backbone is accessible, any aggregation operation is not in demand for HPFL, so the server does not rely on timely responses of clients and will not be disturbed by stale model updates. Clients can finish training and uploading plug-ins at any time. FL plug-in market. HPFL provides the possibility of constructing a more free and transparent model market, and customers can have better confidence knowing the plug-in they are purchasing is able to meet their requirements with a fair plug-in selection mechanism. Plug-in providers can obtain commercial benefits from this market. 8 CONCLUSION In this paper, we explore how to improve the generalization performance when PMs meet test data from other clients. We formalize the SFL to bridge the GFL and PFL together. Then, We propose HPFL to practically solve the SFL. We verify the effectiveness and robustness of HPFL through comprehensive experiments. And we further experimentally verify the remarkable potential of HPFL to resolve other practical FL problems like FCL. Future work can consider to explore new plug-in selection methods, or applying HPFL into more FL related problems. REFERENCES Manoj Ghuhan Arivazhagan, Vinay Aggarwal, Aaditya Kumar Singh, and Sunav Choudhary. Federated learning with personalization layers. CoRR, abs/1912.00818, 2019a. URL http://arxiv.org/abs/1912.00818. Manoj Ghuhan Arivazhagan, Vinay Aggarwal, Aaditya Kumar Singh, and Sunav Choudhary. Federated learning with personalization layers. CoRR, abs/1912.00818, 2019b. 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648Mq6Neuo	Comparison with baseline methods is also quite lacking. The authors identify several AD with type control methods in the literature review but only considers RedPanda among them in the experiments, and that is only in one experiment setting.	GUIDE YOUR ANOMALY WITH LANGUAGE Anonymous authors Paper under double-blind review ABSTRACT Anomaly detection is the task of identifying data that is different from what is considered normal. Recent advances in deep learning have improved the performance of anomaly detection and are used in many applications. However, it can be difficult to create a model that reflects the desired normality due to various issues, including lack of data and nuisance factors. To address this, there have been studies that provide the desired knowledge to the model in various ways, but there are limitations, such as the need to understand deep learning. In this work, we propose a method to guide the desired normality boundary in an image anomaly detection task using natural language. By leveraging the robust generalization capabilities of the vision-language model, we present Language-Assisted Feature Transformation. LAFT transforms image features to suit the task through natural language using the shared image-text embedding space of CLIP. We extensively analyze the effectiveness of the concept on a toy dataset and show that it works effectively on real-world datasets. 1 INTRODUCTION 1.1 MODELING NORMALITY FOR ANOMALY DETECTION Anomaly detection is the task of distinguishing abnormal data that are different from normal data. With the recent development of deep learning, the performance of anomaly detection has improved considerably and is widely used in applications such as industrial anomaly detection and video anomaly detection. To detect abnormalities effectively, deep learning models should be able to learn the concept of normality. Typically, the user provides the model with normal samples to learn from. However, it can be challenging to obtain all the possible variations of the samples and to differentiate anomalies due to nuisance factors in the data (Cohen et al., 2022). In practical applications, there are cases where the model should pay attention to or disregard certain attributes. Here are some motivating examples. (1) When inspecting a product from an image, users may only be interested in the shape of the product, not its position, angle, or setting in which it was taken. In this situation, the model should focus solely on the shape of the product. (2) When performing anomaly detection in CCTV, the change in brightness is irrelevant and only the content such as the movement of the object is important. (3) There are also situations where it is difficult to distinguish anomalies due to entangled attributes. For example, the background and birds are entangled in the Waterbirds dataset (Sagawa et al., 2019). In order to address this issue, there have been attempts to generate additional data through data augmentation or data generation to better learn the decision boundary (Zavrtanik et al., 2021; Li et al., 2021; Du et al., 2021). The aim of these methods is to create samples more diverse than what is available, so that the model can more accurately distinguish between normal and abnormal data. However, these methods should be able to generate the desired normality boundary by adding the characteristics of outliers through appropriate augmentation or generation techniques. Furthermore, there have been attempts to make models learn task-specific feature representations (Chen et al., 2020a;b; Caron et al., 2020) and apply them to anomaly detection to better learn normality at the feature level (Hyun et al., 2023). To make use of pre-trained backbones that are trained task-agnostic, there have been studies that involve fine-tuning the feature extraction backbone or creating task-specific features through feature transformation (Caron et al., 2020; Reiss & Hoshen, 2023; Tack et al., 2020). However, the downside is that it is costly to fine-tune the backbone or train the transformation, and it is difficult to properly learn the desired anomaly at the feature level. 1.2 Vision-Language Models in Anomaly Detection Research in the field of natural language processing has shown the effectiveness of training models with extensive, unlabeled Internet data, and this approach has also been applied to computer vision (Radford et al., 2021; Jia et al., 2021; Desai et al., 2023). They demonstrated the effectiveness of using image-text pairs obtained from the Internet to pre-train models, integrating natural language description to enhance the quality of image representations. Models trained at scale in this manner can establish connections between visual concepts in images and natural language descriptions, aligning image and text features within a shared embedding space. These models can extract remarkably general representations and show impressive performance in downstream tasks. Many researchers are trying to use the powerful performance of vision-language models in the field of image anomaly detection. In particular, there are studies that apply vision-language models to industrial anomaly detection (Jeong et al., 2023; Cao et al., 2023; Chen et al., 2023), or general image out-of-detection tasks by utilizing the characteristics that vision-language models can be applied to downstream tasks with zero-shot using text prompts (Ming et al., 2022; Miyai et al., 2023). The advantage is that human prior knowledge can be fed to the model using text prompts, allowing for zero-shot use without training images. Models that take this approach usually define normality using text prompts and calculate anomaly scores using the similarity of text and image features. However, in some cases, it is difficult to define normality using natural language alone, and it is common to use reference image features in conjunction with text features to define normality. Comparison of the reference image and the target image is done at the feature level, which means that even a slight difference between the two images can cause a drastic change in similarity and reduce scalability. 1.3 Guide Your Anomaly with Language As discussed in subsection 1.1, it is difficult to define the boundary of in-distribution using only images, and a thorough understanding of deep learning is also required. This includes the selection of expertly designed image transformations to reflect user knowledge. In subsection 1.2, we discussed the difficulty in defining normality using only language or a few reference images. In this paper, we propose a method that enables users to define the boundaries of normality for images using language, taking advantage of the properties of CLIP (Radford et al., 2021). Our approach is different from the majority of existing work, as it relies mainly on image features to define normality, with language playing a supporting role. By using language, users can “guide” normality, giving them more flexibility to incorporate their knowledge of what is normal. Additionally, by setting the boundaries of normality with the image features, we can accurately distinguish between normal and abnormal images. We summarize our contributions as follows: 1. We propose Language Assisted Feature Transformation (LAFT), a method that uses natural language to transform image features to suit the task at hand. This is achieved by taking advantage of the strong generalization capabilities of a vision-language model and an image-text aligned embedding space. 2. We introduce LAFT AD, a method for anomaly detection that can focus or ignore image attributes in natural language, using LAFT. 3. We extensively examine the performance of our method on a simple dataset and demonstrate that it is successful on real-world datasets. 2 RELATED WORK Image anomaly detection with vision-language model Starting with Radford et al. (2021) as a basic vision language model, the field of image anomaly detection has seen remarkable progress. Ming et al. (2022) introduced a novel scoring method, which was refined in the updated version, (Miyai et al., 2023), to improve the accuracy of anomaly detection. To address the challenge of out-of-distribution detection, Ming & Li (2023) proposed a parameter-efficient training approach, highlighting the nuances of fine-tuning for this task. In Fort et al. (2021), a method of feeding potential out-of-distribution labels to the CLIP text encoder was introduced. In addition, Esmaeilpour et al. (2022) presented a strategy for training a label generator based on the CLIP image encoder for out-of-distribution detection, although it focused primarily on small inputs. Anomaly detection with type control In addressing anomaly detection with type control problem, several methodologies have been explored. Wang et al. (2022) delves into disentangling the factors of variation in the data. The with-language methodology, as illustrated in El Banani et al. (2023) employs contrastive representation learning guided by a vision-language model, improving feature learning for anomaly detection. Cohen et al. (2022) advocates for labeling all attributes, providing a structured framework to potentially improve the robustness of anomaly detection models. However, Reiss et al. (2023) underscores an inherent limitation, emphasizing that no single method can be universally applied to all anomaly detection problems, thus necessitating a nuanced, problem-specific methodology in this domain. Feature adaptation In the context of feature transformation for anomaly detection, several strategies have been developed to enhance the adaptability and robustness of backbone models. Ruff et al. (2018) have a different strategy, starting with the pre-training of a representation encoder through autoencoding on regular data, creating a basis for subsequent anomaly detection activities. Chen et al. (2020a;b) effectively employs contrastive pre-training to facilitate feature agreement, particularly advantageous for downstream anomaly detection. Caron et al. (2020) utilizes prototype vectors for contrastive training of similar features, leading to the refinement of feature representations. Subsequently, these approaches are adapted for One-Class Classification (OCC) objectives using techniques such as those proposed by Reiss et al. (2021); Hyun et al. (2023); Reiss & Hoshen (2023). However, the adaptation process often faces challenges, including the issue of catastrophic collapse. 3 PRELIMINARIES In our scenario, the training set, represented as $ D_{\text{train}} $, comprises solely of normal samples. We then define the normality within the image features. Our evaluation set $ D_{\text{test}} $ consists of normal and anomalous samples. The attribute labels $(0 \leq j < m)$ for a test image $ x_i $ are denoted as $ y_i = (y_0, \cdots, y_{m-1})_i $. The $ m $ attributes can be divided into relevant $(0 \leq j < n)$ and irrelevant $(n \leq j < m)$ categories, with examples such as the object’s identity, color saturation, and background noise representing different attributes. We assume that $ n $ is not a fixed number but uncertain and that the anomaly label is always a function of (potentially) all relevant attributes $ y_i = f^a(y_0, \cdots, y_{n-1}) $. That is, the nuisance attribute $ y_n, \cdots, y_{m-1} $ never affects the anomaly label $ y_i $. We emphasize that in our described setting, neither the relevant attribute labels nor the anomaly labels are given. Our goal is to transform the feature vector $ T(f_i) = T(f(x_i)) \in \mathbb{R}^d $ using the transformation function $ T $ into a target space that significantly distinguishes between normal and abnormal data, implying that $ T $ distills the information of irrelevant features. That is, we desire our transformation function to represent the relevant attributes in a manner unaffected by the nuisance attributes: \[ p(y_n, \cdots, y_{m-1}) = p(y_n, \cdots, y_{m-1}\|T(f_i)). \] We also wish our code to be informative - to represent sufficient information regarding our relevant attributes ($ I(\cdot; \cdot) $ is the mutual information between its two arguments): \[ I((y_0, \cdots, y_{n-1}); f_i) \sim I((y_0, \cdots, y_{n-1}); T(f_i)). \] In practice, invariance can be measured by the accuracy of predicting $ y_i = (y_{n_1}, \ldots, y_{m-1})_i $ from the transformed code $ T(f_i) $. But, we can assess the informativeness by measuring the accuracy of predicting the relevant attribute utilized to define anomalies. Empirical evaluations of these measures for our datasets can be found in the next section. With such a representation, we may later evaluate anomalies independently, devoid of any bias caused by the irrelevant attribute we aim to disregard. CLIP (Radford et al., 2021) embeds the features in a unit sphere subspace in Euclidean space $ \mathbb{R}^n $. An embedding vector of an image is correlated to the text embedding describing the image. This means that we can construct the transform with the CLIP text encoder $ T_{text} $. We assume that all relevant and irrelevant features can be encoded with the text description, so that natural language assists the manipulation of the vector in the CLIP space: \[ I((y_0, \ldots, y_{n-1}); f_i) \sim I((y_0, \ldots, y_{n-1}); T_{text}(f_i)). \] (3) 4 Method Our main goal is to transform visual features with text guidance without any further training. Typical learning-based methods need to collect data to get a feasible normality space and require many computing powers to train deep neural networks. In this section, we describe a way to distill undesirable attributes with vector projection with the help of provided text prompts. 4.1 Text Prompt To enable the model to focus on or ignore certain attributes of the image, it is necessary to provide the model with proper textual prompts. Similar to (Ming et al., 2022), we assume that the text contains the “concept prototype” for the attributes. So we give the model a list of prompts, each prompt consisting of the following form: TEMPLATE + ATTRIBUTE_VALUE. For example, if we want to ignore the color of the hair, we can construct the prompt as follows: - “a photo of a person with brown hair” - “a photo of a person with black hair” - “a photo of a person with blond hair” - “a photo of a person with gray hair” - ... Using the actual values of the desired attribute in the prompts, we want the model to know the difference between the visual concepts associated with the attribute. Providing values corresponding to this attribute that are not actually in the training set but are likely to appear at test time makes it easier to construct a subspace for that concept. This would be prior user knowledge. As with other language-based methods, you can use multiple types of template to mitigate the bias introduced by the template itself. 4.2 Find Concept Subspace After the user gives a prompt, our method tries to find the subspace of visual concepts in which the attribute exists from the given prompt. Specifically, we find the axis along which the variance between concepts is represented by the concept difference between prompts. For prompts $ t_i $ and $ t_j $ where $ 1 \leq i < j \leq n $, we compute the pairwise difference: \[ v_{ij} = E_{text}(t_i) - E_{text}(t_j) \] where $ n $ is the number of prompts and $ E_{text} $ is the CLIP’s text encoder. We call these vectors “concept axis”. But directly using these vectors as basis is not preferable because the text prompt itself may contain irrelevant information about the target attribute. So, we extract the principle axis from these vectors using PCA: \[ \{g_k\} = \text{PCA}(\{v_{ij}\}, d) \] where $ d $ is the number of components and $ \{g_k\} $ is the $ d $ principal axes and named a set of guidance vectors. Throughout the paper, we typically choose $ d $ from 4 to 32 when guiding an attribute and from 32 to 384 when ignoring an attribute. We construct the attribute subspace using these principal axes as basis vectors. ### 4.3 Feature Transformation with Projection For all image feature vectors $ f_i = f(x_i) $ encoded by the CLIP image encoder, we project the features with the guidance vector $ g_k $: \[ \hat{f}_i = <f_i, g_k> g_k, \] (4) where the notation $ <\cdot, \cdot> $ is the inner product. This projection cancels out the other direction in the context of suppressing irrelevant attributes. Without loss of generality, we can further enlarge the number of relevant attributes to two or more. Then from the attributes, we can generate the guidance vectors $ g_k $. Then we can project on the space generated by the $ g_k $'s or write it: \[ \hat{f}_i = \sum_{k=1}^{m} <f_i, g_k> g_k, \] (5) where $ m $ is the number of guidance vectors. On the other hand, we can also ‘relaxation’ to the irrelevant attributes using orthogonal projection. Let $ \bar{g}_k $ be the guidance vectors generated by the irrelevant attributes. Then we can orthogonally project onto the space generated by $ \bar{g}_k $'s: \[ \hat{f}_i = f_i - \sum_{k=1}^{\bar{m}} <f_i, \bar{g}_k> \bar{g}_k. \] (6) Contrary to the inner project, we can manually cancel out the vectors of irrelevant attributes. By doing so, the specification of the normality can be gained in our anomaly detection task. ### 4.4 Density Based Anomaly Scoring We operate under the assumption that the mapping will place anomalous samples in areas of sparse concentration, while normal data will be allocated to areas of dense concentration, resembling the behavior observed in other anomaly detection methodologies. In a scenario where the representation is exclusively composed of relevant attributes, it’s likely that regions exhibiting low density would correspond to samples characterized by uncommon relevant attributes, which are likely to be classified as anomalies. To numerically estimate the density of the normal data around each test sample, we use the $ k $-nearest-neighbors algorithm ($ k $NN). We begin by extracting the representation for each normal sample: $ f^t_i = T(f(x_i)), \forall x_i \in D_{train} $. Next, for each test sample, we infer its latent $ f^t_{test} = T(f(x_{test})) $. Finally, we score it by the distance $ k $ NN from the normal data: \[ S(x_t) = \frac{1}{k} \sum_{f^t_i \in N_k(f^t_t)} \text{sim}(f^t_i, f^t_t) \] (7) where $ N_k(f^t_t) $ denotes the $ k $ most similar relevant attribute transformed feature vectors in the normal data. We use $ k = 30 $ throughout paper for $ k $NN. We note that the high dimension of the latent space allows us to distinguish between high and low-density areas of the distribution of normal data. Figure 3: The features of the train and test images are mapped linearly onto two sets of axes: (left) the PCA axes and (right) the concept axes. We calculate both axes using image features from the auxiliary train set (not plotted) and reduce attribute values for visualization. 5 EXPERIMENTS 5.1 SETUP Models and Prompts Throughout the paper, we use the CLIP ViT-B/16 model with the pre-trained checkpoint from OpenAI$^1$. For a fair comparison, we also adopted the CLIP ViT-B/16 image encoder as a feature extractor for the baseline methods. We also use the same text prompts for the methods using the CLIP text encoder and our method. See Appendix A for details of the prompts we used in the experiments. Datasets To validate our approach, we employ the colored version of the MNIST (LeCun et al., 2010), Waterbirds (Sagawa et al., 2019), and CelebA (Liu et al., 2015) datasets. We set the normal and abnormal values for each attribute of the dataset and divide the train split into $2^m$ subsets. For instance, in the Colored MNIST dataset, we designate the numbers 0-4 as normal and 5-9 as anomalous, and the color red as normal and the color green and blue as anomalous. We then use one subset as a train set that is considered normal in all $m$ attributes (e.g. 0-4 and red). We consider two scenarios: a standard image anomaly detection task that does not have access to abnormal or external samples, and a more relaxed setting that can use a few abnormal or external samples. In other words, in the latter situation, the model can use a few samples from a different subset in the training set (e.g. 0-4 and blue). This setting is similar to outlier exposure (Liznerski et al., 2022). When using the auxiliary dataset of a few shots, we randomly sample the data of $k$ in each subset. Baselines We use $k$NN for our method and as a baseline. We directly use image features from the CLIP image encoder to compute $k$NN distances. For our method, L.AFT AD, we transform the image features as described in section 4 before $k$NN. Mean-Shifted Anomaly Detection (MSAD; Reiss & Hoshen, 2023) transforms the pre-trained representations to better fit the anomaly detection in an unsupervised manner. We also consider CLIP-based anomaly detection methods. Maximum Concept Matching (MCM; Ming et al., 2022) is a CLIP based out-of-distribution detection method. This only requires prompts for normal images for anomaly scoring. On the other hand, the method proposed by Fort et al. (2021) and Zero-shot OOD detection based on CLIP (ZOC; Esmaeilpour et al., 2022) uses candidate prompts for anomalous images. The main difference from the original ZOC method is that we will provide prompts about unseen candidates of attributes instead of generating them using an image description generator. When we can use auxiliary samples, we train Linear Probe as in Radford et al. (2021). Also, we compared our method with Red PANDA (Cohen et al., 2022) which can learn to ignore specific attributes in the image. Metrics We use three metrics to evaluate the performance of the methods: the Area Under Receiver Operating Characteristics (AUROC), the Area Under the Precision Recall Curve (AUPRC), and the False Positive Rate at the 95% true positive rate (FPR95). AUROC and FPR95 are commonly used for the anomaly detection or out-of-distribution detection task (Ming et al., 2022). And we also use AUPRC because some datasets are imbalanced, with a significant disparity between the number of normal and abnormal examples. --- $^1$https://github.com/openai/CLIP Table 1: The anomaly detection performance on Colored MNIST dataset. We do not use additional data other than normal training set. For details, please refer to the main text. \| Method \| Anom. Prompt \| Number AUPRC ↑ \| Number FPR95 ↓ \| Color AUPRC ↑ \| Color FPR95 ↓ \| \|-----------------\|--------------\|----------------\|----------------\|---------------\|---------------\| \| No guidance \| - \| 0.880 \| 0.879 \| 0.617 \| 0.817 \| \| kNN \| - \| \| \| \| \| \| MSAD \| - \| 0.582 \| 0.551 \| 0.757 \| 1.000 \| \| Guide \| \| \| \| \| \| \| CLIP (MCM) \| X \| 0.549 \| 0.499 \| 0.877 \| 0.892 \| \| CLIP (ZOC) \| O \| 0.981 \| 0.982 \| 0.112 \| 1.000 \| \| LAFT AD (ours) \| O \| 0.989 \| 0.990 \| 0.066 \| 1.000 \| \| \| △ \| 0.984 \| 0.985 \| 0.089 \| 1.000 \| \| Ignore \| \| \| \| \| \| \| LAFT AD (ours) \| O \| 0.938 \| 0.929 \| 0.279 \| 0.989 \| \| \| △ \| 0.935 \| 0.925 \| 0.293 \| 0.991 \| 5.2 Results on Colored MNIST We used the colored version of the MNIST dataset (LeCun et al., 2010), similar to Arjovsky et al. (2019), to demonstrate our concept in the simplest way. We create a dataset that divides each digit of the MNIST and colors each split with red, green, and blue. In this way, the image of a colored MNIST consists of two attributes: number and color. We mark the numbers 0 to 4 as normal and the numbers 5 to 9 as abnormal. In addition, we label red as normal and green and blue as abnormal colors. In this setting, the training set consists of 0 to 4 and red images. We assume that CLIP has learned visual concepts from a sufficiently large variety of image captions so that it can place images according to their degree of concept for a given concept axis. While MCM (Ming et al., 2022) assumes the text feature itself as “concept prototype”, we use the “pairwise difference of concept prototypes” to find this axis. Figure 3 shows the brief overview of our desired transformation using the concept axis. If we choose one axis (number or color), we can just use kNN to detect anomaly with guidance to specific attributes. Table 1 shows the main results on Colored MNIST. The table is divided into three groups: “no guidance”, “guide”, and “ignore”. The “no guidance” group shows the performance of anomaly detection methods that do not provide any guidance and can be thought of as the default performance for that attribute. The “guide” group displays the performance of methods that can be directed to focus on a particular attribute, so it shows the performance when guided to the attribute corresponding to the label. The “ignore” group shows the performance of disregarding attributes other than the one to be evaluated, so it shows the performance of ignoring attributes that do not correspond to the label. For example, performance in the attributes “Number” means ignoring the attribute “Color” and vice versa. And the “Anomaly Prompt” column indicates the method uses the text prompt for anomalous attributes. The O means that the method uses exact anomaly prompts (in this case, ‘green’ and ‘blue’ for color attributes), and the △ means that the method also uses other candidate anomaly prompts (e.g. ‘purple’, ‘orange’, ‘black’, etc.). This format is used throughout the paper. As can be seen from the table, the performance of the guidable methods is generally higher than the performance of the non-guidable methods, and our method has the best performance among them. It is notable that ZOC’s performance drops significantly when given candidate anomaly prompts in addition to the exact anomaly prompts, while our method does not make much difference. This is a problem mentioned in Ming et al. (2022), where the performance of the methods that calculate anomaly scores based on image-text similarity in CLIP is highly affected by inaccurate prompts. In contrast, our method uses the prompt to calculate only the transformation of the image features (Equation 5), and the normality is actually determined through the images, so we can see that the performance is similar even with some inaccurate prompts. The important thing in image anomaly detection is to find anomalies that are different from normal images, so we can verify that our approach works effectively. We can also observe that when we use our method to guide one attribute, the other attributes are actually ignored, which we summarize further in Appendix B. In the “ignore” group of the table, it shows that ignoring one attribute increases the performance for the other attribute. This is because the Colored MNIST very clearly consists of only two attributes. In real-world datasets, the behavior is slightly different, which we discuss further in the following sections. In summary, on the simple Colored MNIST dataset, we demonstrated that our method can leverage language to provide guidance on normality without additional training. Table 2: The anomaly detection performance on Waterbirds dataset. We do not use additional data other than normal training set. For details, please refer to the main text. \| Method \| Anom. Prompt \| Bird AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| Background AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| \|-----------------\|--------------\|--------------\|---------\|---------\|--------------------\|---------\|---------\| \| No guidance \| \| 0.772 \| 0.893 \| 0.618 \| 0.704 \| 0.651 \| 0.849 \| \| kNN \| - \| \| \| \| \| \| \| \| MSAD \| - \| 0.615 \| 0.275 \| 0.833 \| 0.855 \| 0.826 \| 0.504 \| \| Guide \| \| \| \| \| \| \| \| \| CLIP (MCM) \| X \| 0.867 \| 0.946 \| 0.468 \| 0.845 \| 0.836 \| 0.619 \| \| CLIP (ZOC) \| ○ \| 0.927 \| 0.971 \| 0.276 \| 0.961 \| 0.963 \| 0.231 \| \| \| △ \| 0.920 \| 0.966 \| 0.363 \| 0.951 \| 0.952 \| 0.315 \| \| LAFT AD (ours) \| ○ \| 0.945 \| 0.981 \| 0.242 \| 0.970 \| 0.973 \| 0.179 \| \| \| △ \| 0.933 \| 0.972 \| 0.267 \| 0.962 \| 0.966 \| 0.213 \| Table 3: The anomaly detection performance on Waterbirds dataset. We use a few or full additional data other than normal training set for training. For details, please refer to the main text. \| Method \| # Shots / Subset \| Bird AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| Background AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| \|-----------------\|------------------\|--------------\|---------\|---------\|--------------------\|---------\|---------\| \| Guide \| \| \| \| \| \| \| \| \| Linear Probe \| Full \| 0.756 \| 0.555 \| 0.714 \| 0.969 \| 0.973 \| 0.136 \| \| LAFT AD (ours) \| 0 \| 0.945 \| 0.981 \| 0.242 \| 0.970 \| 0.973 \| 0.179 \| \| + CoOp \| 1 \| 0.934 \| 0.974 \| 0.259 \| 0.953 \| 0.961 \| 0.288 \| \| \| 4 \| 0.943 \| 0.982 \| 0.203 \| 0.976 \| 0.976 \| 0.126 \| \| \| 8 \| 0.945 \| 0.980 \| 0.207 \| 0.981 \| 0.983 \| 0.097 \| \| \| 16 \| 0.947 \| 0.980 \| 0.201 \| 0.983 \| 0.984 \| 0.084 \| \| \| 128 \| 0.954 \| 0.983 \| 0.177 \| 0.991 \| 0.992 \| 0.034 \| \| Ignore \| \| \| \| \| \| \| \| \| Red PANDA \| Full \| 0.612 \| 0.610 \| 0.882 \| 0.717 \| 0.722 \| 0.824 \| \| LAFT AD (ours) \| 0 \| 0.773 \| 0.889 \| 0.559 \| 0.693 \| 0.856 \| 0.735 \| \| + CoOp \| 1 \| 0.852 \| 0.941 \| 0.505 \| 0.673 \| 0.609 \| 0.789 \| \| \| 4 \| 0.885 \| 0.947 \| 0.389 \| 0.891 \| 0.897 \| 0.523 \| \| \| 8 \| 0.903 \| 0.958 \| 0.326 \| 0.932 \| 0.935 \| 0.344 \| \| \| 16 \| 0.918 \| 0.969 \| 0.305 \| 0.952 \| 0.955 \| 0.258 \| \| \| 128 \| 0.932 \| 0.976 \| 0.252 \| 0.971 \| 0.974 \| 0.167 \| 5.3 Results on Waterbirds The Waterbirds dataset (Sagawa et al., 2019) is commonly used in studies focused on spurious correlation and representation disentanglement. The dataset consists of two primary attributes: bird (waterbird / landbird) and background (water / land). Naturally, the training set has a very strong correlation between birds and backgrounds, whereas the test set has an equal ratio of birds to backgrounds. We specify waterbirds and water backgrounds as a normal training set. Table 2 summarizes the results for the dataset. The trends observed in the Colored MNIST experiment remain consistent in the results, demonstrating that our method is applicable to real-world datasets. The one difference is that ignoring one attribute does not directly improve the performance of other attributes, as shown in the ignore group of Table 3. To improve performance, we employ the prompt learning technique Context Optimization (CoOp; Zhou et al., 2022), in order to accurately capture the concept difference without prompt bias. See Appendix A for the details of CoOp. To train the prompt, we randomly select a few auxiliary samples from each train subset. In the practical application, we are often able to acquire samples that are not in the training set, and we can benefit from this. The results are summarized in Table 3. To use as a baseline, we trained Linear Probe and Red PANDA using all the data in the train set. Based on our findings, it is clear that using the image features directly from CLIP and training a linear classifier does not outperform our model. This demonstrates that properly transforming the features is more effective for the desired downstream task. Furthermore, we observed that with only four samples for each subset, we are able to effectively learn the prompt to further improve the performance. These findings are consistent with those reported in (Zhou et al., 2022), thus highlighting the effectiveness of combining CoOp with our method. Additionally, as addressed in (Cohen et al., 2022), fine-tuning of the feature extractor is limited in real-world datasets. Table 4: The anomaly detection performance on CelebA dataset. We do not use additional data other than normal training set. For details, please refer to the main text. \| Method \| Anom. Prompt \| Blond \| Eyeglasses \| Young \| \|-----------------\|--------------\|-------\|------------\|-------\| \| \| \| AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| \| No guidance \| - \| 0.865 \| 0.974 \| 0.541 \| 0.778 \| 0.185 \| 0.677 \| 0.701 \| 0.477 \| 0.563 \| \| kNN \| - \| 0.827 \| 0.964 \| 0.637 \| 0.742 \| 0.969 \| 0.659 \| 0.528 \| 0.291 \| 0.974 \| \| MSAD \| - \| \| \| \| \| \| \| \| \| \| \| Guide \| \| \| \| \| \| \| \| \| \| \| \| CLIP (MCM) \| △ \| 0.848 \| 0.972 \| 0.709 \| 0.323 \| 0.044 \| 0.989 \| 0.460 \| 0.234 \| 0.967 \| \| CLIP (ZOC) \| △ \| 0.908 \| 0.980 \| 0.642 \| 0.989 \| 0.963 \| 0.003 \| 0.760 \| 0.592 \| 0.713 \| \| LAFT AD (ours) \| △ \| 0.930 \| 0.987 \| 0.351 \| 0.989 \| 0.923 \| 0.038 \| 0.798 \| 0.634 \| 0.748 \| Table 5: The anomaly detection performance on CelebA dataset. We transform the image features using LAFT to ignore the Male attribute. Then, we use the transformed features for the ZOC anomaly detection. For details, please refer to the main text. \| Method \| Blond \| Eyeglasses \| Male \| \|-----------------\|-------\|------------\|------\| \| \| AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| AUROC ↑ \| AUPRC ↑ \| FPR95 ↓ \| \| CLIP (ZOC) \| 0.908 \| 0.980 \| 0.642 \| 0.989 \| 0.963 \| 0.003 \| 0.996 \| 0.997 \| 0.010 \| \| + LAFT \| 0.916 \| 0.982 \| 0.531 \| 0.989 \| 0.957 \| 0.008 \| 0.508 \| 0.618 \| 0.996 \| 5.4 Results on CelebA To verify that our method works in multi-attribute settings, we use the CelebA dataset (Liu et al., 2015), which contains over 200K celebrity images with 40 attribute labels. For the normal training set, we select three attributes: Blonde_Hair, (No) Eyeglasses, and Young. The results are displayed in Table 4. While the tendencies of Blonde_Hair and Young are similar to previous experiments, the results of Eyeglasses are slightly different. This is because CLIP can classify almost perfectly whether a person is wearing glasses or not. Therefore, using images to define normality provides disturbing information for the attribute. And, notably, the performance for the Young attribute is not good for all models. Similar results are also reported in Gannamaneni et al. (2023). This suggests that CLIP may have difficulty conceptualizing age, a limitation that also affects our method, which relies on CLIP’s image-text alignment. 6 Limitations and Discussion Ignore attributes using LAFT Unlike in a simple Colored MNIST dataset, we observe that ignoring one attribute using LAFT without CoOp does not improve the anomaly detection performance of the other attribute in real-world datasets. However, as seen in Appendix B, the LAFT actually suppresses the attribute to be ignored. We hypothesize this phenomenon because it is difficult to remove all attribute-related information in the embedding space using only text prompts. On the other hand, guiding the attribute is relatively easy, because LAFT only needs to capture the primary information about the attribute. We found that performance improved when we used a genetic algorithm to select the appropriate pair from the given prompt pairs. Selecting the appropriate prompt pair would be our future work. Using LAFT with other methods Our proposed LAFT method can be used as a feature transformation module in other tasks or methods. Basically, we expect that it can be applied to any vision model that requires a feature extractor. As a simple proof of concept, we apply the LAFT method to ZOC for anomaly detection. The results in Table 5 show that we can suppress Male attribute from the image features without significant impact on other attributes. 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fweSF6QplV	The last paragraph of Section 1 is somewhat vague and needs further clarification. While the authors emphasize the significance of adopting the multiple-component method in graph coarsening, the main contribution of this paper is the introduction of a generalized optimization framework that supports four kinds of graph structures simultaneously. The reasons for focusing on the other three structures – sparse graphs, scale-free graphs, and bipartite graphs – merit further explanation.	Structured Graph Reduction for Efficient GNN Anonymous authors Paper under double-blind review Abstract Scalability remains a prominent challenge for Graph Neural Networks (GNNs) when dealing with large-scale graph data. Graph coarsening is a technique that reduces a large graph to a smaller tractable graph. A good quality graph representation with specific properties is needed to achieve good performance with downstream applications. However, existing coarsening methods could not coarsen graphs with desirable properties, such as sparsity, scale-free characteristics, bipartite structure, or multi-component structure. This work introduces a unified optimization framework for learning coarsened graphs with desirable structures and properties. The proposed frameworks are solved efficiently by leveraging block majorization-minimization, log determinant, structured regularization, and spectral regularization frameworks. Extensive experiments with real benchmark datasets elucidate the proposed framework’s efficacy in preserving the structure in coarsened graphs. Empirically, when there is no prior knowledge available regarding the graph’s structure, constructing a multicomponent coarsened graph consistently demonstrates superior performance compared to state-of-the-art methods. 1 Introduction Graph machine learning is a common tool for modeling and analyzing complex systems, such as social networks, biological networks, transportation networks, and computer networks, etc. Battaglia et al. (2018), Wu et al. (2020), Zhou et al. (2020), Bruna et al. (2013), Chen et al. (2020b), Defferrard et al. (2016). Large graphs are becoming increasingly common, requiring significant computational resources for data loading and processing. As a result, analyzing and scaling up graph-based machine learning becomes challenging due to the bottleneck imposed by these large graphs (Rong et al., 2019; Chen et al., 2020a). Techniques such as graph reduction or coarsening (Loukas, 2019; Kumar et al., 2022; Chen et al., 2022), summarization (Liu et al., 2018; Riondato et al., 2017) or condensation (Jin et al., 2021; 2022), graph sparsification (Fung et al., 2011; Spielman & Teng, 2011), etc. have emerged as promising approaches to address this issue. These techniques aim to coarsen or reduce given graphs into smaller ones, allowing for more efficient analysis and processing of the data. For building effective graph-based approaches, the choice of graphs to be used for encoding the relationships is a critical decision and often more important than the particular algorithm or type of loss function to be used. This becomes more critical when the downstream tasks are performed over the reduced graph. For better performance, graphs with additional properties (e.g., structures) are needed for interpretability and precise identification of the relationships in these data sets. There are plenty of works on learning structured graphs from data, for example, for bipartite graph learning (Narang & Ortega, 2012), scale-free (Liu & Ihier, 2011), sparse (Yuan & Lin, 2007), and multicomponent (Hao et al., 2018) graphs. These methods are computationally heavy and can learn only a specific type of structure. Recently, the work in (Kumar et al., 2020) has developed an optimization-based framework, where with a suitable choice of regularization it can learn graphs with a variety of important structures. However, how to enforce desirable structure while learning a reduced graph is not well understood yet. There are two distinct approaches to enforcing structure in a coarsened graph. The first approach involves a two-step process: initially coarsening the graph and subsequently applying existing algorithms designed to enforce structural constraints. In contrast, the second approach simultaneously integrates the coarsening and structural enforcement steps. The joint learning approach works better... because it leverages the synergy between coarsening and structural enforcement, leading to a more adaptive, informed, and data-driven optimization process that ultimately results in a coarsened graph with improved structural properties and, consequently, better overall performance. In this work, we have introduced a novel optimization-based framework for learning coarsened graphs with desirable structure and properties. In this work, we have enforced four structures: sparse, scale-free, multi-component, and bipartite in the coarsened graph. The formulated problems for obtaining sparse and scale-free coarsened graphs are convex optimization problems, while for obtaining multi-component and bipartite coarsened graphs are multi-block non-convex optimization problems which are solved efficiently by leveraging block majorization-minimization, log determinant, Laplacian and adjacency spectral constraints, and regularization frameworks. The developed algorithm is convergent and enforces the desired properties in the learned coarsened graph. We have applied the proposed coarsening algorithms to real datasets for node classification tasks and compared them to the recent graph coarsening techniques. By enforcing structure in the coarsened graph, we observed a notable increase in accuracy compared to the state-of-the-art methods. Furthermore, we have also utilized proposed algorithms to perform classification on various GNN architecture like GCN (Kipf & Welling [2016]), APPNP (Gasteiger et al. [2018]), and GAT (Veličković et al. [2017]), respectively. Also, the proposed structured graph coarsening methods are faster than the state-of-the-art graph coarsening methods. Extensive experiments elucidate the efficacy of the proposed framework for real-world applications. In this work, we have investigated different structure plays a different role in performing the downstream task using graph neural network. Prior knowledge about a graph’s structure can benefit coarsening based on that specific structure. In the absence of prior knowledge about a graph’s structure, the multiple component method typically proves to be a robust approach, and empirical evidence consistently demonstrates the superior performance of the Multicomponent Coarsened Graph Learning (MGC) algorithm compared to state-of-the-art alternatives. Within the framework of multiple component coarsened graphs, the key strategy involves partitioning the graph into components, aligning their number with the classes present in the original graph. This approach effectively simplifies graph analysis and often reveals meaningful insights, making it a valuable choice. 2 BACKGROUND AND PROPOSED FORMULATION In this section, we review the graph coarsening method and formulate the problem of structured graph coarsening. 2.1 GRAPH COARSENING Given an original graph $ G = (V, E, X \in \mathbb{R}^{p \times d}, Y \in \mathbb{R}^{p \times l}) $ with $ p $ nodes, the goal of graph coarsening is to construct an appropriate "smaller" or coarser graph $ G_c = (\tilde{V}, \tilde{E}, \tilde{X} \in \mathbb{R}^{k \times d}, \tilde{Y} \in \mathbb{R}^{k \times l}) $ with $ k << p $ nodes. Given the Laplacian matrix $ \Theta \in \mathbb{R}^{p \times p} $ and adjacency matrix $ A \in \mathbb{R}^{p \times p} $ corresponding to the large graph, the coarsened graph Laplacian and adjacency matrices are $ \Theta_c \in \mathbb{R}^{k \times k} $ and $ A_c \in \mathbb{R}^{k \times k} $ are obtained using the relation $ \Theta_c = C^\top \Theta C $ and $ A_c = C^\top AC $ (Loukas [2019]) where $ C \in \mathbb{R}^{p \times k} $ is the mapping matrix that maps nodes of original graph to supernode of the coarsened graph. For a valid coarsening, the mapping matrix should belong to the following set (Kumar et al. [2023]) \[ C = \left\{ C \geq 0 \| \langle C_i, C_j \rangle = 0 \forall i \neq j, \langle C_i, C_i \rangle = d_i, \\|C_i\\|_0 \geq 1 \text{ and } \\|[C^\top]_i\\|_0 = 1 \right\} \] (1) where $ C_i, C_j $ and $[C^\top]_i$ represents the $ i $-th column, $ j $-th column and $ i $-th row of mapping matrix $ C $ respectively. Many graph coarsening algorithms learn the mapping matrix $ C $, and using $ C $, we can obtain the Laplacian matrix of a coarsened graph $ \Theta_c = C^\top \Theta C $. 2.2 GRAPH COARSENING FOR SCALING UP GNN The coarsened graph serves as the foundation for downstream tasks, such as node classification, where a Graph Neural Network (GNN) is trained using this coarsened graph $ G_c = (\tilde{V}, \tilde{E}, \tilde{X} = C^\top X, \tilde{Y} = \text{argmax}(C^\top Y)) $ (Huang et al. [2021]). The effective training of Graph Neural Networks (GNNs) relies significantly on the labels $ \tilde{Y} $ assigned to the coarsened graph. In this context, each supernode’s label is determined by selecting the most frequent class among its constituent nodes that share the same class. For a given original graph $ G $, there are various possibilities of the coarsened graph $ G_c $ and the loading matrix $ C $. To quantify the quality of a coarsened graph in terms of nodes of the original graph having the same label mapped to the supernode of the coarsened graph relies on the $ \phi $ matrix that is defined as: Definition 2.1. A loading matrix $ C $ is considered balanced, and a coarsened graph is considered informative when, after transforming the one-hot matrix $ Y \in \mathbb{R}^{p \times l} $ of labels from the original graph $ G $ using $ C $, the resulting matrix $ \phi = C^\top Y $ exhibits sparsity in its rows (Ghoroghchian et al., 2021). Moreover, the recent graph coarsening methods for example, (Loukas & Vanderheyden, 2018) is a heuristic-based approach, (Kumar et al., 2023) Dirichlet energy optimization-based approach, and (Jin et al., 2020) is a deep learning method for learning the mapping matrix $ C $. Recent state-of-the-art techniques often encounter challenges when attempting to learn a sparse $ \phi $ matrix, making them less suitable for downstream tasks like node classification. The task of learning a coarsened graph with a sparse $ \phi $ matrix is known to be computationally demanding and falls into the realm of combinatorial hard problems. To address this challenge effectively, our approach involves a two-step process. First, we enforce specific structural characteristics such as multi-component, bipartite, sparsity, or scale-free properties within the coarsened graph. Subsequently, we calculate the $ \phi $ matrix for each of these cases. We train our Graph Neural Network (GNN) using the coarsened graph associated with a sparser $ \phi $ matrix. Subsequently, during the testing phase, we conduct evaluations on the original graph. Importantly, our empirical findings consistently indicate that the coarsened graph with a sparser $ \phi $ matrix outperforms other configurations. For instance, as illustrated in Figure 1, the coarsened graph corresponding to the $ \phi_2 $ matrix exhibits superior performance when training the Graph Neural Network (GNN). The figure below illustrates the workflow of our work. \[ \begin{align} \text{minimize} & \quad -\gamma \log \text{gdet}(\Theta_c) + \frac{\lambda}{2} \\|C^\top\\|_{1,2}^2 + \alpha h(\Theta_c) \\ \text{subject to} & \quad C \in S_c = \{ C \geq 0, \\| [C^\top]_i \\|_2^2 \leq 1 \forall i = 1, 2, 3, \ldots, p \}, \lambda(T(\Theta_c)) \in S_\lambda \end{align} \] (2) Figure 2: The figure shows an illustration of node classification of an original graph $ G $ using coarsened graph $ G_c $. ### 3 Proposed Framework for Structured Graph Coarsening The proposed optimization-based framework for learning a structured coarsened graph is: \[ \begin{align} \text{minimize} & \quad -\gamma \log \text{gdet}(\Theta_c) + \frac{\lambda}{2} \\|C^\top\\|_{1,2}^2 + \alpha h(\Theta_c) \\ \text{subject to} & \quad C \in S_c = \{ C \geq 0, \\| [C^\top]_i \\|_2^2 \leq 1 \forall i = 1, 2, 3, \ldots, p \}, \lambda(T(\Theta_c)) \in S_\lambda \end{align} \] (2) where $\text{gdet}(\Theta_c)$ denotes the generalized determinant defined as the product of the non-zero eigenvalues of the coarsened graph Laplacian matrix $\Theta_c$, $h(\Theta_c)$ is the regularizer, $\lambda(\Theta_c)$ denotes the eigenvalues of $\Theta_c$, $S_\lambda$ is the set containing spectral constraint on the eigenvalues, $T(\cdot)$ is a linear operator used to enforce eigenvalue constrained on other than coarsened Laplacian matrix, and $\gamma, \lambda,$ and $\alpha > 0$ are the hyperparameters. Moreover, $S$ enforces the structure on the coarsened graph to be learned (Kumar et al., 2020). Next, we will introduce different choices of $S_\lambda$ and $h(\Theta_c)$ that will enforce different structures in the resulting coarsened graph. - Sparse coarsened graph can be learned using the following regularizer $$h(\Theta_c) = \\|C^\top \Theta C\\|_F^2.$$ (3) - Multi-component coarsened graph where the super-node set can be partitioned into $n$ disjoint subsets are having the first $n$ eigenvalues of the Laplacian matrix as zeros. Thus the eigenvector constrained of its Laplacian are expressed as (Kumar et al., 2020): $$S_\lambda = \{\{\lambda_j = 0\}_{j=1}^n, c_1 \leq \lambda_{n+1} \leq \ldots \leq \lambda_k \leq c_2\}$$ (4) Where $n \geq 1$ denotes the number of connected components in the learned coarsened graph, and $c_1, c_2 > 0$ are constants that depend on the number of edges and their weights. - Bipartite Coarsened graph: A coarsened graph is bipartite if and only if the eigenvalues of the adjacency matrix of the coarsened graph $A_c = C^\top AC$ is symmetric about the origin (Kumar et al., 2020) such that: $$S_\psi = \{\psi_1 \geq \psi_2 \geq \ldots \psi_{k-1} \geq \psi_k, \psi_i = -\psi_{k-i+1}, i = 1, 2, \ldots, k\}$$ (5) - Scale Free Coarsened Graph: Scale-free graphs represent a class of graphs that follow a power law in their degree distribution (Liu & Ihler, 2011), i.e., the degree of a node $i$ follows a power law degree distribution $p(d) \propto d^{-\alpha}$, where $\alpha > 0$. To enforce the scale-free structure in the resulting coarsen graph, we will use the regularizer $h(\Theta_C)$ at $t$-th iteration (Liu & Ihler, 2011): $$h(\Theta_c) = \sum_{i \neq j} \delta_{ij}^t \\|C^\top \Theta C[ij]\\| + \beta \sum_i \\|C^\top \Theta C[ii]\\| = \\|C^\top AC \delta_{1k \times 1}\\|_1 + \\|C^\top DC \beta_{1k \times 1}\\|_1$$ (6) where $\delta_{ij}^t = \alpha \left( \frac{1}{\sum_{i \neq j} \|\Theta_{cij}\| + \epsilon_i} + \frac{1}{\sum_{j \neq i} \|\Theta_{cij}\| + \epsilon_j} \right)$ (7) where $\beta = 2\alpha$, $\Theta_c^t = [C^\top \Theta C]^t$ is the estimate of $C^\top \Theta C$ found in the $t$-th iteration (Liu & Ihler, 2011), $\epsilon_i$ is the positive quantity, $D = \Theta + A$ is the degree matrix of the original graph, and more details are in (Liu & Ihler, 2011). Using the regularizer $h(\Theta_c)$ for sparsity and scale-free defined in equation (3) and equation (6) the proposed formulation for learning the coarsened graph with structure is $$\min_{C \in S_C} f(C) = -\gamma \log \det(C^\top \Theta C + J) + \frac{\lambda}{2} \\|C^\top\\|_{1,2}^2 + \alpha h(\Theta_C)$$ (8) Lemma 1. Problem (8) is a strictly convex optimization problem. Proof. The function $-\gamma \log \det(C^\top \Theta C + J) + \frac{\lambda}{2} \\|C^\top\\|_{1,2}^2$ is a strictly convex function (Kumar et al., 2023), $h(\Theta_c)$ defined in equation (3) and equation (6) are convex functions and the set $S_C$ is a closed and convex set; thus, problem (8) is a strictly convex optimization problem. Since, there does not exist a closed-form solution to the problem (8). To solve it efficiently, we will use the majorization-minimization (MM) framework to obtain easily solvable surrogate functions for objective functions such that the update rule is easily obtained. The surrogate function $g(C\|C^{(t)})$ is such that it upper-bounds the objective function $f(C)$ and is tangent to it at the current estimate. By using the first-order Taylor series approximation, a majorized function for $f(C)$ at $C^{(t)}$ can be obtained as (Beck & Pan, 2018; Razaviyayn et al., 2012; Sun et al., 2017): $$g(C\|C^{(t)}) = f(C^{(t)}) + (C - C^{(t)}) \nabla f(C^{(t)}) + \frac{L}{2} \\|C - C^{(t)}\\|^2$$ (9) where $ f(C) $ is $ L $-Lipschitz continuous gradient function $ L = \max(L_1, L_2, L_3) $ with $ L_1, L_2, L_3 $ the Lipschitz constants of $ -\gamma \log \det(C^\top \Theta C + J), \\|C^\top\\|_{1,2}, h(\Theta_c) $ respectively. More details are deferred to the supplementary material. After ignoring the constant term, the majorised problem of (8) is \[ \minimize_{C \in S_c} \frac{1}{2} C^\top C - C^\top A. \tag{10} \] where $ A = C(t) - \frac{1}{L} \nabla f(C(t)) $. Note that since $ C \geq 0 $, it implies that $ \|C_{ij}\| = C_{ij} $ hence $ \\|C^\top\\|_{1,2}^2 $ is differentiable. Also, the worst case computational complexity is $ O(p^2k) $, which is due to the matrix multiplication in the gradient of $ f(C) $ in equation (8). Lemma 2. By using KKT optimality condition, the optimal solution of (10) is $ C^{(t+1)} = (C(t) - \frac{1}{L} \nabla f(C(t)))^+ $ where $ (X_{ij})^+ = \max(\frac{X_{ij}}{\\|X^\top\\|_{1,2}}, 0) $ and $[X^\top]_i$ is the $ i $-th row of matrix $ X $ and $ \nabla f(C(t)) = -2\gamma \Theta C(t)(C(t)^\top \Theta C(t) + J)^{-1} + \lambda C(t)1 + \nabla h(\Theta_c) $ where $ 1 $ is all ones matrix of dimension $ k \times k $. Proof: The proof is deferred to the supplementary material. Next, by using the equation (3) and equation (6) as enforcing sparsity and scale-free structure, we obtain sparse coarsened graph(SCG) and scale-free coarsened graph(SFCG), respectively. The gradients $ \nabla h(\Theta_c) $ take the following form, for SCG $ \nabla h(\Theta_c(t)) = 2\alpha \Theta C(t)C(t)^\top \Theta C(t) $ and for SFCG $ \nabla h(\Theta_c(t)) = 2AC(t)\delta I_{k\times k} + 2DC(t)\beta I_{k\times k} $. ### 4 Proposed Framework for Multi-Component & Bipartite Graph Coarsening via Laplacian and Adjacency Spectral Constraints The proposed formulation for learning multi-component coarsened graph via Laplacian spectral constraint(MGC) is \[ \begin{align} \minimize_{C,\Lambda,U} & \quad -\gamma \log \det(\Lambda) + \frac{\lambda}{2} \\|C^\top\\|_{1,2}^2 + \frac{\beta}{2} \\|C^\top \Theta C - U \Lambda U^\top\\|_F^2 \\ \text{subject to} & \quad C \in S_c, \ U^\top U = I, \ \Lambda \in S_\lambda \end{align} \tag{11} \] where $ C^\top \Theta C $ is the desired Laplacian matrix of coarsened graph which seeks to admit the decomposition $ C^\top \Theta C - U \Lambda U^\top $, $ \Lambda \in \mathbb{R}^{k\times k} $ is a diagonal matrix containing $ \{\lambda_i\}_{i=1}^k $ on its diagonal, and $ U \in \mathbb{R}^{k\times k} $ is the matrix satisfying $ U^\top U = I $. We are incorporating multi-component structure by enforcing $ \{\lambda_i\}_{i=1}^k \in S_\lambda $. Furthermore, the term $ \frac{\beta}{2} \\|C^\top \Theta C - U \Lambda U^\top\\|_F^2 $ will keep $ C^\top \Theta C $ close to $ U \Lambda U^\top $ instead of exactly solving the constraint. Note that by choosing sufficiently large $ \beta $ can make this relaxation tight. We consider solving for learning an $ n $ component graph structure utilizing the constraints in equation (4) where the first $ n $ eigenvalues are zero. There are a total of $ q = k - n $ non-zero eigenvalues ordered in the given set $ S_\lambda = \{c_1 \leq \lambda_{n+1} \leq \ldots \leq \lambda_k \leq c_2\} $. Collecting the variables as $ (C \in \mathbb{R}_+^{p\times k}, \Lambda \in \mathbb{R}^{k\times k}, U \in \mathbb{R}^{k\times k}) $, we develop a block MM-based algorithm which updates one variable at a time while keeping the other fixed. Update of $ C $: Treating $ C $ as variable and fixing $ \Lambda $ and $ U $, we obtain the following sub-problem for $ C $: \[ \minimize_{C \in S_c} f(C) = \frac{\lambda}{2} \\|C^\top\\|_{1,2}^2 + \frac{\beta}{2} \\|C^\top \Theta C - U \Lambda U^\top\\|_F^2 \tag{12} \] The function $ f(C) $ in problem (12) is a convex function; more details are in the supplementary material. The set $ S_c $ is a closed and convex set; thus (12) is a strongly convex optimization problem. However, there does not exist a closed-form solution to it. To get a closed-form update rule, we will use the MM framework. We have derived the update of $ C $ in a similar way as in section (3). The update rule is similar to Lemma 2, where the $ \nabla f(C(t)) = \lambda C(t)1 + 2\beta \Theta C(t)(C(t)^\top \Theta C(t) - U \Lambda U^\top) $ where $ 1 $ is all ones matrix of dimension $ k \times k $. More details are deferred to the supplementary material. Update of $ U $: Treating $ U $ as variable and fixing $ C $ and $ \Lambda $, we obtain the following sub-problem for $ U $: \[ \minimize_{U^\top U = I_q} \frac{\beta}{2} \\|C^\top \Theta C - U \Lambda U^\top\\|_F^2 = -\frac{\beta}{2} \text{tr}(U^\top C^\top \Theta C U \Lambda) \tag{13} \] The solution to problem (13) is $ U^{t+1} = \text{eigenvectors}(C^\top \Theta C)[n + 1 : k] $ as solved in [Absil et al., 2008; Benidis et al., 2016]. Update of $ \Lambda $: Treating $ \Lambda $ as variable and fixing $ C $ and $ U $, we obtain the following sub-problem for $ \Lambda $: \[ \min_{\Lambda \in S_\lambda} -\gamma \log \det(\Lambda) + \frac{\beta}{2} \|\|C^\top \Theta C - U \Lambda U^\top\|\|_F^2 \] (14) For ease of notation, we denote the index of non-zero eigenvalues as $ i = 1 $ to $ q $ instead of $ i = n + 1 $ to $ k $. We can rewrite problem (14) as: \[ \min_{c_1 \leq \lambda_1 \leq \lambda_2 \ldots \lambda_q \leq c_2} -\gamma \sum_{i=1}^q \log \lambda_i + \frac{\beta}{2} \|\|\Lambda - d\|\|_F^2 \] (15) where $ \lambda = [\lambda_1, \ldots, \lambda_q] $ and $ d = [d_1, \ldots, d_q] $ with $ d_i $ being the $ i $-th diagonal element of $ U^\top C^\top \Theta C U $. Lemma 3. By using KKT condition we can obtain the solution of convex optimization problem (15) is \[ \lambda_i = \frac{1}{2}(d_i + \sqrt{d_i^2 + 4/\beta}) \quad \forall i = 1, 2, \ldots, q \] (16) Proof: The proof is deferred to the supplementary material. ### 4.1 Bipartite Graph Coarsening via Adjacency Spectral Constraints (BI-GC) The proposed formulation for learning a Bipartite coarsened graph is \[ \begin{align} \min_{C, \Psi, V} & \quad -\gamma \log \det(C^\top \Theta C + J) + \frac{\lambda}{2} \|\|C^\top\|\|_{1,2}^2 + \frac{\beta}{2} \|\|C^\top AC - V \Psi V^\top\|\|_F^2 \\ \text{subject to} & \quad C \in S_c, \quad V^\top V = I, \quad \Psi \in S_\psi \end{align} \] (17) Suppose there are $ z $ number of zero eigenvalues in the set $ S_\psi $. From the symmetry property of the eigenvalues, the zero eigenvalues are positioned in the middle, i.e., in equation (5) the eigenvalues $ \psi_{k+z/2+1} $ to $ \psi_{k+z/2} $ will be zero. Both $ (k+z) $ and $ (k-z) $ must be even by the symmetry property. As a consequence, the zero eigenvalues and the corresponding eigenvectors can be dropped from the formulation. Now $ \psi \in \mathbb{R}^b $ contains $ b $ non-zero eigenvalues and $ V \in \mathbb{R}^{k \times b} $ contains the corresponding eigenvectors that satisfy $ V^\top V = I_b $. The non-zero eigenvalues are required to lie in the set $ S_\psi = \{ \psi_i = -\psi_{b+1-i}, c_1 \geq \psi_1 \geq \psi_2 \ldots \geq \psi_{b/2} \geq c_2, \forall i = 1, 2, \ldots, b/2 \} $. Where $ c_1 $ and $ c_2 > 0 $ are some constants that depend on the graph properties. Collecting the variables as $ (C \in \mathbb{R}_+^{p \times k}, \Psi \in \mathbb{R}^{k \times k}, U \in \mathbb{R}^{k \times k}) $, we develop a block MM-based algorithm which updates one variable at a time while keeping the other fixed. Update of $ C $: Treating $ C $ as variable and fixing $ \Psi $ and $ V $, we obtain the following sub-problem for $ C $: \[ \min_{C \in S_c} -\gamma \log \det(C^\top \Theta C + J) + \frac{\lambda}{2} \|\|C^\top\|\|_{1,2}^2 + \frac{\beta}{2} \|\|C^\top AC - V \Psi V^\top\|\|_F^2 \] (18) The function $ f(C) $ in problem (18) is a convex function, and more details are in the supplementary material. The set $ S_C $ is a closed and convex set; thus (18) is a strongly convex optimization problem. However, there does not exist a closed-form solution to it. To get a closed-form update rule, we will use the MM framework. We have derived the update of $ C $ in a similar way as in section 3. The update rule is similar to Lemma 2 where the $ \nabla f(C(t)) = -2\gamma \Theta C(t)(C(t)^\top \Theta C(t) + J)^{-1} + \lambda C(t)1 + 2\beta AC(t)(C(t)^\top AC(t) - V \Psi V^\top) $ where $ 1 $ is all ones matrix of dimension $ k \times k $. Update of $ V $: Treating $ V $ as variable and fixing $ C $ and $ \Psi $, we obtain the following sub-problem for $ V $: \[ \min_{V^\top V = I_b} \frac{\beta}{2} \|\|C^\top AC - V \Psi V^\top\|\|_F^2 = -\frac{\beta}{2} \text{tr}(V^\top C^\top AC V \Psi) \] (19) The solution to problem (19) is $ V^{t+1} = \text{eigenvectors}(C^\top AC)[1 : \frac{k-z}{2}, \frac{k+z}{2} : k] $ as solved in [Absil et al., 2008; Benidis et al., 2016]. Update of $ \Psi $: The update of $ \Psi $ is similar to the update of $ \lambda $ without log determinant in problem (14). The detailed proof is in the supplementary material. Algorithm 1: Multi-component graph coarsening(MGC) and Bipartite graph coarsening(BI-GC) Input: $ \mathcal{G}(\Theta), \beta, \gamma, \lambda $ while stopping criteria not met do Update $ C^{t+1}, U^{t+1}, $ and $ \Lambda^{t+1} $ for MGC or $ C^{t+1}, V^{t+1}, $ and $ \Psi^{t+1} $ for BI-GC $ t \leftarrow t + 1; $ end Output: $ C $ and $ \Theta_c $ 5 EXPERIMENTS In this section, we demonstrate the effectiveness of the proposed algorithms by a comprehensive set of experiments conducted on real data sets. We compare the proposed methods for structured graph coarsening against the state-of-the-art methods, GCOND (Jin et al., 2021), SCAL (Huang et al., 2021). However, we considered these state-of-the-art algorithms since they are more recent and outperform existing state-of-the-art graph coarsening approaches. The performance is evaluated through classification accuracy (ACC) and time$(\tau)$ required to perform coarsening and classification. It has been experimentally verified that proposed methods for structured graph coarsening outperform in classification tasks and time complexity$(\tau)$. \| Dataset \| Nodes \| Edges \| Features \| Classes \| \|------------------\|---------\|--------\|----------\|---------\| \| CORA \| 2,708 \| 5,429 \| 1,433 \| 7 \| \| CITESEER \| 3,327 \| 9,104 \| 3,703 \| 6 \| \| DBLP \| 17,716 \| 52,867 \| 1,639 \| 4 \| \| COAUTHOR CS \| 18,333 \| 163,788\| 6,805 \| 15 \| \| PUBMED \| 19,717 \| 44,338 \| 500 \| 3 \| \| COAUTHOR PHYSICS \| 34,493 \| 247,962\| 8,415 \| 5 \| Table 1: Datasets used in node classification. \| Data set(ACC) \| r=k/p \| GCOND \| SCAL \| MGC \| BI-GC \| \|---------------\|-------\|-------\|------\|-----\|-------\| \| CORA \| 0.5 \| 81.02 ± 0.37 \| 82.7 ± 0.50 \| 87.20 ± 0.43 \| 86.26 ± 0.04 \| \| \| 0.3 \| 81.56 ± 0.6 \| 79.42 ± 1.71 \| 84.56 ± 1.40 \| 85.15 ± 0.03 \| \| CITESEER \| 0.5 \| 74.28 ± 1.45 \| 72.0 ± 0.5 \| 78.80 ± 1.20 \| 79.69 ± 0.37 \| \| \| 0.3 \| 72.43 ± 0.94 \| 74.54 ± 1.34 \| 74.60 ± 2.31 \| 77.09 ± 0.24 \| \| CO-PHY \| 0.05 \| 93.05 ± 0.26 \| 73.09 ± 7.41 \| 94.52 ± 0.19 \| 91.63 ± 0.45 \| \| \| 0.03 \| 92.81 ± 0.31 \| 63.65 ± 9.65 \| 93.64 ± 0.25 \| 91.39 ± 0.35 \| \| PUBMED \| 0.05 \| 78.16 ± 0.30 \| 72.82 ± 2.62 \| 81.89 ± 0.00 \| 81.72 ± 0.48 \| \| \| 0.03 \| 78.04 ± 0.47 \| 70.24 ± 2.63 \| 80.70 ± 0.00 \| 80.66 ± 0.55 \| \| CO-CS \| 0.05 \| 86.29 ± 0.63 \| 34.45 ± 10.07 \| 87.25 ± 0.90 \| 84.40 ± 0.0 \| \| \| 0.03 \| 86.32 ± 0.45 \| 26.06 ± 9.29 \| 86.38 ± 3.37 \| 83.41 ± 0.06 \| \| DBLP \| 0.05 \| 79.15 ± 0.20 \| 76.52 ± 2.88 \| 78.09 ± 1.88 \| 79.20 ± 0.07 \| \| \| 0.03 \| 78.42 ± 1.26 \| 75.49 ± 2.84 \| 74.81 ± 1.57 \| 78.99 ± 0.71 \| Table 2: The table summarizes the node classification accuracy on real benchmark datasets for the proposed MGC and BI-GC algorithms against the GCOND and SCAL. For small datasets, we have taken coarsening ratio $ r = 0.3 $ and $ 0.5 $, while for large datasets, we have taken $ r = 0.05 $ and $ 0.03 $. It is evident that proposed MGC and BI-GC outperform state-of-the-art methods by a significant amount. Furthermore, across a wide range of datasets, the MGC algorithm consistently outperforms or demonstrates comparable performance to the BI-GC algorithm. Additional results with the SCG and the SFCG are provided in the supplementary material. Furthermore, we have also demonstrated the generalizability of the proposed algorithm by performing node classification on different GNN structures like GCN (Kipf & Welling, 2016), APPNP (Gasteiger et al., 2018), and GAT (Veličković et al., 2017). Moreover, all the experiments were performed on 16GB RAM GPU(NVIDIA P100) processor. Table 1 shows the statistics of the graph datasets used to perform experiments. 5.1 Node Classification In this section, we compute the experiments of node classification on real benchmark datasets. For node classification, a GNN model is trained on coarsened graph data, i.e., $ G_c = (\tilde{V}, \tilde{E}, \tilde{X}, \tilde{Y}) $. The hyperparameters for graph coarsening and the GNN model are tuned using grid search. The learning and decay rates used in the node classification experiments are 0.01 and 0.0001, respectively. The GNN model is tested on full original graph data $ G = (V, E, X, Y) $ to predict the label of every $ p $ node. Then these predicted, and actual labels are used to compute accuracy (ACC). All the results are calculated using 10-fold cross-validation. Also, for the proposed MGC, we have taken components $ n $ as the number of classes of the original graph. It is evident in Table 2 and 3 that enforcing structure on the coarsened graph improves the node classification accuracy. \| Data set \| GCOND \| SCAL \| SCG \| MGC \| GC-BI \| Whole Data \| \|----------\|-------\|------\|-----\|-----\|-------\|------------\| \| CORA \| 79.37 ± 0.4 \| 71.38 ± 3.6 \| 79.81 ± 0.3 \| 76.02 ± 0.9 \| 80.07 ± 1.20 \| 89.50 ± 1.2 \| \| CITESEER \| 70.46 ± 0.4 \| 68.58 ± 2.3 \| 69.49 ± 2.1 \| 70.57 ± 1.2 \| 68.59 ± 0.12 \| 78.09 ± 1.9 \| \| PUBMED \| 78.57 ± 0.2 \| 73.59 ± 3.5 \| 83.13 ± 0.1 \| 84.81 ± 1.5 \| 81.83 ± 0.36 \| 88.89 ± 0.5 \| \| CO-PHY \| 92.98 ± 0.5 \| 86.43 ± 2.4 \| 81.59 ± 5.1 \| 94.71 ± 0.2 \| 91.84 ± 0.06 \| 96.22 ± 0.7 \| \| CO-CS \| 87.13 ± 0.7 \| 55.20 ± 4.3 \| 90.41 ± 0.8 \| 91.67 ± 0.0 \| 88.08 ± 0.0 \| 93.32 ± 0.6 \| \| DBLP \| 80.40 ± 0.9 \| 76.66 ± 1.7 \| 80.49 ± 0.1 \| 81.82 ± 0.6 \| 80.79 ± 0.56 \| 85.35 ± 0.8 \| Table 3: The table summarizes the node classification accuracy on real datasets for the proposed structured graph coarsening algorithms against the GCOND (Jin et al., 2021), SCAL (Huang et al., 2021) for a coarsening ratio of 0.1. It is evident that the proposed structured graph coarsening algorithm outperforms state-of-the-art methods. However, we have compared only with GCOND and SCAL as these are the most recent technique for graph node classification using the coarsened graph. It is evident that the enforcing structure on the coarsened graph improves the performance significantly. \| Data set(ACC) \| r=k/p \| SCG \| TSM-1 \| TSM-2 \| TSBI \| \|---------------\|-------\|-----\|-------\|-------\|------\| \| CORA \| 0.1 \| 79.81 \| 60.6 \| 53.21 \| 28.32 \| \| \| 0.3 \| 85.15 \| 61.6 \| 71.34 \| 85.82 \| \| CITESEER \| 0.1 \| 69.29 \| 51.93 \| 55 \| 20.2 \| \| \| 0.3 \| 74.26 \| 58 \| 63.02 \| 75.01 \| Table 4: The table summarizes node classification accuracy for both single-stage and two-stage structured coarsened graph learning methods. Notably, the single-stage structured coarsened graph learning algorithms consistently outperform their two-stage counterparts. In the two-stage multicomponent coarsened graph learning approaches, which is denoted as TSM-1 and TSM-2, the initial stage entails the construction of the graph using the SCG algorithm. Subsequently, in the second stage, the multi-component structure is imposed through the Louvain method (TSM-1) (De Meo et al., 2017) and the MGAE (Multi-Graph Autoencoder) (TSM-2) (Wang et al., 2017) technique. On the other hand, TSBI represents the two-stage bipartite graph learning approach. The coarsened graph is constructed using the SCG algorithm in the first stage. Then, to enforce the bipartite structure, the SGL (Kumar et al., 2020) (Sparse Graph Learning) algorithm is employed. \| Proposed \| CORA/(n) \| CITESEER/(n) \| PUBMED/(n) \| CO-PHY/(n) \| DBLP/(n) \| \|----------\|----------\|--------------\|------------\|------------\|---------\| \| MGC \| 76.16 ± 0.1/5 \| 69.82 ± 1.6/4 \| 82.28 ± 1.6/1 \| 90.75 ± 0.3/13 \| 80.77 ± 0.8/3 \| \| \| 76.08 ± 0.9/7 \| 69.43 ± 1.3/6 \| 84.81 ± 1.5/3 \| 91.67 ± 0.0/15 \| 80.83 ± 0.9/5 \| \| \| 77.62 ± 2.2/9 \| 69.64 ± 0.4/8 \| 83.65 ± 2.4/5 \| 91.08 ± 0.1/17 \| 81.35 ± 1.1/7 \| Table 5: The table summarizes the node classification accuracy on real datasets for the proposed multi-component graph coarsening (MGC) algorithm for different values of component $ n $ and coarsening ratio 0.1. It is evident that enforcing multi-component structures on the coarsened graph for different values of component $ n $ improves the accuracy. Moreover, we demonstrated the adaptability of the multicomponent graph coarsening (MGC) algorithm by producing coarsened graphs with different component $ n $ values. It is evident that changing the value of $ n $ does not decrease classification accuracy as shown in the Table 5. Next, we will illustrate the generalizability of the learning structured coarsened graph from the proposed algorithms. by using different architectures to train the GNN. Specifically, we have used GNN architectures like GCN (Kipf & Welling, 2016), APPNP (Gasteiger et al., 2018), and GAT (Veličković et al., 2017) to train our GNN and perform the node classification task. Table 6 demonstrates that the proposed methods for learning structured coarsened graph is compatible with different widely used GNN architectures, giving almost similar Node Classification accuracy across all the datasets. \| Data set \| GCN \| GAT \| APPNP \| \|----------\|-----------\|-----------\|-----------\| \| Cora \| 79.81 ± 0.3 \| 76.29 ± 0.1 \| 79.70 ± 0.2 \| \| Citeseer \| 69.49 ± 2.1 \| 67.83 ± 1.3 \| 66.72 ± 1.2 \| \| Pubmed \| 83.13 ± 0.1 \| 83.53 ± 0.3 \| 81.90 ± 0.5 \| \| Co-Phy \| 81.59 ± 5.1 \| 83.43 ± 1.7 \| 93.88 ± 0.1 \| \| Co-CS \| 87.13 ± 0.7 \| 88.32 ± 0.2 \| 90.41 ± 0.8 \| \| DBLP \| 80.09 ± 0.1 \| 79.01 ± 1.1 \| 79.05 ± 0.2 \| Table 6: Node classification accuracy (%) obtained using different GNN structures like GCN, GAT, and APPNP on different datasets using the proposed SCG algorithm for a coarsening ratio of 0.1. It is evident that the proposed SCG method is suitable for all GNN architecture. However, experiments on different GNN architectures for the remaining proposed algorithms are in the supplementary material. Run-time Complexity: The worst-case computational complexity of SCG, MGC and BI-GC are $O(p^2 k)$. This section compares the time ($\tau$) required to perform coarsening and node classification by proposed methods against the state-of-the-art algorithm. It is evident in Table 7 that proposed methods for structured graph coarsening are much faster than state-of-the-art methods. \| Dataset($\tau$) \| r = k/p \| GCOND \| SCAL \| SCG \| MGC \| BI-GC \| Whole dataset \| \|-----------------\|---------\|-------\|------\|-----\|-----\|-------\|---------------\| \| CORA \| 0.05 \| 329.86\| 27.76\| 2.60 \| 2.78 \| 3.10 \| 2.86 \| \| CITESEER \| 0.05 \| 331.33\| 56.21\| 5.08 \| 3.73 \| 4.47 \| 5.24 \| \| PUBMED \| 0.05 \| 202.04\| 54.09\| 27.99 \| 46.09 \| 45.68 \| 58.85 \| \| CO-CS \| 0.05 \| 1600.32\| 180.16\| 44.45 \| 49.80 \| 56.23 \| 72.31 \| Table 7: This table summarizes the time ($\tau$) in sec. required to perform coarsening and node classification for a coarsening ratio of 0.05. It is evident that the proposed SCG, MGC and Bi-GC are faster than state-of-the-art algorithms. Moreover, the time required to perform coarsening and node classification using the proposed methods is less than that required to perform node classification using the given original graph. Figure 3: The figures presented here depict the $\phi$ matrices of three different graph representations: the original graph, the bipartite coarsened graph, and the multi-component coarsened graph. Additionally, a histogram showcases the sparsity levels within each row of the coarsened graphs, focusing on data from the Citeseer dataset. Notably, these visualizations illustrate a critical point: the sparser the heat map of the coarsened graph, the more informative it becomes. To be specific, the node classification accuracy achieved using the bipartite coarsened graph is 77.11%. Meanwhile, the multi-component coarsened graph yields a node classification accuracy of 74.68% for a coarsening ratio of 0.3. The remaining heat maps of $\phi$ matrices are in the supplementary material. 6 CONCLUSION In summary, we have developed an optimization-based framework, structured graph coarsening, that can learn reduced graphs with desirable structures like sparsity, scale-free, bipartite, and multi-component. We have performed a node classification task on the structured coarsened graph, and it is evident that enforcing structure in the coarsened graph increases the accuracy by a significant amount. Moreover, across a wide range of datasets, the MGC algorithm consistently outperforms or demonstrates comparable performance to the BI-GC algorithm. Furthermore, the proposed methods for structured graph coarsening are also suitable for performing tasks on various GNN structures like GCN, APPNP, and GAT. The proposed methods are provably convergent and much faster than the state-of-the-art algorithms. REFERENCES P-A Absil, Robert Mahony, and Rodolphe Sepulchre. 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tf6nR1B8Nt	But from a practical point of view, what can we learn? Do we know what part of the architecture of, say a transformer, is crucial for benign optimization? Which techniques (layer normalization, dropout, learning rate warmup, etc) are playing a role?	No Wrong Turns: The Simple Geometry Of Neural Networks Optimization Paths Anonymous authors Paper under double-blind review Abstract Understanding the optimization dynamics of neural networks is necessary for closing the gap between theory and practice. Stochastic first-order optimization algorithms are known to efficiently locate favorable minima in deep neural networks. This efficiency, however, contrasts with the non-convex and seemingly complex structure of neural loss landscapes. In this study, we delve into the fundamental geometric properties of sampled gradients along optimization paths. We focus on two key quantities, which appear in the restricted secant inequality and error bound. Both hold high significance for first-order optimization. Our analysis reveals that these quantities exhibit predictable, consistent behavior throughout training, despite the stochasticity induced by sampling minibatches. Our findings suggest that not only do optimization trajectories never encounter significant obstacles, but they also maintain stable dynamics during the majority of training. These observed properties are sufficiently expressive to theoretically guarantee linear convergence and prescribe learning rate schedules mirroring empirical practices. We conduct our experiments on image classification, semantic segmentation and language modeling across different batch sizes, network architectures, datasets, optimizers, and initialization seeds. We discuss the impact of each factor. Our work provides novel insights into the properties of neural network loss functions, and opens the door to theoretical frameworks more relevant to prevalent practice. 1 Introduction Despite the theoretical complexity of their loss landscapes, deep neural networks have demonstrated remarkable empirical reliability across a broad range of applications. Blum & Rivest (1992) proved decades ago that neural network training is NP-hard. The intricacy of their loss functions, especially the non-convexity implying potential bad local minima and saddle points (Panageas et al., 2019; Jin et al., 2019). The central hypothesis in these works posits that the efficiency of training arises from the ability of these algorithms to navigate complex loss landscapes adeptly and manage non-convexity. Conversely, other investigations have empirically found loss landscapes to be simpler than their theoretical complexity might suggest (Lucas et al., 2021). Notably, Goodfellow et al. (2015) observed that “in fact, on a straight path from initialization to solution, a variety of state of the art neural networks never encounter any significant obstacles.” Notwithstanding, our current understanding of how neural loss landscapes are empirically simpler than expected remains quite limited. There is yet to emerge a robust mathematical characterization of this empirical simplicity. Consequently, we contend that the theoretical assumptions currently in use fail to accurately capture the objective functions typical in deep learning. This discrepancy is a significant barrier to applying theoretical insights effectively in the optimization of neural networks. One such common assumption, smoothness, is illustrative of this gap. Despite its popularity, smoothness is encumbered by several limitations: it is computationally intensive to approximate for large Figure 1: Cosine similarities between the gradients $G_t$ sampled at step $t$ and the difference $(w_t - w_T)$ between current weights and final weights, averaged over each epoch. The shaded regions denote the range from minimum to maximum values observed at each epoch. The results are presented for a selection of scenarios: (top left) varying depths and widths of ResNet on ImageNet, (top right) different batch sizes on WikiText-2 using a Transformer, (bottom left) a range of optimizers on CIFAR-10 using ResNet-18, and (bottom right) distinct architectures on Vaihingen semantic segmentation. This figure highlights the stability of the cosine similarity throughout most of training, suggesting it as a fundamental characteristic of neural loss landscapes. Neural networks, and necessitates additional assumptions such as bounded gradients for theoretical guarantees in stochastic settings (Qian et al., 2019; Shamir & Zhang, 2013), although recent works have tried to discard them (Nguyen et al., 2018; Loizou et al., 2021). Finally, recent findings suggesting certain directional sharpness in neural networks (Dinh et al., 2017) call into question the suitability of smoothness as a measure of their simplicity. To address these issues, our study undertakes an empirical analysis of the geometric properties of the loss function in regions traversed by first-order optimization algorithms. Our focus is on a variant of the quantities involved in the Restricted Secant Inequality (RSI) (Zhang & Yin, 2013) and Error Bound (EB) (Luo & Tseng, 1993), which pertain to the relationship between sampled gradients, current iterate, and final iterate of the optimization sequence. Our findings indicate that these quantities and their ratio exhibit stable, predictable patterns throughout training across diverse settings, thereby quantitatively characterizing the simplicity of neural loss landscape geometry. Furthermore, these quantities offer several advantages over smoothness, including efficient estimations post-training, inherent compatibility with stochasticity due to direct measurement on sampled gradients, and a well-behaved empirical nature that still allows the derivation of theoretical results such as linear convergence or the prescription of learning rate schedules. Our key contributions are as follows: • We devise an experimental procedure for examining the geometry of optimization paths on common architectures. We assume almost-everywhere differentiability, but not smoothness. • We execute experiments across a range of realistic deep learning settings, identifying consistently verified properties. For instance, the cosine similarity between the negative stochastic gradient and the direction to the final iterate is almost always positive and exhibits remarkable stability across iterations and epochs. • We demonstrate how our empirical investigations can inform the prescription of learning rate schedules, aligning with established empirical knowledge. • We provide an extensive discussion on the implications and limitations of our findings. Collectively, our work quantifies crucial geometric properties of stochastic gradients along deep learning optimization paths, underlining their importance in understanding neural network optimization and enhancing current methodologies. 2 RELATED WORK Our investigation centers on the application of RSI and EB to enhance our comprehension of the geometric principles governing neural network optimization. Consequently, our work intersects with previous research on neural loss landscapes and the utilization of RSI and EB in optimization. RSI and EB: The study of RSI and EB for first-order optimization is not new. RSI (Zhang & Yin, 2013) has been applied in numerous theoretical works (Yi et al., 2019; Schöpfer, 2016; Yuan et al., 2016; Karimi et al., 2016). EB (Luo & Tseng, 1993) has seen less extensive study (Dmitriy Drusvyatskiy, 2018), possibly due to the dominance of smoothness — a condition stronger than EB — in the field. It should not be confused with error bounds on the distance to a set, a term also prevalent in optimization literature (Qian et al., 2023; Zhou & So, 2015). Both RSI and EB, along with other conditions, were analyzed in Guille-Escuret et al. (2021). Furthermore, it was demonstrated in Guille-Escuret et al. (2022) that gradient descent is optimal for the class of functions defined by this pair of conditions. Neural Loss Landscape Geometry: The intricacies of neural loss landscapes have been a focal point of research since the emergence of deep learning. Efforts have ranged from loss landscape visualizations (Li et al., 2018) to investigations of low loss basin connectivity (Garipov et al., 2018) and linear mode connectivity (Frankle et al., 2020). While prior research has noted the seeming simplicity of loss landscape geometry along optimization paths (Lucas et al., 2021; Goodfellow et al., 2015), these observations often involve straightforward phenomena such as monotonic decrease along linear interpolations. Our work takes this approach a step further by studying quantifiable properties with theoretical implications. Additionally, others have examined the geometric properties of neural loss landscapes in the near-infinite width, or Neural Tangent Kernel (NTK), regime (Jacot et al., 2018; Lee et al., 2019). These studies suggest that neural network training can be approximated by linear dynamics or that the loss surface adheres to the Polyak-Łojasiewicz condition (Liu et al., 2022). Unfortunately, this scenario was found to be distinct from empirical settings (Chizat et al., 2019), although recent studies have delved into the evolution of the NTK under more realistic conditions (Fort et al., 2020). We also note the active research direction regarding the influence of BatchNorm on the optimization trajectory (Santurkar et al., 2018b; Ioffe & Szegedy, 2015a). 3 BACKGROUND The training of a neural network on a dataset comprised of $ n $ examples can typically be formulated as the finite-sum optimization problem \[ \min_{w \in \mathbb{R}^d} L(w) := \frac{1}{n} \sum_{i=1}^{n} l_i(w), \] where $ w $ are the parameters of the neural network, $ L $ is the empirical risk, and $ l_i $ corresponds to the loss function for the $ i $-th data sample, for $ i = 1, \ldots, n $. We denote the empirical risk with respect to any minibatch $ B \subseteq [n] $ of size $ m $ as $ L_B := \frac{1}{m} \sum_{i \in B} l_i $. Throughout this work, we assume the loss to be differentiable, but we do not require it to be smooth. We now recall the definitions of RSI$^-$ and EB$^+$ as provided by Guille-Escuret et al. (2021). Given an objective function $ L $ with a convex set of global minima $ X^* $, and letting $ w_p^* $ be the orthogonal projection of $ w $ into $ X^* $. Definition 3.1 (Lower Restricted Secant Inequality). Let $ \mu > 0 $. $ L \in \text{RSI}^- (\mu) $ iff: \[ \forall w \in \mathbb{R}^d, \nabla L(w)^T (w - w_p^) \geq \mu \\| w - w_p^ \\|_2^2. \] Definition 3.2 (Upper Error Bounds). Let $ L > 0 $. $ L \in \text{EB}^+ (L) $ iff: \[ \forall w \in \mathbb{R}^d, \\| \nabla L(w) \\|_2 \leq L \\| w - w_p^* \\|_2. \] The classes of functions RSI$^-$ and EB$^+$ are thus defined in the literature as those respecting the above bounds over the entire parameter space. However, in this work, our focus lies not merely on their extremal values but on the local quantities bounded by RSI$^-$ and EB$^+$. For simplicity, we refrain from introducing new terminology, and henceforth denote these quantities as RSI$(G, w, w^)$ and EB$(G, w, w^)$, where $G$ is an oracle for the gradient at $w$. We do not mandate $G$ to be the full gradient of $L$; it could, for instance, correspond to the gradient $\nabla L_B$ with respect to a minibatch $B$. Similarly, $w^$ is not assumed to be a minimum of the objective function. Formally, for any gradient oracle $G$, and any $w^ \in \mathbb{R}^d, w \neq w^$: \[ \text{RSI}(G, w, w^) := \frac{G(w)^T (w-w^)}{\\|w-w^\\|_2} \quad \text{and} \quad \text{EB}(G, w, w^) := \frac{\\|G(w)\\|_2}{\\|w-w^\\|_2}. \] The ratio between RSI and EB imparts a direct geometrical interpretation: \[ \gamma(G, w, w^) := \frac{\text{RSI}(G, w, w^)}{\text{EB}(G, w, w^)} = \frac{G(w)^T (w-w^)}{\\|G\\|_2 \\|w-w^\\|_2} = \cosine(G(w), w - w^), \] where $\cosine(u_1, u_2)$ is the cosine of the angle between vectors $u_1$ and $u_2$. This ratio, $\gamma$, signifies the alignment between the negative sampled gradient and the direction from $w$ to $w^$. When $\gamma$ approaches 1, it indicates a negative gradient strongly directed toward $w^$. Conversely, a $\gamma$ close to 0 suggests a gradient almost orthogonal to $w - w^$. A negative $\gamma$ indicates a negative gradient directed away from $w^$. Additionally, $\gamma$ can be interpreted as the inverse of a local variant of the condition number, $\kappa := \sup_{w \neq w^, B} \inf_{w \neq w^, B} \text{RSI}$, which is a measure of the complexity of optimizing $L$ in prior works [Guille-Escuret et al., 2021]. RSI and EB are intrinsically connected to the dynamics of stochastic gradient descent (SGD). Indeed, the distance to $w^$ following an SGD step with step size $\eta$ can be precisely articulated using RSI and EB. For all $w \neq w^, B$, \[ \\|w - \eta \nabla L_B(w) - w^\\|_2^2 = \\|w - w^\\|_2^2 - 2\eta \nabla L_B(w)^T (w - w^) + \eta^2 \\|\nabla L_B(w)\\|_2^2 \\ = \left(1 - 2\eta \text{RSI}(\nabla L_B, w, w^) + \eta^2 \text{EB}^2(\nabla L_B, w, w^)\right) \\|w - w^\\|_2^2. \] Consequently, with a step size of \[ \eta^* := \argmin_{\eta} \\|w - \eta \nabla L_B(w) - w^\\|_2 = \frac{\text{RSI}(\nabla L_B, w, w^)}{\text{EB}^2(\nabla L_B, w, w^)}, \] SGD guarantees \[ \\|w_{t+1} - w^\\|_2 = \sqrt{1 - \gamma(\nabla L_B, w, w^)^2} \\|w_t - w^\\|_2. \] Furthermore, if $\inf_{w, B} \text{RSI}(\nabla L_B, w, w^) \geq \mu$ and $\sup_{w, B} \text{EB}(\nabla L_B, w, w^) \leq L$ hold for some $\mu > 0, L > 0$, then equation equation 4 demonstrates that running SGD with a fixed step size of $\eta = \frac{\mu}{L^2}$ will converge to $w^$ at a guaranteed rate: \[ \\|w_t - w^\\|_2^2 \leq (1 - \frac{\mu^2}{L^2})^t \\|w_0 - w^\\|_2^2, \] This holds irrespective of how the minibatches are sampled. Under these assumptions, this rate is, in fact, worst-case optimal among all continuous first-order algorithms [Guille-Escuret et al., 2022]. Experimental Measurement of RSI and EB:* One of the most significant challenges in experimentally measuring RSI and EB lies in the selection of $w^$. Even in cases where the objective function admits an unique global minimum, finding it in the context of deep neural networks is computationally infeasible [Blum & Rivest, 1992]. To navigate this complication, we initially train a neural network and subsequently choose the final iterate $w_T$ of the optimization sequence. Given successful training, the sequence will converge to the vicinity of a (local) minimum, and measuring RSI and EB with respect to this minimum will provide insightful understanding of the training dynamics. Notably, under this procedure, $w^$ is dependent on the optimization sequence rather than being predetermined. Therefore, interpreting the ensuing results warrants care, see Section 6. Considering that saving all gradients and iterates observed during training would be prohibitively resource-intensive, we perform two identical training runs. The first run computes $w^* = w_T$, and the second run computes RSI and EB along the optimization path. A detailed description of our experimental protocol is provided in Algorithm 1 in Appendix A.1 and we share our code at https://anonymous.4open.science/r/LossLandscapeGeometry-B7BD/. 4 Empirical Geometry of Landscapes Along Optimization Paths Figure 2: Depicted are the trends of RSI (top), EB (middle) and $\gamma$ (bottom) across three different scenarios: image classification on CIFAR-10 with a ResNet-18 (left), image classification on ImageNet with a ResNet-50 (middle) and language modeling on WikiText-2 with a transformer model (right). Figure 1 offers an initial glance at our results, outlining the behavior of $\gamma$ across four datasets, with variations across architecture, batch size, and optimization technique. Figure 2 presents a more streamlined view on three of these datasets, exhibiting not only $\gamma$ but also RSI and EB on a single run to preserve clarity. To avoid precision issues when $w_t$ approaches $w^$, the results from the final epoch have been excluded. Our hyperparameters were initially adjusted to optimize validation accuracy, echoing practical conditions. All experiments were coded in PyTorch [Paszke et al., 2019] and detailed descriptions of the specific training configurations, along with final test performances, are available in Appendix A to ensure full reproducibility. CIFAR-10 (ResNet-18):* Across the entire training run, not a single iteration exhibits a negative $\gamma$. Even though there are slight fluctuations across epochs, $\gamma$ predominantly remains within the $[0.0075, 0.02]$ range and does not exhibit substantial shifts. While the variance of RSI and EB across iterations tends to increase as training progresses, their mean values largely remain stable. ImageNet-1K (ResNet-50): Except for a few iterations at the very early stage, $\gamma$ remains positive throughout all of training. Moreover, the variance across iterations is notably low until the last epochs. Epoch-wise, RSI, EB, and $\gamma$ increase monotonically, with a sharp rise observed towards the end. WikiText-2 (Transformer): Throughout training, $\gamma$ remains strictly positive and always exceeds 0.05 after the second epoch. The cosine similarity maintains a remarkable stability, exhibiting only minor variations across iterations and epochs. While RSI and EB show very low variance within epochs, they do increase towards the end of the training period. 4.1 Fundamental Properties Upon careful analysis, we find that the optimization trajectories of deep neural networks exhibit the following major characteristic features: - The cosine similarity, $\gamma$, is almost always positive. - $\gamma$ demonstrates notable stability across both epochs and iterations, rarely departing from its (low) average value. - RSI and EB follow predictable trends, contingent upon whether the model adheres to an interpolation or a non-interpolation regime. Interpolation vs Non-Interpolation Regime: The behavior of RSI and EB are directly tied to how well the final iterate $w^$ interpolates the training data. For CIFAR-10, where the model reaches close to 0 training loss, RSI and EB retain relatively stable mean values up to the last epochs, which is made possible by stochastic gradients decreasing to 0 as $w_t$ approaches $w^$. Conversely, in scenarios where the model fails to interpolate the training data, such as for ImageNet and WikiText-2, stochastic gradients remain significant. In such a scenario, RSI and EB inevitably rise to infinity as $w_t - w^$ approaches zero. This phenomenon is particularly obvious with ImageNet due to the learning rate decay, which induces minuscule distances between $w_t$ and $w^$ in the later stages of training. Additional experimental results supporting this interpretation are provided in Appendix C. Late Training Behavior: The results obtained towards the end of training should be interpreted with caution. Besides the previously described phenomenon in the non-interpolation regime, the correlation between sampled gradients and $w_t - w^$ increases as the sequence nears its termination. Intuitively, $w^$ approximates a minima, and the approximation error becomes significant as iterates get sufficiently close. Further discussion on related implications can be found in Section 6. Low Value of Cosine Similarity: The low values of $\gamma$ empirically encountered are to be expected: if $\gamma$ was stable at reasonably high values, then we would find a near-minima in a small number of steps using SGD, which is notoriously not the case for modern problems. Instead, optimization sequences approach their final iterate at a slow but regular pace. While the stability and positivity of $\gamma$ imply a linear convergence rate, its low value indicate a linear rate close to 1, similarly to a strongly convex and smooth objective being badly conditioned. A plausible cause for $\gamma$ being small is that the useful signal from generalizable features in sampled gradients is dominated by that of spurious and coincidental correlations. Significance: These observations imply that, despite the well-documented non-convexity of the loss landscapes associated with neural networks and the inherent stochasticity introduced by minibatch sampling, the learning process of neural networks remains remarkably consistent. The networks progress steadily towards their destination throughout the training, with each stochastic gradient contributing valuable information to reach the final model state. With very few exceptions, gradients always point toward the right direction, and training trajectories never take a wrong turn when optimizing the loss function. We find these observations to be particularly remarkable on ImageNet. Given the presence of 1000 semantic classes (exceeding the batch size) and in excess of 5000 minibatches per epoch, the consistence of the cosine similarity $\gamma$ throughout entire epochs seems surprising. In addition, Section 6 establishes links between empirically adopted learning rate schedules and RSI and EB. Overall, RSI and EB are powerful tools to capture the elusive simplicity of neural loss landscapes, with empirical properties theoretically guaranteeing linear convergence rates. We thus encourage future works to consider RSI and EB to characterize the classes of objectives encountered in deep learning applications. We further explore the impact of various factors and provide a more comprehensive substantiation of our findings in Section 5. Following this, we discuss the implications and potential limitations of our observations in Section 6. We also discuss plausible causes in Appendix D. ### 5 INFLUENCE OF TRAINING SETTINGS Batch Size: the top right of Figure 1 delineates the cosine similarities corresponding to batch sizes ranging from 32 to 256 on the WikiText-2 dataset. As a complementary experiment, Figure 6 in Appendix B portrays the cosine similarities associated with batch sizes from 64 to 512 on the CIFAR-10 dataset. The outcomes of both these experiments consistently reveal a positive correlation between batch size and cosine similarity. This outcome is foreseeable: for two minibatches $B_i$ and $B_j$, we have $$\text{RSI}(\nabla L_{B_i} + \nabla L_{B_j}) = \text{RSI}(\nabla L_{B_i}) + \text{RSI}(\nabla L_{B_j}), \quad \text{EB}(\nabla L_{B_i} + \nabla L_{B_j}) \leq \text{EB}(\nabla L_{B_i}) + \text{EB}(\nabla L_{B_j}).$$ It should be noted that the selection of batch size not only affects the measurement of RSI and EB, but it also influences the optimization trajectory and the speed of convergence. Therefore, direct numerical comparisons across different batch sizes ought to be interpreted with caution. Nonetheless, our observations suggest that cosine similarities may scale with the square root of the batch size. Optimizer: Figure 1 (bottom left) illustrates the cosine similarity for three distinct optimizers utilized on the CIFAR-10 dataset. Intriguingly, Adam appears to result in lower cosine similarity values, albeit with reduced variance. We hypothesize that Adam, by amplifying the effective step size along directions with lower curvature, traverses further in flat dimensions, thereby leading to a reduced alignment compared to SGD. This conjecture is substantiated by Figure 12 in Appendix C, demonstrating that the journey undertaken by Adam indeed surpasses that of SGD in terms of distance. Notably, the employment of a momentum value of 0.9 with SGD does not significantly impact the value of $\gamma$, compared to not using momentum. Prior works also suggest that the optimization methods may affect the geometry of visited regions (Cohen et al., 2021). Model Depth and Width: Our attention now turns to the impact of depth and width on the geometric characteristics of the optimization trajectory, as depicted in Figure 1 (top left). In this experiment, we trained ResNets of varying depth — 18, 50, and 152 layers — with both standard and doubled width. A salient observation is that an increase in depth slightly enhances the cosine similarities, while an increase in width appears to have a comparatively trivial impact. These findings could potentially shed light on the prevalent trend in contemporary neural network designs favouring increased depth over width (He et al., 2016). ## 6 Key Takeaways and Discussion Geometrically Justified Learning Rate Schedules: As established in Equation equation 5, we define the locally optimal learning rate (IoLR) as the minimizer of $\\|w_t - \eta \nabla L_{B_t} - w^\\|_2$, $\eta^(w) = \frac{RSI(w)}{EB^2(w)}$. It is important to note, however, that $\eta^$ may not necessarily be globally optimal. Indeed, certain methodologies may initiate slower but accumulate more information, ultimately leading to faster convergence over a large number of steps. Furthermore, as the measurement of RSI and EB requires the knowledge of $w^$, which in turn depends on the learning rate (LR), the expression cannot be utilized to dynamically tune it. Despite these limitations, we find intriguing parallels between the evolution of the IoLR derived from our experiments and the shape of empirically validated LR schedules, as demonstrated in Figure 3. For instance, a widely adopted strategy for training on ImageNet involves a linear warm-up phase of the LR for the initial few epochs, followed by a cosine annealing phase. This pattern is mirrored in our empirical observations on ImageNet, except for a sharper decrease immediately after warmup. Moreover, the results on WikiText-2 echo two popular practices: linearly decreasing the LR and increasing the batch size over time. These intriguing observations suggest that the geometry of the loss landscape could potentially inform the design of more effective learning rate schedules. Lastly, the apparent correspondence between IoLR and empirical learning rate strategies implies that the efficiency of fixed learning rates may be contingent upon the stationarity of RSI and EB. Similarly, the existence of straightforward and efficient learning rate schedules can be associated with the predictable evolution of these geometrical properties. This strongly reinforces the view that such geometrical attributes play a substantial role in the widespread practical successes of deep learning. ![Figure 3](image.png) Left panel: The locally optimal learning rate, derived as per Equation equation 5, for various architectures implemented on the CIFAR-10 dataset. Right panel: The locally optimal learning rate, similarly determined, across a spectrum of batch sizes employed on the WikiText-2 dataset. Biases Induced by Using Final Iterates as Reference Points: A critical limitation of our experimental approach is the inescapable correlation between $w^$ and the optimization sequence. This association must be thoroughly addressed to appropriately interpret our findings. Figure 4: Depiction of cosine similarities during the training of a ResNet-18 on the CIFAR-10 dataset, with variations in (left) initialization seed and (right) epoch budget. • Initialization: Firstly, RSI and EB may represent local properties of the loss landscape, and could be dependent on the initialization region. However, this possibility is refuted by the left panel of Figure 4, which demonstrates minimal variation in $\gamma$ measurements across different random seeds. • Epoch Budget: Secondly, our results might be influenced by the particular moment when we terminate the optimization sequence to extract $w^$. The right panel of Figure 4 presents different measurements for epoch budgets ranging from 100 to 280, with all other parameters kept consistent. Our findings indicate a relative similarity in results before the sequence nears $w^$, suggesting that our experiments do not display excessive sensitivity to the epoch budget. • Induced Bias: However, this experiment also underscores the phenomenon detailed in Section 4.1, as the sequence approaches completion, the correlation between sampled gradients and $w_t - w^$ - induced by gradient updates - becomes increasingly significant. This correlation is a by-product of the optimization method, rather than a feature of local geometry, and augments the value of RSI and $\gamma$ by diminishing the impact of stochasticity. Consequently, this correlation should be taken into account when interpreting RSI and EB in the concluding epochs. A compelling illustration of this correlation can be seen in a discrete isotropic random walk with a fixed step size $s$ in a dimension $d$. When dimension $d$ significantly exceeds the number of steps, each pair of steps can be assumed to be nearly orthogonal with high probability. In such a setting, if we denote $(x_t)_{t=0..T}$ as the sequence generated by the random walk, we can calculate that, with high probability, $\forall t$, $$\frac{(x_t - x_{t+1})^T(x_t - x_T)}{\\|x_t - x_T\\|^2} \approx \frac{\\|x_t - x_{t+1}\\|^2}{\\|x_t - x_T\\|^2} \approx \frac{1}{T-t} > 0 \quad \text{and} \quad \\|x_t - x_{t+1}\\|_2 \approx \frac{1}{\sqrt{T-t}}$$ Consequently, the cosine similarity $\gamma(x_t) \approx (T-t)^{-0.5}$ remains strictly positive, and experiences a sharp increase toward the end, exemplifying the effect of the correlation induced by the selection of $w^$. It’s worth noting that in the case of neural networks, $\gamma$ remains approximately constant for the majority of training (as is clearly visible in Figure 1), which marks a distinction in their dynamics. Nonetheless, akin to the random walk scenario, it can be anticipated that the correlation induced by the choice of $w^$ would become increasingly evident as the number of remaining iterations diminishes. Contrasting Examples: Functions Without Beneficial Geometric Properties: We now turn our attention to delineating the behaviors that could potentially manifest in stochastic and non-convex optimization scenarios. To this end, we have engineered two illustrative counter-examples which effectively demonstrate that the consistency observed in Sections 4 and 5 is not a mere byproduct of our experimental paradigm. Our first example, termed Asymmetric Linear Model (ALM), entails the training of a linear model with the objective of consistently yielding outputs that are lower than their corresponding targets. The error between these values is calculated on stochastic minibatches using Root Mean Square Error (RMSE), thereby introducing a substantial degree of stochasticity. Despite this, the objective is a finite sum of convex functions and thus remains convex. The second function, designated Sinusoidal Mixture (SM), is deterministic but exhibits a pronounced degree of non-convexity. The mathematical expressions for both ALM and SM are presented below, with coefficients $a_i$, $x_i$, $y_i$ drawn randomly from normal distributions, $$ALM(w) = \sum_i \left[ \max(0, w^T x_i - y_i) \right]^2; \quad SM(w) = \\|w\\|_2^2 + 100 \sum_i \sin(a_i w_i)^2.$$ Figure 5 presents the measurements of RSI and $\gamma$ for both ALM and SM. Although these functions are characterized by relatively simple functional forms and do not simultaneously exhibit stochasticity and non-convexity, they demonstrate unpredictable trajectories and negative values for RSI and $\gamma$. This evidence compellingly suggests that the observed simplicity associated with neural networks is not a trivial characteristic. 7 CONCLUSION We have conducted an extensive series of experiments, assessing RSI and EB across a broad spectrum of training settings. These experiments reveal that these geometric properties display a collection of desirable characteristics, effectively demonstrating that neural network training proceeds smoothly, maintaining a consistently steady advancement towards its destination throughout the training process. These results contrast starkly with the theoretical complexity of neural landscapes and potentially open new pathways for developing theoretical results tailored to deep learning, or for designing optimization algorithms that exploit the geometry of empirical objective functions. A noteworthy point is that while RSI and EB appear to encapsulate significant beneficial aspects of neural networks, they likely do not encompass the entire scope of these advantages. There may be additional, complementary properties yet to be discovered. An intriguing indication of this is the fact that vanilla gradient descent has been proven to be exactly optimal for functions verifying the lower restricted secant inequality and upper error bound (Guille-Escuret et al., 2022). Given the well-documented efficacy of momentum in training neural networks, we conjecture that momentum exploits additional properties not captured by RSI and EB, which we encourage future works to explore. 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7jUQHmz4Tq	I am curious why the authors dropped the MvTec dataset for comparison since most anomaly detection algorithms are compared in the dataset. It is hard to assert that the method shows state-of-the-art performance without comparing the MvTecAD dataset in my opinion.	D3AD: Dynamic Denoising Probabilistic Model for Anomaly Detection Anonymous authors Paper under double-blind review Abstract Diffusion models have found valuable applications in anomaly detection by capturing the nominal data distribution and identifying anomalies via reconstruction. Despite their merits, they struggle to localize anomalies of varying scales, especially larger anomalies like entire missing components. Addressing this, we present a novel framework that enhances the capability of diffusion models, by extending the previous introduced implicit conditioning approach [Meng et al., 2022] in three significant ways. First, we incorporate a dynamic step size computation that allows for variable noising steps in the forward process guided by an initial anomaly prediction. Second, we demonstrate that denoising an only scaled input, without any added noise, outperforms conventional denoising process. Third, we project images in a latent space to abstract away from fine details that interfere with reconstruction of large missing components. Additionally, we propose a fine-tuning mechanism that facilitates the model to effectively grasp the nuances of the target domain. Our method undergoes rigorous evaluation on two prominent anomaly detection datasets VISA and BTAD, yielding state-of-the-art performance. Importantly, our framework effectively localizes anomalies regardless of their scale, marking a pivotal advancement in diffusion-based anomaly detection. All code will be made public upon acceptance. 1 Introduction Anomaly detection (AD) and related tasks such as identifying out-of-distribution data and detecting novel patterns, holds significant importance within the industrial sector. Applications range from detecting component defects [Roth et al., 2022; Zou et al., 2022] and fraudulent activities [Ahmed et al., 2016] to assistance in medical diagnoses [Baur et al., 2019; Wyatt et al., 2022] through identification of diseases. Overlooked anomalies in these applications could result in adverse financial and safety repercussions. In the manufacturing sector, flawed components which remain undetected lead to high scrap costs or customer complaints. Moreover, manual inspection of defects is a laborious task which often results in visual strain, especially when assessing reflective parts repeatedly. Motivated by these challenges, we explore the intricacies of visual anomaly detection within industrial contexts. In computer vision, anomaly detection entails both classifying images as anomalous or normal and segmenting/localizing anomalous regions. Typically, due to the scarcity of abnormal samples, an unsupervised approach is often employed for AD whereby a one-class classifier is trained on only nominal data. Such approaches can be grouped into representation-based and reconstruction-based methods. The latter reconstructs an anomalous input image, which is anomaly-free since the model is only trained on nominal data; thereby anomalies can be detected by simple comparison of the input with its reconstruction. However, previous generative models [Bergmann et al., 2019c; Gong et al., 2019] are easily biased towards the flawed input image leading to a reconstruction with the anomaly or artifacts. Diffusion models [Sohl-Dickstein et al., 2015; Ho et al., 2020] have shown success in image and video synthesis [Nichol et al., 2022; Rombach et al., 2022; Blattmann et al., 2023], 3D reconstruction [Poole et al., 2023], music generation [Kong et al., 2021], etc. They have also been used for the AD task acquiring promising results [Wyatt et al., 2022; Mousakhan et al., 2023] but their full potential in anomaly detection remain untapped. Figure 1: D3AD segmentation results of anomalies across scales from VisA and BTAD. Figure 2: Dynamic conditioning whereby the amount of added noise is a function of the input image and training dataset dependent on an initial guess of the severity of the anomaly. Anomalies occur in diverse forms from small scratches to complete missing components, see Figure 1. In previous AD diffusion models, we observe that simple application of fixed noise to an anomalous input image, known as static implicit conditioning [Meng et al., 2022], is insufficient to address the entire range of anomaly types and sizes. Therefore, we propose to compute the number of noising steps (noise amount) as a function of the input image and nominal training set, see Figure 2. This dynamic adjustment aids in precise segmentation of anomalies, which is often the weakest attribute of diffusion models in comparison with representation-based methods. To further abstract away from pixel-level details, we adopt a latent diffusion model and show that a latent representation along with the corresponding reconstruction provides state-of-the-art anomaly heatmaps while requiring less computing resources. Finally, our framework does not require noise to be added at inference time whereby a test image is directly denoised into a predicted reconstruction. Our main contributions are as follows: - We propose a dynamic conditioning mechanism where the maximum noise is computed using prior information about the anomaly provided by a KNN model of domain adapted features. - We propose a domain adaptation mechanism that aims to learn the target domain as well as reconstruction errors. - We propose to train a latent diffusion model for the task of anomaly detection to achieve precise anomaly heatmaps. - We perform extensive evaluation and ablation studies on our approach and demonstrate state-of-the-art performance in segmentation of anomalies at all scales. 2 RELATED WORK Reconstruction Methods These methods hinge on the premise that trained models are unable to generate anomalies, resulting in large disparity between an anomalous input and its reconstruction. Autoencoders have been vastly explored [Bergmann et al., 2019c; Gong et al., 2019]. However, the reconstructions often include the anomalous region resulting in erroneous anomaly heatmaps. An improvement has been to combine (variational) Autoencoder [Kingma & Welling, 2014] with adversarial training, leveraging a discriminator, to spot anomalies [Baur et al., 2019; Sabokrou et al., However, these methods still suffer from significant reconstruction error. GANs have also been explored for anomaly detection. For instance, Schlegl et al. (2017) introduced a feature-wise and visual loss. In their approach, nearest latent representation of input images is iteratively sought. In contrast, Akcay et al. (2019) employed an encoder-decoder-encoder architecture, optimizing both image and latent representation reconstructions. A discriminator then compared features from the original and reconstructed images. Alternative techniques, as cited in Haselmann et al. (2018); Zavrtanik et al. (2021b); Ristea et al. (2022), approach the problem as an in-painting task whereby random patches from images are obscured, and neural networks learn to infer the missing data. DRAEM (Zavrtanik et al., 2021a) used an end-to-end approach relying on synthetic data. Though reconstruction-based methods have had some success, they suffer from generated anomalies or artifacts within the reconstructions. Recent innovation have explored the potential of diffusion models in AD making use of an implicit conditioning proposed by SDEdit (Meng et al., 2022). Works by Wyatt et al. (2022); Zhang et al. (2023); Mousakhan et al. (2023) have showcased success in achieving high quality anomaly heatmaps however, these approaches fail in the face of large sized defects. Our D3AD method is agnostic to anomaly size and is capable of detecting a wide range of anomalies with varying severity. Representation Methods These methods gauge the discrepancy between the feature representation of test data and the learned representations of nominal data. This learned representation might either be a prototypical representation or the feature space mapping itself. PaDim (Defard et al., 2021) employs a patch-wise extraction and concatenation of features from multiple CNN layers. An empirical sample mean and covariance matrix for each patch’s feature vector is then computed. Anomalies are pinpointed based on the Mahalanobis distance between patches. Spade (Cohen & Hoshen, 2020) emphasizes this distance principle, computing the average distance of an image to its k-nearest neighbours pixel-wise and thresholding to discover anomalies. Patchcore (Roth et al., 2022) is a synthesis of both PaDim and Spade, employing a patch strategy, with each patch being compared to a coreset of all other patches. The distance comparison mirrors Spade, focusing on the average distance to k-nearest neighbours within the coreset. Similarly CFA (Lee et al., 2022) combines the patch based approach with metric learning. Another line of work utilises normalising flows (Rudolph et al., 2020; Yu et al., 2021; Gudovskiy et al., 2022) to directly estimate the likelihood function whereby sample in the low-density regions can instantly be identified as anomalies. Nonetheless, none of these approaches generate an anomaly-free rendition of the input image. This capability is highly sought after in an industrial context, as it fosters trust and provides valuable insights into the model’s decision-making process. Domain Adaption Most prior approaches employ pretrained feature extractors to map raw images into a latent space. However, these feature extractors often lack adaptation to the target domain, resulting in artifacts for reconstruction-based methods and inaccuracies in representation-based comparisons. To address this, domain adaptation techniques have been explored. For instance, SimpleNet (Liu et al., 2023) enhances a pretrained feature extractor with a domain adaptation layer and uses Gaussian noise to perturb features and training a discriminator to distinguish native from perturbed features. In contrast, RD4AD (Deng & Li, 2022) adopts an encoder-decoder structure, with the student network receiving the teacher’s latent representation instead of the original image. RD++ (Tien et al., 2023) extends this approach by incorporating additional projection layers to filter out anomalous information. Inspired by these successes, we implement a fine-tuning strategy for the pretrained feature extractors in order to leverage the demonstrated benefit. ### 3 BACKGROUND We use a class of generative models called diffusion probabilistic models (Sohl-Dickstein et al., 2015; Ho et al., 2020). In these, parameterized Markov chains with $T$ steps are used to gradually add noise to input data $x_0 \sim q(x_0)$ until all information is lost. The inspiration stems from principles of nonequilibrium thermodynamics (Sohl-Dickstein et al., 2015). Neural networks are then parameterised to learn the unknown reverse process, in effect learning a denoising model. The forward process $q$ is defined as: $$q(x_t \| x_{t-1}) = N(x_t; \sqrt{1 - \beta_t} x_{t-1}, \beta_t I)$$ $$q(x_t \| x_0) = N(x_t; \sqrt{\alpha_t} x_0, (1 - \bar{\alpha}_t) I)$$ \[ x_t = \sqrt{\alpha_t} x_0 + \sqrt{1 - \alpha_t} \epsilon, \quad \text{where} \quad \epsilon \sim \mathcal{N}(0, I) \tag{3} \] Usually the $ \beta_t $ are chosen as hyperparameters of the form $ \beta_t \in (0, 1) $ with a variance schedule $ \beta_0 < \beta_1 < \ldots < \beta_T $ such that the signal of the input gets sequentially disturbed. For direct sampling the $ \beta_t $ parameters are simplified to a compacter notation: $ \alpha_t = 1 - \beta_t $ and $ \bar{\alpha}_t = \prod_{s=1}^{t} \alpha_s $. Furthermore with large $ T $ and small $ \beta_t $, the distribution of $ x_T $ approaches a standard normal which enables sampling from a normal distribution in the reverse process parameterized by $ \theta $. This is defined as: \[ p_\theta(x_{t-1}\|x_t) = \mathcal{N}(x_{t-1}; \mu_\theta(x_t, t), \beta_t I) \tag{4} \] This corresponds to the DDPM [Ho et al. (2020)] formulation, where the variance is equivalent to the forward process while other works found better performance with learning the covariance matrix [Nichol & Dhariwal (2021)]. DDPM is trained by predicting the initially added noise $ \epsilon $ which corresponds to predicting $ \mu_\theta $ and leads to the training objective: \[ L_{\text{simple}}(\theta) = \mathbb{E}_{t,x_0,\epsilon}[\\|\epsilon - \epsilon_\theta(\sqrt{\bar{\alpha}_t} x_0 + \sqrt{1 - \bar{\alpha}_t} \epsilon, t)\\|_2^2] \tag{5} \] The noising and denoising is performed in pixel space which is computationally expensive therefore Rombach et al. (2022) proposed to utilise latent spaces. An encoder $ E $ of a continuous or quantized VAE is used to project an image $ x_0 $ into a lower dimension $ z_0 = E(x_0) $ while a decoder $ D $ aims to reconstruct this such that $ x_0 \simeq \hat{x}_0 = D(z_0) $. The following objective function is used: \[ L_{\text{simple-latent}}(\theta) = \mathbb{E}_{t,E(x_0),\epsilon}[\\|\epsilon - \epsilon_\theta(\sqrt{\bar{\alpha}_t} z_0 + \sqrt{1 - \bar{\alpha}_t} \epsilon, t)\\|_2^2] \tag{6} \] A faster sampling approach is proposed by DDIM [Song et al. (2022)] where a non-Markovian formulation of the DDPM objective is employed allowing sampling steps to be omitted. This implies that a diffusion model trained according to objective Eq. 5 or Eq. 6 can be used to accelerate the sampling without the need for retraining. Their proposed sampling procedure is: \[ x_{\tau_i-1} = \sqrt{\bar{\alpha}_{\tau_i-1}} f_\theta^{(\tau)}(x_\tau) + \sqrt{1 - \bar{\alpha}_{\tau_i-1}} - \sigma_{\tau_i}^2 \epsilon_\theta(x_{\tau_i}, \tau_i) + \sigma_{\tau_i} \epsilon_{\tau_i} \tag{7} \] Here $ \tau_i, i \in [1, \ldots, S] $ acts as an index for subset $ \{x_{\tau_1}, \ldots, x_{\tau_S}\} $ of length $ S $ with $ \tau $ as increasing sub-sequence of $ [1, \ldots, T] $. Moreover, an estimation of $ x_0 $ is obtained at every time step, denoted by $ f_\theta^{(\tau)}(x_t) = \frac{x_t - \sqrt{1 - \bar{\alpha}_t} \epsilon_\theta(x_t, t)}{\sqrt{\bar{\alpha}_t}} $ which utilizes the error prediction $ \epsilon $ according to equation 3. DDIM further demonstrates varying levels of stochasticity within the model with also a fully deterministic version which corresponds to $ \sigma_{\tau_i} = 0 $ for all $ \tau_i $. Guidance and conditioning the sampling process of diffusion models has been recently explored and often requires training on the conditioning with either an extra classifier [Dhariwal & Nichol (2021)] or classifier-free guidance [Ho & Salmans (2021)]. Recent work on AD with diffusion models [Mousakhan et al. (2023)] showed a guiding mechanism which does not require explicit conditional training. Guidance is achieved directly during inference by updating the predicted noise term using $ x_0 $ or respectively $ z_0 $ as: \[ \hat{\epsilon}_t = \epsilon_\theta(x_t, t) - \eta \sqrt{1 - \bar{\alpha}_t} (\hat{x}_t - x_t) \quad \text{with} \quad \hat{x}_t = \sqrt{\bar{\alpha}_t} x_0 + \sqrt{1 - \bar{\alpha}_t} \epsilon_\theta(x_t, t) \tag{8} \] where $ \eta $ controls the temperature of guidance. This updated noise term can then be used in the DDIM sampling formulation [7] to result in the intended reconstruction $ \hat{z}_0 $ and corresponding $ \hat{x}_0 $. ### 4 METHOD Diffusion models for AD learn the distribution of only nominal data such that they are unable to reconstruct anomalous regions leading to a large distance between input image $ x_0 $ and its reconstruction $ \hat{x}_0 $. Previous approaches rely on implicit conditioning [Meng et al. (2022)], whereby the input is noised until a fixed time step $ \hat{T} < T $ such that some input signal remains allowing for targeted reconstruction. We improve on this in two ways, first we discover that an noiseless and only scaled input $ x_{\hat{T}} = x_0 \sqrt{\bar{\alpha}_{\hat{T}}} $ is optimal for anomaly segmentation since it sufficiently reinforces the implicit conditioning applied on the model. Second we propose to choose forward time step $ \hat{T} $ dynamically based on an initial estimate of the anomaly. Furthermore, we adopt the architecture of unconditional latent diffusion model to abstract away from pixel-level representation which allows for improved reconstruction of large anomalies such as missing components in a resource efficient latent space. Our reconstruction and dynamic implicit conditioning frameworks are illustrated in Figure 3. Algorithm 1 describes the reconstruction process where we utilise the error correction (lines 6 and 7), proposed by DDAD Mousakhani et al. (2023), for guidance and the DDIM (Eq. 7) sampling procedure. Algorithm 2 details our dynamic conditioning mechanism for selecting optimum $ \hat{T} $ for the forward process. Training the diffusion model is according to the objective function in Eq. 6 without modifications. 4.1 Dynamic Implicit Conditioning We introduce dynamic implicit conditioning (DIC) into the model’s architecture. Specifically, we set a maximum implicit conditioning level denoted by $ T_{\text{max}} \in \{1, ..., T\} $. This is selected such that the signal-to-noise ratio remains high. We then establish a quantization of the maximum steps into increments ranging up to $ T_{\text{max}} $ with which we compute the dynamic implicit conditioning level $ \hat{T} $ for each image according to an initial estimate of the anomaly. Bin construction Our quantization is founded upon equidistant bins denoted as $ b \in B $. These bins are determined from the average KNN distances of the training set’s feature representations. Given that $ \phi $ is a pretrained domain adapted feature extractor, and $ \phi_j $ outputs the feature map of the $ j $-th layer block, for data point $ x_0 \in X_{\text{train}} $, the features are extracted as $ y_0 = \phi_j(x_0) $ with $ y_0 \in Y_{\text{train}} $. Utilizing $ y_0 $, a KNN-search is executed on the entire feature training set $ Y_{\text{train}} $ using the $ L_1 $-Norm. The K-nearest neighbors of $ y_0 $ are represented by the set $ \{y_{s_1}, ..., y_{s_K}\} $. Subsequently, we compute the mean distance to these KNNs and denote it as $ \bar{y}_0 $. While this method is susceptible to outliers due to its reliance on the arithmetic mean, it is anticipated that anomalous data will be substantially more distinct than regular data. Thus, any outlier within the regular data would be beneficial as it would lead to a wider range for the bins. We compute the average distance for each sample in the training set. Furthermore using the computed average distances, we delineate $ \|B\| $ evenly spaced bins. Dynamic Implicit Conditioning (DIC) We denote DIC by function $ g(x_0, X_{\text{train}}, T_{\text{max}}) $ described in Algorithm 2. A visual representation of this mechanism is illustrated in Figure 3. During inference, for a new image $ x_0 $, we first utilise $ \phi_j $ to extract features of $ x_0 $ and perform a KNN search on $ Y_{\text{train}} $. The distances are averaged to compute $ \bar{y}_0 $ which is then placed into bin $ b $ via a binary search function $ \psi $ on all $ b \in B $. The selected bin $ b $ serves as an initial estimate of the severity of the anomaly in the input image compared to the nominal training data. The dynamic time step $ \hat{T} $ is then simply computed as a fraction of $ T_{\text{max}} $ based on the selected bin. ![Figure 3: Reconstruction Architecture](image) An input $ x_0 $ is fed to the DIC to determine the level it must be perturbed $ \hat{T} $. $ x_0 $ is also projected to a latent representation $ z_0 $. Denoising is performed in the latent space leading to a predicted latent $ \hat{z}_0 $ which is decoded into a reconstruction $ \hat{x}_0 $. DIC: The average distance of extracted features of a test image to the K nearest neighbours from the training set is quantized, using equally sized predefined bins, to then determine the dynamic noising step $ \hat{T} $. Algorithm 1 Dynamic Reconstruction 1: input $ x_0 $ 2: $ \hat{T} = g(x_0, X_{Train}, T_{max}) $ 3: $ z_0 = E(x_0) $ 4: $ z_{\hat{T}} = z_0 \sqrt{\alpha_{\hat{T}}} $ # no noise 5: for $ t = \hat{T}, ..., 1 $ do 6: $ \hat{z}_t = \sqrt{\alpha_t} z_0 + \sqrt{1 - \alpha_t} \epsilon_t(z_t, t) $ 7: $ \hat{\epsilon}_t = \epsilon_0(z_t, t) - \eta \sqrt{1 - \alpha_t} (\hat{z}_t - z_t) $ 8: $ \hat{z}_{t-1} = \sqrt{\alpha_{t-1}} z_{\theta, 0} + \sqrt{1 - \alpha_{t-1}} \epsilon_t $ 9: end for 10: $ x_0 = D(\hat{z}_0) $ 11: return $ \hat{x}_0, \hat{z}_0 $ Algorithm 2 Dynamic Implicit Conditioning $ g $ 1: input $ x_0 $ 2: input $ T_{max} $ 3: $ Y_{Train} = \phi_j(X_{Train}) $ 4: $ y_0 = \phi_j(x_0) $ 5: $ \{y_{s_1}, ..., y_{s_K}\} = KNN(y_0, Y_{train}, K) $ 6: $ b_0 = \frac{1}{K} \sum_{j=1}^{K} \|\|y_0 - y_{s_j}\|\| $ 7: $ b = \psi(b_0) $ # binary search 8: $ \hat{T} = \left\lfloor \frac{b}{T_{max}} \right\rfloor $ 9: return $ \hat{T} $ 4.2 Anomaly Scoring and Map Construction We adopt the convention of comparing the input image with its reconstruction to generate the final anomaly map as illustrated in Figure 4. We compare the latent representation $ z_0 $ with its reconstruction $ \hat{z}_0 $ to construct a latent anomaly map $ l_{map} $. Similarly, we compare the features of the input image $ x_0 $ against its reconstruction $ \hat{x}_0 $ to construct a feature anomaly map $ f_{map} $. A weighted combination generates the final anomaly map $ A_{map} $. The feature anomaly map $ f_{map} $ is determined by first computing the features of an input image $ x_0 $ and its reconstruction $ \hat{x}_0 $ using a pretrained and domain adapted feature extractor $ \phi $ (section 4.3). A cosine distance between the extracted feature blocks at $ J \subseteq \{1, ..., J\} $ layers of a ResNet-54 yields the feature anomaly map. Given that feature blocks at different layers may present divergent dimensionalities, these are upsampled to achieve uniformity. The feature anomaly map $ f_{map} $ is articulated as $ f_{map}(x_0, \hat{x}_0) = \sum_{j \in J} (\cos_d(\phi_j(x_0), \phi_j(\hat{x}_0))) $. Since our approach relies on learning a denoising diffusion model on the latent representation, we further compute distances between the input image latent representation $ z_0 $ and its reconstructed counterpart $ \hat{z}_0 $. Utilizing the $ L1 $-Norm for each pixel, a latent anomaly map is deduced as $ l_{map}(z_0, \hat{z}_0) = \|\|z_0 - \hat{z}_0\|\|_1 $. The final anomaly map $ A_{map} $ is simply a linear combination of the normalized feature-based distance and the latent pixel-wise distance as follows: \[ A_{map} = \lambda * l_{map}(z_0, \hat{z}_0) + (1 - \lambda) * f_{map}(x_0, \hat{x}_0) \tag{9} \] Subsequently, an established threshold facilitates the categorization of every pixel and image, marking them as either anomalous or typical. The global image anomaly score is selected as the maximum pixel-level anomaly score within the entire image. Figure 4: Overview of the Anomaly Map construction. Feature heatmap ($ f_{map} $) are computed as cosine distances of the features of the input $ x_0 $ and its reconstruction $ \hat{x}_0 $ whereas latent heatmap ($ l_{map} $) is calculated using an $ L1 $ distance between the corresponding latent representations of $ x_0 $ and $ \hat{x}_0 $. These combine linearly to form the final anomaly heatmap ($ A_{map} $). 4.3 Domain Adaptation We leverage domain-adapted features for both the dynamic implicit conditioning and the construction of the feature anomaly map $ f_{map} $. Our objective is to grasp the intricacies associated with the target domain. With the use of variational autoencoders (VAEs) having pretrained encoders and decoder introduces artifacts and reconstruction inaccuracies. These are incorrectly flagged as anomalous regions during comparison. To address this, we introduce a loss function to fine-tune the feature extractor $ \phi $ by further training for $ \gamma $ epochs. This function is designed to minimize the feature distance between the input image $ x_0 $ and its reconstruction $ \hat{x}_0 $ as follows where GAP refers to global average pooling: \[ L_{DA}(x_0, \hat{x}_0) = \sum_{j=1}^{J} \text{GAP} \left( 1 - \frac{\phi_j(x_0)^T \phi_j(\hat{x}_0)}{\|\|\phi_j(x_0)\|\| \|\|\phi_j(\hat{x}_0)\|\|} \right). \] 5 Experiments Datasets We employ two widely used benchmarking datasets to evaluate the veracity of our approach, namely VisA [Zou et al., 2022] and BTAD [Mishra et al., 2021] dataset. VisA dataset presents a collection of 10,821 high-resolution RGB images, segregated into 9,621 regular and 1,200 anomalous instances. Comprehensive annotations are available in the form of both image and pixel-level labels. The dataset comprises of 12 different classes with a large variety of scale and type of anomalies. BTAD dataset comprises of RGB images showcasing three unique industrial products. There are 2540 images in total where each anomalous image is paired with a pixel-level ground truth mask. Evaluation Metrics We evaluate our approach using standard metrics for anomaly detection, namely pixel-wise AUROC (P-AUROC), image-wise AUROC (I-AUROC) and the PRO metric. P-AUROC is ascertained by setting a threshold on the anomaly score of individual pixels. A critical caveat of P-AUROC is its potential for overestimation, primarily because a majority of pixels are typically normal. Such skewed distribution occasionally renders a misleadingly optimistic performance portrayal. Addressing this limitation, the PRO metric [Bergmann et al., 2019a] levels the playing field by ensuring equal weighting for both minuscule and pronounced anomalies. This balance is achieved by averaging the true positive rate over regions defined by the ground truth, thereby offering a more discerning evaluative metric making it our primary choice for evaluation. The image-wise AUROC (I-AUROC) is employed to present an evaluation of image-based anomaly detection, where precise segmentation of the anomaly is unimportant. Implementation Details We employ an unconditional Unet from [Rombach et al., 2022] with an 8x downsampling within our diffusion model. For KNN, we set $ K = 20 $ with $ L^1 $ distance. Both dynamic conditioning and anomaly map construction utilize a ResNet-34 pretrained on ImageNet and fine-tuned. Domain adaptation is performed for up to 3 epochs using identical Unet settings. $ T_{max} $ is set at 80 for VisA and remains unchanged for BTAD. We chose $ \|B\| = 10 $ which leads to a percentage-quantization mapping of increments of 10% steps of $ T_{max} $. However, we set the minimum bin to 2, ensuring that we don’t rely solely on prior information. Lastly, the DDIM formulation with 10 steps is adopted for sampling, with the DIC step rounded to the nearest multiple of 10. All experiments were carried out on one Nvidia RTX 8000. Further implementational details are present in Appendix A.1. Anomaly Detection Results We conduct comprehensive experiments on the VisA dataset to evaluate the capability of our proposed method in detecting and segmenting anomalies. Table 1 details the performance of our method. Notably, D3AD excels in 8 of the 12 classes in segmentation accuracy as evident from PRO values, and in 3 of 12 classes for I-AUROC whilst achieving comparable performance in remaining classes. The aggregate performance across all classes yields an I-AUROC of 96.0%, paralleling the performance of the state-of-the-art method, RD4AD. Whereas there is a clear superiority of our method in segmentation achieving an average of 94.1%, outperforming the contemporary state-of-the-art by 2.7% points. In an evaluation alongside other diffusion-based models, as documented in Table 2, D3AD achieves superior anomaly localisation performance on the VisA benchmark. When assessed using PRO and P-AUROC, D3AD demonstrates an enhancement, achieving results higher by at least 0.9% points for both metrics compared to previous diffusion state-of-the-art approaches. Figure 1 offers a teaser of D3AD’s qualitative performance, with a comprehensive evaluation provided in appendix A.2. Significantly, the method excels in precise segmentation and effectively handling large anomalies. Further results from the BTAD benchmark are consolidated in Table 2. Here, D3AD exhibits competitive performance in terms of I-AUROC. More prominently, and following previous trend, segmentation evaluated using PRO highlight our method achieving unparalleled results, surpassing the closest competitors by a margin of 5.9 percentage points. ![Figure 5: Histogram of the binning values for the training set in blue and test set in orange, showing a distribution shift to larger values for the test set. Displayed are categories from VisA and BTAD.](image) Table 1: Anomaly classification and localization performance (I-AUROC, PRO) of various methods on VisA benchmark. The best results are highlighted in bold. \| Method \| Representation-based \| Reconstruction-based \| \|--------------\|----------------------\|----------------------\| \| \| SPADE \| PaDiM \| RD4AD \| PatchCore \| DRAEM \| D3AD (Ours) \| \| Candle \| (91.0,93.2) \| (91.6,95.7) \| (92.2,92.2) \| (98.6,94.0) \| (91.8,93.7) \| (95.6,92.7) \| \| Capsules \| (61.4,36.1) \| (70.7,76.9) \| (90.1,56.9) \| (81.6,85.5) \| (74.7,84.5) \| (88.5,95.7) \| \| Cashew \| (97.8,57.4) \| (93.0,87.9) \| (99.6,79.0) \| (97.3,94.5) \| (95.1,51.8) \| (94.2,89.4) \| \| Chewing gum \| (85.8,93.9) \| (98.8,83.5) \| (99.7,92.5) \| (99.1,84.6) \| (94.8,60.4) \| (99.7,94.1) \| \| Fryum \| (88.6,91.3) \| (88.6,80.2) \| (96.6,81.0) \| (96.2,85.3) \| (97.4,93.1) \| (96.5,91.7) \| \| Macaroni1 \| (95.2,61.3) \| (87.0,92.1) \| (98.4,71.3) \| (97.5,95.4) \| (97.2,96.7) \| (94.3,99.3) \| \| Macaroni2 \| (87.9,63.4) \| (70.5,75.4) \| (97.6,68.0) \| (78.1,94.4) \| (85.0,92.6) \| (92.5,98.3) \| \| PCB1 \| (72.1,38.4) \| (94.7,91.3) \| (97.6,43.2) \| (98.5,94.3) \| (47.6,24.8) \| (97.7,96.4) \| \| PCB2 \| (50.7,42.2) \| (88.5,88.7) \| (91.1,46.4) \| (97.3,89.2) \| (89.8,49.4) \| (98.3,94.0) \| \| PCB3 \| (90.5,80.3) \| (91.0,84.9) \| (95.5,80.3) \| (97.9,90.9) \| (92.0,89.7) \| (97.4,94.2) \| \| PCB4 \| (83.1,71.6) \| (97.5,81.6) \| (96.5,72.2) \| (99.6,90.1) \| (98.6,64.3) \| (99.8,86.4) \| \| Pipe fryum \| (81.1,61.7) \| (97.0,92.5) \| (97.0,68.3) \| (99.8,95.7) \| (100,75.9) \| (96.9,97.2) \| \| Average \| (82.1,65.9) \| (89.1,85.9) \| (96.0,70.9) \| (95.1,91.2) \| (88.7,73.1) \| (96.0,94.1) \| Ablation Studies To understand the significance of each component in our D3AD model, we executed an ablation study using the VisA dataset to evaluate our proposed dynamic implicit conditioning mechanism, domain adapted feature extractor and input scaling without noising method. Table 4 delves into the efficacy of our dynamic implicit conditioning (DIC). The DIC was compared against each quartile of the selected $T_{max}$, ranging from 25% to 100% of 80. The DIC consistently registered superior I-AUROC and P-AUROC scores, surpassing the second-best 80-step static model by margins of 0.6 and 1.2 percentage points, respectively. While PRO scores remained fairly consistent across different maximum step choices, the 20-step model slightly outperformed others with a score of 94.3, a slender 0.2 percentage points above the DIC. Given that PRO evaluates anomalies Table 2: Detection and segmentation performance of diffusion based methods (AnoDDPM [Wyatt et al. (2022)], DiffusionAD [Zhang et al. (2023)], DDAD [Mousakhan et al. (2023)]) on VisA. \| Method \| AnoDDPM \| DiffusionAD \| DDAD \| D3AD (Ours) \| \|--------------\|---------\|-------------\|------\|-------------\| \| I-AUROC \| 78.2 \| 97.8 \| 99.3 \| 96.0 \| \| P-AUROC \| - \| - \| 97.0 \| 97.9 \| \| PRO \| 60.5 \| 93.2 \| 92.0 \| 94.1 \| Table 3: Anomaly classification and localization performance (I-AUROC, PRO) of various methods on BTAD benchmark. The best results are highlighted in bold. \| Method \| FastFlow \| CFA \| PatchCore \| RD4AD \| RD++ \| D3AD (Ours) \| \|--------\|----------\|-----\|-----------\|-------\|------\|-------------\| \| Class 01 \| (99.4,71.7) \| (98.1,72.0) \| (96.7,64.9) \| (96.3,75.3) \| (96.8,73.2) \| (98.9,80.0) \| \| Class 02 \| (82.4,63.1) \| (85.5,53.2) \| (81.4,47.3) \| (86.6,68.2) \| (90.1,71.3) \| (87.0,71.7) \| \| Class 03 \| (91.1,79.5) \| (99.0,94.1) \| (100.0,67.7) \| (100.0,87.8) \| (100.0,87.4) \| (99.7,97.8) \| \| Average \| (91.0,71.4) \| (94.2,73.1) \| (92.7,60.0) \| (94.3,77.1) \| (95.6,77.3) \| (95.2,83.2) \| uniformly across all scales, and P-AUROC is more sensitive to large-scale anomalies, our observations suggest that the DIC adeptly identifies large anomalies, without compromising its efficiency across varying scales. The distribution of the initial signal is depicted in Figure 5 while Figure 6 shows the qualitative effect of DIC. It is apparent that a dynamically computed time step (DIC Mask) provided the most similar anomaly mask prediction to the ground truth (GT) mask, in comparison to fixed time step masks shown from 100% - 25% of $T$. Table 5 illustrates the effects of the domain adaptation in the feature extractor and introducing a scaled, yet noiseless, input. Using a model without domain-adapted feature extraction and conventional noised input as the baseline, we observe notable improvements with the integration of each component. Particularly, the modified implicit conditioning, indicated as "downscaling (DS)" in the table, emerges as the most impactful modification. A detailed qualitative visualisation is shown in appendix Figures 10 to 13 whereas a quantitative study of this effect is present in Figure 14. Figure 6: Overview of prediction masks for different levels of maximum static noise levels and the DIC. DIC tends to segment large anomalies more faithfully Table 4: Impact of Dynamic Implicit Conditioning (DIC) \| Max. Step \| Performance \| \|-----------\|-------------\| \| \| I-AUROC ↑ \| PRO ↑ \| P-AUROC ↑ \| \| 25%(20) \| 95.2 \| 94.3 \| 96.7 \| \| 50%(40) \| 94.7 \| 94.1 \| 96.6 \| \| 75%(60) \| 95.0 \| 94.2 \| 96.7 \| \| 100%(80) \| 95.4 \| 94.0 \| 96.7 \| \| DIC($g(.)$) \| 96.0 \| 94.1 \| 97.9 \| Table 5: Impact of Downscaling (DS) and Domain Adaptation (DA) \| Ablation \| Performance \| \|----------\|-------------\| \| DS \| DA \| \| - \| - \| 89.2 \| 82.0 \| 92.3 \| \| √ \| - \| 95.4 \| 92.0 \| 96.9 \| \| - \| √ \| 90.8 \| 83.8 \| 93.2 \| \| √ \| √ \| 96.0 \| 94.1 \| 97.9 \| 6 CONCLUSION We propose to rethink the convention, of diffusion models for the unsupervised anomaly detection task, of noising all samples to the same time step and instead use prior information to dynamically adjust such implicit conditioning. 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dtFN6T4aMU	The paper utilizes multiple technologies, such as RigL, hybrid TD targets, the Soft Mellowmax operator, and dual buffers, which may make it difficult to discern the specific kernel contribution and novelty.	MAST: A SPARSE TRAINING FRAMEWORK FOR MULTI-AGENT REINFORCEMENT LEARNING Anonymous authors Paper under double-blind review ABSTRACT Deep Multi-agent Reinforcement Learning (MARL) is often confronted with large state and action spaces, necessitating the utilization of neural networks with extensive parameters and incurring substantial computational overhead. Consequently, there arises a pronounced need for methods that expedite training and enable model compression in MARL. Nevertheless, existing training acceleration techniques are primarily tailored for single-agent scenarios, as the task of compressing MARL agents within sparse models presents unique and intricate challenges. In this paper, we introduce an innovative Multi-Agent Sparse Training (MAST) framework. MAST capitalizes on gradient-based topology evolution to exclusively train multiple MARL agents using sparse networks. This is then combined with a novel hybrid TD-(λ) schema, coupled with the Soft Mellowmax Operator, to establish dependable learning targets, particularly in sparse scenarios. Additionally, we employ a dual replay buffer mechanism to enhance policy stability within sparse networks. Remarkably, our comprehensive experimental investigation on the SMAC benchmarks, for the first time, that deep multi-agent Q learning algorithms manifest significant redundancy in terms of Floating Point Operations (FLOPs). This redundancy translates into up to 20-fold reduction in FLOPs for both training and inference, accompanied by a commensurate level of model compression, all achieved with less than 3% performance degradation. 1 INTRODUCTION Multi-agent reinforcement learning (MARL) (Shoham & Leyton-Brown, 2008), combined with deep neural networks, has not only revolutionized the field of artificial intelligence but also demonstrated remarkable success across a diverse spectrum of critical applications. From conquering multi-agent video games like Quake III Arena (Jaderberg et al., 2019), StarCraft II (Mathieu et al., 2021), Dota 2 (Berner et al., 2019), and Hide and Seek (Baker et al., 2019) to guiding autonomous robots through intricate real-world environments (Shalev-Shwartz et al., 2016; Da Silva et al., 2017; Chen et al., 2020b), deep MARL has established itself as an indispensable and versatile tool for addressing complex, multifaceted challenges. Its unique ability to capture intricate interactions and dependencies among multiple agents has generated novel insights and solutions, solidifying its role as a transformative paradigm across various domains (Zhang et al., 2021; Albrecht et al., 2023). Nonetheless, the extraordinary success of deep MARL comes at a substantial computational cost. Training these agents involves the intricate task of adapting neural networks to accommodate an expanded parameter space, especially when the number of agents involved is substantial. For example, the training regimen for AlphaStar (Mathieu et al., 2021), designed for StarCraft II, which spanned an arduous 14-day period, utilizing 16 TPUs per agent. The OpenAI Five (Berner et al., 2019) model for Dota 2 underwent a marathon training cycle, spanning 180 days and tapping into thousands of GPUs. This exponential growth in computational demands as the number of agents increases presents a formidable challenge when deploying MARL in practical problems. The joint action and state spaces swell exponentially, imposing a steep demand on computational resources. Researchers have explored dynamic sparse training (DST) like SET (Mocanu et al., 2018) and RigL (Evci et al., 2020) to address computational challenges. While initial attempts at sparse single-agent deep reinforcement learning (DRL) training have been made in (Sokar et al., 2022; Graesser et al., 2022), DST methods have struggled to achieve consistent model compression across diverse environments. RLx2 (Tan et al., 2022) enables sparse neural network training for DRL but is ineffective... for multi-agent RL (MARL). In a motivating experiment, we tested various sparse training methods on the 3s5z tasks from SMAC (Samvelyan et al., 2019) using a neural network with only 10% of its original parameters, as shown in Figure 1. Classical DST methods, including SET and RigL, as well as RLx2 for single-agent RL, perform poorly in MARL scenarios, not to mention static sparse networks (SS). In contrast, our MAST framework achieves over 90% win rate. The sole prior effort to train sparse MARL agents, as in (Yang et al., 2022), prunes agent networks during training with weight grouping (Wang et al., 2019). However, this approach fails to maintain sparsity throughout training, reaching only about 80% sparsity. Moreover, their experimental evaluation is confined to a two-user environment, PredatorPrey-v2, in MuJoCo (Todorov et al., 2012). These observations underscore the fact that, despite their promise, the application of sparse networks in the context of MARL remains largely uncharted territory. The existing state-of-the-art DST technique, RLx2 (Tan et al., 2022), while effective in single-agent scenarios, demonstrates limitations when confronted with the challenges posed by MARL. MARL introduces unique complexities, including larger system spaces, the non-stationarity inherent in multi-agent training, and the partially observable nature of each agent. Consequently, a critical and intriguing question emerges: Can we train MARL agents using sparse networks throughout? We give an affirmative answer to the question by presenting a novel sparse training framework, MAST, tailored explicitly for value decomposition methods in MARL. It leverages gradient-based topology evolution, offering a powerful tool for the efficient exploration of network configurations in sparse models. Notably, our investigation has unveiled the formidable challenges faced by MARL algorithms in the realm of ultra-sparse models, i.e., inaccurate learning targets and training instability. To surmount these challenges, MAST introduces innovative solutions. We present a novel hybrid TD(λ) target mechanism, coupled with the Soft Mellowmax operator, which facilitates precise value estimation even in the face of extreme sparsity. Additionally, MAST unveils a dual buffer mechanism designed to bolster training stability in sparse environments. As a result, MAST empowers the training of highly efficient MARL agents with minimal performance compromise, employing ultra-sparse networks throughout the training process. Our extensive experiments, conducted across several popular MARL algorithms, validate MAST’s position at the forefront of sparse training. These experiments reveal MAST’s ability to achieve model compression ratios ranging from $5\times$ to $20\times$, all while incurring minimal performance trade-offs, typically under 3%. Moreover, MAST boasts the impressive capability to reduce FLOPs required for both training and inference by up to an astounding $20\times$, showing a large margin over other baselines including SET (Mocanu et al., 2018), RigL (Evci et al., 2020) and RLx2 (Tan et al., 2022). 2 RELATED WORK Sparse networks, initially proposed in deep supervised learning, can train a 90%-sparse network without performance degradation from scratch. However, for deep reinforcement learning, the learning target is not fixed but evolves in a bootstrap way (Tesauro et al., 1995), and the distribution of the training data can also be non-stationary (Desai et al., 2019), which makes the sparse training more difficult. We list some representative works for training sparse models from supervised learning to reinforcement learning. A more comprehensive illustration can be found in Appendix A.1. Sparse Models in Supervised Learning Various techniques have been explored for creating sparse networks, ranging from pruning pre-trained dense networks (Han et al., 2015, 2016; Srinivas et al., 2017), to employing derivatives (Dong et al., 2017; Molchanov et al., 2019). Another avenue of research revolves around the Lottery Ticket Hypothesis (LTH) (Frankle & Carbin, 2019), which posits the feasibility of training sparse networks from scratch, provided a sparse “winning ticket” initialization is identified. Additionally, there is a body of work dedicated to training sparse neural networks from the outset, involving techniques that evolve the structures of sparse networks during training. Examples include SET (Mocanu et al., 2018) and RigL (Evci et al., 2020). Sparse Models in Single-Agent RL Existing research (Schmitt et al., 2018; Zhang et al., 2019) has employed knowledge distillation with static data to ensure training stability and generate small dense agents. Policy Pruning and Shrinking (PoPs) (Livne & Cohen, 2020) generates sparse agents through iterative policy pruning. Another line of investigation aims to train sparse DRL models from scratch, eliminating the necessity of pre-training a dense teacher. (Sokar et al., 2022; Graesser et al., utilize the DST in single-agent RL, achieving a 50% – 80% sparsity level. More recently, RLx2 (Tan et al., 2022) has demonstrated the capacity to train DRL agents with highly sparse neural networks from scratch, yet RLx2 performs poorly in MARL as demonstrated in Section 5.1. Sparse Models in MARL The existing endeavour has made attempts to train sparse MARL agents, such as (Yang et al., 2022), which prunes networks for multiple agents during training. Another avenue of research seeks to enhance the scalability of MARL through sparse architectural modifications. For instance, (Sun et al., 2020) uses a sparse communication graph with graph neural networks to reduce problem scale, and (Kim & Sung, 2023) adopts structured pruning for a deep neural network to extend the scalability. Others focus on parameter sharing between agents to reduce the number of trainable parameters, with representative works including (Li et al., 2021; Christianos et al., 2021). Yet existing methods fail to maintain high sparsity throughout the training process. 3 DEEP MULTI-AGENT REINFORCEMENT LEARNING PRELIMINARIES We model the MARL problem as a decentralized partially observable Markov decision process (Oliehoek et al., 2016), represented by a tuple $(N, S, U, P, r, Z, O, \gamma)$, with detailed specification in Appendix A.2. Deep Multi-Agent Q-learning extends the deep Q learning method (Mnih et al., 2013) to multi-agent scenarios (Sunehag et al., 2018; Rashid et al., 2020b; Son et al., 2019). Each agent encounters partial observability, and the agent-wise Q function is defined over its history $r_i$ as $Q_i$ for agent $i$. Subsequently, the joint action-value function $Q_{tot}(\tau, u)$ operates over the joint action-observation history $\tau$ and joint action $u$. The objective, given transitions $(\tau, u, r, \tau')$ sampled from the experience replay buffer $B$, is to minimize the mean squared error loss $L(\theta)$ on the temporal-difference (TD) error $\delta = y - Q_{tot}(\tau, u)$. Here, the TD target $y = r + \gamma \max_{u'} Q_{tot}(\tau', u')$, where $Q_{tot}$ is the target network for the joint action Q-function, periodically copied from $Q_{tot}$. Parameters of $Q_{tot}$ are updated using $\theta' = \theta - \alpha \nabla_\theta L(\theta)$, with $\alpha$ representing the learning rate. CTDE We focus on algorithms that adhere to the Centralized Training with Decentralized Execution (CTDE) paradigm (Oliehoek et al., 2008; Kraemer & Banerjee, 2016). Within this paradigm, agents undergo centralized training, where they have access to the complete action-observation history and global state information. However, during execution, they are constrained to their individual local action-observation histories. To efficiently implement CTDE, the Individual-Global-Maximum (IGM) property (Son et al., 2019), defined in Eq. (1), serves as a key mechanism. \[ \arg\max Q_{tot}(s, u) = (\arg\max Q_1(s, u_1), \ldots, \arg\max Q_N(s, u_N)). \] Many deep MARL algorithms adhere to the IGM criterion, such as the QMIX series algorithms (Rashid et al., 2020b). These algorithms employ a mixing network $f_s$ with non-negative weights, enabling the joint Q-function to be expressed as $Q_{tot}(s, u) = f_s(Q_1(s, u_1), \ldots, Q_N(s, u_N))$. 4 BOOSTING THE PERFORMANCE OF SPARSE MARL AGENTS This section outlines the pivotal components of the MAST framework. Initially, MAST relies on the gradient-based topology evolution for finding proper sparse network topology. However, as depicted in Figure 1, training ultra-sparse MARL models using topology evolution presents substantial challenges. Consequently, MAST introduces innovative solutions to address the accuracy of value learning in ultra-sparse models by concurrently refining training data targets and distributions. 4.1 TOPOLOGY EVOLUTION The topology evolution mechanism in MAST follows the RigL method (Evci et al., 2020). RigL improves the optimization of sparse neural networks by leveraging weight magnitude and gradient information to jointly optimize model parameters and connectivity. After setting the initial network sparsity, the initial sparsity distribution of each layer is decided by Erdős–Rényi strategy from (Mocanu et al., 2018). As shown in Figure 2, RigL periodically dynamically drops a subset of existing connections with the smallest absolute weight values and concurrently grows an equivalent number of empty connections with the largest gradients. The diminishing update fraction $\zeta_t$ for connections follows $\zeta_t = \frac{\zeta_0}{T_{end}}(1 + \cos(\pi t/T_{end}))$, where $\zeta_0$ is the initial update fraction, and $T_{end}$ is the training steps. This process maintains the network sparsity throughout the training yet with strong evolutionary ability. The topology evolution is detailed in Algorithm 1, where the symbol $\odot$ denotes the element-wise multiplication operator, while $M_\theta$ symbolizes the binary mask that delineates the sparse topology. for the network $\theta$. We set a low topology adjustment rate as prior studies (Evci et al., 2020; Tan et al., 2022), occurring at intervals of 200 gradient updates. This setup minimizes the computational burden of topology evolution, ensuring operational feasibility even on resource-constrained devices. Algorithm 1 Topology Evolution (Evci et al., 2020) 1. $\theta_t, N_l, s_l$: parameters, number of parameters, sparsity of layer $l$. 2. for each layer $l$ do 3. $k = \zeta_t(1 - s_l)N_l$ 4. $I_{\text{drop}} = \text{ArgTopK}(-\|\theta_l \odot M_{\theta_l}\|, k)$ 5. $I_{\text{grow}} = \text{ArgTopK}_{i \notin I_{\text{drop}}}(\|\nabla \theta_l L\|, k)$ 6. Update $M_{\theta_l}$ according to $I_{\text{drop}}$ and $I_{\text{grow}}$ 7. $\theta_l \leftarrow \theta_l \odot M_{\theta_l}$ 8. end for Figure 3 provides an overview of sparse models when MAST is applied to QMIX. MAST introduces three innovative solutions to achieve accurate value learning in ultra sparse models: i) Hybrid TD($\lambda$) targets to mitigate estimation errors from network sparsity. ii) The Soft Mellowmax operator to reduce overestimation in sparse models. iii) Dual replay buffers to stabilize sparse training. ### 4.2 Hybrid TD($\lambda$) Targets In MAST, we utilize hybrid TD($\lambda$) targets to generate reliable learning targets, which achieves a good trade-off between sparse network fitting errors and learning variances. We will first introduce the benefit of TD($\lambda$) targets and then show the necessity of the hybrid scheme. TD($\lambda$) Targets Temporal difference (TD) learning is a fundamental method for determining an optimal policy in reinforcement learning, with the value network iteratively updated by minimizing a squared loss driven by the TD target. Denote the multi-step return $T_t^{(n)}$ at timestep $t$ for deep multi-agent Q learning as $T_t^{(n)} = \sum_{i=t}^{t+n-1} \gamma^{i-t}r_i + \gamma^{n+1} \max_u Q_{\text{tot}}(s_{t+n}, u)$. As evidenced in prior works (Sokar et al., 2022; Tan et al., 2022), sparse networks, denoted by parameters $\hat{\theta} = \theta \odot M_\theta$, where $\odot$ signifies element-wise multiplication, and $M_\theta$ is a binary mask representing the network’s sparse topology, operates within a reduced hypothesis space with fewer parameters. Consequently, the sparse network $\hat{\theta}$ may induce a large bias, such that the learning targets become unreliable. Denote the network fitting error as $\epsilon(s, u) = Q_{\text{tot}}(s, u; \theta) - Q_{\text{tot}}^{\pi_t}(s, u)$, it will be larger under an improper sparsified model compared to a dense network, as evidenced in Figure 1, where improper sparsified models fail in learning good policy. Specifically, Eq. (2) from (Tan et al., 2022) characterises the expected error between the multi-step TD target $T_t^{(n)}$ and the true Q-function $Q_{\pi_t}$ associated with the target policy $\pi_t$, conditioned on transitions from the behaviour policy $b_t$, reveals that introducing a multi-step return target discounts the network fitting error by a $\gamma^n$ factor. $$E_{b_t}[T_t^{(n)}(s, u)] - Q_{\pi_t}(s, u) = \left(E_{b_t}[T_t^{(n)}(s, u)] - E_{\pi_t}[T_t^{(n)}(s, u)]\right) + \gamma^n E_{\pi_t}[\epsilon(s_{t+n}, \pi_t(u_{t+n}))].$$ Thus, employing a multi-step return $T_t^{(n)}$ with a sufficiently large $n$, e.g., $T_t^{(\infty)}$ or Monte Carlo methods (Sutton & Barto, 2018), effectively diminishes the network fitting error by a very small factor of $\gamma^n$ approaching 0 for $\gamma < 1$. However, the Monte Carlo method is susceptible to large variance, which implies that an optimal TD target shall be a multi-step return with a judiciously chosen $n$, striking a balance between network fitting error and variances. This motivates us to introduce the TD($\lambda$) target (Sutton & Barto, 2018) to achieve good trade-off: $T_t^\lambda = (1 - \lambda) \sum_{n=1}^{\infty} \lambda^{n-1} T_t^{(n)}$ for $\lambda \in [0, 1]$, which average all of the possible multi-step returns $\{T_t^{(n)}\}_{n=1}^{\infty}$ into a single return by using a weight that decays exponentially, and is computationally efficient with episode-form data. Hybrid Scheme Previous studies (Fedus et al., 2020; Tan et al., 2022) have highlighted that an immediate shift to multi-step targets can exacerbate policy inconsistency error in Eq. (2). Since the TD($\lambda$) target $T_t^\lambda$ averages all potential multi-step returns $\{T_t^{(n)}\}_{n=1}^{\infty}$, an immediate transition to this target may encounter similar issues. We adopt a hybrid strategy inspired by the delayed approach in Tan et al. (2022). Initially, when the training step is less than $T_0$, we use one-step TD targets. $(T_t^{(1)})$ to minimize policy inconsistency errors. As training progresses and the policy stabilizes, we seamlessly transition to TD($\lambda$) targets to mitigate sparse network fitting errors. Such a hybrid TD($\lambda$) mechanism ensures consistent and reliable learning targets, even within sparse models. Furthermore, we empirically demonstrate the effectiveness of our proposed hybrid TD($\lambda$) targets on the $355z$ task in the SMAC, as illustrated in Figure 4. Our findings underscore the pivotal role of TD($\lambda$) in enhancing the learning process of sparse models. Interestingly, we observe that including a 1-step return target during initial training, although slightly reducing sample efficiency, contributes significantly to the agents’ learning in the final stages. This highlights the necessity of our hybrid approach for sparse networks. Moreover, we examine hybrid multi-step TD targets in RLx2 (Tan et al., 2022) for single-agent sparse training with a fixed $n = 3$, in our experiments on RigL-QMIX. Figure 4 clearly illustrates the superiority of our hybrid TD($\lambda$) mechanism. This suggests the optimal TD target may not always be a fixed multi-step return; instead, an average value is a robust choice, coinciding with Figure 7.2 in (Sutton & Barto, 2018). ### 4.3 Soft Mellowmax Operator We empirically observe that the overestimation issue still arises in sparse MARL models, significantly impacting performance. MAST utilizes a robust operator, i.e., Soft Mellomax operator from (Gan et al., 2021), to alleviate the overestimation and achieve accurate value estimation. Overestimation The max operator in the Bellman operator poses a well-known theoretical challenge, i.e., overestimation, hindering the convergence of various linear or non-linear approximation schemes (Tsitsiklis & Van Roy, 1996), which stands as a significant source of instability in the original deep Q-network (DQN) (Mnih et al., 2015). Deep MARL algorithms, including QMIX (Rashid et al., 2020b), also grapple with the overestimation issue. Recent research efforts (Gan et al., 2021; Pan et al., 2021) have aimed to alleviate overestimation through conservative operators and regularization techniques. Moreover, our empirical investigations reveal that the overestimation issue persists in sparse models, significantly impacting performance. Figure 5 illustrates the win rates and estimated values of QMIX with or without our Soft Mellowmax operator on $355z$ in the SMAC. We derive estimated values by averaging over 40 episodes sampled from the replay buffer every 10,000 timestep. Figure 5(a) shows that the performance of RigL-QMIX-SM outperforms RigL-QMIX, and Figure 5(b) shows that Soft Mellowmax operator does effectively mitigate the overestimation bias. These emphasize that in sparse models, QMIX still faces overestimation issues, highlighting the critical importance of addressing overestimation. Soft Mellow operator For MARL algorithms satisfying the IGM property in Eq. (1), we replace the max operator in $Q_i$ to Soft Mellowmax operator (Gan et al., 2021) in Eq. (3), to mitigate overestimation bias in the joint-action Q function within sparse models. \[ sm_\omega(Q_i(\tau, \cdot)) = \frac{1}{\omega} \log \left[ \sum_{u \in U} \text{softmax}_\alpha(Q_i(\tau, u)) \exp (\omega Q_i(\tau, u)) \right], \] where $\text{softmax}_\alpha(Q_i(\tau, u)) = \frac{\exp(\alpha Q_i(\tau, u))}{\sum_{u' \in U} \exp(\alpha Q_i(\tau, u'))}$, $\omega > 0$ and $\alpha \in \mathbb{R}$. Eq. (3) can be regarded as a specific instance of the weighted quasi-arithmetic mean (Beliakov et al., 2016). The softmax_$\alpha(Q)$ can be interpreted as a representation of policy probability, aligning with the framework of entropy regularization and KL divergence (Fox et al., 2015; Mei et al., 2019). Also note that when $\alpha = 0$, the Soft Mellowmax operator simplifies to the Mellomax operator $mm(\cdot)$ as: \[ mm_\omega(Q_i(\tau, \cdot)) = \frac{1}{\omega} \log \left[ \sum_{u \in U} \frac{1}{\|U\|} \exp (\omega Q_i(\tau, u)) \right]. \] Also, $ \lim_{\omega \to \infty} \min_{\omega} Q_i(\tau, \cdot) = \max_u Q_i(\tau, u), \lim_{\omega \to 0} \min_{\omega} Q_i(\tau, \cdot) = \frac{1}{\|\mathcal{U}\|} \sum_u Q_i(\tau, u) $ according to (Asadi & Littman [2017]). As demonstrated in (Gan et al., 2021), the Soft Mellomax operator extends the capabilities of the Mellomax operator in various aspects, including provable performance bounds, overestimation bias reduction, and sensitivity to parameter settings. ### 4.4 Dual Buffers Training with online data enhances learning stability but sacrifices sample efficiency (Song et al., 2023). Conversely, offline data training boosts sample efficiency at the expense of stability. Figure 6 displays training dynamics for RigL-QMIX and others in SMAC’s 3s5z task, revealing QMIX instability in sparse models. Inspired by (Li et al., 2022), MAST employs a hybrid approach with two replay buffers: $ B_1 $ (offline, large capacity, typically around 5000) and $ B_2 $ (online, smaller capacity, usually around 100). $ B_1 $ follows an off-policy style, while $ B_2 $ aligns with an on-policy style. In each step, MAST samples $ b_1 $ episodes from $ B_1 $ and $ b_2 $ transitions from $ B_2 $, conducting a gradient update based on a combined batch of size $ (b_1 + b_2) $. As seen in Figure 6, dual buffers enhance QMIX’s training stability under sparse models, leading to consistent policy improvements and higher rewards. This mechanism remains insensitive in dense cases where network parameters ensure stable policy improvements. Notably, while prior works have explored prioritized or dynamic-capacity buffers (Schaul et al., 2015; Tan et al., 2022), they may be not applicable here due to data being in episode form, since addressing partial observation issue in MARL using recurrent neural networks. Target Value and Loss Function Combining hybrid TD(λ) with the Soft Mellomax operator, we modify the target $ y $ as follows: \[ y_S = \begin{cases} G_t^{(1)}, & \text{if } t < T_0, \\ (1 - \lambda) \sum_{n=1}^{\infty} \lambda^{n-1} T_t^{(n)}, & \text{Otherwise}. \end{cases} \] Here, $ \lambda \in [0, 1] $ is a hyperparameter, and $ T_t^{(n)} = \sum_{i=t}^{t+n} \gamma^{i-t} r_i + \gamma^{n+1} f_s(\text{sm}_\omega(Q_1(\tau_1, \cdot), \ldots, \text{sm}_\omega(Q_N(\tau_N, \cdot))) $, where $ f_s $ denotes the mixing network and $ Q_i $ is the target network of $ Q_i $. The loss function of MAST is defined as: \[ L_S(\theta) = \mathbb{E}_{(s,u,r,s') \sim B_1 \cup B_2} [(y_S - Q_{tot}(s, u))^2]. \] When $ \lambda = 0 $, it is equivalent to the 1-step TD target. When $ \lambda = 1 $, it can be thought of as the Monte Carlo method. ### 5 Experiments In this section, we conduct a comprehensive performance evaluation of MAST across four tasks: 3m, 2s3z, 3s5z, and 2c_vs_64zq from the SMAC benchmark (Samvelyan et al., 2019). MAST serves as a versatile sparse training framework specifically tailored for value decomposition-based Multi-Agent Reinforcement Learning (MARL) algorithms. In Section 5.1, we integrate MAST with state-of-the-art MARL algorithms, including QMIX (Rashid et al., 2020b), WQMIX (Rashid et al., 2020a), and RES (Pan et al., 2021), with detailed implementation given in Appendix A.3. This integration allows us to meticulously quantify the benefits derived from sparsification. To gain a profound understanding of the individual components that constitute MAST, we present a comprehensive ablation study in Section 5.2. Furthermore, we assess the performance of sparse models generated by MAST in Section 5.3. Detailed experimental configurations can be found in Appendix B. Also note that each reported result is based on the average performance over four independent runs, each utilizing distinct random seeds. #### 5.1 Comparative Evaluation Table 1 presents a comprehensive summary of our comparative evaluation, where MAST is benchmarked against the following baseline methods: (i) Tiny: Utilizing tiny dense networks with a parameter count matching that of the sparse model during training. (ii) SS: Employing static sparse networks with random initialization. (iii) SET (Mocanu et al., 2018): prunes connections based on their magnitude and randomly expands connections. (iv) RigL (Evci et al., 2020): This approach leverages dynamic sparse training, akin to MAST, by removing and adding connections based on magnitude and gradient criteria. (v) RLx2 (Tan et al., 2022): A specialized dynamic sparse training framework tailored for single-agent reinforcement learning. We set the same sparsity levels for both the joint Q function $ Q_{tot} $, and each individual agent’s Q function $ Q_i $. For every algorithm and task, the sparsity level indicated in Table 1 corresponds to... the highest admissible sparsity threshold of MAST. Within this range, MAST’s performance consistently remains within a ±3% margin compared to the dense counterpart, effectively representing the minimal sparse model size capable of achieving performance parity with the original dense model. All other baselines are evaluated under the same sparsity level as MAST. We assess the performance of each algorithm by computing the average win rate per episode over the final 20 policy evaluations conducted during training, with policy evaluations taking place at 10000-step intervals. Identical hyperparameters are employed across all 4 environments for 3 algorithms, detailed in Appendix B.3. Table 1: Comparisons of MAST with sparse training baselines. Sp.: sparsity. Total Size: total model parameters (detailed in Appendix B.4). The data is all normalized w.r.t. the dense model. \| Alg. \| Env. \| Sp. \| Total Size (Train) \| FLOPs (Train) \| FLOPs (Test) \| Tiny (%) \| SS (%) \| SET (%) \| RigL (%) \| RLx2 (%) \| Ours (%) \| \|------\|------\|-----\|-------------------\|---------------\|--------------\|----------\|-------\|--------\|---------\|---------\|---------\| \| Q-MIX \| 3m \| 95% \| 0.066x \| 0.051x \| 0.050x \| 98.3 \| 91.6 \| 96.0 \| 95.3 \| 12.1 \| 100.9 \| \| \| 2s3z \| 95% \| 0.062x \| 0.051x \| 0.050x \| 83.7 \| 73.0 \| 77.6 \| 68.2 \| 45.8 \| 98.0 \| \| \| 3s5z \| 90% \| 0.109x \| 0.101x \| 0.100x \| 68.2 \| 34.0 \| 52.3 \| 45.2 \| 50.1 \| 99.0 \| \| \| 64* \| 90% \| 0.106x \| 0.100x \| 0.100x \| 58.2 \| 40.2 \| 67.1 \| 48.7 \| 9.9 \| 96.4 \| \| \| Avg. \| 92% \| 0.086x \| 0.076x \| 0.075x \| 77.1 \| 59.7 \| 73.2 \| 64.3 \| 29.8 \| 98.6 \| \| WQ-MIX \| 3m \| 90% \| 0.108x \| 0.100x \| 0.100x \| 98.3 \| 96.9 \| 97.8 \| 97.8 \| 98.0 \| 98.6 \| \| \| 2s3z \| 90% \| 0.106x \| 0.100x \| 0.100x \| 89.6 \| 75.4 \| 85.9 \| 86.8 \| 87.3 \| 100.2 \| \| \| 3s5z \| 90% \| 0.105x \| 0.100x \| 0.100x \| 70.7 \| 62.5 \| 56.0 \| 50.4 \| 60.7 \| 96.1 \| \| \| 64* \| 90% \| 0.104x \| 0.100x \| 0.100x \| 51.0 \| 29.6 \| 44.1 \| 41.0 \| 52.8 \| 98.4 \| \| \| Avg. \| 90% \| 0.106x \| 0.100x \| 0.100x \| 77.4 \| 66.1 \| 70.9 \| 69.0 \| 74.7 \| 98.1 \| \| RES \| 3m \| 95% \| 0.066x \| 0.055x \| 0.050x \| 97.8 \| 95.6 \| 97.3 \| 91.1 \| 97.9 \| 99.8 \| \| \| 2s3z \| 90% \| 0.111x \| 0.104x \| 0.100x \| 96.5 \| 92.8 \| 92.8 \| 94.7 \| 94.0 \| 98.4 \| \| \| 3s5z \| 85% \| 0.158x \| 0.154x \| 0.150x \| 95.1 \| 89.0 \| 90.3 \| 92.8 \| 86.2 \| 99.4 \| \| \| 64* \| 85% \| 0.155x \| 0.151x \| 0.150x \| 83.3 \| 39.1 \| 44.1 \| 35.3 \| 72.7 \| 104.9 \| \| \| Avg. \| 89% \| 0.122x \| 0.116x \| 0.112x \| 93.2 \| 79.1 \| 81.1 \| 78.5 \| 87.7 \| 100.6 \| Performance Table 1 unequivocally illustrates MAST’s substantial performance superiority over all baseline methods in all four environments across the three algorithms. Notably, static sparse (SS) consistently exhibit the lowest performance on average, highlighting the difficulty of finding optimal sparse network topologies in the context of sparse MARL models. Dynamic sparse training methods, namely SET and RigL, slightly outperform (SS), although their performance remains unsatisfactory. Sparse networks also, on average, underperform tiny dense networks. However, MAST significantly outpaces all other baselines, indicating the successful realization of accurate value estimation through our MAST method, which effectively guides gradient-based topology evolution. Notably, the single-agent method RLx2 consistently delivers subpar results in all experiments, potentially due to its limited replay buffer capacity, severely hampering sample efficiency. To further substantiate the efficacy of MAST, we conduct performance comparisons across various sparsity levels in 3s5z, as depicted in Figure 7. This reveals an intriguing observation: the performance of sparse models experiences a sharp decline beyond a critical sparsity threshold. Compared to conventional DST techniques, MAST significantly extends this critical sparsity threshold, enabling higher levels of sparsity while maintaining performance. Moreover, RES achieves a higher critical sparsity threshold than the other two algorithms with existing baselines, e.g., SET and RigL, achieving a sparsity level of over 80% on average. However, it is essential to note that the Softmax operator in RES results in significantly higher computational FLOPs (as detailed in Appendix B.4.5), making it incomparable in terms of training and inference acceleration to MAST. FLOPs Reduction and Model Compression In contrast to knowledge distillation or behavior cloning methodologies, exemplified by works such as (Livne & Cohen, 2020; Vischer et al., 2022), MAST maintains a sparse network consistently throughout the entire training regimen. Consequently, MAST endows itself with a unique advantage, manifesting in a remarkable acceleration of training FLOPs. We observed up to 20-fold acceleration in training and inference FLOPs for MAST-QMIX in the 2s3z task, with an average acceleration of 10-fold, 9-fold, and 8-fold for QMIX, WQMIX, and RES-QMIX, respectively. Moreover, MAST showcases significant model... compression ratios, achieving reductions in model size ranging from 5-fold to 20-fold for QMIX, WQMIIX, and RES-QMIX, while incurring only minor performance trade-offs, typically below 3%. 5.2 Ablation Study We conduct a comprehensive ablation study on three critical elements of MAST: hybrid TD(λ) targets, the Soft Mellowmax operator, and dual buffers, specifically evaluating their effects on QMIX and WQMIIX. Notably, since MAST-QMIX shares similarities with MAST-RES, our experiments focus on QMIX and WQMIIX within the 3s5z task. This meticulous analysis seeks to elucidate the influence of each component on MAST and their robustness in the face of hyperparameter variations. The reported results are expressed as percentages and are normalized with respect to dense models. Hybrid TD(λ) We commence our analysis by evaluating various burn-in times $ T_0 $, for hybrid TD(λ). Additionally, we explore the impact of different $ \lambda $ values within hybrid TD(λ). The results are presented in Table 2, revealing hybrid TD(λ) targets achieve optimal performance with a burn-in time of $ T_0 = 0.75M $ and $ \lambda = 0.6 $. It is noteworthy that hybrid TD(λ) targets lead to significant performance improvements in WQMIIX, while their impact on QMIX is relatively modest. \| Alg. \| $ T_0 $ \| $ \lambda $ \| \|------------\|-----------\|---------------\| \| \| 0 \| 0.75M \| 1.5M \| 2M \| 0 \| 0.2 \| 0.4 \| 0.6 \| 0.8 \| 1 \| \| QMIX / RES \| 93.6 \| 97.9 \| 92.5 \| 91.5 \| 91.5 \| 94.7 \| 96.8 \| 96.8 \| 97.9 \| 89.4 \| \| WQMIIX \| 83.5 \| 98.0 \| 76.9 \| 70.3 \| 83.5 \| 83.5 \| 74.7 \| 98.0 \| 96.1 \| 87.9 \| \| Avg. \| 88.5 \| 97.9 \| 84.7 \| 80.9 \| 87.5 \| 89.1 \| 85.7 \| 97.4 \| 97.0 \| 88.6 \| Soft Mellowmax Operator The Soft Mellowmax operator in Eq.(3) introduces two hyperparameters, $ \alpha $ and $ \omega $. A comprehensive examination of various parameter configurations is presented in Table 3. Our analysis reveals that the performance of MAST exhibits robustness to changes in the two hyperparameters associated with the Soft Mellowmax operator. Additionally, it is worth noting that the Softmax operator is also employed in [Pan et al., 2021] to mitigate overestimation in multi-agent Q learning. To examine the effectiveness of various operators, including max, Softmax, Mellowmax, and Soft Mellowmax, we conduct a comparative analysis in Figure 8. Our findings indicate that the Soft Mellowmax operator surpasses all other baselines in alleviating overestimation. Although the Softmax operator demonstrates similar performance to the Soft Mellowmax operator, it is important to note that the Softmax operator entails higher computational costs, as elucidated in Appendix B.4.5. Dual buffers It is worth noting that in each training step, we concurrently sample two batches from the two buffers, $ B_1 $ and $ B_2 $. We maintain a fixed total batch size of 32 while varying the sample partitions $ b_1 : b_2 $ within MAST. The results, detailed in Table 3, reveal that employing two buffers with a partition ratio of 5 : 3 yields the best performance. Additionally, we observed a significant degradation in MAST’s performance when using data solely from a single buffer, whether it be the online or offline buffer. This underscores the vital role of dual buffers in sparse MARL. \| Alg. \| Smaple Partitions \| Soft Mellowmax Operator \| \|------------\|-------------------\|-------------------------\| \| \| 8 : 0 \| 5 : 3 \| 3 : 5 \| 0 : 8 \| $ \alpha = 1 $ \| $ \alpha = 5 $ \| $ \alpha = 5 $ \| $ \alpha = 10 $ \| $ \alpha = 10 $ \| \| \| $ \omega = 10 $ \| $ \omega = 5 $ \| $ \omega = 10 $ \| $ \omega = 5 $ \| $ \omega = 10 $ \| \| QMIX / RES \| 93.6 \| 97.9 \| 97.8 \| 85.1 \| 97.9 \| 100.0 \| 98.9 \| 96.8 \| 97.9 \| \| WQMIIX \| 64.8 \| 98.0 \| 86.8 \| 70.3 \| 98.0 \| 92.3 \| 87.9 \| 92.3 \| 85.7 \| \| Avg. \| 79.2 \| 97.9 \| 92.3 \| 77.7 \| 97.9 \| 96.1 \| 93.4 \| 94.5 \| 91.8 \| 5.3 Sparse Models Obtained by MAST We conduct a comparative analysis of diverse sparse network architectures. With identical sparsity levels, distinct sparse architectures lead to different hypothesis spaces. As emphasized in specific architectures, such as the “winning ticket,” outperform randomly generated counterparts. We compare three architectures: the “random ticket” (randomly sampled topology held constant during training), the “winning ticket” (topology from a MAST or RigL run and kept unchanged during training), and the “cheating ticket” (trained with MAST). Figure 9 illustrates that both the “cheating ticket” and “winning ticket” by MAST achieve the highest performance, closely approaching the original dense model’s performance. Importantly, using a fixed random topology during training fails to fully exploit the benefits of high sparsity, resulting in significant performance degradation. Furthermore, RigL’s “winning ticket” fares poorly, akin to the “random ticket.” These results underscore the advantages of our MAST approach, which automatically discovers effective sparse architectures through gradient-based topology evolution, without the need for pretraining methods like knowledge distillation [Schmitt et al., 2018]. Crucially, our MAST method incorporates key elements: the hybrid TD($\lambda$) mechanism, Soft Mellowmax operator, and dual buffers. Compared to RigL, these components significantly improve value estimation and training stability in sparse models facilitating efficient topology evolution. Figure 10 showcases the evolving sparse mask of a hidden layer during MAST-QMIX training in $3s5z$, capturing snapshots at 0, 5, 10, and 20 million steps. For additional layers, refer to Appendix B.8. The upper section of Figure 10 illustrates the mask, while the lower part presents connection counts for output dimensions, sorted in descending order. Notably, a pronounced shift in the mask is evident at the start of training, followed by a gradual convergence of connections within the layer onto a subset of input neurons. This convergence is discernible from the clustering of light pixels forming continuous rows in the lower segment of the final mask visualization, where several output dimensions exhibit minimal or no connections. This observation underscores the distinct roles played by various neurons in the representation process, emphasizing the prevalent redundancy in dense models and highlighting the effectiveness of our MAST framework. 6 CONCLUSION This paper introduces MAST, a novel sparse training framework for deep MARL, utilizing gradient-based topology evolution to efficiently explore network configurations in sparse models. MARL faces significant challenges in ultra-sparse models, including value estimation errors and training instability. To address these, MAST offers innovative solutions: a hybrid TD($\lambda$) target mechanism combined with the Soft Mellowmax operator for precise value estimation in extreme sparsity, and a dual buffer mechanism for enhanced training stability. MAST enables efficient MARL agent training with minimal performance impact, employing ultra-sparse networks throughout. Our experiments across popular MARL algorithms validate MAST’s leadership in sparse training, achieving model compression of $5\times$ to $20\times$ with minimal performance degradation, and up to a remarkable $20\times$ reduction in FLOPs for both training and inference. Besides, the limitation and future work of the MAST framework are discussed in Appendix A.4. REFERENCES SV Albrecht, F Christianos, and L Schäfer. Multi-agent reinforcement learning: Foundations and modern approaches. Massachusetts Institute of Technology: Cambridge, MA, USA, 2023. Kavosh Asadi and Michael L Littman. 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yKksu38BpM	I find the second sentence in the abstract confusing. I expected this trend to have to do with using kernel-based models for data attribution rather than to “investigate a diverse set of neural network behavior”. Isn’t the goal of your paper exactly to apply kernel models to investigate network behavior?	Faithful and Efficient Explanations for Neural Networks via Neural Tangent Kernel Surrogate Models Andrew Engel¹ Zhichao Wang² Natalie S. Frank³ Ioana Dumitriu² Sutanay Choudhury¹ Anand Sarwate⁴ Tony Chiang¹,⁵,⁶ ¹Pacific Northwest National Laboratory ²University of California, San Diego ³Courant Institute, NYU ⁴Rutgers University ⁵University of Washington ⁶University of Texas, El Paso {andrew.engel,sutanay.choudhury,tony.chiang}@pnnl.gov; {zhw036,idumitriu}@ucsd.edu; nf1066@nyu.edu; ads221@soe.rutgers.edu Abstract A recent trend in explainable AI research has focused on surrogate modeling, where neural networks are approximated as simpler ML algorithms such as kernel machines. A second trend has been to utilize kernel functions in various explain-by-example or data attribution tasks. In this work, we combine these two trends to analyze approximate empirical neural tangent kernels (eNTK) for data attribution. Approximation is critical for eNTK analysis due to the high computational cost to compute the eNTK. We define new approximate eNTK and perform novel analysis on how well the resulting kernel machine surrogate models correlate with the underlying neural network. We introduce two new random projection variants of approximate eNTK which allow users to tune the time and memory complexity of their calculation. We conclude that kernel machines using approximate neural tangent kernel as the kernel function are effective surrogate models, with the introduced trace NTK the most consistent performer. Open source software allowing users to efficiently calculate kernel functions in the PyTorch framework is available here. 1 Introduction Explainability remains a critical open problem for applications of deep neural networks (NNs) (Leavitt & Morcos, 2020). Explain-by-example techniques (Lai et al., 2021; Yang et al., 2020) have emerged as a major category of algorithms for explainability, including prototype examples (Chen et al., 2019), Deep K-Nearest Neighbors (Papernot & McDaniel, 2018; Wang et al., 2021; Dziedzic et al., 2022), and Representer Points (Yeh et al., 2018; Tsai et al., 2023). These techniques explain models by providing example(s) that capture model behavior on new data. Kernel functions (Alvarez et al., 2011) are a natural choice for building explain-by-example algorithms (Yeh et al., 2018); a kernel measures the similarity between individual data points via an inner product in a reproducing kernel Hilbert space (RKHS) (Hilbert, 1912; Ghojogh et al., 2021). A RKHS that faithfully represents a linearized NN feature space can be used in a kernel machine to explain (model) the NN decision as a weighted sum of similarities to training data. In this work, we investigate computationally efficient approximations to the empirical neural tangent kernel (eNTK), which is a kernel function motivated by advances in the theory of deep learning (Jacot et al., 2018). It is well established that NNs trained using gradient descent are equivalent to kernel machines (Schölkopf & Smola, 2002) with a kernel constructed from a sum over eNTK (Lee et al., 2020) computed at each gradient step (Domingos, 2020; Bell et al., 2023). Given this equivalence, we would like to evaluate the eNTK as the kernel function for an explain-by-example algorithm; however, computing eNTK is computationally expensive (Novak et al., 2022; Chen et al., 2022), https://github.com/pnnl/projection_ntk so low computational cost approximations have been developed instead (Mohamadi & Sutherland, 2022). We are the first to define and evaluate one such approximate kernel, the trace neural tangent kernel (trNTK). Additionally, we build from the work of Park et al. (2023) to provide software to compute random-projection variants that can be computed and stored with lower time and memory cost over traditional eNTK. Using these approximations, we build low-cost and faithful surrogate models for neural network classifiers. Our methodology improves over the past evaluation of kernel surrogate models. We measure the faithfulness of a kernel function by assessing how well a kernel generalized linear model (kGLM) (Hofmann et al., 2007) correlates with the softmax probabilities of the original NN using a rank correlation. Previous evaluations relied on test accuracy (Mohamadi & Sutherland, 2022; Long, 2021), or having high similarity to the correct class (Hanawa et al., 2021), which are both flawed. Our approach and accompanying code-repository will allow users to evaluate how close their own NNs are to kernel machines in the PyTorch framework with limited overhead (Paszke et al., 2019). CONTRIBUTIONS We make three major contributions in this work: 1. We define and evaluate new kernel functions for faithful approximation of an underlying neural network; we are the first to analyze random projection variants that permit tuning the computational and memory expense of approximate eNTK. 2. We are the first to show that approximate eNTK kernel surrogate models are consistently correlated to the underlying neural network across experiments including ResNet18 on CIFAR10 and Bert-base on COLA. 3. We compare explanations of NN decisions generated from each kernel function through a data attribution strategy and through an explain-by-example strategy; this is the first such qualitative evaluation between approximate eNTK. RELATED WORK Surrogate Models for Explaining Neural Network Behavior. Recent work in explainable AI has focused on determining when NNs are exactly equivalent to other common ML algorithms (Lee et al., 2018; Balestriero & Baraniuk, 2018; Schmitz et al., 1999), including kernel machines. It has been shown that infinitely wide NNs are equivalent to a kernel machine with kernel function chosen as the neural tangent kernel (Jacot et al., 2018). These infinitely wide models, however, do not replicate the feature learning behavior seen in finite-width networks (Chizat et al., 2018; Yang & Hu, 2021; Wang et al., 2022). Subsequently, researchers turned to investigate properties of finite-width models with NTK computed at various checkpoints (Domingos, 2020; Bell et al., 2023) and/or after training (Long, 2021). This framework was used to explore inductive biases (Ortiz-Jiménez et al., 2021), feature learning (Radhakrishnan et al., 2022), learning dynamics (Fort et al., 2020; Atanasov et al., 2022), and adversarial faithfulness (Tsilivis & Kempe, 2023; Loo et al., 2022). Support vector machines (Vapnik, 1999) using eNTK or approximate eNTK kernels computed after training were shown to achieve the same test accuracy as the underlying NN (Atanasov et al., 2022; Long, 2021; Vyas et al., 2022; Mohamadi & Sutherland, 2022). Our work builds upon this by evaluating whether kernel machines can approximate the underlying neural network function itself, rather than simply reproduce the same test accuracy. Kernels for Explainability. Kernel functions defined from various RKHS have been proposed to explain the behavior of NN in different contexts, (Park et al., 2023; Koh & Liang, 2017; Pruthi et al., 2020; Akyürek et al., 2023), but in each of these works the kernel studied is loss-based and relies upon the availability of labels at inference time. We differ in that our goal is to model/explain the classification behavior on any new data, including unlabeled data where the loss is incalculable. Most relevant to our work, Yeh et al. (2018) (hereafter Presenter Points) used a kernel formed from the NN final embedding in what we call the data attribution task (see section 2). We build from Presenter Points by evaluating their assumptions under new approximate eNTK kernels. Computationally Feasible Approximations of the eNTK The computational cost of the eNTK is prohibitively high for large models and datasets. Advances on this issue have been two-pronged: Some groups focus on algorithmic improvements to calculate the eNTK directly (Novak et al., 2022). An alternative strategy has been to avoid eNTK calculation and instead compute kernel functions that share a similar structure to the eNTK (Mohamadi & Sutherland, 2022). One such approximate kernel was introduced quietly in Chen et al. (2022) which we refer to as the trace-NTK (trNTK). We are the first to explicitly investigate the trNTK’s properties. Finally, Park et al. (2023), hereafter TRAK, utilized random projection matrices to scale the computation of a loss-based kernel function. We modify TRAK to compute projected variants of approximate eNTK. Evaluating Kernel Attribution. In this paper, we use three evaluation strategies. The first focuses on evaluating the faithfulness of the surrogate model through rank correlation. The second evaluates surrogate model performance on a data-attribution task. We follow the methodology in Shan et al. (2022) to evaluate the model via precision and recall in tracing decisions on poisoned test data back to poisoned training data. Finally, we compare kernels qualitatively via explain-by-example. Previous work evaluated kernels through whether the attributions trace to training data of the correct class (Hanawa et al., 2021), whether surrogate models replicate NN test accuracy (Mohamadi & Sutherland, 2022; Long, 2021). These are insufficient: our goal is that kernel functions reflect the neural network behavior, but test accuracy is invariant to the specific classification on individual datapoints. Presenter Points used Pearson correlation as a faithfulness measure, but Pearson correlation can conflate covariance with faithfulness (see Appendix H). We will demonstrate that our methodology is more secure measurement of faithfulness. 2 PRELIMINARIES Neural Networks for Classification. We consider the supervised classification problem with $C$ classes. Consider a data input $x \in \mathcal{X} \subseteq \mathbb{R}^n$ with $n$ the dimensionality of inputs, and a one-hot encoded data label vector $z \in \mathcal{Z} \subseteq \mathbb{R}^C$. We define a neural network $F(x; \theta) : \mathcal{X} \rightarrow \mathcal{Y}$ where the output space $\mathcal{Y} \subseteq \mathbb{R}^C$ is an intermediary step in our classification called a “logit.” The NN $F(x; \theta)$ is parameterized by the vector $\theta$ and was learned via back-propagation to minimize the cross entropy loss between the target label vector $z$ and softmax probability vector $\sigma(F(x; \theta))$, with $\sigma : \mathcal{Y} \rightarrow \mathcal{Z}$ the softmax function. We denote the $c$-th scalar output of the network as $F^c$. We interpret the predicted confidence for the $c$-th class for input $x$ as $\sigma(F(x; \theta))^c$. Kernel Functions. Kernel functions implicitly map the data vector $x$ to a feature vector $\rho(x)$ in a higher dimensional RKHS $\mathcal{V}$ for which the kernel function $\kappa(\cdot, \cdot)$ evaluates the inner product of two feature vectors in $\mathcal{V}$. We will notate the data matrix $X = [x_1, \ldots, x_N] \in \mathbb{R}^{N \times n}$ with $N$ the number of training samples. With some abuse of notation, we will write $\kappa(x, X) \in \mathbb{R}^N$ for the vector whose $j$-th component is $\kappa(x, x_j)$ and $\kappa(X, X) \in \mathbb{R}^{N \times N}$ for the matrix whose $(i, j)$-th entry is $\kappa(x_i, x_j)$. Kernel General Linear Models as Surrogate Models We limit our investigation of surrogate models to kernel general linear models. We define a general kernel linear model kGLM : $\mathcal{X} \rightarrow \mathcal{Y}$ as: $$kGLM(x) := W \kappa(x, X) + b,$$ where $W \in \mathbb{R}^{C \times N}$ is a learnable weight matrix, $\kappa$ is the kernel function, and $b \in \mathbb{R}^C$ is a learnable bias vector. We compute classifications from kGLM by mapping the final activations to softmax confidences. The parameters $W$ and $b$ are learned using an optimizer to minimize the cross entropy loss using the same dataset upon which the NN is trained. Given an input $x$, the softmax activation $\sigma$, and a NN $F(x; \theta)$, the ideal surrogate modeling goal is to find a kGLM that satisfies: $$\sigma(kGLM(x)) = \sigma(F(x; \theta)),$$ for all $x$. Keeping this ideal in mind is useful for building intuition, but in practice, we will relax from this ideal goal for reasons described below. Data Attribution with Kernels. Our main motivation is to explain neural networks through data attribution, i.e., by computing "a score for each training datapoint indicating its importance to the output of interest" (TRAK). Given the choice of kernel function $\kappa$, the scalar valued data attribution for the $c$-th class for a test input $x$ and a training datapoint $x_i$ is given by: $$A(x, x_i)^c := W_{c,i} \kappa(x, x_i) + \frac{b_c}{N}.$$ Where the $ \frac{b}{N} $ term is necessary to ensure that the sum over the attributions for the entire training dataset is equal to the kGLM’s logit for class $ c $, \[ \sum_{i=1}^{N} A(x, x_i)^c = \text{kGLM}(x)^c. \] If the kGLM is an ideal surrogate model Eq. 2, then the softmax function applied to the vector created from each class attribution will equal the NN confidence in each class. Consequently, we will have decomposed the reasoning for the NN’s specific confidence in each class to a linear combination of similarities between $ x $ and each training datapoint $ x_i $. We emphasize that Eq. 3 is our definition of data attribution. Attribution is a weighted sum of kernel/similarity values. ### 3 METHODS We now turn towards the novel work of this research. In the following sections we describe our measure of faithfulness then introduce the kernel functions. Evaluating the Faithfulness of Surrogate Models. Given many choices of kernel functions we require a measure to determine which surrogate models have higher approximation quality (i.e., faithfulness) to the NN. We relax from the ideal surrogate model goal Eq. 2 and instead evaluate kernel functions by how well they are correlated with the neural network using the Kendall-$ \tau $ rank correlation. To assess the faithfulness of a surrogate model, we compute $ \tau_K $ between the softmax probability of the neuron representing the correct class, $ \sigma(F(x; \theta))^c $, and the kGLM softmax probability for the output representing the correct class, $ \sigma(\text{kGLM}(x))^c $. $ \tau_K $ was chosen for two reasons: First, $ \tau_K $ has a range $[-1, 1]$ with $ \pm 1 $ representing a monotonic relationship and a value of 0 representing no correlation. Second, if the relationship between the kGLM and NN is strictly monotonic, then an invertible mapping function exists between the kGLM softmax probabilities and the NN’s (Bartle & Sherbert, 2011). Therefore, for a $ \tau_K = 1 $ we would recover the one-to-one ideal surrogate model relationship given by Eq. 2. In Appendix L, we demonstrate how to find these mapping functions with iterative optimizers (Virtanen et al., 2020). We provide a formal definition of Kendall-$ \tau $ rank correlation in appendix G. We additionally report two more complementary metrics. While we have argued that the test accuracy is flawed to measure faithfulness, we will report the test accuracy differential to be complete with prior works. We define test accuracy differential (TAD) as: \[ \text{TAD} := \text{TestAcc}_{\text{kGLM}} - \text{TestAcc}_{\text{NN}}. \] A fundamental limitation of $ \tau_K $ is that it can only be computed over a set of scalar outputs so does not take advantage of the vectorized output of classification networks. To compensate, we will also report the misclassification coincidence rate, $ R_{\text{miss}} $, which captures whether two models both misclassify the same datapoints as the same class, which is an intuitive property $ \tau_K $ misses. A formal definition of $ R_{\text{miss}} $ is available in appendix G. We now turn to defining the specific kernel functions we evaluate. Trace Neural Tangent Kernel. For any two data inputs $ x_i $ and $ x_j $, we define the Jacobian of the NN’s $ c $-th output neuron with respect to $ \theta $ at datapoint $ x_i $ as $ g^c(x_i; \theta) = \nabla_\theta F^c(x_i; \theta) $. Then, for choice of class $ c $ and $ c' $, the eNTK is a kernel function defined as: \[ \text{eNTK}(x_i, x_j) := \langle g^c(x_i; \theta) \| g^{c'}(x_i; \theta) \rangle. \] For $ C $ classes and $ N $ datapoints, the full eNTK can be evaluated for each choice of $(c, c')$ and $(i, j)$ resulting in a large $ NC \times NC $ total size matrix. This matrix is often too expensive to compute or manipulate in memory, leading researchers to seek approximations. We introduce now the trace neural tangent kernel ($ \text{trNTK} $) approximation, which removes the $ C^2 $ scaling in memory by effectively performing a “block-trace” operation on the original eNTK. The $ \text{trNTK} $ is a kernel function defined as: \[ \text{trNTK}(x_i, x_j) := \frac{\sum_{c=1}^{C} \langle g^c(x_i; \theta) \| g^c(x_j; \theta) \rangle}{\left( \sum_{c=1}^{C} \\| g^c(x_i; \theta) \\|^2 \right)^{\frac{1}{2}} \left( \sum_{c=1}^{C} \\| g^c(x_j; \theta) \\|^2 \right)^{\frac{1}{2}}}. \] The denominator of Eq. 5 is a normalization that makes the trNTK a kernel of cosine-similarity values. It has been suggested that this normalization helps smooth out kernel mass over the entire training dataset (Akyürek et al., 2022). The normalization ensures that two identical inputs always have maximum similarity value 1. Additional intuition about how this kernel relates to the geometry of the neural network function surface is available in Appendix C. We provide additional details about these definitions in Appendix D. In the following section, we relate this kernel to another approximate eNTK kernel, the pseudo neural tangent kernel. Wei et al. (2022) Relationship to the Pseudo Neural Tangent Kernel. We can understand the motivation for the trNTK in the context of another approximate eNTK, called the pseudo neural tangent kernel (pNTK). The pNTK computed between inputs $ x_i $ and $ x_j $ is a kernel function defined as: \[ pNTK(x_i, x_j) := \frac{1}{C} \left( \nabla_\theta \sum_{c=1}^{C} F(x_i; \theta)^c \right)^\top \left( \nabla_\theta \sum_{c=1}^{C} F(x_j; \theta)^c \right). \] Mohamadi & Sutherland (2022) showed that the product of the pNTK($ x_i, x_j $) with the $ C \times C $ identity matrix is bounded in Frobenius norm to the eNTK by $ O(\frac{1}{\sqrt{n}}) $, with $ n $ the width parameter of a feed forward fully connected NN with ReLU activation (Nair & Hinton, 2010; Glorot et al., 2011) and He-normal (He et al., 2015a) initialization, with high probability over random initialization. We can frame the critical differences between the pNTK and trNTK by how each approximate the eNTK. The pNTK approximates the eNTK as a constant diagonal matrix with constant equal to the scalar kernel function given in Eq. 6. In contrast, the trNTK allows the diagonal elements of the eNTK approximation to vary, and in fact, calculates these values directly. Both the pNTK and trNTK perform a simplifying sum over the diagonal elements, which reduces the memory footprint of the approximations by a factor $ C^2 $ compared to the eNTK. We choose not to compare directly with the pNTK because the trNTK is a higher cost, but more precise, approximation of the eNTK. Instead, we focus our comparisons to much lower cost alternatives, including a projection variant of the pNTK. Projection trNTK and Projection pNTK. For large number of parameters $ P $ and large datasets $ N $, computing approximate eNTK remain expensive, therefore, we explore a random projection variant that allows us to effectively choose $ P $ regardless of architecture studied. Let $ P $ be a random projection matrix $ P \in \mathbb{R}^{K \times P}, K \ll P $, with all entries drawn from either the Gaussian $ \mathcal{N}(0, 1) $ or Rademacher (with $ p=0.5 $ for all entries) distribution. $ K $ is a hyperparameter setting the projection matrix dimension. We set $ K = 10240 $ for all experiments. We use $ P $ to project the Jacobian matrices to a lower dimension, which reduces the memory needed to store the Jacobians and reduce the time complexity scaling. The Johnson-Lindenstrauss lemma ensures that most of the information in the original Jacobians is preserved when embedded into the lower dimensional space (Johnson & Lindenstrauss, 1984). We define the proj-trNTK and proj-pNTK as random projection variants of the trNTK and pNTK: \[ \text{proj-pNTK}(x_i, x_j) := \frac{\left\langle P \sum_{c=1}^{C} g^c(x_i, \theta), P \sum_{c=1}^{C} g^c(x_j, \theta) \right\rangle}{\left\\| P \sum_{c=1}^{C} g^c(x_i, \theta) \right\\| \cdot \left\\| P \sum_{c=1}^{C} g^c(x_j, \theta) \right\\|} \] \[ \text{proj-trNTK}(x_i, x_j) := \frac{\sum_{c=1}^{C} \langle Pg^c(x_i; \theta), Pg^c(x_j; \theta) \rangle}{\left( \sum_{c=1}^{C} \\| Pg^c(x_i; \theta) \\|^2 \right)^{\frac{1}{2}} \left( \sum_{c=1}^{C} \\| Pg^c(x_j; \theta) \\|^2 \right)^{\frac{1}{2}}}, \] where both definitions include the cosine-normalization. Random projection variants can improve the time complexity scaling for computing approximate eNTK under large dataset size and large number of parameters. Assuming computation via Jacobian contraction and time $ [FP] $ for a forward pass, the eNTK time complexity is: $ NC[FP] + N^2C^2P $ (Novak et al., 2022). The pNTK computation reduces this to $ N[FP] + N^2P $; while the trNTK computation only reduces to $ NC[FP] + N^2CP $. In contrast, the proj-pNTK costs $ N[FP] + N^2K + $ and the proj-trNTK costs $NC[FP] + CN^2K + CNKP$. The final term in the projection variants is the cost of the extra matrix multiplication with the random projection matrix $P$ and the Jacobian matrix. For $K \ll P$ and $N$ large, projection variants reduce the time complexity. Additional Kernel Functions. We also evaluate the conjugate kernel (CK) formed from the Gram matrix of the final embedding vector (Fan & Wang, 2020; Yeh et al., 2018), the un-normalized trNTK ($\text{trNTK}^0$) which is equal to the numerator of Eq. 5, and the embedding kernel (Akyürek et al., 2023), formed from a sum over the Gram matrices of embedding vectors from various layers in the network architecture. See Appendix B for formal definition of these kernels. ## 4 RESULTS Experiments. Classification NNs with architectures and datasets (MNIST (Lecun et al., 1998), FM-NIST (Xiao et al., 2017), CIFAR10 (Krizhevsky & Hinton, 2009), and COLA (Warstadt et al., 2018)) shown in Table 1 are trained using standard techniques. Additional details regarding datasets are provided in Appendix K.1. Models that have a value of more than 1 in the column ‘# Models’ in Table 1 are trained multiple times with different seeds to generate uncertainty estimates. The ResNet18 (He et al., 2015b), ResNet34, and MobileNetV2 (Sandler et al., 2018) models were trained by an independent research group with weights downloaded from an online repository (Phan, 2021). Bert-base (Devlin et al., 2019) weights were downloaded from the HuggingFace (Wolf et al., 2019) repository then transferred onto the COLA dataset, as is common practice for foundation models (Bommasani et al., 2021). After training, we calculate the trNTK and alternative kernels using PyTorch automatic differentiation (Paszke et al., 2019). We train a kGLM (`sklearn.SGDclassifier`) (Pedregosa et al., 2011) for each $\kappa$ using the same training dataset for training the NN model. All computation was completed on a single A100 GPU with 40GB memory. Details such as specifics of architecture and choice of hyperparameters are available in Appendix K. Faithful Surrogate Modeling via trNTK. We calculate the $\tau_K$ correlation between the surrogate model and underlying NN and report the results in Table 1. We find that the efficacy of our surrogate model as measured by the correlation to the NN changes depending on architecture and dataset; though remarkably, $\tau_K$ is consistently high, with a lower bound value of 0.7 across all experiments, indicating high faithfulness. To demonstrate high $\tau_K$ implies we can achieve a point-for-point linear realization of the NN, we learn a non-linear mapping from the kGLM to the NN (Figure 1 for Bert-base. (Additional visualizations for the remainder of experiments are available in Appendix L.) Finally, we observe that the kGLM with choice of $\kappa = \text{trNTK}$ achieves comparable test accuracy as the underlying NN, which replicates the observations of prior work (Long, 2021; Vyas et al., 2022; Mohamadi & Sutherland, 2022) using our trNTK. Data Attribution with trNTK. Accepting that the trNTK is a faithful kernel function for a kGLM surrogate model, we can use the data attribution formalism to analyze the importance of individual training datapoints to the classification. In Figure 2 we present the visualization of data attribution for one test input and provide additional visualizations in Appendix M.1. The distribution of attribution follows a regular pattern in every visualization generated: the central value of attribution mass for each logit from each class is centered on the distribution of all training data from that class. We emphasize that in no cases have we observed a sparse number of training datapoints dominate the data attribution. Comparison of Faithfulness between Kernels Functions. For ResNet18 and Bert-base models, we evaluate our choice of trNTK against alternative kernel functions, reporting $\tau_K$ and test accuracy differential in Table 2. Across both ResNet18 and Bert-base experiments, we observe that the trNTK forms surrogate models with the highest correlation to the underlying NN decision function and is furthermore consistent in replicating the performance of these networks (TAD nearly 0). The embedding kernel (Em) does not perform as consistently between both tasks, but for its intuitive connection to the internal representation of the neural network may warrant further investigation. Faithful Surrogates in Data Poisoning Regime. Next, we evaluate whether surrogate models can be extended to analyze network behavior on poisoned data. We train a 21-layer CNN (details available in Appendix K.2.5) using BadNet CIFAR10 data (Gu et al., 2019; Shan et al., 2022). We randomly perturb training data by placing a yellow square in a tenth of training images from CIFAR10 and modify the label of these perturbed images to a targeted label (see example in Appendix N). We Table 1: Choice of $\kappa = \text{trNTK}$ faithfully forms a surrogate model of underlying NN. We perform each experiment with ‘# Models’ independent seeds. For each model and dataset we train and extract the trNTK, train a kGLM, then calculate and report the $\tau_K$ correlation between the kGLM softmax probability and NN softmax probability for the correct class. The NN test accuracy column shows that training terminates with a highly performant model, and the test accuracy differential (TAD) columns reports the difference between the kGLM test accuracy and the NN test accuracy. We report the leading digit of error (standard error of the mean) as a parenthetical, when available. \| Model (Dataset) \| # Models \| NN test acc (%) \| TAD (%) \| $\tau_K$ \| \|-----------------------\|----------\|-----------------\|---------\|----------\| \| MLP (MNIST2) \| 100 \| 99.64(1) \| +0.03(5)\| 0.708(3) \| \| CNN (MNIST2) \| 100 \| 98.4(1) \| -0.2(2) \| 0.857(7) \| \| CNN (CIFAR2) \| 100 \| 94.94(5) \| -2.1(5) \| 0.711(3) \| \| CNN (FMNIST2) \| 100 \| 97.95(4) \| -2.2(2) \| 0.882(3) \| \| ResNet18 (CIFAR10) \| 1 \| 93.07 \| -0.28 \| 0.776 \| \| ResNet34 (CIFAR10) \| 1 \| 93.33 \| -0.29 \| 0.786 \| \| MobileNetV2 (CIFAR10) \| 1 \| 93.91 \| -0.4 \| 0.700 \| \| BERT-base (COLA) \| 4 \| 83.4(1) \| -0.1(3) \| 0.78(2) \| Figure 1: Linear Realization of Bert-base Model. Each panel shows a linearization of a Bert-base transfer model, initialized from a different seed. An invertible mapping is fit between the kGLM and NN to transform the kGLM’s final activations to the NN’s, described in Appendix L. Both $\tau_K$ and the Coefficient of Determination ($R^2$) are shown for each model. Table 2: Comparison across surrogate feature spaces. For ResNet18 and Bert-base experiments we report the faithfulness as $\tau_K$, test-accuracy-differential (TAD), and misclassification coincidence rate ($R_{\text{Miss}}$) for each kernel function: the trace-NTK (trNTK), unnormalized trace-NTK (trNTK$^0$), the projection trace NTK (proj-trNTK), the projection pseudo NTK (proj-pNTK), the embedding kernel (Em) and the conjugate kernel (CK). If available, we report leading digit of error (standard error of the mean) as a parenthetical. \| Exp Name \| Metric \| $\kappa$ \| \|----------\|--------\|----------\| \| \| trNTK \| trNTK$^0$ \| proj-trNTK \| proj-pNTK \| Em \| CK \| \| ResNet18 \| $\tau_K$ \| 0.776 \| 0.658 \| 0.737 \| 0.407 \| 0.768 \| 0.630 \| \| \| TAD (%) \| -0.30 \| -0.52 \| -0.20 \| -0.30 \| -0.32 \| -0.20 \| \| \| $R_{\text{Miss}}$ \| 0.75 \| 0.65 \| 0.77 \| 0.71 \| 0.80 \| 0.73 \| \| Bert-base \| $\tau_K$ \| 0.809(9) \| 0.5(1) \| 0.800(9) \| 0.72(2) \| 0.65(2) \| 0.52(4) \| \| \| TAD (%) \| +0.1(3) \| +0.6(2) \| +0.1(2) \| +0.5(2) \| -0.3(5) \| -0.1(1) \| \| \| $R_{\text{Miss}}$ \| 0.67(2) \| 0.71(5) \| 0.61(2) \| 0.86(3) \| 0.86(2) \| 0.91(2) \| create a “clean” test dataset from CIFAR10’s normal test dataset, and a “poisoned” test dataset by placing yellow squares into each image of CIFAR10’s test dataset. At test time, perturbed test data tricks the model into producing labels of the targeted label. We train a model on this poisoned dataset, compute each kernel function, measure faithfulness, and report our results in Table 3. We find that the Analyzing Distribution of Attribution A) An image from the test dataset of CIFAR10 is chosen. B) We propagate the test image through the NN and plot the mean attribution of the training points from each class for each output neuron. C) Zooming into the neuron representing class “dog”, we view the distribution of attributions as a modified box-plot with central lines the mean and outliers shown as flier points. The mean lines are always observed to be within the inner quartile, suggesting that no sparse number of datapoints dominate the central value, and therefore, do not dominate the data attribution. trNTK is most faithful to the NN on the clean test data, but the proj-pNTK is most faithful when evaluated on the poisoned test data. Overall in comparison to the non-poisoned set of experiments each kGLM is less faithful, except for the proj-pNTK. We also point out that the kGLM with overall highest faithfulness are the kernel functions with our cosine-normalization applied. In addition, we show an application of our surrogate modeling approach enabled by kernel-techniques. Forensics models trace NN behavior on unseen poisoned data to the poisoned data source in a training set (Shan et al., 2022). We treat each kernel as a forensic model: for each image in the clean and poisoned test dataset we compute the top 5 most similar training datapoints. If 3/5 of these training datapoints are poisoned we flag the test image as poisoned. In doing so, we can filter poisoned images from clean images. We report the performance of our forensic models using precision and recall (see Appendix G) in table 3. Each kernel, except for the conjugate kernel, are all comparable in performance as forensics models. Appendix N provides examples of multiple forensic models acting on poisoned and clean versions of CIFAR10 data. Table 3: Poisoned data attribution forensics. We compute each kernel function between all poisoned training data and the clean test dataset. We report $\tau_K$, TAD, and $R_{Miss}$ between the kGLM and NN for both the poisoned (poi.) and clean set of unseen test images. Finally, we evaluate each kernel as a filter for identifying unseen poisoned data through high similarity to poisoned training data and report the performance as Precision and Recall. \| Method \| Precision (%) \| Recall (%) \| $\tau_K$ \| TAD (%) \| $R_{Miss}$ \| poi. $\tau_K$ \| poi. TAD(%) \| poi. $R_{Miss}$ \| \|------------\|---------------\|------------\|----------\|---------\|-----------\|----------------\|--------------\|----------------\| \| trNTK \| 99.99 \| 100.00 \| 0.643 \| +0.45 \| 0.44 \| 0.569 \| +0.09 \| 0.12 \| \| trNTK$^0$ \| 99.99 \| 99.97 \| 0.344 \| +0.87 \| 0.20 \| 0.125 \| +0.13 \| 0.01 \| \| proj-trNTK \| 99.99 \| 99.97 \| 0.565 \| +0.09 \| 0.45 \| 0.418 \| +1.3 \| 0.12 \| \| proj-pNTK \| 99.99 \| 100.00 \| 0.554 \| +0.07 \| 0.59 \| 0.665 \| -1.3 \| 0.11 \| \| Embedding \| 99.71 \| 100.00 \| 0.430 \| -2.73 \| 0.07 \| 0.261 \| -13.98 \| 0.22 \| \| CK \| 1.65 \| 50.61 \| 0.552 \| -3.50 \| 0.38 \| 0.454 \| -81.25 \| 0.00 \| 5 SUMMARY AND CONCLUSIONS Impact of Linear Surrogate Modeling for Explainability. We have shown evidence supporting the choice of the trNTK as a consistently faithful choice of kernel function for a surrogate model (table 1). We made this determination by measuring the correlation between the kGLM surrogate and the NN, which is an improvement over past methodologies. Our choice of a linear model as surrogate model allows us to separate the attribution terms from each training datapoint, and ensures the central value of the attribution distribution is coupled to the kGLM’s logit, and therefore the NN which it approximates (Section 2). We observed that the highest attributed images from the trNTK have relatively small mass compared to the bulk contribution, suggesting that the properties of the bulk, rather than a few outliers, are the main source driving decision making. We believe this is a result of the cosine normalization we apply in our definition of the trNTK, as the unnormalized trNTK shows a much tighter IQR of attribution (see appendix M.1.2), and in fact, this pattern exists between all normalized vs un-normalized kernel functions. This directly visualizes the intuition that the cosine normalization “smooths-out” the attribution (Akyürek et al., 2022). Because the properties of the bulk drive classification, we conclude that presenting the top highest attribution training images without the context of the entire distribution of attribution is potentially misleading as a form of explanation, i.e., the assumption of sparsity in explain-by-example strategies is misguided. Comparison of Kernel Functions for Surrogate Models. Our quantitative experiments showed the trNTK as more consistently correlated to the NN model compared to the unnormalized trNTK, Embedding kernel, and CK. We observe qualitative differences between these kernel’s attributions (Appendix M.1) and which training datapoints have highest similarity (Appendix N). As a qualitative comparison between kernel functions, in Appendix M.2 we visualize the top-5 most similar datapoints evaluated by each kernel function. This further reveals the similarities and differences between kernel functions. Overall, we observe that the trNTK is more sensitive to conceptual similarities between test and train examples than the CK. The embedding kernel is consistently sensitive to background pixel values, though this may be an artifact from our specific choice of layers to sample from. The proj-trNTK, as expected, follows closely with the regular trNTK. These differences could be used to tie to interesting phenomena: for example, because the CK is computed from the final embedding it is likely more sensitive to the effects of neural-collapse (Papyan et al., 2020) than the NTK, which is computed from Jacobians of weight tensors across the entire architecture. We believe this fact explains why the highest similar images measured by the trNTK are more conceptually tied to the specific test image, while the CK has collapsed that inner-class variance away. Computational Feasibility. Finally, we comment on the computational feasibility of each of the kernel functions. Table 4 reports the time to compute each kernel, and Appendix F shows that the empirical residual distribution between the trNTK and proj-trNTK falls exponentially. The projection-trNTK and projection-pNTK have efficient computation thanks to software made available in Park et al. (2023). The full trNTK is by far the slowest. As implemented, our trNTK computation was layerwise (see Appendix D), except in the Poisoning experiment, which we now believe is sub-optimal. Both the trNTK and projection-trNTK computation scales with the number of output neurons linearly, so for models with large output space the projection-pNTK may remain the only feasible option. Finally, because the residuals between the trNTK and proj-trNTK are small and decay rapidly, we believe using the projected variants are well justified. In total, we believe the differences between the trNTK and proj-trNTK are small enough that for small number of outputs, our recommendation is to utilize the proj-trNTK. Finally, see Appendix A for limitations. Table 4: Computational Complexity of Large Model Experiments. We report time to compute each of the trNTK, proj-trNTK, and proj-pNTK for the large model large dataset experiments are shown. \| Exp Name \| trNTK \| proj-trNTK \| proj-pNTK \| \|------------\|-------\|------------\|-----------\| \| ResNet18 \| 389h \| 1.12h \| 7.4m \| \| BertBase \| 1200h \| 22m \| 12m \| \| Poisoning \| 50h \| 9.3m \| 1m \| ACKNOWLEDGMENTS The authors thank Panos Stinis, Mark Raugas, Saad Qadeer, Adam Tsou, Emma Drobina, Amit Harlev, Ian Meyer, and Luke Gosink for varied discussions while preparing the draft. This work would not have been possible without the help from Wendy Cowley in helping navigate the release protocol. The authors thank Davis Brown for discussions regarding TRAK. A.W.E., Z.W., S.C., N.F., and T.C. were partially supported by the Mathematics for Artificial Reasoning in Science (MARS) initiative via the Laboratory Directed Research and Development (LDRD) Program at PNNL and A.D.S. and T.C. were partially supported by the Statistical Inference Generates kNowledge for Artificial Learners (SIGNAL) Program at PNNL. A.D.S. was partially supported by the US NSF under award CNS-2148104. PNNL is a multi-program national laboratory operated for the U.S. Department of Energy (DOE) by Battelle Memorial Institute under Contract No. DE-AC05-76RL0-1830. REFERENCES Ekin Akyürek, Tolga Bolukbasi, Frederick Liu, Binbin Xiong, Ian Tenney, Jacob Andreas, and Kelvin Guu. Towards tracing knowledge in language models back to the training data. In Findings of the Association for Computational Linguistics: EMNLP 2022, pp. 2429–2446, Abu Dhabi, United Arab Emirates, December 2022. Association for Computational Linguistics. 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Qwq4cpLtoX	Thus, each train and test example is a unique learning problem, but of a consistent type (e.g. linear regression)* : In this particular setup, “consistent type” does not imply linear regression but just regression in general. Would that be correct?	IS ATTENTION REQUIRED FOR ICL? EXPLORING THE RELATIONSHIP BETWEEN MODEL ARCHITECTURE AND IN-CONTEXT LEARNING ABILITY Ivan Lee, Nan Jiang, Taylor Berg-Kirkpatrick University of California, San Diego {iylee,n3jiang,tberg}@ucsd.edu ABSTRACT What is the relationship between model architecture and the ability to perform in-context learning? In this empirical study, we take the first steps toward answering this question. We evaluate thirteen model architectures capable of causal language modeling across a suite of synthetic in-context learning tasks. These selected architectures represent a broad range of paradigms, including recurrent and convolution-based neural networks, transformers, state space model inspired, and other emerging attention alternatives. We discover that all the considered architectures can perform in-context learning under a wider range of conditions than previously documented. Additionally, we observe stark differences in statistical efficiency and consistency by varying the number of in-context examples and task difficulty. We also measure each architecture’s predisposition towards in-context learning when presented with the option to memorize rather than leverage in-context examples. Finally, and somewhat surprisingly, we find that several attention alternatives are sometimes competitive with or better in-context learners than transformers. However, no single architecture demonstrates consistency across all tasks, with performance either plateauing or declining when confronted with a significantly larger number of in-context examples than those encountered during gradient-based training. 1 INTRODUCTION In-context learning (ICL) refers to the ability to learn new tasks at inference time, using only input-output pair exemplars as guidance. Radford et al. (2019) demonstrate early signs of this ability in GPT-2, a causal transformer (Vaswani et al., 2017). ICL was further popularized by GPT-3 (Brown et al., 2020), a large language model with the same architectural foundation but augmented with greater capacity and trained on large-scale data. By simply adjusting a natural language prompt, it was shown that GPT-3 could adapt to new tasks, such as translation and question answering, without updating any of its parameters. These findings spurred significant interest in the research community to investigate this curious behavior (Zhao et al., 2021; Min et al., 2022; Liu et al., 2022). Yet, a prevailing uncertainty remains: are large language models genuinely learning from their prompts or simply being conditioned to surface relevant aspects of their training data? To address this, a new line of research emerged that examines ICL in controlled, synthetic environments where task resolution fundamentally depends on prompt utilization (Xie et al., 2021; von Oswald et al., 2022; Garg et al., 2023; Akyürek et al., 2023). However, most of these studies anchor their investigations on the assumption that models utilize an internal attention mechanism (as is the case for transformers). Whether attention mechanisms are necessary for in-context learning to emerge remains an open question. Notable exceptions to this assumption include Xie et al. (2021) and Chan et al. (2022) who consider recurrent neural networks alongside transformers. The former finds RNNs and LSTMs fail to learn image classification in the ICL setting. In contrast, the latter demonstrate that LSTMs possess ICL abilities in a synthetic language modeling task, where hidden Markov models generate the data. Table 1: Examples of our synthetic in-context learning tasks. \| Task \| Prompt \| Target \| \|-----------------------\|------------------------------------------------------------------------\|------------------------------------------------------------------------\| \| Associative Recall \| a, 1, b, 3, c, 2, b \| 3 \| \| Linear Regression \| $x_1, y_1, x_2, y_2, x_3, y_3, x_4$ \| $y_i \exists w \text{ such that } \forall i, y_i = x_i \cdot w$ \| \| Multiclass Classification \| $x_1, b, x_2, a, x_3, a, x_4$ \| $b$ $x_1, x_4 \sim N(y_b, I_d)$ $x_2, x_3 \sim N(y_a, I_d)$ \| \| Image Classification \| ![Image](image.png) \| 4 bursty training prompt \| \| \| ![Image](image.png) \| 2 non-bursty training prompt \| \| \| ![Image](image.png) \| 0 evaluation prompt \| \| Language Modeling \| Colorless green ideas sleep \| furiously \| However, whether both findings are specific to their task or indicative of more general behavior remains uncertain. The community’s focus on attention is understandable given the success of transformers. However, the architecture comes with a number of limitations, such as quadratic time and memory complexity. These limitations spurred research into alternative architectures such as efficient self-attention models (Tay et al., 2022a) and state space models (Gu et al., 2021). If these alternatives are to replace transformers as the dominant model architecture, it is natural to wonder if they are capable of ICL. Moreover, some are designed to handle prompts of arbitrary length, potentially introducing a novel ICL form, constrained only by dataset size rather than inherent architectural limitations. Furthermore, classic architectures such as recurrent neural networks and convolutional neural networks were once the backbone of machine learning research before the introduction of transformers and ICL as a concept. Do these classic architectures inherently lack ICL capabilities, or were they simply constrained by the compute and data available during their heyday. In this study, we set out to address the aforementioned questions. Specifically, we aim to answer the following research questions: Which architectures are capable of ICL, and which exhibit superior ICL performance? Our primary focus lies on the former question. While the latter is more challenging to assess, our experiments provide insights into which families of architectures tend to perform well, even if they do not offer definitive answers. To advance our objectives, we evaluate a diverse range of model architectures that span several design paradigms. This includes both the classical methods previously mentioned and modern approaches such as the transformer and those inspired by state space models. Our assessment covers the ICL capabilities of each architecture over a wide array of synthetic tasks, spanning different modalities and including both classification and regression, as depicted in Table 1. Our specific contributions are as follows: - Large-scale empirical study: We conduct the first large-scale empirical study comparing ICL performance across diverse model architectures, shedding light on their relative strengths and weaknesses. Code is available at [https://github.com/ivnle/synth-icl](https://github.com/ivnle/synth-icl). - Universality of ICL: We discover that all the considered architectures can perform in-context learning under a wider range of conditions than previously documented, lending support to the position that ICL is not exclusive to attention-based models. - Empirical success of attention alternatives: Our findings demonstrate that some attention alternatives not only compete with but, in certain cases, surpass transformers at in-context learning. This suggests that efficiency gains in these architectures do not necessarily come at the expense of performance. 2 Synthetic In-context Learning Tasks Studying in-context learning in large language models presents inherent challenges. One fundamental question is whether these models are truly learning new predictors during the forward-pass, or whether in-context examples simply focus the model on specific aspects of the knowledge already acquired during gradient-based pretraining. While from a Bayesian perspective this dichotomy represents endpoints of a spectrum (Xie et al., 2021), it nonetheless clouds interpretation of ICL experimental results. To address this concern, a new line of research has emerged that examines ICL in controlled, synthetic environments where task resolution depends fundamentally on prompt utilization (von Oswald et al., 2022; Garg et al., 2023; Akyürek et al., 2023). In these settings, models must rely on their prompts to solve tasks, eliminating the possibility of memorization: Models are trained from scratch to take a labeled dataset as input and then predict the result of learning from this data directly in the forward-pass of the resulting model. Thus, each train and test example is a unique learning problem but of a consistent type (e.g. linear regression). In addition to offering a clearer perspective on in-context learning, synthetic tasks have low computational requirements. These decreased barriers allow for more equitable comparisons across model architectures. Utilizing publicly available pretrained models may introduce confounding variables, stemming from disparities in model capacity, training durations, and data quality. By training models from scratch on synthetic tasks, we are given greater control over these factors. Furthermore, a suite of such tasks is a valuable tool for the research community, enabling rapid benchmarking of emerging architectures without the intensive computational overhead typically associated with large language models. For these reasons, we curate a suite of synthetic in-context learning tasks and summarize them in Table 1. The majority of our tasks take the form \[ x_1, f(x_1), x_2, f(x_2), \ldots, x_n, f(x_n) \] where the goal is to learn function $ f $ by observing a prompt, a sequence of input-output pairs $(x_i, f(x_i))$, which ends with a query. The model’s objective is to produce an appropriate completion based on the given prompt. We train model $ M_\theta $ parameterized by $ \theta $ to minimize the expected loss over all prompts \[ \min_\theta \mathbb{E} [\ell(M_\theta(P), f(x_n))], \] where $ \ell(\cdot, \cdot) $ is the appropriate loss function for a given task. Associative recall (Ba et al., 2016; Fu et al., 2023) is the task of learning key-value mappings from a prompt and can be viewed as the simplest form of in-context learning. Let $ V $ be a discrete vocabulary of size $ k $. We consider the class of functions \[ F = \{ f \| f : V \rightarrow V \} \] where $ f $ is a bijective mapping. These mappings are created by randomly pairing elements of $ V $ without replacement, ensuring each element maps to a unique counterpart. We uniformly sample $ f $ from $ F $ and $ x_1, \ldots, x_n $ from $ V $ to construct the prompt as $ P = (x_1, f(x_1), x_2, f(x_2), \ldots, x_n) $. Elements of $ P $ are mapped to vectors with a simple lookup table, as is standard in language modeling. Linear regression (Garg et al., 2023) is the task of learning a linear function from a prompt. We consider the class of functions \[ F = \{ f \| f(x) = w^\top x, w \in \mathbb{R}^d \} \] We sample $ x_1, \ldots, x_n $ and $ w $ from the isotropic Gaussian distribution $ \mathcal{N}(0, I_d) $. We then compute each $ y_i = w^\top x_i $ and construct the prompt as $ P = (x_1, y_1, x_2, y_2, \ldots, x_n) $. Since $ y_i $ is a scalar, we represent it as a $ d $-dimensional vector, with its first index set to $ y_i $ and remaining indices set to zero. Multiclass Classification is a clustering task in which the items to be clustered are sampled from $ k $ distinct Gaussians. For this task, we use the procedure \[ \mu_i \sim U(-1, 1)^d, \text{ for } i = 1, \ldots, k \] \[ y_j \sim U(\{1, \ldots, k\}), \text{ for } j = 1, \ldots, n \] \[ x_j \sim N(\mu_{y_j}, I_d), \text{ for } j = 1, \ldots, n \] to construct the prompt as $ P = (x_1, y_1, x_2, y_2, \ldots, x_n) $. Since $ y_j \in \{1, \ldots, k\} $, we map each cluster label to a $ d $-dimensional vector with a simple lookup table. We set $ d $ to 16 in all experiments. To facilitate a clearer understanding, we defer detailed discussions of Image Classification and Language Modeling to Sections 5 and 6, respectively. ### 3 MODEL ARCHITECTURES Recurrent We consider three common variations of recurrent neural networks: Elman (Rumelhart et al., 1986, RNN), long short-term memory (Hochreiter & Schmidhuber, 1997, LSTM), and gated recurrent unit (Cho et al., 2014, GRU). Recurrent neural networks are characterized by their length-invariant inference cost and theoretically infinite context size, though empirical findings suggest an upper limit on this context size (Khandelwal et al., 2018). Furthermore, since the introduction of transformers, this class of architecture has seen diminished focus within the community, particularly in the ICL setting. We believe revisiting approaches that have fallen out of favor helps counterbalance the community’s potential over-reliance on a select few contemporary methodologies. Convolutional Representing the class of convolutional neural networks (CNN), we focus on the architectures proposed by Wu et al. (2019): lightweight convolutions (LIGHTCONV) and dynamic convolutions (DYNAMICCONV). These architectures, derived as special cases of depthwise convolutions (Sifre & Mallat, 2014), have demonstrated competitive performance with transformers in specific contexts (Tay et al., 2022b). LIGHTCONV is simply a depthwise CNN with weights normalized across the temporal dimension via a softmax. This design means that, unlike in self-attention, its context window is fixed and the importance placed on context elements does not change across time. To remedy this shortcoming, DYNAMICCONV predicts a different convolution kernel at every time-step. However, the kernel is a function of the current time-step only as opposed to the entire context as in self-attention. Similar to the recurrent class, CNNs exhibit length-invariant inference costs. However, they trade infinite context size for training parallelism. Structured State Space Sequence Models (SSMs) We also examine a category of recently proposed architectures inspired by state space models (Kalman, 1960). These architectures attempt to merge the efficient inference capabilities of RNNs with the parallel training attributes of transformers and CNNs. S4 (Gu et al., 2021) set a new state-of-the-art on long-range sequence modeling, but falls short in language modeling compared to transformers. Subsequently, H3 (Fu et al., 2023), HYENA (Poli et al., 2023), and Mamba (Gu & Dao, 2023) were proposed, each progressively improving upon this language modeling gap. We also include architectures inspired by linear attention (Katharopoulos et al., 2020; Zhai et al., 2021). Specifically, we examine RETNET (Sun et al., 2023) and RWKV (Peng et al., 2023). While not necessarily inspired by state space models, these architectures also strive for efficient inference, parallelizable training, and can be viewed as variants of SSMs. Transformers Finally, we consider two popular autoregressive transformer designs: GPT2 (Radford et al., 2019) and LLAMA2 (Touvron et al., 2023). Their primary differences lie in choice of positional embeddings and activation functions. GPT2 utilizes learned absolute positional embeddings and ReLU activation while LLAMA2 incorporates rotary positional embedding (Su et al., 2022) and SWIGLU activation (Shazeer, 2020). Rotary embeddings endow transformers with both absolute and relative positional information through rotations in complex space. We also perform an ablation study across positional embeddings (or lack thereof) and show our results in Appendix E. Note that we train all models from scratch, adopting only the architectural design choices made by the named models’ authors. In the following sections, we delve into our experimental methods and findings. Section 4 presents our results for linear regression, associative recall, and multiclass classification. We discuss image classification outcomes in Section 5, and conclude with our language modeling results in Section 6. 4 LEARNING TO LEARN (IN-CONTEXT) In our initial experiments, we evaluate the capacity of various architectures to in-context learn associative recall, multiclass classification, and linear regression. Results are shown in Figure 1 and experimental details are shown in Appendix A.1. Besides confirming the existence of ICL ability, we are particularly interested in measuring statistical efficiency—which models make better use of a fixed amount of data (in-context examples)—and in determining if our trained models demonstrate consistency, i.e., whether their performance converges in probability to some ceiling. ![Figure 1](image) (a) Associative recall (b) Linear regression (c) Multiclass classification Figure 1: Evaluating various architectures on associative recall, linear regression, and multiclass classification. We plot test accuracy and mean squared error as a function of the number of in-context examples. A query index of $2^5 = 32$ implies 31 in-context examples, which is also the highest number of in-context examples seen during training (vertical dotted line). Task difficulty increases from left to right. Each line represents the single run that achieved the best validation accuracy or mean squared error at query index $2^5$. See Tables 9, 7, 11 for a tabular view of the same data. See Figure 5 for average performance across training runs. See Appendix B.1 for linear regression experiments with Gaussian noise where we observe trends are largely unchanged relative to the non-noisy setting. Classical baselines (black) are shown for linear regression (ridge regression) and multiclass classification (logistic regression). Why is consistency of interest? First, a proficient learner, irrespective of the ICL setting, is expected to improve its performance given more i.i.d. training data. Consequently, a rise in in-context examples should lead to regular performance improvements. However, it is unclear if this is true in the in-context setting, a query we offer clarity on shortly. Second, the emergence of length-invariant inference architectures, rivaling transformers in task performance, paves the way for ICL with a substantially larger number of in-context examples than what is typically used today. One can imagine a new paradigm to replace finetuning: adapting pretrained language models to new tasks by utilizing a precomputed (previous) hidden state without parameter updates. All architectures can in-context learn. We first turn our attention to the left most plots in Figure 1, and specifically the region left of the dashed vertical line. Clearly, all architectures successfully in-context learn the three tasks. This provides an existence proof that ICL is not a unique property of transformers. Differences among the architectures becomes more evident as we increase difficulty and take into account their ability to extrapolate to large data sizes than seen in training (right of the dotted vertical line). Which architectures are consistent? Initially, all architectures appear consistent when considering only prompt lengths encountered during training. However, this perception changes when we introduce prompt lengths well beyond those seen during training. Specifically, the performance degradation is most pronounced in the four state space model inspired architectures and the two transformers. Note that this behavior is expected for GPT2 which uses learned positional embeddings, but not for LLAMA2 which uses rotary embeddings. Interestingly, other architectures with recurrent formulations (such as the RNNs, RetNet, and RWKV) do not exhibit such drastic declines. This also holds true for the CNNs, which are inherently limited to finite context lengths. This behavior in CNNs makes intuitive sense, as long range information that may “confuse” this architecture class are discarded over time. It is possible that, similar to RNNs (Khandelwal et al., 2018), RetNet and RWKV exhibit stronger preference to nearby context relative to the state space model inspired architectures (originally motivated by long sequence modeling) and transformers (which have random access to their entire context). This preference may explain why these architectures are more robust to unseen prompt lengths. Variations in statistical efficiency. The following summary assumes the most difficult setting for all tasks. For associative recall, the top performers were the transformers, H3, HYENA, MAMBA, RetNet, and RWKV when given 31 in-context examples (the longest prompt length seen during training). When extrapolating to longer prompt lengths, HYENA, MAMBA, and RWKV achieved near perfect accuracy, but performance degraded as the number of in-context examples grew. Our ablation over positional embeddings in Table 15 reveal that transformers without positional embeddings and transformers with sinusoidal embeddings are the best at associative recall regardless of prompt length. For linear regression, the transformers, MAMBA, and RetNet achieve near perfect MSE when given 31 in-context examples. Interestingly, these four architectures match the performance of ridge regression. Beyond 31 examples, however, performance quickly deteriorates, with RetNet showing the most robustness to this deterioration. Surprisingly, GRU and LSTM demonstrated competitive performance when extrapolating to unseen prompt lengths. We saw improved extrapolation ability in transformers without positional embeddings (Table 16), but its performance still degraded as the number of examples increased. For multiclass classification, the transformers, all the state space model inspired architectures (except for S4), RetNet and RWKV achieved the best accuracy, surpassing logistic regression. In particular, MAMBA scored the highest accuracy when given 255 in-context examples. We also note that LSTM was competitive with the other architectures but did not achieve a top score. Hyperparameter sensitivity. We now consider average performance for each architecture (Figure 5). Earlier, we found that some RNNs, despite not achieving the best scores, were competitive with modern architectures. However, these performances were difficult to replicate and were isolated to a few lucky combinations of hyperparameters. For associative recall, the transformers, HYENA, MAMBA, and RetNet were consistently strong performers. In particular, MAMBA achieved an average accuracy of 0.96 when given 63 examples. For linear regression, LLAMA2 was the clear leader for prompt lengths seen during training, followed by RetNet. For multiclass classification, LLAMA2, MAMBA, and RWKV were the top performers, followed by H3 and HYENA. Both RWKV and MAMBA improved in performance as prompt lengths increased beyond those seen during training. Interestingly, multiclass classification was the sole task where GPT2 did not perform well on average. ## 5 THE INFLUENCE OF TRAINING DATA DISTRIBUTIONAL PROPERTIES We now study how the distributional properties of training data can influence ICL. We follow the image classification experiments of Chan et al. (2022) who show ICL emerges when training data exhibits particular properties such as burstiness and having large numbers of rarely occurring classes. To manage the number of experiments in this study, we focus exclusively on burstiness, a feature of natural data not found in typical supervised datasets. For example, natural language is temporally “bursty”. That is, a given entity (e.g., word, person) may appear in clusters rather than uniformly across time (Altmann et al., 2009). We train models on a mixture of bursty and non-bursty prompts. See Table 1 and Figure 7 for examples. In bursty prompts, the query class appears 3 times. To prevent the model from simply outputting the most common class in the prompt, a second class also appears 3 times. Bursty prompts can be solved by either leveraging query-label pairs across different training prompts (i.e. memorization) or referring to the in-context examples within prompts (i.e., ICL). For non-bursty prompts, the image-label pairs are drawn randomly and uniformly. This implies there is no incentive for a model to utilize the in-context examples. Note that models now have two options to learn how to classify images: memorization or ICL. This stands in contrast to our experiments in Section 4 where ICL was the only option to solve a task. We want to understand if certain architectures are predisposed towards adopting one of these modes. We evaluate models with standard few-shot sequences containing images from two holdout classes and randomly assign one class to label 0 and the other to label 1. To solve this evaluation task, the model must utilize ICL. Images are sourced from Omniglot (Lake et al., 2019), a dataset of handwritten characters with 1623 classes. We follow Chan et al. (2022) and embed images using a randomly initialized ResNet (He et al., 2015) that trains alongside the evaluated model. Their corresponding labels are mapped to vectors with a simple lookup table. We perform the same sweep outlined in Section 4 resulting in 1512 training runs. We show our results in Figure 2 with supplementary results in Appendix C. We note that all training runs achieved near perfect training accuracy, confirming that models have indeed learned at least one of the two methods of image classification. ![Figure 2](image) Figure 2: Measuring the effects training data distributional properties on in-context learning. We plot average (over training runs) test accuracy as a function of training steps. $ P(\text{bursty}) $ indicates the proportion of training prompts that were bursty (with the remainder non-bursty). See Table 14 for a tabular view of the same data. See Figure 8 for training runs that achieved max validation accuracy. Can ICL emerge given purely non-bursty examples? As shown in the first column of Figure 2, no architectures demonstrate ICL ability when all prompts are non-bursty. This is not surprising given that i.i.d in-context examples rarely provide useful information for classifying the query image. Are some architectures predisposed towards ICL? After increasing $ P(\text{bursty}) $ to 0.5, we find that LLAMA2 and HYENA demonstrate a strong preference towards ICL. It is surprising that GPT2 did not share this predisposition as it is similar in design to LLAMA2. We hypothesize that the rotary positional embeddings employed by LLAMA2 provide a stronger inductive bias towards ICL than the absolute learned positional embeddings used by GPT2. Further increasing $ P(\text{bursty}) $ to 0.9 reveals that ICL ability emerges consistently in GPT2, MAMBA, H3, and RWKV. Are some architectures predisposed towards memorization? Setting $ P(\text{bursty}) $ to 1 reveals that a subset of architectures strongly prefer memorization over ICL. In particular, RETNET, S4, the two CNNs and all three RNNs strongly favor memorization. This is not to say that these architectures are incapable of solving this task which we address shortly. We were particularly surprised at the resistance of RETNET to develop ICL ability given that it was one of the top performers in Section 4. ICL emerged in only 2 of 108 training runs for RETNET, and notably, this development occurred after 30K training steps, a window similar to that of the three RNNs. In contrast, the other high-performing architectures from Section 4 developed ICL capabilities in fewer than 10K steps. Does ICL emerge in all architectures? While average accuracy across training runs is depicted in Figure 2, we also present the training runs that achieved the best validation accuracy in Figure 8. In these analyses, we observe that ICL emerges in all evaluated architectures, except for LIGHTCONV. We hypothesize that the absence of a time-step dependent kernel, a feature present in DYNAMICCONV, might be responsible for this outcome. Interestingly, ICL emerges in all three RNNs when P(bursty) is set to 0.9 and 1.0, a finding that contradicts those reported by Chan et al. (2022). Moreover, GRU exhibits the ability to perform ICL even with P(bursty) set as low as 0.5. Given that the RNNs fail at this task on average, we credit this finding to luck with our hyperparameter sweep. 6 TOWARDS IN-CONTEXT LEARNING IN THE REAL WORLD Up until now, our experiments have fallen under the few-shot learning concept of ICL where models are prompted with several in-context examples in a next-token-prediction format. We now consider an alternative perspective on ICL, represented in Kaplan et al. (2020) and Olsson et al. (2022). This approach focuses on observing loss at different token indices to measure improvements in language modeling performance as context length grows. Indeed, this is simply what language models are designed to do. However, as their ability to predict later tokens based on earlier ones improves, they can be utilized in increasingly interesting ways, such as instruction following. We report both in-context learning score and validation loss in Figure 3. Olsson et al. (2022) define in-context learning score as “the loss of the 500th token in the context minus the average loss of the 50th token in the context, averaged over dataset examples.” One can view ICL score as a simple heuristic to measure the statistical efficiency of a given model. Note that this task is distinct from the large language model setting of in-context learning, where models are trained on language modeling and undergo evaluation with few-shot prompts. We assess models on the same task they were trained on: next-token prediction. See Appendix A.2 for experiment details. ![Figure 3](image) Figure 3: Evaluating architectures on language modeling. Left: Validation loss during training. Middle: ICL score as training progresses. Right: Validation loss as a function of context length. Most architectures exhibit an abrupt improvement in ICL score. This same phenomenon was noted by Olsson et al. (2022) in transformers. They discover that induction heads, which they hypothesize as the key mechanism behind ICL, form during the same window where ICL score abruptly improves. Since most architectures considered do not incorporate the concept of an attention head, an intriguing question emerges: What mechanism, analogous to induction heads in transformers, exists in these alternative architectures that facilitate a similar role in ICL? Does ICL score correlate with our previous experiments? In Section 4, our top performers included the two transformers, RWKV, RetNet, H3, Hyena, and Mamba. Section 5 shares this list (except for RetNet). Consistently, these architectures also achieved the highest ICL scores, led by the transformers and Mamba. We noted that DynamicConv and LSTM, despite sharing similar validation loss, exhibited a significant gap in ICL score. We find that, when considering their best training runs, LSTM consistently outperformed DynamicConv in all prior tasks and demonstrated superior extrapolation abilities. We observe the same relationship between GRU and LightConv. While ICL score does appear to correlate with performance in the previous sections, it should not be considered in isolation. For example, S4 and H3 share almost identical ICL scores. However, S4 did not perform as well in our prior tasks as H3 and achieved a lower validation loss on language modeling. Lastly, it is worth mentioning that RNN, despite its poor ICL score, outperformed the two CNNs in image classification when looking at their best training runs (see... Table 13). This suggests that RNN might be more effective at ICL than the CNNs in scenarios with shorter prompt lengths, as our image classification experiments used prompt lengths of 17 versus 512 in language modeling. We also observe that ICL ability in Section 5 appears to emerge during the same window where ICL score dramatically improves, lending credibility to Olsson et al. (2022)’s use of the metric. 6.1 A SIMPLE FEW-SHOT NATURAL LANGUAGE TASK An interesting property of the dataset we use for language model training (Appendix A.2) is that we can produce relatively small models that still result in fluent language generation. To take advantage of this property, we evaluate architectures on a final ICL task that more resembles those used with large language models: in-context examples are composed using only natural language. Specifically, we compose 200 sentence pairs of the following form: “Lilly scrapped her knee. Lily is sad.” Given a target number of in-context examples, for each of the 200 pairs, we randomly sample from the remaining 199 pairs without replacement to assemble 200 prompts. We ensure the two classes (happy and sad) are balanced. For example: “Lilly scrapped her knee. Lily is sad. Lilly played with her friends. Lilly is happy. Lilly ate ice cream. Lilly is ____”. This procedure is repeated 10 times yielding 2000 prompts for each target number of in-context examples. ![Figure 4](image) Figure 4: Evaluating various architectures on a simple natural language ICL task. We report accuracy as a function of the number of in-context examples. We use the open sourced weights for Llama2-7B and do not fine-tune. All other models are trained from scratch and are approximately 33M parameters (excluding embedding layers). Right: Flipped label setting, i.e., “happy” is replaced with “sad” and vice versa. See Figure 9 for normalized accuracy. We also repeat the experiment but flip the classes, i.e., all instances of “sad” are replaced with “happy” and vice versa, testing if the model can override semantic priors (Wei et al., 2023). We show our results in Figure 4. Note that we include Llama2-7B as a reference point. We use the open sourced weights for this model as is and do not further train it on TinyStories. Accuracy improves with more examples, but quickly plateaus in the unflipped setting. This pattern held true for all architectures, with the exception of HYENA which showed an initial peak in accuracy, followed by a decline. This decay was also noted in Section 4, when HYENA encountered prompt lengths unseen during training. However, the prompt lengths in the current context fall well within the sequence lengths encountered during their language model training. 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RqUMWdDg52	It is known that fine-tuning can hurt the generalization ability of LLMs. But the authors claimed that the proposed “agent fine-tuning” method is “more generalizable to novel tasks”, which is hard to understand.	FIREACT: TOWARD LANGUAGE AGENT FINE-TUNING Anonymous authors Paper under double-blind review ABSTRACT Recent efforts have augmented language models (LMs) with external tools or environments, leading to the development of language agents that can reason and act. However, most of these agents rely on few-shot prompting techniques with off-the-shelf LMs. In this paper, we investigate and argue for the overlooked direction of fine-tuning LMs to obtain language agents. Using a setup of question answering (QA) with a Google search API, we explore a variety of base LMs, prompting methods, fine-tuning data, and QA tasks, and find language agents are consistently improved after fine-tuning their backbone LMs. For example, fine-tuning Llama2-7B with 500 agent trajectories generated by GPT-4 leads to a 77% HotpotQA performance increase. Furthermore, we propose FireAct, a novel approach to fine-tuning LMs with trajectories from multiple tasks and prompting methods, and show having more diverse fine-tuning data can further improve agents. Along with other findings regarding scaling effects, robustness, generalization, efficiency and cost, our work establishes comprehensive benefits of fine-tuning LMs for agents, and provides an initial set of experimental designs, insights, as well as open questions toward language agent fine-tuning. 1 INTRODUCTION Recent work has explored grounding language models (LMs; Brown et al., 2020; Chowdhery et al., 2022; Touvron et al., 2023a) to interact with external tools or environments, leading to a new class of language agents (Nakano et al., 2021; Yao et al., 2022b; Park et al., 2023) that could obtain new knowledge from environmental feedback, make sequential decisions via language reasoning, and improve task solving using self-reflection (Shinn et al., 2023; Wang et al., 2023a). Beyond research, industrial developments such as ChatGPT Plugins (OpenAI, 2023c) have indicated the great potential of language agents for real-world applications. So far, most language agents prompt off-the-shelf LMs for convenience and flexibility. However, existing LMs were not developed for agentic usecases (e.g., generating actions or self-evaluations), for which few-shot prompting only offers limited learning support. As a result, most LMs have poor performance and robustness when used for agents, and some advanced agents (Yao et al., 2023; Wang et al., 2023a) can only be supported by GPT-4 (OpenAI, 2023b), resulting in high costs and latencies, along with issues like controllability and reproducibility. Fine-tuning is an appropriate solution for these issues: it has been shown that fine-tuned smaller LMs could outperform prompted larger LMs for specific reasoning (Zelikman et al., 2022; Huang et al., 2022a) and acting (Yao et al., 2022b) needs, while enjoying reduced inference time and expense. But the study of LM fine-tuning for agents has been very limited, despite the large amount of studies around language agents and LM fine-tuning respectively (Figure 1). Only a few prior works have fine-tuned LMs for web navigation (Nakano et al., 2021; Yao et al., 2022a) or API tool use (Schick et al., 2023; Patil et al., 2023; Qin et al., 2023), with preliminary scaling analysis specific to a type of models (Yao et al., 2022b; Schick et al., 2023; Nakano et al., 2021). In this work, we take an initial step toward a more systematic study of language agent fine-tuning. We propose FireAct, a novel way to fine-tune LMs with agent trajectories generated from multiple tasks and prompting methods, and unified in the ReAct (Yao et al., 2022b) format (Figure 2). We implement FireAct using open-domain question answering (QA) tasks with access to a Google search API, and GPT-4 (OpenAI, 2023b) for fine-tuning data generation. By thoroughly investigating a variety of base LMs (OpenAI, 2023a; Touvron et al., 2023a; Rozière et al., 2023), prompting... Figure 1: While language agents and language model fine-tuning are both popular topics, their intersection is understudied. This work takes an initial step to show multiple advantages of fine-tuning LMs for agentic uses, and opens up various new questions toward language agent fine-tuning. methods (Yao et al., 2022b; Wei et al., 2022b; Shinn et al., 2023), fine-tuning data, and tasks (Yang et al., 2018; Press et al., 2022; Hendrycks et al., 2021; Geva et al., 2021), our experiments illustrate various advantages of fine-tuning and the importance of fine-tuning data diversity. For example, while few-shot ReAct prompting GPT-3.5 on HotpotQA achieves an exact match (EM) score of 31.4, fine-tuning with 500 ReAct trajectories improves the EM to 39.2 (25% increase), and fine-tuning with a mix of ReAct and CoT trajectories further improves the EM to 41.0 (31% increase). Furthermore, fine-tuning reduces inference time by 4x, and improves performances by 64% in face of distracting tool outputs. Such benefits can be even more visible for smaller open-source LMs where few-shot prompting performs poorly, e.g., fine-tuning Llama2-7B (Touvron et al., 2023a) leads to a 77% EM increase on HotpotQA. Besides showcasing these benefits, our experiments also explore complex interactions among various factors of fine-tuning and provide actionable insights for practitioners. As for the base LM, we find GPT-3.5 significantly outperforms other open-source LMs when fine-tuning with less than 500 samples, but the gap can be gradually caught up by scaling to more fine-tuning samples. As for the prompting methods to generate fine-tuning data, we find different LMs benefit from different mix ratios, and present trajectory statistics and oracle analyses for further understanding. As for the tasks to generate fine-tuning data, our preliminary results show that adding a task might not improve downstream performances on significantly different tasks, but also does not hurt performances. This suggests the potential for massive multi-task fine-tuning to obtain a single LM as the agent backbone for various applications. Along with various other findings, discussions, and the release of FireAct code, data, and model checkpoints, we hope our work ignites and inspires future efforts toward more capable and useful fine-tuned language agents. 2 RELATED WORK Language agents. Language agents (Weng, 2023; Wang et al., 2023b) represent an emerging kind of AI systems that use language models (LMS) to interact with the world. While earliest language agents simply used LMs to generate action commands (Nakano et al., 2021; Huang et al., 2022b; Ahn et al., 2022; Schick et al., 2023), learning direct observation-action mappings from few-shot demonstrations is challenging when the domain is complex or involves long-horizon activities. ReAct (Yao et al., 2022b) proposed to use LMs to generating both reasoning traces (Wei et al., 2022b; Nye et al., 2021; Kojima et al., 2022) and actions, so that reasoning can flexibly guide, track, and adjust acting, leading to substantial improvements over act-only methods. Follow up work has applied LM-based reasoning for more purposes in agent design, such as reflection (Shinn et al., 2023; Park et al., 2023), planning (Yao et al., 2023; Dagan et al., 2023; Liu et al., 2023a), program synthesis (Liang et al., 2023; Wang et al., 2023a), etc. The forms of external grounding have also diversified, ranging from digital games (Huang et al., 2022b; Wang et al., 2023a), APIs (“tools”; Schick et al., 2023; Patil et al., 2023; Qin et al., 2023), webpages (Yao et al., 2022a; Deng et al., 2023; Zhou et al., 2023b), to physical (Bharadhwaj et al., 2023; Vemprala et al., 2023; Driess et al., Figure 2: Illustration of FireAct. (a) During fine-tuning, a large LM (e.g., GPT-4) generates task-solving trajectories based on questions from different datasets and prompts from different methods. The successful trajectories are then converted into the ReAct format to fine-tune a smaller LM. (b) During inference, the fine-tuned LM could operate without few-shot prompting, and could implicitly select an prompting method to complete a ReAct trajectory with flexible lengths, adapting to different question complexities. For example, a simple question could be solved using only one thought-action-observation round, without using tools. Language model fine-tuning. Adapting pre-trained LMs to downstream tasks is another active field of study [Zhang et al., 2023b], including various instruction-based fine-tuning datasets [Mishra et al., 2022; Sanh et al., 2022; Köpf et al., 2023; Wang et al., 2023d; Honovich et al., 2023; Longpre et al., 2023], models [Taori et al., 2023; Chiang et al., 2023; Xu et al., 2023; Muennighoff et al., 2023; Ouyang et al., 2022], parameter-efficient fine-tuning methods [Hu et al., 2022; Ding et al., 2023; Lv et al., 2023; Dettmers et al., 2023; Ivison et al., 2023], and data selection principles [Zhou et al., 2023a; Gunasekar et al., 2023]. Additionally, there are various studies on fine-tuning specific types of LMs, such as coding LMs [Li et al., 2023; Luo et al., 2023; Rozière et al., 2023], multimodal LMs [Zhang et al., 2023c; Gong et al., 2023; Dai et al., 2023; Zhang et al., 2023a; Brooks et al., 2023; Su et al., 2023], and retrieval-augmented LMs [Guu et al., 2020; Wang et al., 2023c]. However, fine-tuning LMs for language agents that reason and act has been limited. Language agent fine-tuning. Despite the vast interests in language agents and fine-tuning, their intersection has received limited attention, with only some initial study about how performances scale with the model size for a particular model family [Nakano et al., 2021; Schick et al., 2023; Yao et al., 2022b], how to incorporate more tools via retrieval [Patil et al., 2023; Qin et al., 2023], and some task-specific ablations [Yao et al., 2022a; Le et al., 2022]. This paper takes on a more systematic investigation, proposing and answering new questions toward language agent fine-tuning. 3 FireAct: Fine-tuning LMs with diverse ReAct trajectories Our work is largely based on ReAct [Yao et al., 2022b], a popular approach to language agents. A ReAct task-solving trajectory (Figure 5) consists of multiple thought-action-observation rounds, where an LM generates free-form “thoughts” for versatile purposes (e.g., extract information from observations, propose and adjust action plans, track task progress), and structured “actions” to interact with environments (tools) and receive “observation” feedback. ReAct outperforms reasoning or acting only baselines, as reasoning can guide acting, and acting can support reasoning with new information. The ReAct format has thus been a basis of many follow-up language agents, such as Reflexion [Shinn et al., 2023], SwiftSage [Lin et al., 2023], and AutoGPT [Richards, 2023]. Also shown in Yao et al. (2022b) was a preliminary PaLM (Chowdhery et al., 2022) fine-tuning experiment on HotpotQA (Yang et al., 2018), where a fine-tuned PaLM-62B outperforms a prompted PaLM-540B. But it remains unknown if such a finding generalizes to other types of LMs, prompting methods, or tasks. Follow-up studies on language agent fine-tuning have been sparse (see Section 2). Thus we propose FireAct, a novel fine-tuning approach to language agents. As shown in Figure 2(a), FireAct also leverages few-shot prompting of a strong LM to generate diverse ReAct trajectories to fine-tune a smaller LM (i.e., distillation (Hinton et al., 2015)). But different from Yao et al. (2022b), FireAct explicitly promotes data diversity by mixing multiple training tasks and prompting methods. Here we consider two other methods compatible with the ReAct format: - Chain of Thought (CoT) (Wei et al., 2022b) generates intermediate reasoning to bridge the question-answer gap. Each CoT trajectory can be turned into a simple one-round ReAct trajectory, with “thought” being the intermediate reasoning and “action” being returning the answer. CoT is useful for simple questions without tool needs (Figure 2(b)). - Reflexion (Shinn et al., 2023) mostly follows the ReAct trajectory, but incorporates extra feedback and self-reflections. In this work, we simply prompt for reflections at the 6th and 10th ReAct round, so that long ReAct trajectories could pivot the strategy for solving the current task (e.g., “film search has not been helpful yet, I should search directors now”). During inference (Figure 2(b)), a FireAct agent alleviates the need for few-shot prompting, which makes inference more efficient and convenient. It could also implicitly select the suitable method adaptive to the task complexity, and show stronger generalization and robustness than prompting as a result of a wider and more diverse learning support. ### 4 EXPERIMENTAL SETUP Tasks. Following prior work (Wei et al., 2022b; Yao et al., 2022b; Shinn et al., 2023), we train and test on well-established question answering (QA) tasks, which enjoy abundant and high-quality training data plus easy and faithful evaluation (answer exact match). We use four datasets: - HotpotQA (Yang et al., 2018) is a QA dataset challenging multi-step reasoning and knowledge retrieval. The answer is usually a short entity or yes/no. We use 2,000 random training questions for fine-tuning data curation, and 500 random dev questions for evaluation. - Bamboogle (Press et al., 2022) is a test set of 125 multi-hop questions with similar formats as HotpotQA, but carefully crafted to avoid direct solving with Google search. - StrategyQA (Geva et al., 2021) is a yes/no QA dataset requiring implicit reasoning steps. - MMLU (Hendrycks et al., 2021) covers 57 multi-choice QA tasks in various domains such as elementary mathematics, history, and computer science. Tool. Following Press et al. (2022), we use SerpAPI\footnote{https://serpapi.com} to build a Google search tool that returns the first existent item from “answer box”, “answer snippet”, “highlight words”, or “first result snippet”, which ensures the response is short and relevant. We find such a simple tool sufficient for basic QA needs across tasks, and increases our fine-tuned models’ ease of use and generality. LMS. We investigate three families of LMs: - OpenAI GPT. We prompt GPT-4 (OpenAI, 2023b) to generate all fine-tuning data, and use GPT-3.5 for fine-tuning (OpenAI, 2023a) as well as prompting. We used both models in ChatCompletion mode from July to Sept 2023. - Llama-2 (Touvron et al., 2023b) with 7B and 13B parameters in “chat” mode. - CodeLlama (Roziere et al., 2023) with 7B, 13B, and 34B parameters in “instruct” mode, which help further understand model size scaling and the importance of code fine-tuning for agentic tasks. Fine-tuning methods. We use Low-Rank Adaptation (LoRA) \cite{Hu2022} for most fine-tuning experiments, but also use full-model fine-tuning for some comparison. Given the various factors underlying language agent fine-tuning, we split experiments into three parts with increasing complexities: - Fine-tuning using a single prompting method on a single task (Section 5); - Fine-tuning using multiple methods on a single task (Section 6); - Fine-tuning using multiple methods on multiple tasks (Section 7). ## 5 SINGLE-TASK, SINGLE-METHOD FINE-TUNING In this section, we focus on fine-tuning with data from a single task (HotpotQA) and a single prompting method (\textit{ReAct}). Using such a simple and controlled setup, we confirm various benefits of fine-tuning over prompting (performance, efficiency, robustness, generalization), and study effects of different LMs, data sizes, and fine-tuning methods. By default, we use 500 successful few-shot prompting trajectories generated by GPT-4 for training and a random subset of 500 HotpotQA dev questions for evaluation. Other experimental details can be found in the Appendix B. ### 5.1 PERFORMANCE AND EFFICIENCY \| Prompt \| EM \| ReAct \| FireAct \| abs./rel. diff \| \|--------\|------\|-------\|---------\|----------------\| \| IO \| 37.2 \| Llama-2-7B \| 14.8 \| 26.2 \| +11.4 / 77% \| \| CoT \| 45.0 \| Llama-2-13B \| 21.2 \| 34.4 \| +13.1 / 62% \| \| ReAct \| 42.0 \| CodeLlama-7B \| 17.4 \| 27.8 \| +10.4 / 60% \| \| IO \| 22.4 \| CodeLlama-13B \| 20.8 \| 29.0 \| +8.2 / 39% \| \| CoT \| 28.0 \| CodeLlama-34B \| 22.2 \| 27.8 \| +5.6 / 25% \| \| ReAct \| 31.4 \| GPT-3.5 \| 31.4 \| 39.2 \| +7.8 / 25% \| Fine-tuning significantly increases agent performances. As shown in Table 2, fine-tuning consistently and significantly improves the HotpotQA EM from prompting. While weaker LMs benefit more from fine-tuning (e.g., Llama-2-7B increases by 77%), even strong LMs such as GPT-3.5 could improve performances by 25%, clearly showing the benefit of learning from more samples. When compared to strong prompting baselines in Table 1, we find fine-tuned Llama-2-13B could outperform all GPT-3.5 prompting methods (Input-Output prompting, IO; Chain-of-thought, CoT; \textit{ReAct}). It is a promising signal that fine-tuning small open-source LMs could outperform prompting stronger commercial LMs. Finally, fine-tuned GPT-3.5, which is the strongest fine-tuned LM, could outperform GPT-4 + IO prompting but still lags behind GPT-4 + CoT/\textit{ReAct} prompting, suggesting room for improvement. More results (e.g., standard error) are in Appendix A.1. Fine-tuning is cheaper and faster during agent inference. Since few-shot in-context examples are not needed for fine-tuned LMs, their inference becomes more efficient, especially for agentic applications where the context is iteratively accumulated. For example, the first part of Table 3 compares costs of fine-tuned vs. prompted GPT-3.5 inference, and finds the inference time is reduced by 70% (9.0s to 2.7s per trial), and the inference cost is reduced even though fine-tuned inference is charged 8× expensive. While these costs will vary by conditions (e.g., parallelism implementation), the advantage of having a much smaller context is clear. ## 5.2 ROBUSTNESS AND GENERALIZATION Robustness to noisy tools. The tools or environments that language agents interact with are not always trustworthy, which has led to safety concerns like jailbreaking \cite{Liu2023b} or prompt injection \cite{Willison2023}. Here we consider a simplified and harmless setup, where the search API has a probability of 0.5 to return 1) “None” or 2) a random search response (from all previous experiments and trials), and ask if language agents could still robustly answer questions. As shown... Table 3: Comparison of costs, robustness, and generalization for fine-tuned vs. prompted GPT-3.5. \| \| Cost per trial \| Obs. Robustness (EM) \| Generalization \| \|----------\|----------------\|----------------------\|----------------\| \| \| Money ($) \| Time (s) \| Normal \| “None” \| Random \| Bamboogle (EM) \| \| FireAct \| $2.2 \times 10^{-3}$ \| 2.7 \| 39.2 \| 33.6 \| 37.2 \| 44.0 \| \| ReAct \| $2.6 \times 10^{-3}$ \| 9.0 \| 31.4 \| 20.8 \| 22.6 \| 40.8 \| Figure 3: Data scaling. Figure 4: Results across different LMs and data types. In the second part of Table 3, the “None” setup turns out the more challenging one, which lowered ReAct EM by 33.8% and FireAct EM only by 14.2%. Interestingly, random observations hurt ReAct by a similar degree (28.0% drop), but does not hurt FireAct much (only 5.1% drop), possibly because the fine-tuning trajectories already contain examples of noisy search queries and how GPT-4 “reacts” to such noises successfully. These initial results hint at the importance of a more diverse learning support for robustness. More results on robustness can be found in Appendix A.2. Generalization to new tasks. Table 3’s third part shows EM results of fine-tuned and prompted GPT-3.5 on Bamboogle (Press et al., 2022), a test set of 125 multi-hop questions carefully crafted such that searching the questions on Google cannot directly give answers. While HotpotQA fine-tuned or prompted GPT-3.5 both generalize to Bamboogle reasonably, the former (44.0 EM) still beats the latter (40.8 EM), suggesting generalization advantages of fine-tuning. Similarly, combined with the few-shot prompts, fine-tuning on HotpotQA greatly improves the performance on Bamboogle, while slightly improving on MMLU and downgrading on StrategyQA compared to vanilla models (Appendix A.9). Since fine-tuning on HotpotQA could hardly generalize to StrategyQA (yes/no questions) or MMLU (multi-choice questions), two other QA datasets with different question styles and answer formats, it motivates our multi-task fine-tuning experiments in Section 7. 5.3 ANALYSIS OF VARIOUS FINE-TUNING FACTORS Effect of fine-tuning method (LoRA vs. Full model). For Llama-2-7B, we observe that full-model fine-tuning (30.2 EM) outperforms LoRA fine-tuning (26.2 EM) by 15.3% (see Appendix A.5). However, LoRA training is much more affordable, which can train 5.4 examples per second on a single RTX 4090 with 24GB GPU memory, while training 19.7 examples by full fine-tuning requires four A100 GPUs with 80GB GPU memory. Hence, running most experiments with LoRA allows us to explore more training settings with a limited budget and time frame. Effect of fine-tuning data scale. Figure 5 shows how FireAct performances scale with the number of fine-tuning trajectories ($n \in \{100, 200, 500, 1000\}$). GPT-3.5 appears very sample-efficient, requiring only 100 samples to reach an EM around 35, and the gain after 200 samples is marginal. On the other hand, Llama models cannot even learn the ReAct format using 100 or 200 samples, but non-trivial scores “emerge” with 500 samples, and most models (except CodeLlama-13B) further improve with 1,000 samples. Such a data scaling trend suggests that smaller open-source LMs could potentially catch up with stronger LMs on a particular agentic task given enough fine-tuning data (e.g., Llama-2-13B fine-tuned on 1,000 samples can match GPT-3.5 fine-tuned on 100 samples). Effect of base LM type. Table 2 reveals that GPT-3.5 is superior to all Llama-based models in both prompting and fine-tuning configurations. Additionally, CodeLlama-7B outperforms Llama-2-7B, while CodeLlama-13B does not perform as well as Llama-2-13B, suggesting that coding fine-tuning may not always be beneficial for agentic use cases. CodeLlama performs slightly better when using the default CodeLlama tokenizer instead of the Llama tokenizer (Appendix A.6). Effect of base LM scale. As can be seen in Table 2 or the blue bars of Figure 4, CodeLlama models with 13B parameters always outperform ones with 7B parameters, but CodeLlama-34B seems worse than CodeLlama-13B when fine-tuned purely on ReAct trajectories. But as we will see in Section 6 (and hinted in the rest of Figure 4), other factors such as the fine-tuning data type might affect the conclusion and make CodeLlama-34B outperforming CodeLlama-13B. In general, multiple components (LM type, LM scale, fine-tuning data and method) might influence fine-tuning results jointly, so different dimensions of scaling trends and LM/data types should also be considered jointly for agent design. ## 6 Multi-method Fine-tuning Next we integrate CoT (Wei et al., 2022b) and Reflexion (Shinn et al., 2023) with ReAct for multi-method fine-tuning on HotpotQA. For both methods, we generate 500 few-shot prompting trajectories via GPT-4, and use 47 long Reflexion trajectories that incorporated self-reflections after 6 or 10 ReAct rounds, and 187 successful CoT trajectories reformatted as single-round ReAct trajectories, on top of 500 existing ReAct trajectories. More details are in Appendix B. Multi-method fine-tuning increases agent flexibility. Before quantitative results, we present two example questions in Figure 5 and some fine-tuned GPT-3.5 trajectories to illustrate the benefit of multi-method FireAct fine-tuning. The first question (a) is simple, but the ReAct-only fine-tuned agent (a1) searched an over-complicated query that led to the distraction and wrong answer. In contrast, an agent fine-tuned with both CoT and ReAct chose to solve the task within one round relying on confident internal knowledge. The second question (b) is harder, and the ReAct-only fine-tuned agent (b1) kept searching queries ending in “during the Libyan Civil War” without useful information. In contrast, an agent fine-tuned with both Reflexion and ReAct reflected upon this problem, and pivoted the search strategy to change the time constraint to “during his rule”, which led to the right answer. The flexibility to implicitly choose methods for different problems is another key advantage of fine-tuning over prompting. Multi-method fine-tuning affect different LMs differently. Despite the intuitive benefit, Figure 4 shows mixing more methods does not always improve results, and the optimal mix of methods depends on the base LM. For example, React+CoT outperforms React for GPT-3.5 and Llama-2 models, but hurts for CodeLlama models. React+CoT+Reflexion is the worst for CodeLlama-7/13B, but is the best for CodeLlama-34B. These non-trivial results call for further studies of the interaction of base LMs and fine-tuning data. Can multi-method agents choose suitable methods? Table 4 displays HotpotQA test results of various FireAct agents based on GPT-3.5, as well as the mean ($\mu$) and standard deviation ($\sigma$) of the number of React rounds across their trajectories. Compared to React-only fine-tuning, React+CoT improves the EM and reduces the trajectory length, while React+Reflexion hurts the EM and increases the trajectory length. This suggests the two method mixes shift the method selection to two different directions, and CoT is perhaps more helpful for HotpotQA questions. To further understand if multi-method agents could choose the suitable methods, we calculate the result of randomly choosing a method during inference. The result of 32.4 is much lower than all multi-method agents, suggesting the method selection is non-trivial. But applying the best method for each instance leads to an “oracle” result of 52.0, suggesting room for improving prompting method selection. Future work could explore more systematic grid search or connections between trajectory statistics and performances to set up better method mix ratios. ### 7 MULTI-TASK FINE-TUNING So far fine-tuning has only used HotpotQA data, but empirical studies on LM fine-tuning have shown the benefit of mixing different tasks (Longpre et al., 2023). Here we fine-tune GPT-3.5 using a mix of training data from three datasets: HotpotQA (500 React samples, 277 CoT samples), StrategyQA (388 React samples, 380 CoT samples), and MMLU (456 React samples, 469 CoT samples). These samples are picked from successful React/CoT few-shot prompting trajectories generated via GPT-4. As shown in Table 5, when StrategyQA/MMLU data is added (“Multi-task”), HotpotQA/Bamboogle performances almost remain the same. On one hand, StrategyQA/MMLU trajectories contain very different questions (e.g., MMLU questions are multi-choice) and tool use strategies (e.g., MMLU React trajectories tend to search answer choices), which makes transfer hard. On the other hand, despite the distribution shift, adding StrategyQA/MMLU does not hurt HotpotQA/Bamboogle performances, which hints at the promise of fine-tuning one multi-task agent to replace multiple single-task agents, capturing the performance improvement of fine-tuning based agents, without sacrificing the flexibility of prompting based agents or worrying about negative cross-task influences. When we switch from multi-task, single-method fine-tuning to multi-task, multi-method fine-tuning, we find increased performances across all tasks, again reinforcing the value of multi-method agent fine-tuning. Intriguingly, all fine-tuned agents (plus CoT/React prompting) underperform naive input-output (IO) prompting on MMLU. One possible explanation is these questions might be too easy to require reasoning and acting, and another explanation could be answer choice memorization. This urges efforts for better prompting methods as well as for better agent datasets. \| Prompting method \| #Turns \| EM \| $\mu$ \| $\sigma$ \| \|------------------\|--------\|----\|------\|--------\| \| React \| \| 39.4 \| 3.2 \| 1.4 \| \| React + CoT \| \| 41.0 \| 2.7 \| 1.7 \| \| React + Reflexion\| \| 38.8 \| 3.8 \| 2.8 \| \| React + CoT + Reflexion \| \| 40.0 \| 3.0 \| 4.8 \| \| Random method choice \| \| 32.4 \| - \| - \| \| Oracle method choice \| \| 52.0 \| - \| - \| \| Method \| HotpotQA \| StrategyQA \| Bamboogle \| MMLU \| \|-----------------\|----------\|------------\|-----------\|------\| \| Prompting \| 10 \| 22.4 \| 48.0 \| 7.2 \| 68.6 \| \| CoT \| \| 28.0 \| 49.0 \| 41.6 \| 50.8 \| \| React \| \| 31.4 \| 61.0 \| 40.8 \| 58.6 \| \| Fine-tuning \| \| \| \| \| \| \| HotpotQA \| \| 39.2 \| - \| 44.0 \| - \| \| Multi-task \| \| 39.2 \| 55.5 \| 43.2 \| 63.2 \| \| + CoT \| \| 39.6 \| 72.9 \| 50.4 \| 65.8 \| 8 DISCUSSION When to fine-tune vs. prompt for language agents? While most existing language agents use prompting, our work calls for a re-thinking of best practices by showing multi-facet benefits of fine-tuning as a result of more diverse learning support. Thus, prompting and fine-tuning seem more suitable for exploration and exploitation usecases respectively. To develop new agents or solve new tasks, prompting off-the-shelf LMs could provide flexibility and convenience. On the other hand, when the downstream task is known (e.g., QA), effective prompting methods for agents have been explored (e.g., ReAct), and enough data can be collected (e.g., via GPT-4), fine-tuning can provide better performances, stronger generalization to new tasks, more robustness to noisy or adversarial environments, as well as cheaper and more efficient inference. These features make fine-tuning especially attractive when used for large-scale industrial solutions. Which LM to fine-tune? Of all models we considered, GPT-3.5 consistently outperforms other Llama-based LMs in various setups, which is not surprising given its much larger model size and continued training from GPT-3. It also has better sample efficiency and a reasonable cost (around $10 per fine-tuning experiment in our case). However, we have also shown that open source Llama models could be fine-tuned to catch up with GPT-3.5 performances, given enough fine-tuning data with the right mixing of prompting methods and tasks. Practitioners should balance the tradeoff between the convenience and performance of GPT-3.5 versus the controlability and reproducibility of open-source LMs for agent fine-tuning. When to use tools or reflect for language agents? Prompting-based language agents can only imitate a small and fixed set of successful task-solving trajectories. This could lead to tool overuse (e.g., search for knowledge already stored in LMs), and inabilities to recover when the trajectory deviates from the “successful” patterns (e.g., keep searching similar queries with useless observations). FireAct’s multi-method fine-tuning helps increase a language agent’s flexibility and robustness, but the problem of knowing when to get help (tool use) and feedback (reflection) is still far from being solved. Work on calibration (Ren et al., 2023) and meta-reasoning (Griffiths et al., 2019) might shed light into better agent designs in this regard. Limitations and future directions. This work is an initial step toward language agent fine-tuning, and is constrained to a single type of task (QA) and a single tool (Google search). Future work could apply the research questions raised by FireAct to more tasks and grounding setups (e.g., more API tools, the web, the physical world). Also, we focused on three methods (ReAct, CoT, Reflexion) that maintain a single autoregressive trajectory context, which makes fine-tuning straightforward. It remains underexplored how to fine-tune more advanced agents involving multiple prompts, roles, and contexts (Wang et al., 2023a; Park et al., 2023; Yao et al., 2023), model multi-agent interaction and orchestration, or best combine prompting and fine-tuning in a complex agent system. These are exciting future directions for fine-tuning based language agents. Finally, the multi-task setup in this work is limited to three QA tasks, and the best LM we could fine-tune is GPT-3.5. A large-scale multi-task fine-tuning (Wei et al., 2022a) using the state-of-the-art LM backbone will test the limit of language agent fine-tuning, but more suitable and diverse benchmarks to develop and evaluate agents should be explored first. REPRODUCIBILITY STATEMENT Our main experiments are performed on API-based GPT4[^1] and GPT-3.5-Turbo[^2] and open source Llama[^3] and CodeLlama[^4]. Details of the experiment setting are in Appendix B[^5] and all used prompts are in Appendix C[^6]. The codebase is released at: https://anonymous.4open.science/r/FireAct-DC39/ [^1]: https://openai.com/research/gpt-4 [^2]: https://openai.com/blog/gpt-3-5-turbo-fine-tuning-and-api-updates [^3]: https://huggingface.co/meta-llama [^4]: https://huggingface.co/docs/transformers/main/model_doc/code_llama ETHICS STATEMENT This research focuses on language agents, and we are aware of the potential risks associated with uncontrolled autonomous interactions. Thus, we have chosen a setup of open-domain question answering with access to a Google search API, where the API is read-only and does not cause any changes to the Internet. 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Qlora: Efficient finetuning of quantized llms. CoRR, abs/2305.14314, 2023. doi: 10.48550/arXiv.2305.14314. URL https://doi.org/10.48550/arXiv.2305.14314 Ning Ding, Yujia Qin, Guang Yang, Fuchao Wei, Zonghan Yang, Yusheng Su, Shengding Hu, Yulin Chen, Chi-Min Chan, Weizhe Chen, Jing Yi, Weilin Zhao, Xiaozhi Wang, Zhiyuan Liu, Hai-Tao Zheng, Jianfei Chen, Yang Liu, Jie Tang, Juanzi Li, and Maosong Sun. Parameter-efficient fine-tuning of large-scale pre-trained language models. Nat. Mac. Intell., 5(3):220–235, 2023. doi: 10.1038/s42256-023-00626-4. URL https://doi.org/10.1038/s42256-023-00626-4
JfcLYCqOkQ	In Figure 4, it can be seen that the difference between layer 0-3 and layer 0-6 is significant, and according to your statement, there should be an improvement. However, the experimental results show a decrease from 84.1 to 84.0. On the other hand, the improvement from layer0-6 to layer0-9 is significant, but the graph shows that the difference from the baseline is not as significant. This has caused some confusion, and it would be helpful if the author could explain this discrepancy.	CONDITIONAL MAE: AN EMPIRICAL STUDY OF MULTIPLE MASKING IN MASKED AUTOENCODER Anonymous authors Paper under double-blind review ABSTRACT This work aims to study the subtle yet often overlooked element of masked autoencoder (MAE): masking. While masking plays a critical role in the performance of MAE, most current research employs fixed masking strategies directly on the input image. We introduce a masked autoencoder framework with multiple masking stages, termed Conditional MAE, where subsequent maskings are conditioned on previous unmasked representations, enabling a more flexible masking process in masked image modeling. By doing so, our study sheds light on how multiple masking affects the optimization in training and performance of pretrained models, e.g., introducing more locality to models, and summarizes several takeaways from our findings. Finally, we empirically evaluate the performance of our best-performing model (Conditional-MAE) with that of MAE in three folds including transfer learning, robustness, and scalability, demonstrating the effectiveness of our multiple masking strategy. We hope our findings will inspire further research in the field and code will be made available. 1 INTRODUCTION Self-supervised learning (Chen et al., 2020; Caron et al., 2020; Zbontar et al., 2021; Caron et al., 2021; He et al., 2021; Chen et al., 2021; Grill et al., 2020; Chen & He, 2021) has great potential to leverage substantial unlabeled data, and the learned representation is beneficial to downstream tasks. Among them, one promising approach is masked image modeling (MIM), which partitions an image into visible patches and masked patches and predicts the masked patches from visible patches (Bao et al., 2021; He et al., 2017; Chen et al., 2022a). As a representative, masked autoencoder (He et al., 2021) (MAE) first masks an image, then feeds visible patches into a vision transformer encoder, and finally reconstructs the masked patches in RGB space via a shallow decoder. After MAE, numerous masked-based methods have been proposed, leading to an explosion of research in MIM and quickly spreading to other fields (Tong et al., 2022; Baevski et al., 2022b; Wang et al., 2023a; Baevski et al., 2022a; Pang et al., 2022; Zhang et al., 2022a), e.g., video and 3D. A crucial component of the masked autoencoder is the mask ratio, which directly impacts the model’s performance. For instance, in MAE, the performance gap for fine-tuning accuracy may vary by up to 2% with different mask ratios (He et al., 2021). However, current methods, including MAE, mostly ablate the mask ratio only on the input image: they mask the input image with various ratios and select the best-performing ratio after training those model variants. Considering that masking is an important and flexible operation that can be performed at different stages (e.g., the input image and different levels of representations) and with different ratios, these approaches may fail to fully exploit the potential of the autoencoder. Hence, a question naturally raises: Can the masked autoencoder handle multiple rounds of masking at different levels, and how does multiple masking affect its optimization in training and performance? To answer the above question, this work presents a framework called Conditional MAE, which aims to explore the impact of multiple rounds of masking in the training process and performance. In Conditional MAE, subsequent maskings are conditioned on previous unmasked representations, enabling more flexible masking on different granularities of inputs. Based on it, we progressively conduct a thorough empirical study about multiple masking to address three critical questions: 1) where to mask, 2) how much to mask, and 3) what’s the impact? In our experiments, we investigate one, two, and three-shot masking$^1$, where each round of masking is considered a shot. Our results highlight several key takeaways from each shot, which are summarized below: - In the one-shot case, we find that masking at the beginning is always beneficial for task performance. Moreover, it is critical to find a suitable mask ratio. Generally, though the model size is different, e.g., ViT-S and ViT-B, 75% mask ratio is firstly recommended. - In the two-shot case, building on the best one-shot setting, increasing the interval of two-shot masking with a large ratio followed by a small ratio is helpful for fine-tuning. Additionally, our experiments strongly suggest that there may not exist a positive relationship between linear probing and fine-tuning. Finally, the second masking brings locality bias into the model and helps capture low-level features, especially for finer-grained classification. - In the three-shot case, we find that using a greedy-like masking selection strategy, which uses the best two-shot setting as a starting point, is superior to other three-shot strategies. Simultaneously, the third masking brings more locality into models than two-shot case. Based on the above results of our empirical experiments, we select the best-performing model (Conditional-MAE) and evaluate its transferability to downstream tasks, including image classification, object detection, and semantic segmentation. We also verify its robustness to noisy inputs, e.g., random occlusion and shuffling, and empirically demonstrate its scalability. Note that in this research, we are not to propose a state-of-the-art method, but to enhance both the understanding and performance of MAE by exploring the potential of masking and to inspire future research. Our contributions are three-fold: - Building on our proposed flexible framework, i.e., Conditional MAE, we are the first to make an in-depth analysis of multiple masking and reveal its impact on masked autoencoder’s optimization in training and performance. - Through extensive empirical experiments on multiple masking, we provide several key takeaways from each shot as shown above. More importantly, we observe a key phenomenon that multiple masking is capable of introducing locality bias to models. - We demonstrate the superiority of our Conditional-MAE over MAE in downstream transfer, robustness against occlusion and shuffling, and scalability. 2 CONDITIONAL MAE 2.1 PRELIMINARIES Given an image, MAE first partitions it into $N$ patches $P = \{P^1, P^2, \ldots, P^N\}$ that are randomly categorized into two parts, i.e., visible patches $P_v = \{P^1_v, P^2_v, \ldots, P^{N_1}_v\}$ and masked patches $P_m = \{P^1_m, P^2_m, \ldots, P^{N_2}_m\}$, with a pre-define ratio $\eta_1$ ($N_2 = \eta_1 * N$ and $N_1 + N_2 = N$). Then, $P_v$ are feed into Encoder that outputs corresponding patch representations $Z_v = \{z^1_v, z^2_v, \ldots, z^{N_1}_v\}$. Finally, $Z_v$ along with learnable mask token [MASK]$^2$ are sent into Decoder to predict masked patches in RGB space. $P_m$ is served as the supervision signal. The whole process is formulated as: \[ Z_v = \text{Encoder}(P_v), \] \[ \hat{P}_m = \text{Decoder}(Z_v, [\text{MASK}]), \] \[ L = \text{MSE}(\hat{P}_m, P_m), \] where MSE is the mean square error loss function. 2.2 CONDITIONAL MAE Our Conditional MAE is derived from MAE and able to perform multiple shots masking on MAE as shown in Fig 1. We take two-shot masking for example to elaborate why we call it Conditional MAE. The first masking is implemented on RGB space with a pre-defined mask ratio $\eta_1$ on image patches, $^1$Note that we do not study more shots as it is inferior to three-shot masking in our preliminary experiments. $^2$We omit the operation of adding position embedding for a better description. Figure 1: An overview of our Conditional MAE compared with MAE. $N_1, N_3,$ and $N_5$ indicate the number of unmasked patches or representations. which is what MAE does. Afterward, the second masking is conditioned on previous unmasked representations on a given layer of the encoder, e.g., $j$. Thus, for visible patch representations $Z_{j^}^v$ (output from the $j^$-th layer of the encoder, $j^* = j - 1$), Conditional MAE mask part of them with another pre-defined masking ratio $\eta_2$. We denote the left visible patch representations as $Y_{j^}^v = \{y_1^v, y_2^v, \ldots, y_{N_3}^v\}$ and the masked patch representations as $Y_{j^}^m = \{y_1^m, y_2^m, \ldots, y_{N_4}^m\}$ ($N_3 + N_4 = N_1$ and $N_4 = \eta_2 \times N_1$). Additionally, we collect the visible patches corresponding to $Y_{j^}^m$ from $P_v$, denote them as $P_{j^}^m = \{P_1^m, P_2^m, \ldots, P_{N_4}^m\}$, and merge them with $P_m$ as $\{P_m, P_{j^}^m\}$ ($\|\{P_m, P_{j^}^m\}\| = N_2 + N_4$) as our new reconstruction target. Therefore, for two-shot masking, the whole process can be formulated as: \[ Z_{j^}^v = \text{Encoder}_{0 \to j^}(P_v), \] \[ Y_{j^}^v, Y_{j^}^m = \text{Mask}(Z_{j^}^v, \eta_2), \] \[ Z_v = \text{Encoder}_{j \to 11}(Y_{j^}^v), \] \[ \hat{P}_m = \text{Decoder}(Z_v, [\text{MASK}]), \] \[ L = \text{MSE}(\hat{P}_m, \{P_m, P_{j^}^m\}), \] where $\text{Encoder}_{0 \to j^}$ means that the input passes through 0-th layer of the encoder and is outputted from $j^$-th layer. Compared with MAE, due to the $\text{Mask}$ function, the main discrepancies lie in Eq (6) and Eq (8). We need to reconstruct two targets, i.e., $P_m$ and $P_{j^}^m$, with less visible patch representations. Note that this process cannot be bridged by increasing mask ratio $\eta_1$ of MAE to remove more visible patches. We explain it below. For $P_m$, similar to MAE, it has never been seen by the encoder and thereby we need infer it via visible patch representations $Y_{j^}^v$. For $P_{j^}^m$, it has been seen by partial encoder (i.e., layers before $j$), resulting in its information involved in $Y_{j^}^v$ via attention-manner interaction between $Y_{j^}^v$ and $Y_{j^}^m$ before $j$-th layer. We reconstruct the patches $P_{j^}^m$ primarily conditioned on the “borrowed” information involved in $Y_{j^}^v$ via the interaction above. This is easily generalized to multiple shots. Particularly, in the two-shot case, if $j$ is set to 0 or $\eta_2$ is 0, Conditional MAE is reduced to MAE, if $\eta_1$ (the first mask ratio) is 0, our Conditional MAE is still established with only reconstruction of $P_m$ removed. 3 EXPERIMENT 3.1 MULTIPLE SHOTS MASKING In our study, we investigate the Conditional MAE in three different settings by pretraining on ImageNet100: one-shot masking, two-shot masking, and three-shot masking. We do not explore settings with more shots, as preliminary experiments have shown them to be inferior to three-shot. For ease of description, we denote the three mask ratios as $\eta_1, \eta_2, \eta_3$, and the corresponding layer indexes as $i, j, k$, respectively, where masking is applied before inputting. Considering our Conditional MAE Figure 2: Results of one-shot masking on ViT-S/16. \| Model Size \| Mask Ratio \| Linear Probe \| Fine-tune \| \|------------\|------------\|--------------\|-----------\| \| ViT-S/16 \| 0.75 \| 45.0 \| 82.5 \| \| \| 0.90 \| 44.9 \| 81.3 \| \| ViT-B/16 \| 0.75 \| 62.9 \| 86.9 \| \| \| 0.90 \| 57.9 \| 85.6 \| Table 1: Comparisons on ViT-S/16 and ViT-B/16 with different mask ratio. is derived from MAE, we fix $i = 0$ to match with MAE. Through exhaustive experiments conducted below, we aim to address three key questions: where to mask, how much to mask, and what is the impact? For training details, please refer to the Appendix A.1. ### 3.1.1 ONE-SHOT MASKING In the one-shot setting, we only mask patch tokens in the encoder once, allowing us to examine the impact of different mask positions and mask ratios on encoder performance. Specifically, for mask positions, we consider four positions at equal intervals: the 0-th, 3-th, 6-th, and 9-th layer of encoder blocks, denoted as $(i, j, k) = (0, 0/3/6/9, 0)$. We exclude the 12-th layer as it would cause the denoise autoencoder to degenerate into a vanilla autoencoder. Regarding mask ratios, we carefully select two representative ratios used in MAE (He et al., 2021), namely 0.75 and 0.9, denoted as $(\eta_1, \eta_2, \eta_3) = (0, 0.75/0.9, 0)$. The reasons are two-fold: 0.75 is widely used in MAE; For 0.9, previous work (Riquelme et al., 2021) has shown that even using 10% patch features can still yield competitive performance in visual recognition. The results on ViT-S/16 are illustrated in Fig 2. It has been observed that masking at the beginning position ($j = 0$) is beneficial for both linear probing and fine-tuning. Conversely, we also notice a significant drop in performance for linear probing when masking is applied at the other positions. This indicates that the representations encoded by the fixed encoder at $j = 0$ are relatively more distinguishable and implies that these encoders learn comparatively less knowledge compared to the encoder at $j = 0$. To support this observation, we visualize the training loss curves of pretraining and linear probing and t-SNE of output representation in Appendix A.2.1. Finally, to investigate the impact of mask ratio on models of different sizes, we also conduct experiments on ViT-B/16 and present the results in Tab 1. Interestingly, we observe that a mask ratio of 0.75 enhances the performance of ViT-B/16 compared to a mask ratio of 0.9, which is similar to ViT-S/16. Moreover, our results are consistent with MAE (He et al., 2021) trained on ImageNet1k (Russakovsky et al., 2015) whose best mask ratio is also 75%. Conclusion.* For one-shot masking, we summarize two useful tips: ① Masking at the beginning is always beneficial for task performance; ② Finding a suitable mask ratio is critical. Generally, though the model size is different, e.g., ViT-S and ViT-B, a 75% mask ratio is firstly recommended. --- 3We set $\eta_1$ to 0 as its layer index $i = 0$ is fixed as described at beginning while our mask position should be flexible. 3.1.2 TWO-SHOT MASKING Two-shot masking means we can mask twice in the encoder. We use a step-by-step scheme by following the conclusion from one-shot and first mask patch tokens at the beginning also with two representative mask ratios, i.e., 0.75 and 0.9. Therefore, it is critical to figure out where the second shot masking should be and how much it should mask. The experiments on ViT-S/16 are shown in Fig 3. $L(i,j)$ (k is omitted) indicates we mask the $i$-th and $j$-th Layers $(i = 0$ and $i < j < 12)$. We use $(\eta_1, \eta_2)$ ($\eta_3$ is omitted) to denote the mask ratio of two-shot masking. For example, $L(0, 5; 0.75, 0.5)$ means that we mask the 0-th layer with mask ratio 0.75 and mask the 5-th layer with mask ratio 0.5. The dashed line denotes the one-shot baseline with masking ratios of 0.75 respectively. For $\eta_1 = 0.75$, we ablate five combinations of mask layers for two-shot masking. Three involve an equal interval for the second masking layer indexes following the one-shot masking scheme: $L(0, 3)$, $L(0, 6)$, and $L(0, 9)$; Two are continuous combinations: $L(0, 10)$ and $L(0, 11)$. We initially set a larger mask ratio of $\eta_2$ (0.5). Considering that the performance is inferior to the baseline in both linear probing and fine-tuning, we replace $\eta_2 = 0.5$ with three relatively smaller ones containing 0.25, 0.15, and 0.1. As shown in Fig 3 (a), the performance of two-shot masking is inferior to the baseline for linear probing. However, as opposed to linear probing, one can see in Fig 3 (b) that our two-shot masking shows potential to outperform the baseline in fine-tuning: An apparent trend for fine-tuning is that the second masking performed at the last several layers (i.e., increasing the interval of two-shot masking) with a smaller $\eta_2$ leads to significant improvement compared to baseline, especially at $L(0, 10)$. The contradictory experiment results imply that there may not exist a positive correlation between linear probing and fine-tuning. Hence, following (Woo et al., 2023), we would like to pay more attention to end-to-end fine-tuning because of its practical relevance in transfer learning. We put two-shot results for $\eta_1 = 0.9$ in Appendix A.3.1 In light of the superior performance, a question arises: What two-shot masking brings to the encoder? We dive deep into two-shot masking and analyze its layer representation and attention map. Layer Representation Analyses. We first leverage Centered Kernel Alignment (CKA) (Cortes et al., 2012; Nguyen et al., 2020) to analyze the layer representation similarity across pretrained models. As shown in Fig 4, we visualize the layer representation similarity between several two-shot masking pre-trained models and baseline $(0, 0.75)$ as heatmaps. It is seen an increasing discrepancy between the representations of two-shot models and that of baseline, especially between the high layer of two-shot models and shallow layer of baseline. This implies that the second masking introduce certain bias into pretrained models, rendering the representations varying from that of baselines. Attention Map Analyses. We then analyze the attention maps that reveal the behaviors for aggregating information in the attention mechanism of ViTs. Following (Wang et al., 2023c) we use two metrics, i.e., attention distance and attention entropy, to analyze two-shot masking and baseline models. We pick $L(0, 10; 0.75, 0.1)$ as it performs best and illustrate its attention distance and entropy variation before/after fine-tuning and compare with that of baseline $L(0; 0.75)$ in Fig 5. We see that the second --- 4 In our preliminary experiments, we found that $L(0, 9)$ performs the best in fine-tuning among these three combinations. To provide a more comprehensive analysis, we include $L(0, 10)$ and $L(0, 11)$. We do not include $L(0, 8)$ as it performs worse than $L(0, 9)$. 5 CKA computes the normalized similarity in terms of the Hilbert-Schmidt Independence Criterion (HSIC (Song et al., 2012)) between two feature maps or representations. 6 Note that the disparity in the heatmap does not necessarily imply whether the learned representation is advantageous or detrimental. It only reflects how the representation learned by our two-shot masking model varies from that of the baseline. Hence, it would be unreasonable to use the significance of heatmap to assess the performance after fine-tuning. 7 The attention distance reveals how much local vs. global information is aggregated, and a lower distance means each token focuses more on neighbor tokens. The attention entropy reveals the concentration of the attention distribution, and lower entropy means each token attends to fewer tokens. We refer the reader of interest to (Wang et al., 2023c) for detailed formula. Figure 4: Layer representation similarity between pretrained two-shot masking model and baseline. Figure 5: Comparison of two-shot masking $L(0, 10; 0.75, 0.1)$ and baseline model $L(0; 0.75)$ on attention distance and attention entropy before/after fine-tuning. The lp means pretrained model. The ft means fine-tuned models. Masking decreases the attention distance and entropy to some extent during pretraining in Fig 5 (a), bringing locality inductive bias into model and thereby rendering the representations varying from that of baselines. From the view of reconstruction, we conjecture such adjustment is because the second masking requires the unmasked patches to recover their parallel neighbor (masked ones) of a forward. In Fig 5 (b) and (c), compared to pretraining, fine-tuning decreases the attention distance and entropy in low layer and also elevates attention distance in high layer for both models. Finally we compare the attention distance and entropy between the two models after fine-tuning in Fig 5 (d) to figure out what makes $L(0, 10; 0.75, 0.1)$ have potential to outperform baseline $L(0; 0.75)$. We see that $L(0, 10; 0.75, 0.1)$ has similar attention distance and entropy in high layers while more concentrated and lower attention distance and entropy in low and middle layers. We attribute it to locality inductive bias brought by the second masking that captures better low-level features. Similar observations can be found in other two-shot model variants ($\eta_1 = 0.75$ and 0.9) which we put in Appendix A.3.2. Information Leakage and Locality. In the two-shot setting, the second masked patches have been seen by previous layers, potentially resulting in information leakage. However, it’s important to note that this leakage does not cause a trivial solution as the presence of $\eta_1$ and its substantial gap in magnitude compared to $\eta_2$ necessitates the model to acquire the ability to infer the masked patch in the first masking. In contrast, the presence of the second masking necessitates that patches that interacted in previous layers must recover their corresponding masked neighbors in the forward pass. Figure 6: Visualization of reversed attentions (showing how much information a second-masked patch sends to others) in layer 9 of models. Top: single masking model $L(0; 0.75)$ (vanilla MAE). Bottom: two-shot masking model $L(0, 10; 0.75, 0.1)$. It is evident that these second-masked tokens tend to send and store the information to their neighbors just prior to being masked, resulting in more localized and even attention. As a result, the model needs to dedicate a portion of its capacity to learn how to infer local neighbors. This would introduce a certain degree of locality bias, which can be advantageous under some task conditions (Jiang et al., 2022). To illustrate this, we visualize the reversed attention (Ding et al., 2023) of pretrained model $L(0, 10; 0.75, 0.1)$ as shown in Fig 6 (bottom), containing the information flow of second masking, i.e., how much information a second-masked patch sends to other. It clearly demonstrates that the attention head retains object-related local information. In this way, the information leakage is controllable, and information of the second-masked patch flows and is stored in the neighboring patches, to be reconstructed after the second masking. Also, compared with single masking in Fig 6 (top), the locality of the attention head is enhanced, potentially benefiting some downstream tasks that require low-level or local representations. Potential Application. Considering the derived locality on two-shot masking models, it would enable models to learn local fine-grained features. To verify it, we use $L(0, 10; 0.75, 0.1)$ and $L(0; 0.75)$ conduct fine-grained classification on three widely-used fine-grained datasets including Flower102 (Nilsback & Zisserman, 2008), Stanford Dog (Khosla et al., 2011), and CUB-200 (Wah et al., 2011), and compare the results with that of ImageNet100 (generic classification) in Tab 2. We find $L(0, 10; 0.75, 0.1)$ obtains more enhancement than $L(0; 0.75)$ in fine-grained classification. Additionally, a subtle and interesting phenomenon is captured during our experiments. We take $L(0, 10; 0.75, 0.15)$ and $L(0, 10; 0.9, 0.1)$ for example and in Fig 7, the second reconstruction loss (orange) of masked patches (2nd shot) unanimously decreases faster than that of the first (blue) (1st shot). This result indicates the second reconstruction task is relatively easier to optimize than the first. To some extent, using the same loss weights for them is unreasonable and wastes model’s capability. Hence, intuitively, we adopt their mask ratios as their new loss weights during training to force the model to concentrate more on the first reconstruction task. In Tab 3, we find that this adjustment significantly improves the performance of linear probing but has limited enhancement on fine-tuning. Since our focus is primarily on the performance of finetuning, we did not adopt this strategy in our experiments and leave it as a potential avenue for future exploration. Finally, we apply our findings in ViT-S/16 on ViT-B/16, hoping to further improve its performance as well. Since the performance of $\eta_1 = 0.9$ for ViT-B/16 in Tab 1 is inferior to that of $\eta_1 = 0.75$, we focus primarily on $\eta_1 = 0.75$ for ViT-B/16 in the experiment. Specifically, we employ the three best two-shot settings of finetuning performance of ViT-S/16 on ViT-B shown in Tab 4 and compare the results with MAE. Our two-shot masking strategy unanimously outperforms MAE. And among them, $L(0, 10; 0.75, 0.1)$ performs best, which also performs best for ViT-S/16. Conclusion. For two-shot masking, we summarize four useful findings: ① building on one-shot, increasing the interval of two-shot masking with a large $\eta_1$ and a small $\eta_2$ is helpful for fine-tuning in both ViT-S/16 and ViT-B/16, e.g., $L(0, 10)$ in our experiments; ② it strongly suggests that there may not exist a positive relationship between linear probing and fine-tuning; ③ the second masking brings locality bias into model and help capture low-level features, especially for finer-grind classification; ④ adopting a weighted reconstruction loss for different shot masking is helpful for linear probing. \| Dataset \| $L(0; 0.75)$ \| $L(0, 10; 0.75, 0.1)$ \| \|---------------\|-------------\|-----------------------\| \| ImageNet100 \| 82.5 \| 84.6 (+2.1) \| \| Flower102 \| 34.7 \| 37.3 (+2.6) \| \| Standford Dog \| 51.6 \| 54.3 (+2.7) \| \| CUB-200 \| 48.2 \| 51.1 (+2.9) \| \| $\eta_1$, $\eta_2$ \| $w_1$, $w_2$ \| LP \| FT \| \|---------------------\|--------------\|----\|----\| \| 0.75, 0.15 \| 0.5, 0.5 \| 31.0 \| 83.9 \| \| 0.9, 0.1 \| 0.5, 0.5 \| 35.2 \| 83.9 \| \| 0.75, 0.15 \| 0.9, 0.1 \| 35.5 \| 81.9 \| \| 0.9, 0.1 \| 0.9, 0.1 \| 36.3 \| 82.0 \| \| $i$, $j$ \| $\eta_1$, $\eta_2$ \| FT \| \|----------\|---------------------\|----\| \| L(0) \| 0.75 \| 86.88 \| \| L(0,10) \| 0.75, 0.1 \| 87.66 \| \| L(0,10) \| 0.75, 0.15 \| 87.46 \| \| L(0,11) \| 0.75, 0.1 \| 87.26 \| Table 5: The best results of our step-by-step shots masking. \| Different shots masking \| $i$, $j$, $k$ \| $\eta_1$, $\eta_2$, $\eta_3$ \| FT \| \|-------------------------\|--------------\|-----------------------------\|----\| \| One-shot \| 0, -, - \| 0.75, -, - \| 82.5 \| \| Two-shot \| 0, 10, - \| 0.75, 0.1, - \| 84.6 \| \| Three-shot \| 0, 10, 11 \| 0.75, 0.1, 0.1 \| 81.9 \| Table 6: Downstream performance of Conditional-MAE compared to MAE. CF means CIFAR (Krizhevsky et al., 2009). Tiny indicates TinyImageNet (Le & Yang, 2015). \| Model \| Classification \| Obj Det \| Sem Seg (mIoU) \| \|----------------\|----------------\|---------\|----------------\| \| \| DTD \| CF10 \| CF100 \| Tiny \| APb \| APm \| \| MAE \| 57.9 \| 84.5 \| 62.5 \| 63.4 \| 38.9 \| 35.1 \| 38.3 \| \| Conditional-MAE\| 59.1 \| 85.5 \| 63.4 \| 64.1 \| 39.5 \| 35.5 \| 38.9 \| ### 3.1.3 Three-shot Masking We further explore the three-shot masking. Specifically, we leverage a greedy-algorithm-like strategy by using the best two-shot setting $L(0, 10; 0.75, 0.1)$ and add the third masking on the last layer of encoder ($k = 11$) with a small masking ratio $\eta_3 = 0.1$. We verify the effectiveness of our three-shot masking by comparing it with various strategies including “Equal interval”, “Prefer front layer”, and “Unbalanced interval”. Additionally, by visualizing the attention distance and entropy and comparing with that of two-shot and one-shot masking, we find the third masking introduces a more prominent locality bias as shown in Fig 20. Similarly, we conduct fine-grained classification in Tab 9 and find that though the model outperforms the baseline but the enhancement is inferior to that of two-shot. Intuitively, we speculate that this would be due to the over-locality introduced by the third shot masking. Due to the limited space, we put all the results in Appendix A.4. Conclusion. In three-shot masking, we find that a greedy-like masking selection strategy is superior over a wide range of strategies. And more prominent locality is brought into models. ### 3.2 Transfer Learning To conduct transfer learning in downstream tasks, we compare the best results of one-shot, two-shot, and three-shot in Tab 5. It is shown that two-shot performs the best. Hence, we pick up the best two-shot masking pretrained ViT-B/16 model (Conditional-MAE). To verify its effectiveness in transfer learning, we perform classification on four datasets, object detection on COCO (Lin et al., 2014), and semantic segmentation on ADE20K (Zhou et al., 2017) following previous works (He et al., 2021; Chen et al., 2022a; Zhou et al., 2021). As shown in Tab 6, Conditional-MAE generally produces better performance than MAE in downstream tasks, showing its great transfer capability. ### 3.3 Robustness Analysis Considering Conditional-MAE suffers extra masking, it should be intuitively more robust than MAE. To verify it, we use a fine-tuned model to conduct two kinds of perturbation schemes, i.e., occlusion and shuffling, aiming to simulate the real circumstances. For occlusion, we randomly mask half of the patches following (Zhou et al., 2021) before inputting the model. For shuffling, we randomly shuffle the patches as well. As presented in Tab 7, compared to Tab 6, Conditional-MAE suffers less performance drop than MAE, indicating more excellent robustness. Table 7: Robustness analysis (occlusion and shuffling) of Conditional-MAE and MAE with four classification datasets. \| Model \| occlusion \| shuffling \| \|----------------\|-----------\|-----------\| \| \| DTD \| CF10 \| CF100 \| Tiny \| DTD \| CF10 \| CF100 \| Tiny \| \| MAE \| 56.3 \| 71.6 \| 48.4 \| 49.9 \| 47.7 \| 68.8 \| 45.6 \| 42.9 \| \| Conditional-MAE\| 57.8 \| 72.8 \| 49.5 \| 51.2 \| 49.1 \| 70.2 \| 47.1 \| 44.0 \| ### 3.4 Scalability To verify the scaling capability, we pre-train Conditional-MAE on ImageNet1K (Russakovsky et al., 2015), scaling on large model, i.e., ViT-L, and longer pretraining times, e.g., 1600 epoch. The result is presented in Fig 8 where the left is training with 300 epochs for both models and the right uses ViT-B/16. It is shown that Conditional-MAE meets the scaling law: pretraining with a longer time and increasing model size can significantly improve performance. Also, Conditional-MAE generally outperforms MAE, verifying its superiority. 4 RELATED WORK Masked image modeling. Masked image modeling is the task of predicting the masked part of an image from the visible part. Inspired by masked language modeling in natural language processing, BEiT (Bao et al., 2021) is the first to employ this paradigm in computer vision. PeCo (Dong et al., 2021) further improves the performance of BEiT by involving more semantics in visual tokens. MAE (He et al., 2021) removes the need for a tokenizer (e.g., d-vae (Ramesh et al., 2021) in BEiT) by directly predicting the masked part in RGB space. This greatly simplifies the whole pipeline and improves the model performance simultaneously. CAE (Chen et al., 2022a) adds a regressor between the encoder and decoder to align masked and visible representations in the same representation space. iBOT (Zhou et al., 2021) combines masked image modeling with contrastive learning, showing great potential. Recently, with more effort devoted to this field, numerous works (Dong et al., 2022a; Gao et al., 2022; Zhang et al., 2022b; Chen et al., 2022b; Kakogeorgiou et al., 2022; Li et al., 2021; El-Nouby et al., 2021; Liu et al., 2022; Tao et al., 2022; Wei et al., 2022a; Zhang et al., 2022a; Yu et al., 2022; Assran et al., 2022; Fang et al., 2022; Bachmann et al., 2022; Shi et al., 2022; Wei et al., 2022b; Huang et al., 2022a;b; Dong et al., 2022b) are proposed including BootMAE (Dong et al., 2022a), MCMAE (Gao et al., 2022), CAE v2 (Zhang et al., 2022b), SdAE (Chen et al., 2022b), MST (Li et al., 2021), SplitMask (El-Nouby et al., 2021), dBOT (Liu et al., 2022), SIM (Tao et al., 2022), etc. Understanding masked image modeling. Xie et al. shows that masked image modeling brings rich diversity to the self-attention head and pays more attention to locality compared to supervised one (Xie et al., 2021b). Additionally, Xie et al. also demonstrates that larger models, more data, and longer training times are beneficial for masked image modeling (Xie et al., 2021a). CAE (Chen et al., 2022a) illustrates its attention map and speculates that masked image modeling cares more about the global including both foreground and background. Kong & Zhang (Kong & Zhang, 2022) point out that masked image modeling brings occlusion invariant to the model representation. Cao et al. (Cao et al., 2022) deliver a mathematical understanding of masked image modeling. More recently, Zhu et al. (Zhu et al., 2023) speculate that masked image modeling is a part-to-part process: the masked representations are hallucinated from the visible part of an image, thereby leading to self-supervised models with strong part-aware capability. In this work, we attempt to reveal the impact of multiple shots masking on masked autoencoder. Masking in generation modeling. Chang et al. introduce MaskGIT (Chang et al., 2022), which employs a bidirectional transformer decoder and is capable of learning to predict randomly masked tokens via attending to tokens in all directions during training. When inference, MaskGIT first generates all tokens of an image and then refines the generated image iteratively based on the previous generation. Recently, Chang et al. propose Muse (Chang et al., 2023) and train it to predict randomly masked image tokens given the text embedding extracted from a pre-trained large language model (LLM). Leveraging LLM enables Muse to understand fine-grained language, translate to high-fidelity image generation, etc. Moreover, Muse directly enables inpainting, outpainting, and mask-free editing without the need to fine-tune or invert the model. Li et al. (Li et al., 2023) propose to use semantic tokens learned by a vector-quantized GAN at inputs and outputs and combine this with masking to unify representation learning and image generation. Bandara et al. propose an adaptive masking strategy called AdaMAE (Bandara et al., 2023). AdaMAE samples visible tokens based on the semantic context using an auxiliary sampling network and empirically demonstrates the efficacy. Xiao et al. introduce a simple yet effective adaptive masking over masking strategy called AMOM (Xiao et al., 2023) to enhance the refinement capability of the decoder and make the encoder optimization easier. Masking. Masking is a key operation in masked image modeling. Traditional masked strategies include random masking used in MAE (He et al., 2021), and block masking used in BEiT (Bao et al., 2021) and CAE (Chen et al., 2022a). Besides, previous works also explore extra masking strategies. MST (Li et al., 2021) masks low-attended patches to enhance the performance without additional cost. AttMask (Kakogeorgiou et al., 2022) further proves the usefulness of masking highly attended portions. AMT (Gui et al., 2022) uses the attention map in the last layer of the vision transformer to guide the masking. SemMAE (Li et al., 2022) leverages a masking with semantics provided by an additional pretrained model. However, it is worth noticing that almost all of them mask an image just at the beginning. (Choe & Shim, 2019) leverages self-attention mechanism to hide the most discriminative part and highlight the informative region to improve the accuracy of weakly supervised object Localization (WSOL). (Shi et al., 2022) uses an adversarial objective to consistently improve on state-of-the-art self-supervised learning (SSL) methods. MaskFeat (Wei et al., 2022a) uses Histograms of Oriented Gradients (HOG), a hand-crafted feature descriptor, as reconstruction target, and verifies its effectiveness on video recognition. These studies primarily focus on studying how to further improve the performance. VideoMAE v2 (Wang et al., 2023b) is the first to propose to mask MAE and primarily focuses on the benefits obtained in computational cost, memory consumption, etc., by introducing dual masking. But it is not the first work to dive deep into the impact of multiple shots masking in MAE. In contrast, our work reveals the secret of multiple masking on masked autoencoder with different masking positions and ratios. 5 CONCLUSION In this paper, we reveal how multiple masking affects masked autoencoder’s optimization in training and performance by using a flexible framework called Conditional MAE. Based on our findings, we summarize several takeaways from each shot and find that multiple masking can bring locality bias to models. We also show the superiority of Conditional-MAE over MAE in downstream tasks, robustness again occlusion and shuffling, and scalability. We hope our findings can inspire more future work. 6 LIMITATION AND BROADER IMPACT One limitation of our study is the limited computational resources available. We conducted our experiments using small, base, and large ViT. Therefore, it would be interesting to extend this study to larger models e.g., Huge ViT. Our empirical study primarily focuses on the masked autoencoder. There may not exist any negative effects on itself but on how it is used. REFERENCES Mahmoud Assran, Mathilde Caron, Ishan Misra, Piotr Bojanowski, Florian Bordes, Pascal Vincent, Armand Joulin, Mike Rabbat, and Nicolas Ballas. Masked siamese networks for label-efficient learning. In Computer Vision–ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part XXXI, pp. 456–473. Springer, 2022. Roman Bachmann, David Mizrahi, Andrei Atanov, and Amir Zamir. Multimae: Multi-modal multi-task masked autoencoders. In Computer Vision–ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part XXXVII, pp. 348–367. Springer, 2022. Alexei Baevski, Arun Babu, Wei-Ning Hsu, and Michael Auli. Efficient self-supervised learning with contextualized target representations for vision, speech and language. arXiv preprint arXiv:2212.07525, 2022a. Alexei Baevski, Wei-Ning Hsu, Qiantong Xu, Arun Babu, Jiatao Gu, and Michael Auli. Data2vec: A general framework for self-supervised learning in speech, vision and language. In International Conference on Machine Learning, pp. 1298–1313. PMLR, 2022b.
4Ua4hKiAJX	For the necessity of sequential rewiring, authors claimed that instantaneous rewiring easily violates either locality or sparsity constraint. In Figure 3, authors conducted an ablation study on the number of snapshots. Is there any comparison with an instantaneous rewiring, i.e., snapshot being 1?	LOCALITY-AWARE GRAPH REWIRING IN GNNs Federico Barbero\textsuperscript{1,}, Ameya Velingker\textsuperscript{2}, Amin Saberi\textsuperscript{3}, Michael Bronstein\textsuperscript{1}, Francesco Di Giovanni\textsuperscript{1} \textsuperscript{1}University of Oxford, Department of Computer Science \textsuperscript{2}Google Research \textsuperscript{3}Stanford University, Department of Management Science and Engineering ABSTRACT Graph Neural Networks (GNNs) are popular models for machine learning on graphs that typically follow the message-passing paradigm, whereby the feature of a node is updated recursively upon aggregating information over its neighbors. While exchanging messages over the input graph endows GNNs with a strong inductive bias, it can also make GNNs susceptible to \textit{over-squashing}, thereby preventing them from capturing long-range interactions in the given graph. To rectify this issue, \textit{graph rewiring} techniques have been proposed as a means of improving information flow by altering the graph connectivity. In this work, we identify three desiderata for graph-rewiring: (i) reduce over-squashing, (ii) respect the locality of the graph, and (iii) preserve the sparsity of the graph. We highlight fundamental trade-offs that occur between \textit{spatial} and \textit{spectral} rewiring techniques; while the former often satisfy (i) and (ii) but not (iii), the latter generally satisfy (i) and (iii) at the expense of (ii). We propose a novel rewiring framework that satisfies all of (i)–(iii) through a locality-aware sequence of rewiring operations. We then discuss a specific instance of such rewiring framework and validate its effectiveness on several real-world benchmarks, showing that it either matches or significantly outperforms existing rewiring approaches. 1 INTRODUCTION Graph Neural Networks (GNNs) (Sperduti, 1993; Goller & Kuchler, 1996; Gori et al., 2005; Scarselli et al., 2008; Bruna et al., 2014; Defferrard et al., 2016) are widely popular types of neural networks operating over graphs. The majority of GNN architectures act by locally propagating information across adjacent nodes of the graph and are referred to as Message Passing Neural Networks (MPNNs) (Gilmer et al., 2017). Since MPNNs aggregate messages over the neighbors of each node recursively at each layer, a sufficient number of layers is required for distant nodes to interact through message passing (Barceló et al., 2019). In general, this could lead to an explosion of information that needs to be summarized into fixed-size vectors, when the receptive field of a node grows too quickly due to the underlying graph topology. This phenomenon is known as \textit{over-squashing} (Alon & Yahav, 2021), and it has been proved to be heavily related to topological properties of the input graph such as curvature (Topping et al., 2022) and effective resistance (Black et al., 2023; Di Giovanni et al., 2023). Since over-squashing is a limitation of the message-passing paradigm that originates in the topology of the input-graph, a solution to these problems is \textit{graph rewiring} (Topping et al., 2022), in which one alters the connectivity of the graph to favor the propagation of information among poorly connected nodes. \textit{Spatial rewiring} techniques often connect each node to any other node in its $k$-hop (Brüel-Gabrielsson et al., 2022; Abboud et al., 2022), or in the extreme case operate over a fully-connected graph weighted by attention – such as for Graph-Transformers (Kreuzer et al., 2021; Mialon et al., 2021; Ying et al., 2021; Rampasek et al., 2022). \textit{Spectral rewiring} techniques instead aim to improve the connectivity of the graph by optimizing for graph-theoretic quantities related to its expansion properties such as the spectral gap, commute time, or effective resistance (Arnaiz-Rodríguez et al., 2022; Karhadkar et al., 2022; Black et al., 2023). While graph rewiring is a promising direction, it also introduces a fundamental trade-off between the preservation of the original topology and the ‘friendliness’ of the graph to message passing. Spatial rewiring techniques partly preserve the graph-distance information (i.e. its ‘locality’) by Correspondence to federico.barbero@cs.ox.ac.uk. Figure 1: Difference between spectral (left), spatial (middle), and LASER (right) rewirings in green with respect to the blue node of reference. Spectral rewirings are sparse and connect distant nodes. Spatial rewirings are able to retain local inductive biases at the cost of sparsity. LASER remains both local and sparse by optimizing over the edges to be added. only adding edges within a certain radius or by relying on positional information. However, these methods often result in a dense computational graph that increases memory complexity and can cause issues such as over-smoothing (Ni & Maehara, 2019; Oono & Suzuki, 2020; Rusch & Mishra, 2020; Di Giovanni et al., 2022). Conversely, spectral rewiring approaches add fewer edges according to some optimization criterion and hence better preserve the sparsity of the input graph. However, these methods ‘maximally’ destroy the locality induced by the graph since they typically insert very ‘long’ edges among distant nodes (see Figure 1). The following natural question then arises: Can we design a general graph rewiring framework that leverages the inductive bias of spatial methods but in a more edge-efficient way characteristic of spectral methods? Contributions and outline. In this work, we address the above question by proposing a general framework for graph-rewiring that improves the connectivity, while preserving locality and sparsity: • In Section 3 we review existing rewiring approaches and classify them as either spatial or spectral, highlighting their limitations. We then provide a general list of desiderata for rewiring that amounts to (i) reducing over-squashing, and preserving both (ii) the graph-locality and (iii) its sparsity. • In Section 4 we introduce a paradigm for rewiring that depends on arbitrary connectivity and locality measures. We argue that in order to satisfy (i)–(iii) above, a single rewiring is not enough, and instead propose sequential rewiring, where multiple graph snapshots are considered. Building on Karhadkar et al. (2022), we also draw an important equivalence between graph-rewiring on one side, and multi-relational GNNs and temporal-GNNs on the other. • In Section 5 we present a specific instance of the aforementioned paradigm termed Locality-Aware SEquential Rewiring (LASER). Our framework leverages the distance similarly to spatial rewiring while also guaranteeing the efficiency of spectral techniques by sampling edges to add according to equivariant, optimal conditions. We show that LASER reduces over-squashing and better preserves the locality of the graph compared to spectral rewiring techniques. • In Section 6 we validate LASER on different tasks, attaining performance that is on par or superior to existing rewiring techniques. In particular, we present extensive ablation studies to support our claim that LASER is more efficient than spatial methods while being better at preserving graph-distance information in comparison to spectral approaches. 2 BACKGROUND Preliminaries on graphs. Let $G = (V, E)$ be an undirected graph with $n$ nodes $V$ and edges $E$, which are encoded by the non-zero entries of the adjacency matrix $A \in \mathbb{R}^{n \times n}$. Let $D$ be the diagonal degree matrix such that $D_{uv} = d_u$. We recall that the normalized graph Laplacian $\Delta = D^{-1/2}(D - A)D^{-1/2}$ is a symmetric positive semi-definite operator with eigenvalues $0 = \lambda_0 \leq \lambda_1 \leq \cdots \leq \lambda_{n-1}$. We assume that $G$ is connected, so that $\lambda_1 > 0$ and refer to it as the spectral gap. From the Cheeger inequality, it follows that a larger $\lambda_1$ generally means better connectivity of $G$. We denote by $d_G(u, v)$ the shortest-path distance between the nodes $u, v$. We finally recall that a random walk on $G$ is a Markov chain on $V$ with transition matrix $D^{-1}A$ and that the commute time $CT$ is defined as the expected number of steps required for a random walk to commute between two nodes. Note that the commute time $ \text{CT}(v, u) $ between two nodes $ v $ and $ u $ is proportional to their effective resistance $ R(v, u) $ (Chandra et al., 1996) as $ \text{CT}(v, u) = 2\|E\|R(v, u) $. The message-passing paradigm. We consider the case where each node $ v $ has a feature $ x_v^{(0)} \in \mathbb{R}^d $. It is common to stack the node features into a matrix $ X^{(0)} \in \mathbb{R}^{n \times d} $ consistently with the ordering of $ A $. GNNs are functions defined on the featured graph that can output node, edge, or graph-level values. The most common family of GNN architectures are Message Passing Neural Networks (MPNN), which compute latent node representations by stacking $ T $ layers of the form: \[ x_v^{(t)} = \text{up}^{(t)}(x_v^{(t-1)}, a^{(t)}(\{x_u^{(t-1)} : (v, u) \in E\})), \] for $ t = 1, \ldots, T $, where $ a^{(t)} $ is some permutation-invariant aggregation function, while $ \text{up}^{(t)} $ updates the node’s current state with aggregated messages from its neighbors. Over-squashing and long-range interactions. While the message-passing paradigm usually constitutes a strong inductive bias, it is problematic for capturing long-range interactions due to a phenomenon known as over-squashing. Given two nodes $ u, v $ at distance $ d_G(u, v) = r $, an MPNN will require $ T \geq r $ layers to exchange messages between them. When the receptive fields of the nodes expand too quickly (due to volume growth properties characteristic of many real-world scale free graphs), the MPNN needs to aggregate a large number of messages into fixed-size vectors, leading to some corruption of the information (Alon & Yahav, 2021). This effect on the propagation of information has been related to the Jacobian of node features decaying exponentially with $ r $ (Topping et al., 2022). More recently, it was shown that the Jacobian is affected by topological properties such as effective resistance (Black et al., 2023; Di Giovanni et al., 2023). 3 EXISTING GRAPH-REWIRING APPROACHES AND THEIR LIMITATIONS The main principle behind graph rewiring in GNNs is to decouple the input graph $ G $ from the computational one. Namely, rewiring consists of applying an operation $ R $ to $ G = (V, E) $, thereby producing a new graph $ R(G) = (V, R(E)) $ on the same vertices but with altered connectivity. We begin by generalizing the MPNN formalism to account for the rewiring operation $ R $ as follows: \[ x_v^{(t)} = \text{up}^{(t)}(x_v^{(t-1)}, a^{(t)}_G(\{x_u^{(t-1)} : (v, u) \in E\}), a^{(t)}_{R(G)}(\{x_u^{(t-1)} : (v, u) \in R(E)\})), \] where a node feature is now updated based on information collected over the input graph $ G $ and the rewired one $ R(G) $, through (potentially) independent aggregation maps. Many rewiring-based GNN models simply exchange messages over $ R(G) $, i.e., they take $ a_G = 0 $. The idea of rewiring the graph is implicit to many GNNs, from using Cayley graphs (Deac et al., 2022), to virtual nodes (Cai et al., 2023) and cellular complexes (Bodnar et al., 2021). Other works have studied the implications of directly changing the connectivity of the graph to de-noise it (Klicpera et al., 2019), or to explore multi-hop aggregations (Abu-El-Haija et al., 2019; Ma et al., 2020; Wang et al., 2020; Nikolentzos et al., 2020). Ever since over-squashing was identified as an issue in MPNNs (Alon & Yahav, 2021), several novel rewiring approaches have been proposed to mitigate this phenomenon. Related work on spatial rewiring. Most spatial rewiring models attempt to alleviate over-squashing by adding direct connections between a node and every other node within a certain distance (Brüel-Gabrielsson et al., 2022; Abboud et al., 2022) — with (dense) Graph Transformers being the extreme case (Ying et al., 2021; Mialon et al., 2021; Kreuzer et al., 2021; Rampasek et al., 2022). These frameworks follow equation 2, where $ a_G $ and $ a_{R(G)} $ are learned independently, or the former is zero while the second implements attention over a dense graph. Spatial rewiring reduces over-squashing by creating new paths in the graph, thus decreasing its diameter or pairwise effective resistances between nodes. The rewired graph still preserves some information afforded by the original topology in the form of distance-aware aggregations in multi-hop GNNs, or positional encoding in Graph-Transformers. A drawback of this approach, however, is that we end up compromising the sparsity of the graph, thereby impacting efficiency. Thus, a natural question is whether some of these new connections introduced by spatial rewiring methods may be removed without affecting the improved connectivity. We also mention spatial rewiring methods based on improving the curvature of $ G $ by only adding edges among nodes at distance at most two (Topping et al., 2022; Nguyen et al., 2022). Accordingly, these models may fail to significantly improve the effective resistance of the graph unless a large number of local edges is added. Related work on spectral rewiring methods. A different class of approaches consist of rewiring the graph based on a global spectral quantity rather than using spatial distance. Two prototypical measures that have been explored in this regard are spectral gap (Karhadkar et al., 2022) and effective resistance (Arnaiz-Rodríguez et al., 2022; Banerjee et al., 2022; Black et al., 2023). It has recently been shown that a node $ v $ is mostly insensitive to information contained at nodes that have high effective resistance (Black et al., 2023; Di Giovanni et al., 2023); accordingly, spectral rewiring approaches alleviate over-squashing by reducing the effective resistance. Moreover, they achieve that adding only a few edges by optimally increasing the chosen measure of connectivity, hence maintaining the sparsity level of the input graph. However, the edges that are added in the graph typically end up connecting very distant nodes (since the distance between two nodes is at least as large as their effective resistance), hence rapidly diminishing the role of locality provided by distance on the original graph. An ideal rewiring approach. Given a graph $ G $, an ideal rewiring map $ R $ should satisfy the following desiderata: (i) Reduce over-squashing: $ R $ increases the overall connectivity of $ G $—according to some topological measure—in order to alleviate over-squashing; (ii) Preserve locality: $ R $ preserves some inductive bias afforded by $ G $, e.g., nodes that are “distant” should be kept separate from nodes that are closer in the GNN architecture; (iii) Preserve sparsity: $ R $ approximately preserves the sparsity of $ G $, ideally adding a number of edges linear in the number of nodes. While condition (i) represents the main rationale for rewiring the input graph, criteria (ii) and (iii) guarantee that the rewiring is efficient and do not allow the role played by the structural information in the input graph to degrade too much. As discussed above and summarized in Table 1, spatial methods typically satisfy only (i) and (ii), but not (iii), while spectral-methods meet (i) and (iii) but fail (ii). Main idea. Our main contribution is a novel paradigm for graph rewiring that satisfies criteria (i)–(iii), leveraging a key principle: instead of considering a single rewired graph $ R(G) $, we use a sequence of rewired graphs $ \{R_\ell(G)\}_\ell $ such that for smaller $ \ell $, the new edges added in $ R_\ell(G) $ are more ‘local’ (with respect to the input graph $ G $) and sampled based on optimizing a connectivity measure. ### 4 A GENERAL PARADIGM: DYNAMIC REWIRING WITH LOCAL CONSTRAINTS In this Section, we discuss a general graph-rewiring paradigm that can enhance any MPNN and satisfies the criteria (i)–(iii) described above. Given a graph $ G $, consider a trajectory of rewiring operations $ R_\ell $, starting at $ G_0 = G $, of the form: \[ G = G_0 \xrightarrow{R_1} G_1 \xrightarrow{R_2} \cdots \xrightarrow{R_L} G_L. \] Since we think of $ G_\ell $ as the input graph evolved along a dynamical process for $ \ell $ iterations, we refer to $ G_\ell $ as the $ \ell $-snapshot. For the sake of simplicity, we assume $ R_\ell = R $, though it is straightforward to extend the discussion below to the more general case. In order to account for the multiple snapshots, we modify the layer form in equation 2 as follows: \[ x_v^{(t)} = \text{up}(t)\left(x_v^{(t-1)}, \left(a_{\mu_\ell}\left(\{x_u^{(t-1)} : (v, u) \in E_\ell\}\right)\right)_{0 \leq \ell \leq L}\right). \] Below we describe a rewiring paradigm based on an arbitrary connectivity measure $ \mu : V \times V \to \mathbb{R} $ and locality measure $ \nu : V \times V \to \mathbb{R} $. The measure $ \mu $ can be any topological quantity that captures how easily different pairs of nodes can communicate in a graph, while the measure $ \nu $ is any quantity that penalizes interactions among nodes that are ‘distant’ according to some metric on the input graph. In a nutshell, our choice of $ R $ samples edges to add according to the constraint $ \nu $, prioritizing those that maximally benefit the measure $ \mu $. By keeping this generality, we provide a universal approach to do graph-rewiring that can be of interest independently of the specific choices of $ \mu $ and $ \nu $. \| Property \| Spatial \| Spectral \| LASER \| \|---------------------------\|---------\|----------\|-------\| \| Reduce over-squashing \| ✓ \| ✓ \| ✓ \| \| Preserve locality \| ✓ \| ✗ \| ✓ \| \| Preserve sparsity \| ✗ \| ✓ \| ✓ \| Improving connectivity while preserving locality. The first property we demand of the rewiring sequence is that for all nodes $ v, u $, we have $ \mu_{G_{\ell+1}}(v,u) \geq \mu_{G_\ell}(v,u) $ and that for some nodes, the inequality is strict. If we connect all pairs of nodes with low $ \mu $-value, however, we might end up adding non-local edges across distant nodes, hence quickly corrupting the locality of $ G $. To avoid this, we constrain each rewiring by requiring the measure $ \nu $ to take values in a certain range $ I_\ell \subset [0, \infty) $: an edge $(v,u)$ appears in the $ \ell $-snapshot (for $ 1 \leq \ell \leq L $) according to the following rule: \[ (v,u) \in E_\ell \text{ if } (\mu_{G_0}(v,u) < \epsilon \text{ and } \nu_{G_0}(v,u) \in I_\ell) \text{ or } (v,u) \in E_{\ell-1}. \] To make the rewiring more efficient, the connectivity and locality measures are computed once over the input graph $ G_0 $. Since the edges to be added connect nodes with low $ \mu $-values, the rewiring makes the graphs $ G_\ell $ friendlier to message-passing as $ \ell $ grows. Moreover, by taking increasing ranges of values for the intervals $ I_\ell $, we make sure that new edges connect distant nodes, as specified by $ \nu $, only at later snapshots. Sequential rewiring allows us to interpolate between the given graph and one with better connectivity, creating intermediate snapshots that progressively add non-local edges. By accounting for all the snapshots $ G_\ell $ in equation 2, the GNN can access both the input graph, and more connected ones, at a much finer level than ‘instantaneous’ rewirings, defined next. Instantaneous vs sequential rewiring. As discussed in Section 3, existing rewiring techniques — particularly those of the spectral type — often consider the simpler trajectory $ G_0 \rightarrow R(G_0) := G_1 $ (“instantaneous rewiring”). The main drawback of this approach is that in order to improve the connectivity in a single snapshot, the rewiring map $ R $ is bound to either violate the locality constraint $ \nu $, by adding edges between very distant nodes, or compromise the graph-sparsity by adding a large volume of (local) edges. In fact, if that were not the case, we would still be severely affected by over-squashing. Conversely, sequential rewiring allows a smoother evolution from the input graph $ G_0 $ to a configuration $ G_L $ which is more robust to over-squashing, so that we can more easily preserve the inductive bias afforded by the topology via local constraints under equation 5. An equivalent perspective: multi-relational GNNs. In Karhadkar et al. (2022) the notion of relational rewiring was introduced for spectral methods. We expand upon this idea, by noticing that the general, sequential rewiring paradigm described above can be instantiated as a family of multi-relational GNNs (Battaglia et al., 2018; Barcelo et al., 2022). To this aim, consider a slightly more specific instance of equation 4, which extends common MPNN frameworks: \[ x_v^{(t)} = \text{up}^{(t)} \left( x_v^{(t-1)}, \sum_{\ell=0}^{L} \sum_{(v,u) \in E_\ell} \psi_\ell^{(t)}(x_v^{(t-1)}, x_u^{(t-1)}) \right), \] where $ \psi_\ell^{(t)} $ are learnable message functions depending on both the layer $ t $ and the snapshot $ \ell $. It suffices now to note that each edge set $ E_\ell $, originated from the rewiring sequence, can be given its own relation, so that equation 6 is indeed equivalent to the multi-relation GNN framework of Battaglia et al. (2018). In fact, since we consider rewiring operations that only add edges to improve the connectivity, we can rearrange the terms and rename the update and message-function maps, so that we aggregate over existing edges once, and separately over the newly added edges i.e. the set $ E_\ell \setminus E_{\ell-1} $. Namely, we can rewrite equation 6 as \[ x_v^{(t)} = \text{up}^{(t)} \left( x_v^{(t-1)}, \sum_{u : (v,u) \in E} \psi_0^{(t)}(x_v^{(t-1)}, x_u^{(t-1)}) + \sum_{\ell=1}^{L} \sum_{(v,u) \in E_\ell \setminus E_{\ell-1}} \psi_\ell^{(t)}(x_v^{(t-1)}, x_u^{(t-1)}) \right). \] Accordingly, we see how our choice of sequential rewiring can be interpreted as an extension of relational rewiring in Karhadkar et al. (2022), where $ L = 1 $. Differently from Karhadkar et al. (2022), the multiple relations $ \ell \geq 1 $ allow us to add connections over the graph among increasingly less local nodes, meaning that the edge-type $ \ell $ is now associated to a notion of locality specified by the choice of the constraint $ \nu(v,u) \in I_\ell $. We finally observe that the connection between graph-rewiring and relational GNNs is not surprising once we think of the sequence of rewiring in equation 3 as snapshots of a temporal dynamics over the graph connectivity. Differently from the setting of temporal GNNs (Rossi et al., 2020) though, here the evolution of the connectivity over time is guided by our rewiring procedure rather than by an intrinsic law on the data. In fact, Gao & Ribeiro (2022) studied the equivalence between temporal GNNs and static multi-relational GNNs, which further motivate the analogy discussed above. 5 LOCALITY-AWARE SEQUENTIAL REWIRING: THE LASER FRAMEWORK We consider an instance of the outlined sequential rewiring paradigm, giving rise to the LASER framework used in our experiments. We show that LASER (i) mitigates over-squashing, (ii) preserves the inductive bias provided by the shortest-walk distance on $G$ better than spectral approaches, while (iii) being sparser than spatial-rewiring methods. The choice of locality. We choose $\nu$ to be the shortest-walk distance $d_G$. In particular, if in equation 5 we choose intervals $I_\ell = \delta_{\ell+1}$, then at the $\ell$-snapshot $G_\ell$ we only add edges among nodes at distance exactly $\ell + 1$. Our constraints prevent distant nodes from interacting at earlier snapshots and allows the GNN to learn message functions $\psi_\ell$ in equation 7 for each hop level $\ell$. If we choose $E_\ell \setminus E_{\ell-1}$ to be the set of all edges connecting nodes whose distance is exactly $\ell + 1$, then equation 7 is equivalent to the $L$-hop MPNN class studied in Feng et al. (2022). This way though, we generally lose the sparsity of $G$ and increase the risk of over-smoothing. Accordingly, we propose to only add edges that satisfy the locality constraint and have connectivity measure ‘small’ so that their addition is optimal for reducing over-squashing. The choice of the connectivity measure $\mu$. Although edge curvature or effective resistance $R$ are related to over-squashing (Topping et al., 2022; Black et al., 2023; Di Giovanni et al., 2023), computing these metrics incur high complexity – $O(\|E\|d_{max}^2)$ for the curvature and $O(n^3)$ for $R$. Because of that, we propose a more efficient connectivity measure: $$\mu_k(v,u) := (\tilde{A}^k)_{vu}, \quad \tilde{A} := A + I.$$ Because of the self-loops, the entry $(\tilde{A}^k)_{vu}$ equals the number of walks from $v$ to $u$ of length at most $k$. Once we fix a value $k$, if $\mu_k(v,u)$ is large, then the two nodes $v,u$ have multiple alternative routes to exchange information (up to scale $k$) and would usually have small effective resistance. In particular, according to Di Giovanni et al. (2023, Theorem 4.1), we know that the number of walks among two nodes is a proxy for how sensitive a pair of nodes is to over-squashing. LASER focus. We can now describe our framework. Given a node $v$ and a snapshot $G_\ell$, we consider the set of nodes at distance exactly $\ell + 1$ from $v$, which we denote by $N_{\ell+1}(v)$. We introduce a global parameter $\rho \in (0, 1]$ and add edges (with relation type $\ell$ as per equation 7) among $v$ and the fraction $\rho$ of nodes in $N_{\ell+1}(v)$ with the lowest connectivity score – if this fraction is smaller than one, then we round it to one. This way, we end up adding only a percentage $\rho$ of the edges that a normal multi-hop GNNs would have, but we do so by prioritizing those edges that improve the connectivity measure the most. To simplify the notations, we let $N_{\ell+1}^\rho(v) \subset N_{\ell+1}(v)$, be the $\rho$-fraction of nodes at distance $\ell + 1$ from $v$, where $\mu_k$ in equation 8 takes on the lowest values. We express the layer-update of LASER as $$x_v^{(t)} = \text{up}^{(t)} \left( x_v^{(t-1)}, \sum_{u: (v,u) \in E} \psi_0(x_v^{(t-1)}, x_u^{(t-1)}) + \sum_{\ell=1}^L \sum_{u \in N_{\ell+1}^\rho(v)} \psi_\ell(x_v^{(t-1)}, x_u^{(t-1)}) \right).$$ We note that when $\rho = 0$, equation (9) reduces to a standard MPNN on the input graph, while for $\rho = 1$ we recover multi-relational $L$-hop MPNNs (Feng et al., 2022). Although the framework encompasses different choices of the message-functions $\psi_\ell$, in the following we focus on the LASER-GCN variant, whose update equation is reported in Appendix (Section A). We now show that the LASER framework satisfies the criteria (i)–(iii) introduced in Section 3. Let $J^{(r)}(v,u) := \partial x_v^{(r)} / \partial x_u^{(0)}$ be the Jacobian of features after $r$ layers of GCN on $G$, and similarly we let $\hat{J}^{(r)}(v,u)$ be the Jacobian of features after $r$ layers of LASER-GCN in equation 10. In the following, we take the expectation with respect to the Bernoulli variable ReLU' which is assumed to have probability of success $\rho$ for all paths in the computational graph as in Xu et al. (2018); Di Giovanni et al. (2023). We recall that given $i \in V$ and $1 \leq \ell \leq L$, $d_{i,\ell}$ enters equation 10. Proposition 5.1. Let $v,u \in V$ with $d_G(v,u) = r$, and assume that there exists a single path of length $r$ connecting $v$ and $u$. Assume that LASER adds an edge between $v$ and some node $j$ belonging to the path of length $r$ connecting $v$ to $u$, with $d_G(v,j) = \ell < r$. Then for all $m \leq r$, we have $$\|\|\mathbb{E}[\hat{J}^{(r-\ell+1)}(v,u)]\|\| \geq \frac{(d_{min})^\ell}{\sqrt{d_{v,\ell-1}d_{j,\ell-1}}} \|\|\mathbb{E}[J^{(m)}(v,u)]\|\|.$$ The result is not surprising and shows that in general, the LASER-rewiring can improve the Jacobian sensitivity significantly and hence alleviates over-squashing, satisfying desideratum (i). Next, we validate that the effects of the local constraints when compared to unconstrained, global spectral methods. Below, we let $ D_G $ be the matrix of pairwise distances associated with the graph $ G $, i.e. $(D_G)_{vu} = d_G(v, u)$. We propose to investigate $ \\|D_G - D_{R(G)}\\|_F $, where $ \\| \cdot \\|_F $ is the Frobenius norm and $ R(G) $ is either a baseline spectral rewiring, or our LASER-framework. We treat this quantity as a proxy for how well a rewiring framework is able to preserve the inductive bias given by the input graph. In fact, for many graphs (including molecular-type with small average degree), spectral rewirings incur a larger Frobenius deviation even if they add fewer edges, since these edges typically connect very distant nodes in the graph. To this aim, we show a setting where LASER preserves more of the locality inductive bias than spectral-based methods provided we choose the factor $ \rho $ small enough. Below, we focus on a case that, according to Di Giovanni et al. (2023); Black et al. (2023), we know to be a worst-case scenario for over-squashing considering that the commute time scales cubically in the number of nodes. Put differently, the graph below represents a prototypical case of ‘bottleneck’ encountered when information has to travel from the end of the chain to the clique. Proposition 5.2. Let $ G $ be a ‘lollipop’ graph composed of a chain of length $ L $ attached to a clique of size $ n $ sufficiently large. Consider a spectral rewiring $ R $ which adds an edge between nodes with the highest effective resistance. We can choose the factor $ \rho \in (0, 1) $ as a function of $ L $ so that LASER with a single snapshot, on average, adds a number of edges that guarantees: \[ \\|D_G - D_{R(G)}\\|_F \geq \\|D_G - D_{LASER}\\|_F. \] We refer to the Appendix (Section A) for an explicit characterization on how large $ n $ needs to be depending on $ L $ and the proofs of the statements above. Finally, as desired in (iii), we observe that compared to dense multi-hop GNNs, LASER is more efficient since it only adds a fraction $ \rho $ of edges for each node $ v $ and each orbit-level $ N_{\ell+1}(v) $. In fact, for many sparse graphs (such as molecular ones) the model ends up adding a number of edges proportional to the number of nodes (see Section C.2 in the Appendix for a discussion and ablations). ### 6 EXPERIMENTS In this section, we validate our claims on a range of tasks and benchmarks. Beyond comparing the performance of LASER to existing baselines, we run ablations to address the following important questions: (1) Does LASER improve the graph’s connectivity? (2) Does LASER preserve locality information better than spectral rewiring approaches? (3) What is the impact of the fraction $ \rho $ of edges sampled? (4) What if we sample edges to be added from $ N_{\ell+1}(v) $ randomly, rather than optimally according to $ \mu $ in equation 8? (5) Is LASER scalable to large graphs? In the Appendix (Section C), we provide a density comparison between LASER and Multi-Hop GNNs, discuss our tie-breaking procedure that guarantees equivariance in expectation and further improves performance, provide an ablation using different underlying MPNNs, and discuss additional motivation for the need for locality. We also provide, in Section D, a more thorough scalability analysis. Benchmarks. We evaluate on the Long Range Graph Benchmark (LRGB) (Dwivedi et al., 2022) and TUDatasets (Morris et al., 2020). In the experiments, we fix the underlying model to GCN, but provide ablations with different popular MPNNs in the Appendix (Section C.3). For spatial curvature-based rewirings, we compare against SDRF (Topping et al., 2022) and BORF (Nguyen et al., 2023). For spectral techniques, we compare against FOSR (Karhadkar et al., 2022), a spectral gap rewiring technique, and GTR (Black et al., 2023), an effective resistance rewiring technique. We also compare to DiffWire (Arnaiz-Rodriguez et al., 2022), a differentiable rewiring technique. \| Rewiring \| Peptides-func Test AP ↑ \| Peptides-struct Test MAE ↓ \| PCQM-Contact Test MRR ↑ \| \|----------\|-------------------------\|---------------------------\|------------------------\| \| None \| 0.5930±0.0023 \| 0.3496±0.0013 \| 0.3234±0.0006 \| \| SDRF \| 0.5947±0.0035 \| 0.3404±0.0015 \| 0.3249±0.0006 \| \| GTR \| 0.5075±0.0029 \| 0.3618±0.0010 \| 0.3007±0.0022 \| \| FOSR \| 0.5947±0.0027 \| 0.3078±0.0026 \| 0.2783±0.0008 \| \| BORF \| 0.6012±0.0031 \| 0.3374±0.0011 \| TIMEOUT \| \| LASER \| 0.6440±0.0010 \| 0.3043±0.0019 \| 0.3275±0.0011 \| Based on Karhadkar et al. (2022) and the parallelism we draw between rewiring and multi-relational GNNs, for all techniques, we report results tuned over both a ‘standard’ and relational (Schlichtkrull et al., 2018) model for the baselines, where we assign original and rewired edges distinct relational types. In particular, R-GCN in these cases is then a special instance of equation 2. For additional details on the tasks and hyper-parameters, we refer to the Appendix (Section B). LRGB. We consider the Peptides (15,535 graphs) and PCQM–Contact (529,434 graphs) datasets, from the Long Range Graph Benchmark (LRGB). There are two tasks associated with Peptides, a peptide function classification task Peptides–func and a peptide structure regression task Peptides-struct. PCQM–Contact is a link-prediction task, in which the goal is to predict pairs of distant nodes that will be adjacent in 3D space. We replicate the experimental settings from Dwivedi et al. (2022), with a 5-layer MPNN for each of the rewirings as the underlying model. We choose the hidden dimension in order to respect the 500k parameter budget. In Table 2, we report the performance on the three tasks. LASER convincingly outperforms all baselines on the three tasks, while the other rewiring baselines frequently perform worse than the standard GCN model. On PCQM–Contact, the rewiring time for BORF surpasses the 60 hour limit enforced by Dwivedi et al. (2020) on our hardware, so we assign it a TIMEOUT score. TUDatasets. We evaluate LASER on the REDDIT–BINARY, IMDB–BINARY, MUTAG, ENZYMES, PROTEINS, and COLLAB tasks from TUDatasets, which were chosen by Karhadkar et al. (2022) under the claim that they require long-range interactions. We evaluate on 25 random splits, fixing the hidden dimension for all models to 64 and the number of layers to 4, as in Karhadkar et al. (2022). We avoid the use of dropout and use Batch Norm (Ioffe & Szegedy, 2015). We refer to the Appendix (Section B.2) for further details on the hyper-parameters and a discussion on some drawbacks of these tasks. Table 3 shows the results on the aforementioned benchmarks. LASER most consistently achieves the best classification accuracy, attaining the highest mean rank. Table 3: Accuracy ± std over 25 random splits for the datasets and rewirings. Colors highlight First, Second, and Third; we report the mean rank achieved on the valid runs. OOM is Out of Memory. \| Rewiring \| REDDIT–BINARY \| IMDB–BINARY \| MUTAG \| ENZYMES \| PROTEINS \| COLLAB \| Mean Rank \| \|----------\|---------------\|-------------\|-------\|---------\|----------\|--------\|-----------\| \| None \| 81.000±2.717 \| 64.280±1.990 \| 74.737±5.955 \| 28.733±5.297 \| 64.286±2.004 \| 68.960±2.284 \| 4.83 \| \| DiffWire \| OOM \| 59.000±3.847 \| 80.421±9.707 \| 28.533±4.475 \| 72.714±2.946 \| 65.440±2.177 \| 4.83 \| \| GTR \| 85.700±2.786 \| 52.560±4.104 \| 78.632±6.201 \| 26.333±5.821 \| 72.303±4.658 \| 68.024±2.299 \| 4.67 \| \| SDRF \| 84.420±2.785 \| 58.290±3.201 \| 74.526±5.355 \| 30.567±6.188 \| 68.714±4.233 \| 70.222±2.571 \| 4.50 \| \| FOSR \| 85.930±2.793 \| 60.400±5.855 \| 75.895±7.211 \| 28.600±5.253 \| 71.643±3.428 \| 69.848±3.485 \| 3.67 \| \| BORF \| 84.920±2.534 \| 60.820±3.877 \| 81.684±7.964 \| 30.500±6.593 \| 68.411±4.122 \| OOM \| 3.60 \| \| LASER \| 85.458±2.827 \| 64.333±3.298 \| 82.204±6.728 \| 34.333±6.936 \| 74.381±3.443 \| 70.923±2.538 \| 1.37 \| Ablation studies. In the following, we choose FOSR as a typical spectral rewiring approach, while taking LASER with $ \rho = 1 $ as an instance of a dense, multi-hop GNN (i.e. classical spatial rewiring). For the purpose of these ablations, we conduct experiments on the Peptides dataset. We start by investigating questions (1) and (2), namely, how well LASER improves connectivity while respecting locality. To this end, we increment the number of snapshots from 2 to 5 given densities $ \rho = 0.1 $ and $ \rho = 1 $ for LASER and instead vary the number of edge additions of FOSR spanning the values 10, 20, 50, and 100. To assess the connectivity, we report the mean total effective resistance — which is a good proxy for over-squashing (Black et al., 2023; Di Giovanni et al., 2023) — while for the locality, we evaluate the norm of the difference between the original graph distance matrix and that of the rewired graph $ \\| D_G - D_{R(G)} \\|_F $ as per Proposition 5.2. Figure 2 shows the results of this ablation. We validate that the sparse LASER framework decreases the mean total effective resistance consistently over increasing snapshots as well as other rewiring techniques. Moreover, we find that LASER with $ \rho = 0.1 $ is better than dense spatial methods and especially surpasses spectral approaches at preserving information contained in the distance matrix. Next, we investigate question (3), i.e. the impact of the fraction $ \rho $ of edges being sampled, by increasing the number of snapshots from 2 to 5 and varying the density $ \rho $ ranging 0.1, 0.25, 0.5, and 1, with results reported in Figure 3. The majority of the performance gains are obtained through a sparse rewiring, as even with $ \rho = 0.1 $ Table 4: Comparison between LASER and random sampling, with $ L = 3 $ and $ \rho = 0.1 $. \| Model \| Peptides–func ↑ \| Peptides–struct ↓ \| \|----------------\|-----------------\|------------------\| \| Random \| 0.4796±0.0067 \| 0.3382±0.0019 \| \| LASER \| 0.6414±0.0020 \| 0.3119±0.0005 \| the performance is greatly increased over the baseline. The additional density in the orbits does seem to help with performance, but this comes at the cost of density. Finally, we address question (4), by evaluating how sampling edges uniformly over the nodes at distance $ l + 1 $ given a density $ \rho $, compares to our choice of prioritizing edges with lowest connectivity score $ \mu $ as per equation 8. We report the results in Table 4. We see that LASER greatly outperforms the random rewiring, verifying our claim that guiding the rewiring through $ \mu $ is a more sound approach. Scalability. The operations required to compute $ \mu $ and $ \nu $ in LASER are designed to be efficiently implemented on modern hardware accelerators, mostly relying on matrix multiplication. Furthermore, the rewiring operation is done once and stored for future runs. The $ \rho $ factor can be tuned to calibrate the density of the rewiring, giving further control on the training efficiency. LASER does not seem to significantly impact the run-time compared to the standard baseline models and we found through a synthetic benchmarking experiment that our implementation of LASER is able to rewire graphs with 100k nodes and a million edges in 2 hours. This is in contrast to FOSR and SDRF that failed to finish the computation within 24 hours. We report a large number of benchmarking experiments, alongside a theoretical complexity analysis in the Appendix (Section D). ### 7 CONCLUSION In this work, we have identified shortcomings of rewiring techniques and argued that a rewiring must: (i) improve connectivity, (ii) respect locality, and (iii) preserve sparsity. Unlike current spectral and spatial rewirings that compromise some of these properties, we have outlined a general rewiring paradigm that meets criteria (i)–(iii) by interpolating between the input graph and a better connected one via locally constrained sequential rewiring. We have then proposed a specific instance of this paradigm — LASER — and verified, both theoretically and empirically, that it satisfies (i)-(iii). Limitations and Future Work. In this paper, we considered a simple instance of the general rewiring paradigm outlined in Section 4, but we believe that an interesting research direction would be to explore alternative choices for both the connectivity and locality measures, ideally incorporating features in a differentiable pipeline similar to Arnaiz-Rodríguez et al. (2022). Furthermore, the identification between graph-rewiring on the one hand, and multi-relational GNNs and temporal-GNNs on the other, could lead to interesting connections between the two settings, both theoretically (e.g., what is the expressive power of a certain rewiring policy?) and practically, where techniques working in one case could be effortlessly transferred to the other. Finally, we highlight that, as is customary in rewiring approaches, it is always hard to pinpoint with certainty the reason for any performance improvement, including whether such an improvement can be truly credited to over-squashing and long-range interactions. 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1GdAJ3GsOw	The cost model uses `beta` to somehow model the arithmetic intensity, and the tuning of `beta` was presented in the evaluation (Sec4.2). The authors claim that jointly considering computation+communication cost is better than considering computational cost alone. However, this is not supported in Fig.3, when computational cost alone performs very well (where `beta=max`). Does it mean that the sum of input/output tensor alone can model the performance of the DNN models? Why is the performance of DNN models consistent across a very large range of beta? Why is the unit of the y-axis in MB/sec?	DistPar: Tensor Partitioning for Distributed Neural Network Computing Anonymous authors Paper under double-blind review Abstract Existing distributed training systems suffer from the difficulties of adapting to diverse model architectures and balancing the trade-off between computational and communication costs. We introduce Distributed Partitioning (DistPar), a framework that allows users to develop parallel models with the ease of writing single-device programs. We establish the basic properties of tensor partitioning, which significantly expand the search space for optimal parallel strategies. The process of distributing global tensors from a single-device perspective is driven by the innovative use of collective communication primitives and their extensions which represent conversions between arbitrary tensor distribution properties. To further address the challenge of parallel scheme optimization, we carry out a cost function that considers both computational and communication costs. Guided by the cost function, the best-performing parallel scheme is automatically selected with configurable parameters, thus simplifying the process of developing parallel models. We demonstrate state-of-the-art results on extensive experiments. Moreover, DistPar reaches 50% higher throughput in large-scale face recognition tasks and a 20% improvement in language modeling tasks compared with data parallelism provided by PyTorch. This performance improvement aligns with the expected speedup and is particularly notable as the number of computing devices increases. The code will be released at https://github.com/DistPar. 1 Introduction In recent years, deep learning has been widely applied in many fields such as image, speech, and natural language processing (Angelova et al., 2015; Ba et al., 2015; Frome et al., 2013; Gonzalez-Dominguez et al., 2015; Hinton et al., 2012; Heigold et al., 2013; Karpathy et al., 2014; Le, 2013; Maddison et al., 2015). With the increasing demand for training efficiency and data processing capabilities of deep learning, single-device training systems, although useful in certain scenarios, may struggle to meet the requirements. Hence, the distributed training approach has become an effective way to improve computing power constantly. Distributed deep learning’s performance relies primarily on efficient collective communication to adapt to different given computational devices (Yuan et al., 2022; Lepikhin et al., 2020). Existing deep learning parallelism libraries have made great efforts on it. Typically, parallelization strategies in the context of distributed deep learning include two main aspects: data parallelism and model parallelism. Data parallelism, the former, entails the further subdivision of a mini-batch of data, subsequently distributed across computational nodes, which facilitates the training of substantial volumes of data (Baruah et al., 2022; Shallue et al., 2018; Nguyen & Wahbi, 2021; Herlihy et al., 2021; Krizhevsky, 2014). Model parallelism, the latter, is conventionally applied to partition neural networks into segments that are subsequently deployed across computational nodes (Dean et al., 2012; Narayanan et al., 2021; Huang et al., 2018; Harlap et al., 2018; Shoeybi et al., 2020; Xu et al., 2021; Wang et al., 2021; Bian et al., 2021a). Based on the parallelism strategies mentioned, we believe a comprehensive approach that aggregates them with each other, enables faster computation and efficient utilization of computational devices. Existing parallelism libraries like Pytorch, its DistributedDataParallel interface is challenging to users, because it requires users to design the communicative module of parallelism strategies manually. Hence, it’s necessary for us to design a set of parallel operation semantics from the bottom to achieve an end-to-end structure so that users can handle parallel training tasks on multiple devices with the same ease as a single device. Our unified strategy, DistPar, introduces a set of tensor partitioning attributes aimed at instructing the allocation of global logical tensors to specific physical devices—referred to as physical tensors for simplicity. DistPar merges these devices into a coherent logical supercomputer, allowing developers to handle parallel training tasks on multiple devices as simply as a single device. This enhanced accessibility for individual users, so they can focus on more top-level design. The process of distributing global tensors from a single-device perspective is driven by the innovative use of collective communication primitives and their extensions which represent conversions between arbitrary tensor distribution properties. This capability is integrated into DistPar through the inclusion of pass layers. Therefore, DistPar effectively enhances the extensibility, enabling to be adaptive to different model structure and computational device. To further address the challenge of parallel scheme optimization, DistPar assesses the cost in a comprehensive manner, which combines the conversion of parallel attributes across various parallelization strategies. At the meantime, to simplify the process of designing and selecting the best scheme, we provide a configurable parameter so that users can easily optimize computational cost and communication cost collaboratively and automatically. Evidently, the cost design helps users to adapt to different computational devices and design their own parallelism program easily. The overall contributions are as follows: • We present a novel tensor partitioning strategy, DistPar, aimed at generating a comprehensive range of parallelization strategies. • We employ meticulously designed intermediate primitives to facilitate the automatic transformation of distributed properties within the context of physical tensors. These mechanisms naturally support arbitrary parallelization combinations. • We introduce cost hyperparameter to generate different parallelization strategies, enabling the user to evolve the selection of optimal parallelization schemes. • We prove that DistPar attains state-of-the-art performance in standard benchmark assessments. 2 RELATED WORKS Numerous distributed parallelism strategies exist, with data parallelism and model parallelism being as the most widely adopted approaches. Data parallelism involves dividing a mini-batch of data into smaller segments and distributing them to different computational nodes (Baruah et al., 2022; Shallue et al., 2018; Nguyen & Wahib, 2021; Herlihy et al., 2021; Krizhevsky, 2014). In data parallelism (Krizhevsky, 2014), each device retains a complete copy of the distributed neural network (DNN) model and processes a portion of the entire training dataset. This approach enables the training of large datasets, thereby enhancing both the scale and speed of training. However, data parallelism introduces inter-device communication overhead during the synchronization process when model weights are updated. This issue can become more apparent as the model size increases, which poses some challenges to the scalability and compatibility of data parallelism. Model parallelism offers an alternative to data parallelism by directly partitioning DNN models across devices. With model parallelism (Kingma & Ba, 2017; Fang et al., 2023), weight parameters within the model are distributed among the available workers, which are typically GPUs. This approach consists of two main components: tensor parallelism and pipeline parallelism. Tensor parallelism involves splitting tensors across an array of devices, typically occurring between the forward and backward propagation phases (Shoeybi et al., 2020; Xu et al., 2021; Wang et al., 2021; Bian et al., 2021a; Wang et al., 2021b; Bian et al., 2021b; Cannon, 1969; Berntsen, 1989; van de Geijn & Watts, 1995; Solomonik & Demmel, 2011). Megatron-LM (Shoeybi et al., 2020) introduced 1D tensor parallelism, which divides the linear layer along either the column or row dimensions. When employing tensor parallelism, communication tends to be frequent, and the data volume transferred during these communications is often substantial. Pipeline parallelism divides the model on a layer basis, occurring at the junction of adjacent stages (Huang et al., 2018; Harlap et al., 2018; Li & Hoefler, 2021). Recent developments, such as GPipe (Huang et al., 2018), have introduced pipeline parallelism, which involves synchronous weight updates. In this case, communication remains frequent but typically involves smaller data volumes. Due to the inherent characteristics of pipeline parallelism, amounts of device idle time called bubbles are generated. Comparison. To reduce communication volume, tensor parallelism is preferred. Meanwhile, to improve peer-to-peer communication, pipeline parallelism is a suitable choice. However, it is equally important to note that bubbles cost a significant amount of time. To mitigate this, it is recommended to limit the number of pipeline stages to the number of micro-batches. In practice, when the level of tensor parallelism matches the number of devices, performance tends to reach its peak. Other optimized strategies, as demonstrated in previous studies (Jia et al., 2018a,b), concentrate on tensor-related refinements along multiple axes to determine the most optimal parallelization strategy. Achieving high throughput at a large scale demands innovative and intricate design across various facets. This includes the intelligent partitioning of computational graphs onto devices to minimize data transfer over the network while minimizing device idle time. It also involves the implementation of communication optimizations specific to the domain. Unified strategy. Based on the comparisons mentioned earlier, we conclude there is an imperative need for a unified strategy that amalgamates various advantages. A commonality observed in existing parallelization strategies is the shared goal of optimizing the utilization of computational resources and enhancing overall computational efficiency. However, it is crucial to acknowledge that a single parallelization strategy often struggles to meet the efficiency requirements of complex business models. These individual parallelization strategies fall short in planning and executing the global logical computational graphs effectively. Therefore, a holistic approach to the entire process is necessary. We have identified three key indicators—accessibility, compatibility, and communication cost—as crucial elements to facilitate comprehensive considerations. ### 3 METHODOLOGY This section establishes the theoretical foundation for subsequent experiments detailed in Section 4. We also introduce the proposed intermediate primitives designed to optimize model communication cost. Moreover, we illustrate complex operations using intermediate primitives. To be clear, we induce the transformations of distributed properties, offering a comprehensive perspective on distributed computation and collective communication. Finally, we employ partition analysis to quantitatively assess associated expenses in the theory. #### 3.1 DISTRIBUTED PROPERTIES Many parallelism strategies suffer from the bottleneck to be adaptive to different model structures and computational devices, so we need to design parallelism operation semantics from the bottom of the distributed training system. In this way, we can satisfy arbitrary parallelism strategies and their extensions. Distributed properties involve various parallel-related terms, with the goal of modeling global distributed computation by parameterizing operator deployment schemes. Within the modeling framework, developers have access to flexibly construct algorithmic models and configure distributed attributes according to their preferences. Formally, distributed properties are defined as a set of parameters associated with primitive operators. Their core framework involves the registration of operators along with their distributed attribute signatures. Here, we define the framework and further explain it with a qualitative analysis. Specifically, we discuss four key distributed properties: Placement, Scatter, Broadcast, and PartialReduce. Placement of each operator in the logical graph specifies the devices where logical operators will be deployed. In the case of common data parallelism, all operators are deployed to all devices. Logically, all operators are designed to run on a single device, but in practice, they operate on different devices based on their placement configuration. Broadcast is a procedure that involves sending the complete data of a logical tensor to all other computational nodes in the cluster, resulting in the creation of physical tensors that are copies of the logical tensors. Its process ensures that each physical operator has access to the entire dataset stored in the logical tensor. For convenience, we denote the Broadcast attribute as B. Scatter involves splitting data from a logical tensor into chunks and sending these chunks to devices in a certain order. This creates local physical tensors. The Scatter property is characterized by a single parameter for partitioning, denoted as $S(0)$ for horizontal slicing and $S(1)$ for vertical-axis slicing. Scatter represents a one-to-multiple distribution similar to Broadcast. Their distinction is that Broadcast sends identical copies to all devices, whereas Scatter sends different chunks to each device. For simplicity, we denote Scatter as $S$. PartialReduce signifies that the physical and logical tensors have matching shapes, but the values in the physical tensors constitute a subset of those in the logical tensors. Figure 1(a) illustrates the characteristics of PartialReduce. The complete global logical tensor can be reconstructed by reducing the physical tensor at the target location across all devices. Logically, the global logical tensor $Y$ is obtained by the logical tensors $U$ and $V$. However, in the physical implementation, component $U_0$ of logical tensor $U$, sliced by $S(1)$, and component $V_0$ of logical tensor $V$, with $S(0)$, are deployed on device 0. They are utilized to execute the corresponding operator, yielding the local physical tensor $Y_0$. Meanwhile, we use the same operation to obtain $Y_1$. Consequently, $Y$ can be reconstructed by reducing $Y_0$ and $Y_1$. Furthermore, $Y_0$, $Y_1$, and $Y$ share an identical shape. 3.2 Conversions of Distributed Properties This section derives the intermediate primitives and their variants, such as complex operation construction, and conversions between distributed properties, and also mentions the crucial intermediate primitives for converting diverse distributed attributes and evaluating the associated communication cost. The optimal parallel strategy selection relies on minimizing communication overhead. Converting tensor distributed attributes between devices incurs overhead, except when executed on the same device, in $S2P$, which eliminates communication costs. However, cross-device communication cost in conversions is proportional to the size of the logical tensor $T$. Furthermore, induced from the modeling, we introduce existing intermediate primitives. The combinations of primitives and various conversions between distributed properties have been shown in Appendix A.1, and the complex operations are included in Appendix A.2. ![Figure 1](image) Figure 1: An example of a PartialReduce procedure(a), where PartialReduce is denoted as $P$, and the behavior of $12P$(b), $12P$ is an atomic operation deploying a global logic tensor to a local reduction, where one device places a physical tensor, a copy of the global logic tensor, other devices only place physical tensors that have the same shape as the global logic tensor but with all values set to zero. 3.3 Immediate Inference Immediate inference involves deducing the distributed properties of the output from the attributes of the input tensor. Table 1 in Appendix A.1 illustrates the process of directly inferable distribution using the matmul operator, where each case of the input’s properties is specified, and the valid output’s distributed properties are inferred. It takes a global logical tensor as input and infers the distributed attributes of local physical tensors across all devices. If the inference depends on the assistance of intermediate primitives, we select the most cost-effective primitive to insert between the input and the local physical tensor beforehand. When two adjacent operators establish a producer-consumer relationship and the distributed properties of the output tensor from the producer operator do not align with the properties required by the consumer operator, DistPar needs to dynamically derive intermediate transformation primitives. These primitives are automatically inserted between the producer and consumer operators through the pass layers to ensure alignment. We present an example of inferring the intermediate primitive AllGather in Appendix A.1.2 3.4 COST DESIGN The overall cost is evaluated based on both computational cost and communication cost. To be specific, in order to optimize computational cost and communication cost collaboratively, we need to characterize the trade-off between them. Therefore, we introduce the ratio of computational cost to communication cost, which is denoted by beta. Computational Cost in DistPar is simplified to the sum of the elements of the input and output tensors corresponding to different parallelization strategies, due to the fact that DistPar assumes all parallelization strategies use the same operator library. Communication Cost is defined as the total communications across multiple devices. In our implementation, communication cost is estimated using the conversion cost that results from the conversions of distributed properties. Details are revealed in Appendix A.1. 4 EXPERIMENTS In this section, we conduct a comparative analysis of DistPar, TensorFlow, and Pytorch to demonstrate the effectiveness of DistPar. 4.1 SYSTEM PERFORMANCE Setup. We conducted a comparative evaluation, analyzing ResNet-50 pre-trained on the ImageNet-2012 dataset (Heigold et al., 2013) for image recognition and the BERT-Base model (Karpathy et al., 2014) for query answering in natural language processing tasks. We assessed the throughput and speedup of these models implemented with DistPar, as well as the data parallelism libraries of PyTorch and TensorFlow. It is worth noting that our emphasis is on system performance metrics rather than learning objectives. ![Graphs showing training speed for 2 models using 32-bit floats. Throughput is measured in images per second for the ResNet-50 and in sentences per second for the BERT Base model. The fastest speed for each model is shown in the group of green rectangles in subplots (a) and (c). Larger batch sizes narrow the distance between DistPar’s speedup curve and the ideal curve, indicating that DistPar can effectively leverage system scalability with large-scale datasets in subplots (b) and (d).](image-url) Figure 2: Training speed for 2 models using 32-bit floats. Throughput is measured in images per second for the ResNet-50 and in sentences per second for the BERT Base model. The fastest speed for each model is shown in the group of green rectangles in subplots (a) and (c). Larger batch sizes narrow the distance between DistPar’s speedup curve and the ideal curve, indicating that DistPar can effectively leverage system scalability with large-scale datasets in subplots (b) and (d). Analysis. We analyze the system performance in view of throughput and speedup. On mainstream models for various tasks, namely ResNet-50 in Figure 2(a)(b) and BERT in Figure 2(c)(d), we conducted a comparative evaluation on the performance of DistPar’s automatically selected parallelism strategy against data parallelism in PyTorch and TensorFlow frameworks. • Throughput Comparison Figure 2(a) and (c) illustrate the variation in the throughput performance of the three libraries as the number of computational devices changes. When comparing the throughput of DistPar-implemented ResNet-50 models with 16 and 32 computational devices, it is observed that they outperform the suboptimal PyTorch implementation by 1500 and 2300 images/second, respectively. In the case of BERT-base models, the respective throughput improvements are 500 and 750 sentences/second. As depicted in Figure 2(a) and (c), which illustrate the throughput of DistPar across various numbers of computational devices, it’s evident that DistPar consistently outperforms the comparative frameworks. Furthermore, this advantage becomes more obvious as the scale of computational devices increases. These findings underscore the superior overall throughput performance of DistPar, owing to its designed and selected global parallelization strategy in comparison to the data parallelism strategy employed by the comparative frameworks. • Speedup Comparison Figure 2(b) and (d) illustrate the variation in the speedup performance of the three libraries as the number of computational devices changes. With the increase in the number of devices, it becomes more evident that both the ResNet-50 model(b) and the BERT model(d) implemented with DistPar(blue curve) closely approach the ideal system(black curve), while TensorFlow (green curve) follows DistPar as the next best option. For ResNet-50 model(b) and BERT model(d), when the number of computational devices reaches 32, they achieve speedups 2 and 5 times higher than PyTorch(red curve), respectively. This indicates that when dealing with a larger number of computational devices, the performance improvement of DistPar over PyTorch’s data parallelism strategy becomes more notable. These results collectively highlight that, in comparison to the baselines, DistPar exhibits enhanced system scalability. From the figure, it’s clear that DistPar outperforms the existing TensorFlow and PyTorch. When batch sizes get larger, the distance between DistPar’s speedup curve and the ideal curve is narrowed, indicating that DistPar can effectively leverage system scalability with large-scale datasets, showcasing its promising adaptability. In summary, DistPar can boost the system’s overall performance including throughput and speedup, and achieve promising results compared with popular deep learning parallelism libraries. 4.2 Hyperparameter Optimization Setup. This experiment demonstrates DistPar’s optimization of parallelization strategies, as Figure 3 shows. The definition of overall cost can be found in Section 3.4. Specifically, the evaluating environment is configured with 4 * NVIDIA GeForce GTX 1080 GPU. Figure 3: Results of the hyperparameter optimization experiment. Since the values of beta corresponding to the maximum throughput vary on different models, we can select the optimal parallelism strategy for each model by adjusting the value of beta (a). Compared with the cost design of baselines that only takes communication cost into account, DistPar has notably better performance due to its collaborative optimization on both computational cost and communication cost (b). Analysis. DistPar exhibits varying parallelization strategies based on the ratio of computational cost to communication cost, denoted as the hyperparameter beta. This leads to different distribution characteristics of input and output tensors for the operators comprising the model. For different models, the beta value corresponding to the maximum throughput varies. For LeNet, AlexNet, Vgg16, and MobileNetV2, the beta values corresponding to their respective maximum throughputs are 10, 1, 0.1, and 0.01, with the corresponding speedup percentages being 7.48%, 64.75%, 2.83%, and 8.41%. The results highlight that DistPar adapts its parallelization strategy based on beta, resulting in different throughput outcomes. It is worth noting that the beta value corresponding to the maximum throughput is not consistent with the baseline which only considers the communication cost. This implies that, compared to a baseline approach that only considers communication cost, DistPar effectively leverages both computational and communication costs to guide its parallelization strategy selection. In summary, DistPar empowers users to optimize parallelization strategies for different models by fine-tuning the hyperparameter beta. This enables the selection of the parallelization strategy that corresponds to the maximum throughput for each model. 4.3 Scalability Analysis Setup. In order to observe the DistPar’s implementation of the large-scale face recognition insightface model, we conduct a series of separate experiments. The throughput on the insightface model was evaluated on different batch sizes and the number of categories. The configured with 8 GPUs of NVIDIA Tesla V100, FP32. Moreover, data parallelization with Broadcast and model parallelization with S1. To explore more cases, we vary the batch size and parallelization options for the fully connected layer of the last layer of the insightface model. As shown in Figure 4. ![Figure 4](image) (a) ![Image](image) (b) ![Image](image) Figure 4: Performances of DistPar, data parallelization, and model parallelization, with batch_size fixed to 8 and 64. As the number of categories and the batch size vary, DistPar shows an identical pattern of prioritizing data parallelism when the number of categories is small and tends to select model parallelism when it is gradually increasing. DistPar can outperform data parallelism by 120% and 50% within batchsize fixed to 8 and 64 respectively, which confirms that DistPar is able to automatically plan and select the better parallelization scheme that is adaptive to different computational resources according to different tasks. Analysis. Based on the Insightface model structure for face recognition tasks, we analyze the impact of changes in the number of categories on the selection of DistPar parallelization strategies. When the number of categories is small, data parallelism performs similarly to model parallelism and maintains a relatively good performance. However, as the number of categories increases, the throughput of data parallelism decreases. On the other hand, the performance of the model parallelism strategy remains stable. For DistPar, when the number of categories is low, it favors data parallelism. However, as the number of categories increases, DistPar tends to choose model parallelism as the overall strategy. These experimental results confirm that DistPar has the capability to select the optimal parallelization strategy that matches different numbers of categories effectively. Furthermore, we analyze the impact of batch size on the selection of DistPar parallelization strategies. When the batch size is small, DistPar exhibits better compared to data parallelism and model parallelism. As the batch size increases, the performance of DistPar remains competitive with model parallelism. It’s worth noting that when the batch size is 128, DistPar’s performance is slightly lower than that of model parallelism. However, by adjusting the hyperparameter beta, DistPar can be fine-tuned to match the performance of model parallelism. These experimental results confirm that DistPar can adapt to different batch sizes and select the optimal parallelization strategy accordingly. 4.4 OPTIMIZATION SPACE Setup. We conducted comparative experiments on the last three fully connected layers of the VGG16 network using DistPar with the manual configuration strategy provided by PyTorch, which involves a potential combination of all parallel strategies, DistPar implements an optimal parallelization strategy suitable for the last three layers. Analysis From the experiments, the throughput of the data parallelism strategy DDD configured in PyTorch is the lowest, as shown in Figure 5. By introducing some degree of model parallelism, the overall performance of VGG16 is improved. Considering the large dimension of the first fully connected layer, configuring it with the S0 parallelization strategy yields favorable results. The results indicate that the manually configured optimal parallelization strategy in PyTorch is RCR, confirming that the S0 parallelization strategy is best suited for the first fully connected layer. ![Figure 5: Performance evaluation of all possible parallelism strategies. where "Auto" describes the DistPar strategy, "R" represents S0 parallelism, "C" represents S1 parallelism, and "D" represents data parallelism. Specifically, the PyTorch configuration using the RCR parallel strategy, as illustrated in the figure, describes the optimal setup: the first fully connected layer employs S0 parallelism, the second layer utilizes S1 parallelism, and the third layer again adopts S0 parallelism.](image) Compared to the manually configured PyTorch parallelization strategy, the DistPar strategy exhibits significant performance improvements. In PyTorch’s manual configuration approach, only the distributed attributes affecting variable operations are determined, while the parallelization strategy for intermediate tensors remains undetermined. Meanwhile, DistPar has the capability to comprehensively select and optimize parallelization strategies for intermediate tensors, analyzing operators within the backward computation graph to determine the best parallelization strategy. In contrast to Pytorch’s manual configuration approach, DistPar has a larger search space. In summary, compared to manually configured PyTorch parallelization strategies, DistPar yields superior performance, resulting from DistPar’s larger search space and its optimization capabilities. 4.5 PRIMITIVE-LEVEL OPTIMIZATION Setup. DistPar offers multiple implementations for the same parallelization strategy. For example, as shown in Figure 3(b) (see Appendix A.3), the S2B transformation can be realized using both the AllGather approach and a combination of Gather and Broadcast. In order to investigate how DistPar’s use of different implementations for the same parallelization strategy affects system throughput performance, we evaluated the throughput performance of various collective communication operations, including ReduceScatter, AllGather, and AllReduce, as they vary with the scale of computational devices, using the Enflame-CloudBlazer T10-16GB DCU in the same environment. Analysis. In Figure 6, the results indicate that different communication primitives exhibit various throughput performances at the same number of computational devices. The overall throughput trends for all primitives show a pattern of initial decline followed by stabilization as the scale of computational devices increases. When there are 8 devices, the throughput of AllGather is 10.36 and 12.40 times higher than ReduceScatter and AllReduce, respectively. This suggests that when the number of computational devices is relatively low, significant performance differences exist among different communication primitives. As the number of devices increases to 320, these differences are reduced to 1.03 and 1.0, respectively, indicating that the performance gap between different primitives gradually narrows with the growth in the number of computational devices. This experiment confirms that, when the number of computational devices is low, DistPar exhibits significant performance variations based on different communication primitives, expanding the candidate space for selecting the optimal strategy for the same parallelization strategy. When the number of computational devices is high, DistPar’s implementations based on different communication primitives for the same parallelization strategy tend to have stabilized performance differences, highlighting DistPar’s ability to select the most stable and highest throughput implementation when there are a significant number of computational devices. ![Graph showing throughputs for data parallelism with different tensor partition options in DistPar.](image) Figure 6: The throughputs for data parallelism with different tensor partition options in DistPar. This figure illustrates throughputs of varied intermediate primitives are different under the same device. Notably, throughputs for all primitives initially drop before plateauing. This decline is due to the reduced communication bandwidth between devices as the parallel width of collective communication widens, leading to less bandwidth utilization by individual intermediate primitives. ## 5 CONCLUSIONS AND FUTURE WORK In this paper, we propose DistPar, a unified approach for efficient tensor partitioning in parallel computation of neural networks, and describe the methodology of determining solution spaces for attribute conversions in distributed training systems. The results indicate that the proposed tensor partitioning approach of DistPar supports flexible combinations of various parallelism strategies. Furthermore, under the collaborative guidance of computational cost and communication cost, DistPar enables users to select the parallelism strategy that yields the maximum throughput corresponding to different models. Hence, we believe DistPar is very promising in related domains. However, there are potential limitations that need to be considered. 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LYS3RhIYCq	The authors mention that environments that do not fit the story of the paper are excluded -- Double dunk proved too simple an environment -- and games that were selected were picked due to intuitions that their score would lead to scaling laws. This drastically reduces any conclusions that can be drawn from the paper -- it shows only there exist _some_ environments where scaling laws hold for _some_ portion of the compute space.	SCALING LAWS FOR IMITATION LEARNING IN SINGLE-AGENT GAMES Anonymous authors Paper under double-blind review ABSTRACT Imitation Learning (IL) is one of the most widely used methods in machine learning. Yet, many works find it is often unable to fully recover the underlying expert behavior (Wen et al., 2020; Jacob et al., 2022), even in constrained environments like single-agent games (De Haan et al., 2019; Hambro et al., 2022). However, none of these works deeply investigate the role of scaling up the model and data size. Inspired by recent work in Natural Language Processing (NLP) (Kaplan et al., 2020; Hoffmann et al., 2022) where “scaling up” has resulted in increasingly more capable LLMs, we investigate whether carefully scaling up model and data size can bring similar improvements in the imitation learning setting for single-agent games. We first demonstrate our findings on a variety of Atari games, and thereafter focus on the extremely challenging game of NetHack. In all games, we find that IL loss and mean return scale smoothly with the compute budget (FLOPs) and are strongly correlated, resulting in power laws for training compute-optimal IL agents. Finally, we forecast and train several NetHack agents with IL and find they outperform prior state-of-the-art by 2x in all settings. Our work both demonstrates the scaling behavior of imitation learning in a variety of single-agent games, as well as the viability of scaling up current approaches for increasingly capable agents in NetHack, a game that remains elusively hard for current AI systems. 1 INTRODUCTION While conceptually simple, imitation learning has powered some of the most impressive feats of AI in recent years. AlphaGo (Silver et al., 2016) used imitation on human Go games to bootstrap its Reinforcement Learning (RL) policy. Cicero, an agent that can play the challenging game of Diplomacy, used an IL-based policy as an anchor to guide planning (Jacob et al., 2022). Go-Explore, a method for hard-exploration problems which solved all previously unsolved Atari games, used self-imitation learning in its robustification phase (Ecoffet et al., 2021). Despite its prevalence, several works have pointed out some of the limitations of IL. De Haan et al. (2019) and Wen et al. (2020) call out the issue of causal confusion, where the IL policy relies on spurious correlations to achieve high training and held-out accuracy, but performs far worse than the data-generating policy, even in single-agent Atari games. Jacob et al. (2022) have mentioned similar issues for policies learning from human games: they consistently underperform the data-generating policy. However, in many of these works, the role of model and data size is not deeply investigated. This is especially striking considering the increasingly impressive capabilities that recent language models have exhibited, mostly as a consequence of scale. In a series of papers trying to characterize these improvements with scale starting with Hestness et al. (2017) and Rosenfeld et al. (2019), it has been shown language modeling loss (i.e. cross-entropy) scales smoothly with model size and number of training tokens (Kaplan et al., 2020; Hoffmann et al., 2022). If we think of language models as essentially performing “imitation learning” on text, then a natural next question is whether some of these results extend to IL-based agents in games, and whether scale could provide similar benefits and alleviate some of the issues mentioned earlier on. In this paper, we ask the following question: How does compute in terms of model and data size affect the performance of agents trained with imitation learning in the single-agent game setting? We first focus on several Atari games with dense rewards which allows us to demonstrate our findings on a variety of games. However, since Atari games have all been solved at this point, there is not much room for further improvement in terms of scaling up. To demonstrate the potential of scaling up IL, our core focus will be on the extremely challenging game of NetHack, a roguelike video game released in 1987. NetHack is an especially well-suited and interesting domain to study for several reasons. First, it is procedurally generated and highly stochastic, disqualifying approaches relying heavily on memorization instead of generalization, such as Go-Explore (Ecoffet et al., 2021). Second, the game is partially observed, requiring the use of memory, potentially for thousands of steps due to the game’s long-term dependencies. Finally, the game is extremely challenging for current AI systems, with current agents reaching scores nowhere close to average human performance.\footnote{The average overall human performance is around 127k (Hambro et al., 2022b), while the current best performing NetHack agent gets a score of 10k.} The best agent on NetHack is a purely rule-based system called AutoAscend (Hambro et al., 2022a), with RL approaches lagging behind (Hambro et al., 2022b; Küttler et al., 2020; Mu et al., 2022; Mazoure et al., 2023). Even just recovering this system is hard, with Hambro et al. (2022b) reporting that the best neural agents achieve less than 10% of the system’s mean return in the environment, causing the authors to call for significant research advances. We instead investigate whether simply scaling up BC can help close some of this gap. Contributions. We train a suite of neural Atari and NetHack agents with different model sizes using BC to imitate expert policies and analyze the loss and mean return isoFLOP profiles. We find the optimal cross-entropy loss scales as a power law in the compute budget, and we use two different methods to derive scaling laws for the loss-optimal model and data sizes. We then relate the cross-entropy loss of our trained BC agents to their respective mean return when rolled out in the environment, and find that the mean return follows a power law with respect to the optimal cross-entropy loss, showing improvements in loss predictably translate in better performing agents. We use our two scaling law derivations to forecast the training requirements of a compute-optimal neural BC agent for NetHack. These forecasts are then used to train an agent which outperforms prior neural NetHack agents by 2x in all settings, showing scale can provide dramatic improvements in performance. We briefly extend our results to the RL setting, where we also train a suite of NetHack agents using IMPALA (Espeholt et al., 2018) and again find that model and data size scale as power laws in the compute budget. Our results demonstrate that the improvements in imitation learning performance for single-agent games with dense rewards\footnote{Please refer to section 6 for a discussion of why we need this requirement.} can be described by clean power laws. This suggests carefully scaling up model and data size can provide a promising path towards increasingly capable game agents, as well as potentially boost performance in other imitation learning settings. ## 2 Preliminaries We now introduce the formal setup for behavioral cloning. We assume the environment can be described by a Partially Observable Markov Decision Process (POMDP) $ \langle S, T, A, O, R, \gamma \rangle $, with states $ S $, transition function $ T $, action set $ A $, possible observation emissions $ O $, reward function $ R(s, a) $, and discount factor $ \gamma $. In the behavioral cloning setup, we don’t assume access to the rewards but instead assume access to a dataset $ D $ consisting of trajectories $ \tau = (s_0, a_0, s_1, a_1, \ldots) $ of states and actions. These trajectories can be generated by multiple (possibly sub-optimal) demonstrators acting in the environment. However, in this work, they are assumed to all come from the same expert policy $ \pi $. The goal is to recover this expert policy. To do this, a learner $ \pi_\theta $ will optimize the following cross-entropy loss: \[ L(\theta) = -\mathbb{E}_{(h_t, a_t) \sim D} [\log \pi_\theta(a_t\|h_t)], \] where $ h_t $ can include part or the entirety of the history of past states and actions. ## 3 Experimental Setup We analyzed the scaling behavior of agents trained with BC in two domains: (1) Atari and (2) NetHack. The former serves to test the validity of the scaling laws in a range of games, while the latter tests the performance gains of scaling in an extremely challenging and unsolved game. Whenever we report FLOP or parameter counts, we are referring to their effective counts, which we define as only including the parts of the network that are being scaled, similar to Hilton et al. (2023) (see Appendix E for full details). Please see Appendix F for details on all hyperparameters. ### 3.1 Atari We chose the following set of 8 Atari games: Battle Zone, Qbert, Bank Heist, Boxing, Breakout, Name This Game, Phoenix, and Space Invaders. We chose these games either because they were... part of the Atari-5 subset\footnote{We also experimented with Double Dunk, which is part of the Atari-5, but found even our smallest models could perfectly learn the policy with very few samples for the expert we trained. Therefore, we left it out.} (Aitchison et al., 2023), a reduced dataset that aims to be representative of the full set, or they were games where the reward is at least somewhat dense (see section 6 for more discussion on this). We then perform the following steps for each game. First, we train a CNN-based agent with PPO (Schulman et al., 2017) in order to get an expert agent. Second, we gather a dataset of about 1B samples consisting of rollouts of the expert agent. We then train a family of CNN-based agents on this dataset using BC, varying the width of the core CNN and the final linear layer (see Appendix E). The total number of parameters ranged from 1k to 5M. ### 3.2 NetHack We train LSTM-based agents on the NLD-AA dataset (Hambro et al., 2022b), mainly varying the width of the LSTM (see Appendix E). The total number of parameters ranged from 10k to 500M. While the original NLD-AA dataset already contains around 3B samples, we extended the dataset to around 60B samples (NLD-AA-L) and 150B samples (NLD-AA-XL) by collecting more rollouts from AutoAscend (i.e. the data-generating policy). NLD-AA-L is used for the results in Figure 1a, while NLD-AA-XL is used for all our forecasting-based experiments (see section 5). ### 4 SCALING UP IMITATION LEARNING This section is structured as follows. We first investigate the role of model size and number of samples with respect to cross-entropy loss (subsection 4.1). While intuitively it feels like a lower loss should result in a better agent, we verify this by directly investigating the role of model size and number of samples with respect to the environment return (subsection 4.2), and relating these results to the loss results. Finally, we also show a possible extension of our analysis to the RL setting (subsection 4.3). #### 4.1 SCALING LAWS FOR BC LOSS To investigate the role of model size and number of samples with respect to cross-entropy loss, we follow similar approaches to the ones used in Hoffmann et al. (2022). Approach #1: isoFLOP profiles.* “IsoFLOP” refers to constant FLOP budget contour lines. For Atari, we train up to 12 different model sizes, ranging from 1k to 5M. For NetHack, we train 14 different model sizes, ranging from 10k to 500M. For all domains, we train FLOP budgets of at least $1e^{13}$ and up to $1e^{18}$. In Figure 1, we plot the loss evaluated on a held-out set of about 100 trajectories against the parameter count for each FLOP budget. Similarly to Hoffmann et al. (2022), we observe clear parabolas with well-defined minima at the optimal model size for a given compute budget in all games. We take these loss-optimal data points to fit three regressions: one that regresses the log parameters on the log FLOPs, another that regresses the log samples on the log FLOPs, and a final one that regresses the log loss on the log FLOPs. These regressions give rise to the following power laws (Figure 1c, Figure 1d, and Figure 1b): $$N_{\text{opt}} \propto C^{\alpha}, \quad D_{\text{opt}} \propto C^{\beta}, \quad L_{\text{opt}} \propto C^{\gamma},$$ where $N_{\text{opt}}$ indicates the loss-optimal model size, $D_{\text{opt}}$ the loss-optimal number of training samples, $L_{\text{opt}}$ the minimal validation loss, and $C$ the compute budget in FLOPs. We refer to the legends of Figure 1c, Figure 1d, and Figure 1b for sample values of $\alpha$, $\beta$, and $\gamma$, respectively. Approach #2: parametric fit. Instead of only fitting the loss-optimal points as was done in approach #1 above, one can also fit all points from Figure 1a to the following quadratic form: $$\log \hat{L}(N, D) = \beta_0 + \beta_N \log N + \beta_D \log D + \beta_{N^2} (\log N)^2 + \beta_{ND} \log N \log D + \beta_{D^2} (\log D)^2.$$ If we only look at the linear terms here, we notice that this loss has the form of a Cobb-Douglas production function: $$\hat{L}(N, D) = \exp(\beta_0) \times N^{\beta_N} \times D^{\beta_D},$$ Figure 2: BC return scaling. We train a wide range of model sizes across several orders of magnitudes of FLOP budgets (same models as in Figure 1a) and plot their average return in the environment (a). We left off the first four FLOP budgets as we found them to be especially noisy. We then regress the optimal returns (b), the return-optimal number of parameters (c), and the return-optimal number of samples (d) on their corresponding FLOP budgets. We find mostly clear power law trends for Nethack (left), Battle Zone (middle), and Qbert (right). Full Atari results can be found in Appendix I. where we can think of parameters $N$ and samples $D$ as inputs that affect how much output (i.e. loss) gets produced. We then take the functional form in Equation 3 and minimize the loss subject to the constraint that $\text{FLOPs}(N, D) \approx 6ND$. To do this, we used the method of Lagrange multipliers to get the following functional forms for $N_{\text{opt}}$ and $D_{\text{opt}}$ (see Appendix A for full derivation): --- 4 Note that this FLOPs equation is only valid for our NetHack experiments, since the model there is LSTM-based. To carry out a similar analysis for Atari, where the models are CNN-based, this formula needs to be adjusted. We only perform the analysis for NetHack due to the simplicity of the FLOPs equation. Table 1: Fitted power law coefficients in NetHack. We list the scaling coefficients for model size ($\alpha$) and number of samples ($\beta$) for all three settings. 95% CIs are noted in parentheses, where the delta method was used for the parametric fit parameters (see Appendix G). \| Setting \| IsoFLOP profiles \| Parametric fit \| \|---------------\|------------------\|----------------\| \| \| $\alpha$ \| $\beta$ \| $\alpha$ \| $\beta$ \| \| 1. BC Loss \| 0.57 (0.50, 0.64)\| 0.43 (0.36, 0.50)\| 0.48 (0.47, 0.49)\| 0.52 (0.51, 0.53)\| \| 2. BC Return \| 0.35 (0.18, 0.52)\| 0.65 (0.48, 0.82)\| 0.34 (0.33, 0.35)\| 0.66 (0.65, 0.67)\| $$N_{\text{opt}} = G \left( \frac{C}{6} \right)^{\alpha}, \quad D_{\text{opt}} = G^{-1} \left( \frac{C}{6} \right)^{\beta}, \quad \text{where} \quad G = \exp \left( \frac{\beta_D - \beta_N}{2\beta_D^2 - 2\beta_ND + 2\beta_N^2} \right).$$ We find that $\alpha = \frac{2\beta_D^2 - \beta_ND}{2\beta_D^2 - 2\beta_ND + 2\beta_N^2}$ and $\beta = \frac{2\beta_N^2 - \beta_ND}{2\beta_D^2 - 2\beta_ND + 2\beta_N^2}$. We compare the two approaches for NetHack in Table 1. 4.2 Scaling laws for BC return Note that the analysis in the previous section was all in terms of cross-entropy loss. However, in the imitation learning setting, we almost never care directly about this quantity. Instead, we care about the average return of the resulting agent in the environment. To investigate how this quantity scales, we roll out every model from Figure 1a in the corresponding Atari or NetHack environment and average their score across 100 (Atari) and 1k (NetHack) rollouts each. We show the results in Figure 2a. We then follow a similar procedure as in subsection 4.1 and perform the same three regressions, giving rise to the following power laws (Figure 2c, Figure 2d, and Figure 2b): $$N_{\text{opt}} \propto C^\alpha, \quad D_{\text{opt}} \propto C^\beta, \quad R_{\text{opt}} \propto C^\gamma,$$ where $N_{\text{opt}}$ indicates the return-optimal model size, $D_{\text{opt}}$ the return-optimal data size, $R_{\text{opt}}$ the maximal return, and $C$ the compute budget in FLOPs. We refer to the legends of Figure 2c, Figure 2d, and Figure 2b for sample values of $\alpha$, $\beta$, and $\gamma$, respectively. When looking at Figure 2b, we find that for the Atari games the power laws hold all the way until expert performance. For NetHack, we find more FLOPs will be required to reach the expert score of 10k. Additionally, we can take the functional form in Equation 3 and simply replace loss with mean return. We can then solve the same constrained optimization problem resulting in the exact same expressions as found in Equation 5. We list the resulting coefficients for NetHack in Table 1. To investigate the relationship between loss and mean return, we regress the loss-optimal log returns on the corresponding log loss values. We find a power law of the form $R_{\text{opt}} \propto L_{\text{opt}}^\delta$, as shown in Figure 3. The fit in the figure shows optimal loss and mean return are highly correlated in all games, indicating we can expect return to increase smoothly as we make improvements in loss, rather than showing sudden jumps. 4.3 Extension to reinforcement learning Given the stark trends we found for BC in the previous sections, we investigate whether similar trends can be found for RL. We explore this briefly for the game of NetHack since several works in the past years have attempted RL-based approaches for NetHack (Küttler et al., 2020; Hambro et al., 2022b), without too much success, unlike in Atari. We investigate the role of model size and environment interactions using approaches 1 and 2 from subsection 4.1 applied to IMPALA (Espeholt et al., 2018). While learning curves in RL tend to have high variance, Figure 4 suggests that compute-optimal agents should increase both the number of parameters and number of environment interactions as the --- 5 Note that the breaking down of our scaling laws after reaching expert performance is expected. This is similar to other scaling laws such as those of Kaplan et al. (2020) breaking down at the entropy of language. Figure 3: BC return vs. optimal loss. We investigate the relationship between the optimal loss of a BC agent and the mean return. We find they are highly correlated for all games. Figure 4: RL return scaling. We train a wide range of model sizes across several orders of magnitude of FLOP budgets and plot the average return when rolled out in the environment at the end of training (a). We then regress the return-optimal average returns (b), parameters (c), and samples (d) on their corresponding FLOP budgets. We run 1 seed per point on the isoFLOP profile. FLOP budgets are scaled up. We also find that the NetHack game score varies smoothly with FLOPs and hence can be seen as a natural performance metric (Hilton et al., 2023). We provide complete details of our setup and results in Appendix H. 5 FORECASTING COMPUTE-OPTIMAL BC AGENTS The isoFLOP profiles and power laws shown in Figure 1 and Figure 2 allow us to estimate the compute-optimal number of samples and parameters needed to train an agent that recovers the expert’s behavior. For all our Atari games except for Space Invaders, we already found such an agent! It is simply the first dot that reaches the expert score in Figure 2b. However, for NetHack, even the largest FLOP budget models don’t come close to expert performance (10k). To attempt to close this gap, we forecast and train a compute-optimal NetHack agent aimed at getting a score of 10k. To do this, we follow two approaches: 1. Using loss isoFLOP profiles. We first plug in $R = 10k$ into the regression in Figure 3a to solve for $L_{10k}$, the loss needed for a score of 10k. Then, we plug $L_{10k}$ into the regression in Figure 1b to get $C_{10k}$, the FLOPs needed to recover a score of 10k. Then we find the optimal parameters and samples using Figure 1c and Figure 1d. This way, we find that the model size should be 43M, and the data size should be 144B. 2. Using parametric fit. We take $C_{10k}$ from above, and use Equation 5 found by the parametric fit to solve for the parameters $N$ and samples $D$. This way, we find that the model size should be 17M, and the data size should be 362B. We find that the second approach predicts a smaller model size but more samples, similar to findings in Hoffmann et al. (2022). Based on early forecasting fits, we train a 30M parameter model for 115B samples, which took 11 days on 8 GPUs. The results can be found in Table 2. While we do not recover the underlying expert behavior (score of 10k), we do find the resulting model gets a big boost in performance and outperforms prior state-of-the-art by 2x, both when using a random initial character (hardest setting) as well as when its kept fixed to human monk. Discussion The gap with the expert could have several explanations. Uncertain power law exponents may have caused substantial extrapolation error when predicting model and data sizes for FLOP budgets much larger than those in the isoFLOP profile. In Appendix J, we perform a rolling cross-validation to evaluate one-step-ahead forecasting performance, which we do find to be accurate. Table 2: Forecasting results. We compare a model trained with BC using 30M parameters on 115B samples with previous models in the NetHackChallenge-v0 environment and find it outperforms all of them on both randomized character initialization (harder) as well as on human monk (easier). Exact scores not reported. Scores from Hambro et al. (2022b) were adjusted to account for an error in their evaluation code. See Appendix B for full results with standard errors. \| Models \| All Random \| Human Monk \| \|-------------------------------\|------------\|------------\| \| Offline only \| \| \| \| DQN-Offline (Hambro et al., 2022b) \| 0.0 \| 0.0 \| \| CQL (Hambro et al., 2022b) \| 352 \| 366 \| \| IQL (Hambro et al., 2022b) \| 171 \| 267 \| \| BC (CDGPT5) (Hambro et al., 2022b,a) \| 554 \| 1059 \| \| BC (Transformer) (Piterbarg et al., 2023) \| 1318 \| - \| \| Scaled-BC (ours) \| 2740 \| 5218 \| \| Offline + Online \| \| \| \| Kickstarting + BC (Hambro et al., 2022b) \| 962 \| 2090 \| \| APPO + BC (Hambro et al., 2022b) \| 1282 \| 2809 \| \| APPO + BC (Piterbarg et al., 2023) \| 1551 \| - \| \| LDD* (Mu et al., 2022) \| - \| 2100 \| 6 LIMITATIONS Natural performance metrics. There is no reason in general to expect game scores to scale smoothly. If they do, Hilton et al. (2023) define them as natural performance metrics. We expect that for any game score to be a natural performance metric, it needs to be at least somewhat dense so it tracks learning progress, which is why we focused on environments with relatively dense rewards in this paper. It’s possible our results extend to highly sparse reward settings as well, but one may need to introduce alternative proxy metrics (e.g. intrinsic performance (Hilton et al., 2023)) in that case. Experimental setup. Previous works have pointed to the importance of tuning hyperparameters (e.g. learning rate, batch size, adam optimizer parameters, etc.) for every run on the isoFLOP profile. Since we didn’t find any major sensitivities to hyperparameters during some initial tuning, and to limit computational cost, we kept all hyperparameters fixed for all isoFLOP profiles (Figure 1a), Figure 2a, and Figure 4a) and used “snapshots” of the same run to evaluate different FLOP budgets for the same model size. Therefore, we would like to point out there is considerable uncertainty in the exact values of the reported power law coefficients. Nevertheless, we expect the overall trends to still hold. 7 RELATED WORK NetHack Work on NetHack has been quite limited so far, with early work establishing the NLE benchmark (Küttler et al., 2020), evaluating symbolic vs. neural agents (Hambro et al., 2022a), and creating large-scale datasets based off of rule-based and human playthroughs for methods aiming to learn from demonstrations (Hambro et al., 2022b). More recent work has either focused on better reward signal supervision and sample efficiency through proxy metrics and contrastive pre-training (Mazoure et al., 2023; Bruce et al., 2023) or leveraged dynamics models with language descriptions in order to improve sample efficiency and generalization (Mu et al., 2022). Concurrent 6This, however, does not guarantee we will observe scaling laws in these environments when using IL! work also investigates the gap between neural methods and AutoAscend, but focuses on leveraging an action hierarchy, improvements in architecture, and fine-tuning with RL (Piterbarg et al., 2023). Scaling laws (Hestness et al., 2017) and Rosenfeld et al., 2019 are one of the earliest works that try to characterize empirical scaling laws for deep learning. Kaplan et al., 2020 and Hoffmann et al., 2022 specifically focus on training compute-optimal language models, finding similar trends as presented in this paper. While in the imitation learning setting, our agents also minimize cross-entropy loss, we additionally show that the eventual performance of the agent as measured by the average return in the environment scales smoothly with the loss. Other works focus more broadly on generative modeling (Henighan et al., 2020), or analyze specific use cases such as acoustic modeling (Droppo & Elbou, 2021). Clark et al., 2022 investigate scaling laws for routing networks, and Hernandez et al., 2021 study scaling laws for transfer, finding the effective data transferred (the amount of extra data required to match a pre-trained model from scratch) follows a power-law in the low-data regime. More recent works have also tried to extend these scaling law results to multi-modal learning (Cherti et al., 2022; Aghajanyan et al., 2023). Caballero et al., 2022 introduce broken neural scaling laws, which allow modeling of double descent and sharp inflection points. Finally, scaling laws relate to sample complexity theory, which shows that increases in the number of samples (i.e. dataset size) can improve the suboptimality of IL (Rajaraman et al., 2020; Xu et al., 2020; Rajaraman et al., 2021) and is applicable to all architectures and datasets. Perhaps the closest work to our paper is that of Hilton et al., 2023, who characterize scaling laws in RL. However, they don’t consider IL, and they do not evaluate on Atari or NetHack, the latter of which we consider an especially interesting environment because of its extremely challenging nature. ### 8 DISCUSSION Extensions beyond single-agent games. We have shown that in the imitation learning (and to some extent in the reinforcement learning setting), scaling up model and data size provides predictable improvements, and a promising path to improving performance, as demonstrated in a variety of Atari games and in the full game of NetHack. While we do not extend our analysis beyond single-agent games in this paper, we believe these results could be suggestive of similar findings across many imitation learning domains, where oftentimes model and data sizes are not carefully picked. Leveraging human data. In this work, we did not consider analyzing the scaling relationships when using human trajectories (e.g. from NLD-NAO (Hambro et al., 2022b)) instead of those from AutoAscend (NLD-AA (Hambro et al., 2022b)). This is because extra care must be taken to handle the lack of actions in the human dataset, requiring techniques such as BCO (Torabi et al., 2018). Investigating scaling laws here could be especially interesting since: (1) the human dataset is more diverse, containing trajectories from many different players with varying level of skill, and (2) it contains many examples of trajectories that ascend (i.e. win the game). (1) could shed perspective on the role of scaling when the data includes many different and potentially suboptimal demonstrations, similar to Beliaev et al., 2022. (2) could provide insight into the viability of methods such as Video PreTraining (Baker et al., 2022) since these rely heavily on being able to clone the expert data well. ### 9 CONCLUSION In this work, we find that imitation learning loss and mean return follow clear power law trends with respect to FLOPs, as demonstrated in Atari and in the challenging game of NetHack. In addition, we find loss and mean return to be highly correlated, meaning improvements in loss predictably translate in improved performance in the environment. Using the found power laws, we forecast the compute requirements (in terms of model and data size) to train compute-optimal agents aimed at recovering the underlying expert. In NetHack, we find the performance improves dramatically, surpassing prior SOTA by 2x in all settings. We also briefly extend our results to the reinforcement learning setting, and find similar power laws for model size and number of interactions in NetHack. Our results demonstrate that scaling up model and data size is a promising path towards training increasingly capable agents for single-agent games. More broadly, they also call for work in the larger imitation learning and reinforcement learning community to more carefully consider and study the role of scaling laws, which could provide large improvements in many other domains. 10 Ethics Statement While we do not see a direct path towards any negative applications, we note that scaling up could have unknown unintended consequences. As scaling results in imitation and reinforcement learning agents that are increasingly more capable and influential in our lives, it will be important to keep them aligned with human values. 11 Reproducibility Statement Due to legal reasons, we unfortunately cannot release the code for the NetHack results. However, we included the code for all Atari results as part of the supplementary material. In addition, we plan to release the pretrained weights of the forecasted NetHack agent (section 5). Finally, we have dedicated several sections in the appendix to ensure reproducibility of our results. Appendix F provides a complete account of all training details, including hyperparameters, dataset information, GPU types, training times, etc. Appendix D provides complete details of our architectures for both domains. Appendix E provides details on how we scale our networks and how we do FLOP counting. 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xnhvVtZtLD	Could authors elaborate why they chose to work with (4) instead of the simpler form in (3)? It appears that the trick of applying instance-level weighting function $r$ could be applied as well to solving (3).	ON THE FAIRNESS ROAD: ROBUST OPTIMIZATION FOR ADVERSARIAL DEBIASING Vincent Grari∗,1,2,4, Thibault Laugel∗,1,2,4, Tatsunori Hashimoto2, Sylvain Lamprier3, Marcin Detyniecki1,4,5 1 AXA Group Operations 2 Stanford University 3 LERIA, Université d’Angers, France 4 TRAIL, Sorbonne Université, Paris, France 5 Polish Academy of Science, IBS PAN, Warsaw, Poland {grari,lauge1}@stanford.edu code: https://github.com/axa-rev-research/ROAD-fairness/ ABSTRACT In the field of algorithmic fairness, significant attention has been put on group fairness criteria, such as Demographic Parity and Equalized Odds. Nevertheless, these objectives, measured as global averages, have raised concerns about persistent local disparities between sensitive groups. In this work, we address the problem of local fairness, which ensures that the predictor is unbiased not only in terms of expectations over the whole population, but also within any subregion of the feature space, unknown at training time. To enforce this objective, we introduce ROAD, a novel approach that leverages the Distributionally Robust Optimization (DRO) framework within a fair adversarial learning objective, where an adversary tries to predict the sensitive attribute from the predictions. Using an instance-level re-weighting strategy, ROAD is designed to prioritize inputs that are likely to be locally unfair, i.e., where the adversary faces the least difficulty in reconstructing the sensitive attribute. Numerical experiments demonstrate the effectiveness of our method: it achieves, for a given global fairness level, Pareto dominance with respect to local fairness and accuracy across three standard datasets, as well as enhances fairness generalization under distribution shift. 1 INTRODUCTION The increasing adoption of machine learning models in various applications such as healthcare or criminal justice, has raised concerns about the fairness of algorithmic decision-making processes. As these models are often trained on historical data, they have been shown to unintentionally perpetuate existing biases and discrimination against certain vulnerable group (Obermeyer et al., 2019). Addressing fairness in ML has thus become an essential aspect of developing ethical and equitable systems, with the overarching goal of ensuring that prediction models are not influenced by sensitive attributes. One of its most common concepts, group fairness, entails dividing the population into demographic-sensitive groups (e.g., male and female) and ensuring that the outcomes of a decision model are equitable across these different groups, as measured with criteria like Demographic Parity (DP) (Dwork et al., 2012) and Equal Opportunity (EO) (Hardt et al., 2016). However, focusing solely on these group fairness criteria, along with predictive performance, has been increasingly questioned as an objective: besides being shown to poorly generalize to unseen, e.g., drifted, environments (Kamp et al., 2021), it has been more generally criticized for being too simplistic (Selbst et al., 2019; Binns, 2020), leading to arbitrariness in the bias mitigation process (Krco et al., 2023) and the risk of having some people pay for others (Mittelstadt et al., 2023). Recognizing these issues, some researchers have long focused on exploring more localized fairness behaviors, proposing to measure bias sectionally within predefined demographic categories, in which comparison between sensitive groups is deemed meaningful for the considered task. For instance, using Conditional Demographic Disparity (Zliobaite et al., 2011), fairness in predicted ∗Equal contribution salaries between men and women shall be evaluated by comparing individuals within the same job category and seniority level, rather than making a global comparison across sensitive groups. Nevertheless, predefining these comparable groups to optimize their local fairness is often difficult: for instance, which jobs should be deemed legally comparable with one another? (Wachter et al., 2021) In this paper, we therefore propose to address the difficult problem of enforcing fairness in local subgroups that are unknown at training time (Sec. 2). For this purpose, we leverage the Distributionally Robust Optimization (DRO) framework, initially proposed to address worst-case subgroup accuracy (see e.g. Duchi & Namkoong, 2021). Our approach ROAD (Robust Optimization for Adversarial Debiasing, described in Sec. 3) combines DRO with a fair adversarial learning framework, which aims to minimize the ability of an adversarial model to reconstruct the sensitive attribute. By boosting attention on feature regions where predictions are the most unfair in the sense of this sensitive reconstruction, ROAD is able to find the best compromise between local fairness, accuracy and global fairness. Such dynamic focus is done by relying on a weighting process that respects some locality smoothness in the input space, in order to mitigate bias in any implicit subgroup of the population without supervision. Experiments, described in Section 4, show the efficacy of the approach on various datasets. 2 Problem Statement Throughout this document, we address a conventional supervised classification problem, trained using $ n $ examples $(x_i, y_i, s_i)$ for $ i = 1, \ldots, n $, where each example is composed of a feature vector $ x_i \in \mathbb{R}^d $, containing $ d $ predictors, a binary sensitive attribute $ s_i $, and a binary label $ y_i $. These examples are sampled from a training distribution $ \Gamma = (X, Y, S) \sim p $. Our goal is to construct a predictive model $ f $ with parameters $ w_f $ that minimizes the loss function $ L_Y(f(x), y) $ (e.g. log loss for binary classification), whilst adhering to fairness constraints based on specific fairness definitions relying on the sensitive attribute $ S $. In this section, we present the fairness notions and works that are necessary to ground our proposition. 2.1 Group Fairness One key aspect of algorithmic fairness is group fairness, which aims to ensure that the outcomes of a decision model are equitable across different demographic groups. In this paper, we focus on two of the most well-known group fairness criteria: Demographic Parity and Equalized Odds. Demographic Parity: Demographic parity (DP) (Dwork et al., 2012) is achieved when the proportion of positive outcomes is equal across all demographic groups. Using the notations above, the learning problem of a model $ f $ under demographic parity constraints can be expressed as follows: \[ \arg\min_{w_f} \mathbb{E}_{(x,y,s) \sim p} L_Y(f_{w_f}(x), y) \quad \text{s.t.} \quad \|\mathbb{E}_p[f_{w_f}(x)\|s = 1] - \mathbb{E}_p[f_{w_f}(x)\|s = 0]\| < \epsilon \tag{1} \] Where $ \hat{f} $ represents the output prediction after threshold (e.g., $ \hat{f}_{w_f}(x) = \mathbb{I}_{f_{w_f}(x) > 0.5} $). The parameter $ \epsilon $ represents the deviation permitted from perfect statistical parity, allowing for flexibility in balancing accuracy and fairness. In the following, this deviation is noted as Disparate Impact (DI), representing the absolute difference in positive outcomes between the two demographic groups. Although numerous methods exist to solve the problem described in Equation 1, we focus in this work on the family of fair adversarial learning, which has been shown to be the most powerful framework for settings where acting on the training process is an option (i.e., in-processing method) (Louppe et al., 2017; Wadsworth et al., 2018; Zhang et al., 2018; Grant, 2022). One of the most well-known fair adversarial approaches by Zhang et al. (2018) is framed as follows: \[ \min_{w_f} \mathbb{E}_{(x,y,s) \sim p} L_Y(f_{w_f}(x), y) \quad \text{s.t.} \quad \min_{w_g} \mathbb{E}_{(x,y,s) \sim p} L_S(g_{w_g}(f_{w_f}(x)), s) > \epsilon' \tag{2} \] Where $ L_S $ represents a loss for sensitive reconstruction (e.g. a log loss for a binary sensitive attribute). In this adversarial formulation, the goal is to learn a model $ f $ that minimizes the traditional loss of the predictor model, while simultaneously ensuring that an adversary $ g $ with parameters $ w_g $ cannot effectively distinguish between the two sensitive demographic groups based on the predictor’s output $ f_{w_f}(x) $. The fairness constraint is thus imposed here as the adversary’s ability to reconstruct the sensitive attribute, which should be limited, i.e., the value of the loss function $ L_S(g_w(f_w(x)), s) $ should be above a minimum value $ \epsilon' $. In practice, to achieve a balance between the predictor’s and the adversary’s performance, a relaxed formulation of Equation 2 is used: \[ \min_{w_f} \max_{w_g} \mathbb{E}_{(x,y,s) \sim p} L_Y(f_w(x), y) - \lambda \mathbb{E}_{(x,y,s) \sim p} L_S(g_w(f_w(x)), s). \] The coefficient $ \lambda \in \mathbb{R}^+ $ controls the trade-off between the predictor’s performance on the task of predicting $ Y $ and the adversary’s performance on reconstructing the sensitive attribute. A larger value of $ \lambda $ emphasizes the importance of restricting the adversary’s ability to reconstruct the sensitive attribute, while a smaller value prioritizes the performance of the predictor on the main task. Equalized Odds: Equalized Odds (EO) (Hardt et al., 2016) is another group fairness criterion that requires the classifier to have equal true positive rates (TPR) and false positive rates (FPR) across demographic groups. This criterion is especially relevant when misclassification can have significant impacts on individuals from different groups. To achieve EO, Zhang et al. (2018) employs an adversarial learning approach by concatenating the true outcome $ Y $ to the input of the adversary. ### 2.2 The Local Fairness Problem The global aspect of these group fairness criteria begs the question of the emergence of local undesired behaviors: by enforcing constraints on global averages between sensitive groups, we still expect that some local differences may persist (Krco et al., 2023). We illustrate this phenomenon through a simple experiment, shown in Fig. 1. On two datasets, Adult and Compas (described in App. A.8.1), two models are trained: an unconstrained model solely optimizing for accuracy (called Biased, in red), and the adversarial model from Zhang et al. (2018) (in blue) optimizing for Demographic Parity for the sensitive attributes gender (Adult) and race (Compas). For each model, two types of Disparate Impact (DI) values are shown: the global DI values, calculated over all the test set (dashed lines); and the local ones, calculated in subgroups of the population (full lines). The subgroups are defined here as age categories: discretized bins of the continuous attribute age. Although local DI values are generally lower for the fair model, they vary a lot across subgroups, sometimes remaining unexpectedly high. This is especially true for less populated segments (e.g., higher age values), and segments where the sensitive attribute distribution is extremely unbalanced: as the fairness constraint only concerns global averages, more attention is put on densely populated regions. On the other hand, less populated segments are more likely to be ignored during the training. These local differences echo the long-asserted claim that the blunt application of group fairness metrics bears inherent inequalities through their failure to account for any additional context (Selbst et al., 2019; Binns, 2020). Here, although reductive, the additional context we refer to is the information already available in the dataset $ X $, in which comparable subgroups (Wachter et al., 2021) can be drawn to evaluate fairness. This helps defining the notion of Local Fairness that is the focus of this paper: a locally fair model thus guarantees minimal differences in expectations within these comparable subgroups of $ X $. Contrary to works on intersectional fairness (Kearns et al., 2018), the desired behavior in Fig. 1 is thus not to treat age as a sensitive attribute: predictions $ f(x) $ are expected to vary along age. However, in the Compas dataset for instance, equality between race groups is expected to hold regardless of the age category considered. It is important to note that the notion studied here is also different from the one of individual fairness, which aims to treat similarly individuals who are close w.r.t. some predefined similarity measure (see, e.g., Dwork et al. (2012)), without any notion of sensitive data, rather than minimize DI among subgroups of individuals. In the same vein of fairness without demographics, Hashimoto et al. (2018), Duchi et al. (2023) consider the case of unknown subgroups via the Distributionally Robust Optimization (DRO) framework. While their goal is to train models that perform uniformly well across all partitions of the population, our goal is to train a model that is uniformly fair (regarding a sensitive attribute) across all subregions of the feature space, which is quite different. Having knowledge of these subgroups at training time would mean that it could be included as an additional constraint in the learning objective, akin to the work of Žliobaite et al. (2011). The criterion they propose, Conditional Demographic Disparity, measures Demographic Disparity across user-defined subcategories. However, several issues make this difficult, if not impossible, in practice. Besides that such expert knowledge is generally unavailable, or costly to acquire, the subgroups definitions might even be inconsistent across different testing environments (e.g. conflicting legal definitions of job categories or gender (Wachter et al., 2021)), making its optimization futile. Furthermore, exploring multiple categories is problematic in a combinatorial perspective. In this paper, we propose to optimize accuracy while adhering to a worst-case fairness constraint, an objective that was originally introduced to enhance fairness generalization capabilities in scenarios involving distribution drift or noisy labels (cf. Sec. 2.3). We implicitly define the subpopulations of interest, for which we aim to optimize fairness, using distributions $ q $ within an uncertainty set $ Q $, and present the DRO framework for the Demographic Parity criterion as follows: \[ \min_{w_f} \mathbb{E}_{(x,y,s) \sim p} L_y(f_{w_f}(x), y) \quad \text{s.t.} \quad \max_{q \in Q} \left\| \mathbb{E}_q \left[ f_{w_f}(x) \| s = 1 \right] - \mathbb{E}_q \left[ f_{w_f}(x) \| s = 0 \right] \right\| < \epsilon \tag{3} \] The constraint ensures that the Disparate Impact remains less than a predefined threshold $ \epsilon $ under the worst-case distribution $ q \in Q $. Working with distribution $ q $ allows us to enforce local fairness by targeting subpopulations of interest, thus creating a more focused and adaptable model that addresses fairness problems both globally and at a granular level. ### 2.3 Related Work and Positioning Several works have proposed to address the objective in Eq. 3 either to ensure better fairness generalization capabilities in drift scenarios (Rezaei et al., 2021; Ferry et al., 2022; Wang et al., 2023) or when facing noisy labels (Mandal et al., 2020; Wang et al., 2020; Roh et al., 2021). The uncertainty set $ Q $ then represents the perturbations that might affect the data at test time, and can therefore take several forms. While we expect $ Q $ contains the distribution of test data, leaving too much freedom to $ q $ may lead to trivial solutions that degenerate as uniform classifiers (Martínez et al., 2021). To do so, the uncertainty set $ Q $ is commonly defined as a ball centered on $ p $ using distribution distances or similarities. Examples include maximal Total Variation distance (Wang et al., 2020), Wasserstein distance (Wang et al., 2021) or Jaccard index (Ferry et al., 2022). From the fairness without demographics literature (Duchi et al., 2023), it is known that the maximal allowed divergence is connected to the risk of the smallest component of the training distribution, seen as a mixture of distributions. This observation also holds for worst-case fairness using DRO, as defined in Eq. 3. To the best of our knowledge, our work is the first one to address the topic of local fairness with unknown subgroups. This different objective implies additional constraints on the set $ Q $ considered in Eq. 3. Notably, under our local fairness objective, we also want that the discrepancies of $ q $ w.r.t. $ p $ are smooth in the feature space, so that the fairness constraint does not increase mitigation on specific disconnected individuals, but rather on local areas of the space. This will guide the design of our approach in the next section. Moreover, due to the discrete nature of the problem expressed in Eq. 3 (the constraint is applied on $ \hat{f} $ which is binary), most existing works restrict to linear models (Wang et al., 2020; Rezaei et al., 2020; Mandal et al., 2020; Taskesen et al., 2020), or rule-based systems (Ferry et al., 2022). This allows them to look for analytical solutions using linear programming. Although Rezaei et al. (2021) is an exception in this regard, they suffer from several drawbacks, namely requiring knowledge about the target distribution at train time and about the sensitive attribute at test time. Solving Equation 3 using a wider class of models remains therefore, to the best of our knowledge, unexplored. 3 ROAD: ROBUST OPTIMIZATION FOR ADVERSARIAL DEBIASING 3.1 FORMALIZATION To overcome the limitations of previous works, we introduce our proposition to address the fairness generalization problem by combining adversarial optimization and the DRO framework. In order to learn a predictor $ f_{w_f} $ that is fair both globally and for any subregion of the feature space, the idea is therefore to boost, at each optimization step, the importance of regions $ q $ for which the sensitive reconstruction is the easiest for an optimal adversary $ g_{w_g^} $ given the current prediction outcomes. Rewriting the fairness constraint of Equation 3 with an adversary $ g_{w_g} : Y \rightarrow S $, we thus focus on the following problem for Demographic Parity: \[ \min_{w_f} \mathbb{E}_{(x,y,s) \sim p} L_Y(f_{w_f}(x), y) \] subject to \[ \min_{q \in Q} \mathbb{E}_{(x,y,s) \sim q} L_S(g_{w_g^}(f_{w_f}(x)), s) > \epsilon' \] with $ w_g^* = \arg \min_{w_g} \mathbb{E}_{(x,y,s) \sim p} L_S(g_{w_g}(f_{w_f}(x)), s) $ A major challenge with this formulation is that exploring all possible distributions in $ Q $ is infeasible in the general sense. Worse, modeling distribution $ q $ directly over the whole feature space as support is very difficult, and usually highly inefficient, even for $ Q $ restricted to distributions close to $ p $. This motivates an adversarial alternative, which relies on importance weighting of training samples from $ p $. We therefore restrict $ Q $ to the set of distributions that are absolutely continuous with respect to $ p $, inspired by Michel et al. (2022). This allows us to write $ q = rp $, with $ r : X \times S \rightarrow \mathbb{R}^+ $ a function that acts as a weighting factor. Given a training set $ \Gamma $ sampled from $ p $, we can thus reformulate the overall objective, by substituting $ q $ with $ rp $ and applying its Lagrangian relaxation, as an optimization problem on $ r \in R = \{ r \| rp \in Q \} $: \[ \min_{w_f} \max_{r \in R} \frac{1}{n} \sum_{i=1}^{n} L_Y(f_{w_f}(x_i), y_i) - \lambda_q \frac{1}{n} \sum_{i=1}^{n} r(x_i, s_i) L_S(g_{w_g^}(f_{w_f}(x_i)), s_i) \] with $ w_g^ = \arg \min_{w_g} \frac{1}{n} \sum_{i=1}^{n} L_S(g_{w_g}(f_{w_f}(x_i)), s_i) $ With $ \lambda_q $ a regularization parameter controlling the trade-off between accuracy and fairness in the predictor model. In the following, we describe two constraints, inspired from the DRO literature, that we consider to ensure $ q $ keeps the properties of a distribution and avoids pessimistic solutions. Validity Constraint To ensure $ q $ keeps the properties of a distribution (i.e., $ r \in R $), previous works in DRO (e.g., Michel et al., 2022) enforce the constraint $ \mathbb{E}_{(x,s) \sim p} r(x, s) = 1 $ during the optimization. In the context of local fairness using our adversarial formulation from Eq.5, we argue that this constraint is not sufficient to ensure a safe behavior with regard to the fairness criterion, as it allows disturbances in the prior probabilities of the sensitive (i.e., $ q(s) \neq p(s) $). As discussed more deeply in Appendix A.2.2, this may lead to a shift of the optimum of the problem, by inducing a stronger mitigation emphasis on samples from the most populated demographic-sensitive group. To avoid this issue, we propose to further constrain $ r $ by considering a restricted set $ \tilde{R} = \{ r \in R \| rp \in \tilde{Q} \} $, with $ \tilde{Q} \subset Q $ such that: $ \forall s, q(s) = p(s) $. To achieve this, we rely on the following constraint: $ \forall s, \mathbb{E}_{p(x\|s)} r(x, s) = 1 $. Besides guaranteeing the desired property $ q(s) = p(s) $ (proof in Sec. A.2.1), we also note that ensuring these constraints still imply the former one: $ \mathbb{E}_{p(x,s)} r(x, s) = 1 $, which guarantees that $ q(x, s) $ integrates to 1 on its support. We further discuss the benefits of this conditional constraint in Section A.2.3. Shape Constraint As discussed in Section 2.3, the definition of $ Q $ heavily impacts the desired behavior of the solution. In particular, controlling the shape of the allowed distributions $ q $ is especially --- 1 Adapting our work to EO is straightforward: as described in Sec. 2.1, adapting the adversarial method of Zhang et al. (2018) to the EO task simply requires to concatenate the true outcome $ Y $ to the prediction $ f(x) $ as input of the adversarial classifier. The same process can be followed for ROAD. 2 In the situation where all distributions in $ Q $ are absolutely continuous with respect to $ p $ all measurable subset $ A \subset X \times Y $, all $ q \in Q, q(A) > 0 $ only if $ p(A) > 0 $. crucial in a setting such as ours, where the focus of the mitigation process is done dynamically. Without any constraint (as proposed by Mandal et al. (2020)), the mitigation could indeed end up focusing on specific points of the dataset where the sensitive reconstruction from $ f_{w_f}(X) $ is the easiest, using very sharp distributions $ q $ close to a Dirac. This may turn particularly unstable and, more critically, could concentrate the majority of fairness efforts on a relatively small subset of samples. To control the shape of the bias mitigation distribution $ q $, we therefore choose to consider $ Q $ as a KL-divergence ball centered on the training distribution $ p $. However, similarly to Michel et al. (2022), we do not explicitly enforce the KL constraint (due to the difficulty of projecting onto the KL ball) and instead use a relaxed form. Using previous notations, the KL constraint takes the simple form \[ KL(q\|\|p) = KL(pr\|\|p) = E_p r \log \frac{pr}{p} = E_p r \log r. \] The spread of $ Q $ can then be controlled with a temperature weight $ \tau $ in the overall optimization process, which can be seen as the weight of a Shannon entropy regularizer defined on discrepancies of $ q $ regarding $ p $. Setting $ \tau = 0 $ means that no constraint on the distribution of $ r $ is enforced, thus encouraging $ r $ to put extreme attention to lower values of $ L_S $. On the other hand, higher values of $ \tau $ favors distributions $ q $ that evenly spreads over the whole dataset, hence converging towards a classical globally fair model for highest values (cf. Section 2.1). Note that setting this hyper-parameter is strongly related to implicitly tuning the size of the smallest subgroup of the population for which we ensure fairness (cf. section 2.3). ROAD Formulation The overall optimization problem of our Robust Optimization for Adversarial Debiasing (ROAD) framework can thus finally be formulated as (full derivation given in A.1): \[ \min_{w_f} \max_{r \in \mathbb{R}} \frac{1}{n} \sum_{i=1}^{n} L_Y(f_{w_f}(x_i), y_i) - \lambda_g \left[ \frac{1}{n} \sum_{i=1}^{n} r(x_i, s_i) L_S(g_{w_g^}(f_{w_f}(x_i)), s_i) + \tau \frac{1}{n} \sum_{i=1}^{n} r(x_i, s_i) \log(r(x_i, s_i)) \right] \] with $ w_g^ = \arg \min_{w_g} \frac{1}{n} \sum_{i=1}^{n} L_S(g_{w_g}(f_{w_f}(x_i)), s_i) $ (6) ### 3.2 TWO IMPLEMENTATIONS FOR ROAD #### 3.2.1 BROAD: A NON-PARAMETRIC APPROACH Let us first introduce a non-parametric approach, called Boltzmann Robust Optimization Adversarial Debiasing (BROAD), where each $ r(x_i, s_i) $ value results from the inner maximization problem from Eq. 13. As described below, this inner optimization accepts an analytical solution, whenever $ r $ values respect the aforementioned conditional validity constraints (proof in Appendix A.3). Lemma 3.1. (Optimal Non-parametric Ratio) Given a classifier $ f_{w_f} $ and an adversary $ g_{w_g} $, the optimal weight $ r(x_i, s_i) $ for any sample from the training set, is given by: \[ r(x_i, s_i) = \frac{e^{-L_S(g_{w_g}(f_{w_f}(x_i)), s_i)/\tau}}{\sum_{s_j \in \Gamma, s_j = s_i} e^{-L_S(g_{w_g}(f_{w_f}(x_j)), s_j)/\tau}} \] With $ n_{s_i} = \sum_{i=1}^{n} 1_{s=s_i} $. This expression allows us to set optimal weights for any sample from the training dataset, at no additional computational cost compared to a classical adversarial fairness approach such as Zhang et al. (2018). However, this may induce an unstable optimization process, since weights may vary abruptly for even very slight variations of the classifier outputs. Moreover, it implies individuals weights, only interlinked via the outputs from the classifier, hence at the risk of conflicting with our notion of local fairness. We therefore propose another - parametric - implementation, described in the next section, that improves the process by introducing local smoothness in the fairness weights. 3.2.2 Parametric Approach To introduce more local smoothness in the fairness weights assigned to training samples, we propose an implementation of the $r$ function via a neural network architecture. Our goal is to ensure that groups of similar individuals, who might be neglected in the context of group fairness mitigation (e.g., due to their under-representation in the training population, cf. Fig. 1), receive a similar level of attention during the training process. However, solely relying on adversarial accuracy, as done in BROAD, may induce many irregularities in such groups. The lipschitzness of neural networks can add additional implicit locality smoothness assumptions in the input space, thus helping define the distributions $q$ as subregions of the feature space. Note that, in this approach, the network architecture therefore plays a crucial role in how local the behavior of $r_{w_r}$ will be: more complex networks will indeed tend to favor more local solutions, for a same value of $\tau$. In particular, a network of infinite capacity that completes training will have, in theory, the same behavior as BROAD. To enforce the conditional validity constraint presented earlier, we employ an exponential parametrization with two batch-level normalizations, one for each demographic group. For each sample $(x_i, y_i, s_i)$ in the mini-batch, we define the normalized ratio as: $$\forall i, r_{w_r}(x_i, s_i) = \frac{e^{h_{w_r}(x_i, s_i)}}{\sum_{(x_j, s_j) \in \Gamma, s_j = s_i} e^{h_{w_r}(x_j, s_j)}}$$ with $h : X \times \{0; 1\} \rightarrow \mathbb{R}$ a neural network with weights $w_r$. To train ROAD, we use an iterative optimization process, alternating between updating the predictor model’s parameters $w_f$ and updating the adversarial models’ parameters $w_\phi$ and $w_r$ by multiple steps of gradient descent. This leads to a far more stable learning process and prevents the predictor classifier from dominating the adversaries. More details are provided in the appendix (see Alg. 1). 4 Experiments 4.1 Assessing Local Fairness In this first experiment, we assess how effective ROAD is for generating predictions that are locally fair for unknown subpopulations, while guaranteeing a certain level of global accuracy and global fairness. For this purpose, we use 3 datasets often used in fair classification, described in Appendix A.8.1: Compas (Angwin et al., 2016), Law (Wightman, 1998) and German Credit (Hofmann, 1994). Each dataset is split into training and test subsets, and the models described below are trained to optimize accuracy while mitigating fairness with respect to a sensitive attribute $S$. To assess fairness at a local level, various subpopulations chosen among features of $X$, i.e. excluding $S$, are selected in the test set. As an example on the Compas dataset, in which $S$ is Race: to create the subgroups, Age is discretized into buckets with a 10-year range. These intervals are then combined with the Gender feature, identifying 12 distinct subgroups. As measuring DI in segments of low population is highly volatile, we filter out subgroups with less than 50 individuals (see App. A.8.3). These subgroups are unknown at training time, and chosen arbitrarily to reflect possible important demographic subgroups (see Sec. 4.3.2 for further discussion). Given these subgroups $G$, the local fairness is then assessed on the worst Disparate Impact value across these subgroups: $$\text{Worst-1-DI} = \max_{g \in G} \|\mathbb{E}_{(x,s) \in g}(\hat{f}_{w_f}(x)\|s = 1) - \mathbb{E}_{(x,s) \in g}(\hat{f}_{w_f}(x)\|s = 0)\|.$$ To evaluate our approach, we compare our results with the globally fair adversarial models from Zhang et al. (2018) and Adel et al. (2019), and 3 approaches that address fairness generalization: FairLR (Rezaei et al., 2020), RobustFairCORELS (Ferry et al., 2022) and CUMA (Wang et al., 2023) (cf. App. A.8.2). As local fairness can only be measured against global accuracy and fairness, we evaluate the approaches by plotting the tradeoffs between global accuracy and worst-1-DI subject to a global DI constraint (we choose $DI \leq 0.05$, following the fairness literature (Pannekoek & Spigler, 2021)). To ensure a thorough exploration of these tradeoffs, we sweep across hyperparameter values for each algorithm (hyperparameter grids in App. A.8.4). Fig. 2 shows the resulting Accuracy-Worst-1-DI Pareto curves for each method. Overall, ROAD mostly outperforms all other methods. This tends to show how our method efficiently maximizes local fairness, without sacrificing any other desirable criterion too much. On the other hand, BROAD does not always perform as effectively as ROAD, illustrating the benefit from the local smoothness induced by the use of a neural network. Interest- Figure 2: Results for the experiment on Local Fairness. For all datasets, the X-axis is Worst-1-DI, Y-axis is Global accuracy. The curves represented are, for each method, the Pareto front for the results satisfying the imposed global fairness constraint (here, Global DI < 0.05 for all datasets). Figure 3: Pareto front results on distribution drift using the Adult dataset. For all figures, the X-axis is Equalized Odds; the Y-axis is Accuracy. Left: in-distribution (i.e. Adult UCI in 1994) test dataset; Center and Right: resp. 2014 and 2015 test datasets from Folktables (Ding et al., 2021). ingly, despite not including any robustness component, globally fair methods of Zhang et al. (2018) and Adel et al. (2019) still manage to slightly reduce local bias through their global mechanisms. 4.2 Experiments on Distribution Drift As discussed in Section 2.3 DRO-based techniques have been considered before to help with the generalization of fairness. In this section, we therefore aim to show how our approach also leads to a better generalization of fairness in the face of distribution shift in addition to better-protecting subpopulations. For this purpose, we replicate the experimental protocol of Wang et al. (2023): after training classifiers on the training set of the classical Adult dataset (1994), we evaluate the tradeoff between accuracy and global fairness (measured with Equalized Odds (EO)) on the 2014 and 2015 Folktables datasets (Ding et al., 2021), containing US Census data from corresponding years, thus simulating real-world temporal drift. The same approaches as in the previous section, adapted to optimize for EO (details in Appendix A.8.2), are tested. Once again, the hyperparameters of every method are adjusted to maximize the two considered criteria, and the Pareto front is shown in Fig. 3. Results on the classical Adult test set (in-distribution, left figure) are somewhat similar for most methods, with CUMA (Wang et al., 2023) slightly out-performing other methods. However, on drifted test sets (center and right figures), ROAD seems to achieve significantly better results than other methods, including other DRO-based fairness approaches. This suggests that the parametric implementation proposed in the paper is better suited to ensure robust behavior. 4.3 Ablation Studies 4.3.1 Behavior of $\tau$ and Impact of $\tau$ The behavior of ROAD depends on $\tau$, which controls the extent to which the distributions $q \in Q$ are allowed to diverge from $p$. The impact of $\tau$ can be observed in the left figure of Fig. 4 for the Compas dataset. As values of $\tau$ increase, the variance of the distribution of $r$ decreases, going from having most weights close to 0 and very high importance on a few others, to having most weights $r_i$ lying around 1. Choosing the right value of $\tau$ thus helps control the emphasis put on some subpopulations. Figure 4: Analysis of the behavior of ROAD on Compas. Left: distribution of $r$ for several values of $\tau$ at epoch 200 (truncated at $r > 5$). Center: Relationship between Local DI and the average value of $r$ assigned to instances belonging to the corresponding subgroups. Each dot is a subgroup. Right: Worst-1-DI as a function of $\tau$ for different values for $\lambda_g$ (quartiles between 0.0 and 10.0). Figure 5: Worst-1-DI scores for subgroups of the Law dataset of various definitions, built by varying age bin width and splits along gender. Full description or the subgroups is available in Sec. A.8.3. A critical assumption ROAD relies on is that the adversary $r$ puts more attention on locally unfair regions. We test this assumption on the Compas dataset (same subgroups as in Sec. 4.1) and observe the results in the middle of Fig. 4. For each subgroup $k \in G$ (blue dots), we measure its local fairness (y-axis) and the average weight $\mathbb{E}_{(x,s) \sim k}(r_i(x,s))$ associated to instances of $k$. The graph reveals a correlation between these two notions, suggesting that more emphasis is indeed put on more unfair regions. As a consequence of these two results, setting $\tau$ helps control local bias, as shown in the right of Fig. 4 for various values of $\lambda_g$. The perfect local fairness score achieved when $\tau = 0$ is due to a constant model $f_{w_j}$: with no shape constraint, $r$ concentrates all the fairness effort on each training sample successively, which finally leads to $f(X) = \mathbb{E}[Y]$ for any input. Choosing a higher value of $\tau$ helps regularizing the process by inducing a distribution $q(x\|s)$ closer to $p(x\|s)$. 4.3.2 How important is the definition of subgroups? The main motivation for ROAD is its ability to maximize local fairness when the definition of the local subgroups is unknown. To assess the veracity of this claim, we conduct another experiment where we measure the local fairness of ROAD when the definition of these subgroups vary. Concretely, we train once a biased model, a globally fair model (Zhang et al., 2018) and ROAD (with resp. accuracy scores 0.72, 0.60, and 0.60), and measure the local fairness for these models in subgroups of various definitions. These subgroups are defined successively as age bins with a width of 5, 10, 15 and 20, first across the whole population and then across subpopulations of other, non-sensitive, variables. Fig. 5 shows the local fairness results for the Law dataset (sensitive attribute is Race, subgroup attributes are Age and Gender). As expected, although the worst local DI for ROAD varies when the subgroup definition changes, it is almost consistently below the values reached by the globally fair model (except Def. 3 corresponding to the largest subgroups). This suggests that its tuning is not over-reliant on one subgroup definition, showcasing the flexibility of the approach. 5 Conclusion In this work, we introduced the problem of enforcing local fairness in unknown subpopulations. By leveraging the strengths of adversarial learning and Distributionally Robust Optimization, our proposed framework ROAD provides a powerful approach for this setting, addressing the shortcomings of previous DRO-based approaches. Future works include extending our work to settings where the sensitive attribute is not available, to other differentiable penalties (e.g., Mutual Information in Ragonesi et al., 2021), and further exploring the optimization of a 3-network adversarial approach. 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kuTZMZdCPZ	I am not sure that (Izacard et al., 2019) is the right reference for sparse seismic networks and small earthquakes (or even for sparse spatial coverage in scientific data; it addresses a super-resolution problem)	Continuous Field Reconstruction from Sparse Observations with Implicit Neural Networks Xihaier Luo, Wei Xu, Yihui Ren, Shinjae Yoo Brookhaven National Laboratory {xluo,xuw,yren,sjyoo}@bnl.gov Balasubramanya Nadiga Los Alamos National Laboratory {balu}@lanl.gov Abstract Reliably reconstructing physical fields from sparse sensor data is a challenge that frequently arises in many scientific domains. In practice, the process generating the data often is not understood to sufficient accuracy. Therefore, there is a growing interest in using the deep neural network route to address the problem. This work presents a novel approach that learns a continuous representation of the physical field using implicit neural representations (INRs). Specifically, after factorizing spatiotemporal variability into spatial and temporal components using the separation of variables technique, the method learns relevant basis functions from sparsely sampled irregular data points to develop a continuous representation of the data. In experimental evaluations, the proposed model outperforms recent INR methods, offering superior reconstruction quality on simulation data from a state-of-the-art climate model and a second dataset that comprises ultra-high resolution satellite-based sea surface temperature fields.[Project Website: data & code] 1 Introduction Achieving accurate and comprehensive representation of complex physical fields is pivotal for tasks spanning system monitoring and control, analysis, and design. However, in a multitude of applications, encompassing geophysics (Reichstem et al., 2019), astronomy (Gabbard et al., 2022), biochemistry (Zhong et al., 2021), fluid mechanics (Deng et al., 2023), and others, using a sparse sensor network proves to be the most practical and effective solution. In meteorology and oceanography, variables such as atmospheric pressure, temperature, salinity/humidity, and wind/current velocity must be reconstructed from sparsely sampled observations. Currently, two distinct approaches are used to reconstruct full fields from sparse observations. Traditional physics model-based approaches are based on partial differential equations (PDEs). These approaches draw upon theoretical techniques to derive PDEs rooted in conservation laws and fundamental physical principles (Hughes, 2012). Yet, in complex systems such as weather (Brunton et al., 2016) and epidemiology (Massucci et al., 2016), deriving comprehensive models that are both sufficiently accurate and computationally efficient remains elusive. Moreover, integrating field data into these derived PDEs for validation and calibration poses significant challenges (Raissi et al., 2019). Concurrently, machine-learning-based approaches emerge as an alternative avenue for nonlinear field reconstruction (Mescheder et al., 2019; Sitzmann et al., 2020; Mildenhall et al., 2021). In contrast to standard image and video data, scientific data describing complex physical systems present unique challenges. For example, sparse seismic networks (sparse spatial coverage) can lead to smaller earthquakes being unnoticed or their epicenters being misestimated (Myers & Schultz, 2000). Meanwhile, fluid dynamics in turbulent flows, an example of high nonlinearity, exhibit nonlinear behavior due to interactions between vortices and eddies (Stachenfeld et al., 2021). Other examples include sensor mobility, e.g., ocean waves and currents that transport floating buoys (Rodrigues et al., 2021), and on-off dynamics, e.g., cloud cover impacting solar panels that cause power fluctuations and grid instability (Paletta et al., 2022). These factors are driving the advancement of novel machine learning models, aiming to enhance and refine current approaches for field reconstruction. In this work, we introduce the first implicit neural representation (INR)-based model for global field reconstruction of scientific data from sparse observations with the following contributions: • We introduce a context-aware indexing mechanism that compared to standard time index $(t)$-based INR models, incorporates additional semantic information. • The presented network factorizes target signals into a set of multiplicative basis functions, subsequently applying element-wise shift and scale transformations to amalgamate latent information. • Empirical validation demonstrates the proposed model achieves an average relative error reduction of 39.19% compared to other state-of-the-art INR models. 2 RELATED WORK 2.1 CLASSICAL METHODS Regression Methods. Field reconstruction resembles regression, predicting new outcomes at new locations. The most straightforward approach is to construct a linear model using available data. To account for nonlinearity, inverse distance weighting (IDW) employs distance weighting with a power parameter for weighted averaging in spatial interpolation (Shepard, 1968). While popular, IDW assumes isotropy and operates within the convex hull of data points. A more potent alternative is Gaussian Process (Rasmussen et al., 2006). In practice, the application of Gaussian Process can be hindered by the computational complexity of inverting the covariance matrix, which scales with a time complexity of $O(n^3)$ and renders it infeasible for extremely large datasets (Angell & Sheldon, 2018; Yadav et al., 2021). Model Reduction Methods. Model reduction techniques address the reconstruction task by converting the continuous spatial representation $u$ into a composite of basis functions, often referred to as modes: $u(x) \approx \hat{u}(x) = \sum_{i=1}^{m} a_i \phi_i(x)$. Coefficients $a_i$ and modes $\phi_i(x)$ typically are determined through optimization or regression models. Extensions of existing model reduction techniques for reconstructing full-field from partial-field measurements typically incorporate a mask function $m(u, x)$ that is defined as 0 where data are missing and 1 where data are present, such as in the case of Gappy proper orthogonal decomposition (Everson & Sirovich, 1995; Bui-Thanh et al., 2004). Recently, deep learning techniques have been employed to construct a nonlinear manifold, enhancing model performance for slowly decaying Kolmogorov $n$-width problems (Lee & Carlberg, 2020; Lusch et al., 2018; Kim et al., 2019). 2.2 DEEP LEARNING METHODS Super Resolution (SR). SR typically focuses on upscaling low-resolution images $u_{low}$ to higher resolution $u_{high}$. Yet in the context of field reconstruction, we lack such paired data. Our primary focus is on generating a continuous representation $u$ from a discretized dataset. Recent progress in deep-learning-based SR enables continuous magnification through diverse techniques, including innovative training approaches, such as scale-consistent positional encodings (Ntavelis et al., 2022) and variable-size training (Chai et al., 2022); local conditioning methods that use deep surrounding features (Chen et al., 2021) or neighborhood-based interpolation (Luo et al., 2023); and global conditioning methods that leverage continuous coordinates and latent variables, e.g., Mesh-freeFlowNet (Esmailzadeh et al., 2020) and Neural Implicit Flow (NIF) (Pan et al., 2023). Neural Inpainting. Image inpainting techniques can be broadly classified into two categories: traditional and learning-based methods. Traditional methods primarily rely on low-level features, employing approaches like diffusion- (Bertalmio et al., 2000) or patch-based (Barnes et al., 2009) methods to extend information from surrounding regions into the missing areas. On the other hand, learning-based methods, particularly those employing GANs (Yu et al., 2018; Lee et al., 2020) and probabilistic diffusion models (Song et al., 2023b; Chung et al., 2023), have achieved more precise and semantically meaningful inpainting results. However, these learning-based methods may require post-processing and can be computationally demanding. Implicit Neural Representations. INR models rely on coordinate-based neural networks (Xie et al., 2022). INRs can serve two main purposes: they either parameterize the sensor domain or the density domain directly. In the former case, INRs map sensor coordinates to predicted sensor activations, which can be used to enhance real measurement data. For the latter, INRs directly predict the density value at a two- or three-dimensional (2D/3D) spatial coordinate. Typically, raw sensor mea- measurements are derived from spatially varying density using transformations. Therefore, such direct prediction is supervised by mapping the model’s output back to the sensor domain through various transformations, such as Radon in computed tomography (Reed et al., 2021), Fourier in magnetic resonance imaging (Song et al., 2023a), or convolution in cryo-electron microscopy (Zhong et al., 2021). 3 METHODOLOGY Overview. The objective is to accurately reconstruct a spatiotemporal continuous physical field, denoted as $ u $, representing quantities such as temperature, velocity, or displacement. This field $ u $ is inherently a function of both spatial coordinates ($ x $) and time ($ t $). Directly modeling such complex spatiotemporal physical fields poses significant challenges. Consequently, methodologies, e.g., functional separation of variables (Donà et al., 2021), have been devised to mitigate complexity, enhance tractability, and improve physical interpretability. Using these approaches, the underlying physical process $ u(x,t) $ can be decomposed, for example, in the form of the product as $ f_1(x) \cdot f_2(t) $. Proposed Method. Conventional spatiotemporal disentangled representation utilizes the time index ($ t $) primarily as a reference to indicate a specific time instance. Motivated by the desire for a more context-aware indexing mechanism, we pose the question: Can the pointing process be improved? A natural approach to incorporate available context information in field reconstruction involves using measurements of the underlying physical process at time $ t $. As the number and positions of available measurements change over time, we propose a design wherein an encoder extracts a latent representation from actual measurements. This latent representation is subsequently employed to guide the model to the target time instance. When coupled with an INR-based decoder, this proposed method achieves continuous field reconstruction. Figure 1 provides a comparison between the conventional scalar-index-based INR and our context-aware INR. ![Figure 1](image) Figure 1: Field reconstruction from sparse observations: The Prior approach uses the time index ($ t $) as a reference to indicate a specific time instance. Our approach is context-aware, leveraging available context information by incorporating measurements at time $ t $. Overall Architecture. We introduce a neural network reconstruction method, MMGN (Multiplicative and Modulated Gabor Network), that features an encoder-decoder architecture. The encoder extracts features from available measurements $ U_t = \{u^{(1)}_t, u^{(2)}_t, \ldots\} $ at time $ t $, while the decoder, guided by spatial coordinates $ x $ and a context-aware latent code $ z_t $, performs inference for the specific point at time $ t $ and location $ x $. Overall, the model is defined in Equation (1). \[ z_t = E_\phi(U_t), \quad \hat{u}(x,t) = D_\psi(z_t, x), \quad \forall t \in T \text{ and } \forall x \in \Omega, \] where $ \Omega \subset \mathbb{R}^{d_x} $ and $ T = \{t_i \in \mathbb{R}_+ \}_{i=1}^{N_t} $ are the spatial and temporal domains, respectively. 3.1 ENCODER Autoencoders (AE) and their probabilistic version, variational autoencoders (VAE) (Kingma & Welling, 2014), are commonly used for representation learning due to their natural latent variable formations. However, vanilla AE and VAE struggle with the randomness of sensor locations and numbers over time. While their graph counterparts handle spatial randomness and modified versions manage dynamic graphs, they become computation-intensive for large graphs and struggle with long-range dependencies (Pfaff et al., 2021). In contrast to AE and VAE, auto-decoder, as Figure 2: Architecture of the MMGN Model. The MMGN model employs auto-decoding to infer the latent variable $z$. Consequently, only the decoder is explicitly defined, and encoding takes place through stochastic optimization. More precisely, the latent code $z = \arg\min_z L(z, \Theta)$ is obtained by minimizing a loss function $L$ calculated as an expectation over a dataset. demonstrated in Park et al. (2019), exhibits reduced underfitting and enhanced flexibility. It accommodates free-formed observation grids, including irregular ones or those on a manifold, without necessitating a specialized encoder architecture—as long as the decoder possesses the same property. Auto-decoder. The aim of an auto-decoder is to compress essential information into $z_t$, enabling the reconstructed value $\hat{u}_t^{(i)}$ to closely approximate the original value $u_t^{(i)}$ for any point within the domain. This is accomplished through an iterative process $z_t^{(0)} \rightarrow z_t^{(1)} \rightarrow \ldots$, employing gradient descent optimization. To initialize the trainable latent codes, we can assume the prior distribution over codes $p(z)$ follow a zero-mean multivariate Gaussian with a spherical covariance $\sigma^2 I$ (Xie et al., 2022). In practice, we empirically notice that initializing $z_t^{(0)}$ to 0 yields slightly better results than Gaussian initialization: $$z_t^{(0)} = 0; \quad z_t^{(i+1)} = z_t^{(i)} - \alpha \nabla_{z_t} L(\hat{u}_t^{(i)}, u_t^{(i)}) \quad \text{for} \quad i = 0, \ldots, N - 1,$$ where $\alpha$ is the learning rate, $N$ is the number of iteration steps, and $L(\cdot)$ is the loss function. ### 3.2 Decoder The decoder inputs consist of two parts: spatial coordinates $x$ and latent codes $z$. Subjecting $x$ to fully connected feed-forward layers yields a coordinate-based multilayer perceptron (MLP). While such a coordinate-based MLP can offer a continuous representation, it struggles to learn high-frequency signals, a phenomenon known as spectral bias. Recent research indicates this issue can be mitigated using positional encoding with Fourier features (Tancik et al., 2020) or periodic nonlinearities in the first hidden layer (Sitzmann et al., 2020). In lieu of Fourier bases, we use Gabor filters to transform the coordinates as Fourier transforms emphasize a global frequency representation, making them less suitable for capturing varying frequency and orientation across different parts of the signal and more susceptible to noise and amplitude fluctuations. Specifically, we employ $N_g$ shift-invariant Gabor filters in the following form: $$g_i(x) = \exp \left( -\frac{\gamma^{(i)}}{2} \\| x - \mu^{(i)} \\|_2^2 \right) \sin \left( W_g^{(i)} x + b_g^{(i)} \right), \quad i = 1, \ldots, N_g,$$ where $\mu^{(i)} \in \mathbb{R}^{d_h}$ and $\gamma^{(i)} \in \mathbb{R}^{d_h \times d_x}$ denote the respective mean and scale term of the $i$th Gabor filter. The former is associated with the central frequency of the sinusoidal waveform that $g_i(x)$ is designed to identify, while the latter parameter corresponds to the standard deviation of the Gaussian envelope that modulates the sinusoidal waveform. This filter exhibits a multiplicative property (Fathony et al., 2020), which implies the product of the outcomes can be written from any pair of Gabor filters $ g_1(x), g_2(x) $ into a summation of Gabor bases $ g_1(x) \circ g_2(x) = \sum_{i=1}^{N} \beta_i g_i(x) $ with $ \beta_{1:N} $ denoting the coefficients. This decomposability property facilitates the construction of hierarchical features, allowing the model to capture different levels of abstraction in the data and enhancing the interpretability of learned features. After transforming the coordinates $ g(x) $, we introduce a modulation step where the transformed coordinates are modulated through a multiplicative layer, thereby integrating $ x $ and $ z $. Multiplicative Filter Network. Similar to the multiplicative filter network approach (Fathony et al., 2020), the modulation layer involves the iterative application of nonlinear Gabor filters to the network’s input. These filters are then multiplied by the linear transformations of both $ x $ and $ z $. Specifically, in a decoder comprising $ L $ layers, the decoding process is defined iteratively as follows: \[ h^{(1)} = g_1(x) \] \[ h^{(i+1)} = g_i(x) \odot \left( W_h^{(i)} h^{(i)} + W_z^{(i)} z + b_h^{(i)} \right), i = 1, \ldots, L - 1 \] \[ D_\phi(z, x) = W_h^{(L)} h^{(L)} + b_h^{(L)} \] where $ W_h^{(i)} \in \mathbb{R}^{d_i \times d_i}, W_z^{(i)} \in \mathbb{R}^{d_i \times d_z}, $ and $ b^{(i)} \in \mathbb{R}^{d_i} $ denote the weights and bias of the $ i $th layer; $ h^{(i)} \in \mathbb{R}^{d_i} $ marks the hidden unit at layer $ i $; and $ \odot $ indicates the element-wise multiplication. An intriguing aspect of the multiplicative filter network is that the final output can be expressed as a linear composition of Gabor filters: \[ F_\theta(U_k, x) = \sum_{m=1}^{M} c_k^{(m)} g(x; \tau_k^{(m)}) + \text{bias}, \] where $ M \gg L \in \mathbb{N}, \{c_k^{(m)}\}_{m=1}^{M} $ represents coefficients that depend on $ \{W_h, W_z, b_h\} $ individually, and $ \{\tau_k^{(m)}\}_{m=1}^{M} $ a set of filter parameters, i.e., $ \{\gamma, \mu\} $. We jointly train both the encoder and decoder. Additional information about training procedures and empirical observations related to the initialization of the encoder and decoder can be found in Appendix B.2. ### 3.3 Temporal Adaptability Owing to its coordinate-based architecture, the proposed model generates a continuous representation in the spatial domain. However, replacing the conventional time index ($ t $)-based INR with a context-aware indexing mechanism introduces a consideration: the direct support for continuous representation in the temporal dimension may be affected. For example: - Reconstruction. For time instances where $ U_t $ is not available. This can be accomplished through the application of interpolation techniques, such as linear or spline interpolation, to the latent codes. Subsequently, the interpolated latent code, in conjunction with the trained decoder, is employed to make inferences. - Nowcasting. Typically involves predicting current or very-near-future weather conditions within the next few hours, using real-time observational data. In this context, we assume access to observations for the nowcasting model. During the inference stage, the model generates a new latent code based on these observations while keeping all decoder parameters fixed. - Forecasting. As the decoder remains fixed after training, the extrapolation for forecasting tasks involves predicting the latent code. This can be achieved through two types of methods: autoregressive and neural operator. Autoregressive methods iteratively update the latent code $ z(t + \Delta t) = M(\Delta t, z(t)) $, where $ M : \mathbb{R}_{>0} \times \mathbb{R}^{d_z} \rightarrow \mathbb{R}^{d_z} $ is the temporal update (e.g., recurrent neural network (RNN) (Sherstinsky, 2020)), while neural operator methods map a stack of latent codes to a future state/states: $ z_t = M(t, z_1, z_2, \ldots, z_{t-1}) $ (e.g., Fourier neural operator (FNO) (Li et al., 2021)). In summary, the proposed model’s temporal adaptability is contingent on the study’s objectives. Detailed temporal interpolation and extrapolation can be seamlessly integrated into the proposed model as needed. 4 EXPERIMENTS 4.1 EXPERIMENTAL SETUP Datasets. For these experiments, we evaluate the model’s performance on two challenging datasets (additionally detailed in Appendix A): • Simulation-based data. The Community Earth System Model version 2 (CESM2) (Danabasoglu et al., 2020), a fully coupled global climate model, is used to simulate Earth’s climate states. This work uses monthly averaged global surface temperature data, representing an atmospheric field, for model testing. Note that seasonal cycles have been removed to augment complexity, and the dataset dimensions are 1024 (time), 192 (lat), and 288 (lon). • Satellite-based data. Sea surface temperature data are derived from both a retrospective dataset with a four-day latency and a near-real-time dataset with a one-day latency (Martin et al., 2012). Wavelets are employed as basis functions for optimal interpolation on a global 0.01-degree grid. We analyze one year of daily data, spanning from August 20, 2022 to August 20, 2023, at the provided resolution of one-hundredth of a degree, focusing on the Gulf Stream region. The dataset dimensions are 360 (time), 901 (lat), and 1001 (lon). Tasks. We assess the performance of models across diverse field reconstruction tasks to gauge their efficacy in various scenarios, including: • Randomness. Four tasks of increasing complexity are defined to evaluate the models’ capability in handling sensor-related randomness. In the first task, models are trained with a consistent number of data points located at fixed positions. The second task introduces variability by randomly varying the number of data points. In the third task, we maintain a fixed number of data points but introduce randomness by randomly sampling grid maps, while the fourth task combines both aspects of variability, involving random sampling of the number of data points and the grid map itself. • Sparsity. In each task, we define three sparsity levels. Specifically, the training procedure involves using partial observations sampled from the complete state, i.e., $ s \in \{5\%, 25\%, 50\% \} $ for simulation-based data and $ s \in \{0.1\%, 0.3\%, 0.5\% \} $ for satellite-based data. Testing employs the complete state ($ s = 100\% $). In Appendix B.1, Figure 8 provides a visual representation of the four tasks with additional task definitions available. Baselines. Our model is assessed against a series of state-of-the-art implicit neural networks for field reconstruction. • ResMLP (Huang & Hoefer, 2023) contains a sequence of six fully connected \| Model \| Task 1 \| Task 2 \| Task 3 \| Task 4 \| Task 1 \| Task 2 \| Task 3 \| Task 4 \| \|-------\|--------\|--------\|--------\|--------\|--------\|--------\|--------\|--------\| \| ResMLP \| 1.951e-2 \| 1.672e-2 \| 1.901e-2 \| 1.468e-2 \| 1.717e-3 \| 1.601e-3 \| 1.179e-3 \| 1.282e-3 \| \| SIREN \| 2.483e-2 \| 2.457e-2 \| 2.730e-1 \| 2.455e-2 \| 3.129e-1 \| 4.398e-2 \| 1.304e-2 \| 9.338e-2 \| \| FFN+P \| 2.974e-2 \| 1.121e-2 \| 1.495e-2 \| 8.927e-3 \| 2.917e-3 \| 2.392e-3 \| 7.912e-4 \| 7.565e-4 \| \| FFN+G \| 2.943e-2 \| 1.948e-2 \| 1.980e-2 \| 1.426e-2 \| 4.904e-3 \| 7.969e-3 \| 1.005e-3 \| 1.044e-3 \| \| MMGN \| 4.244e-3 \| 4.731e-3 \| 3.148e-3 \| 3.927e-3 \| 1.073e-3 \| 1.131e-3 \| 6.309e-4 \| 6.298e-4 \| \| Promotion \| 78.24% \| 57.79% \| 78.94% \| 56.01% \| 37.51% \| 29.35% \| 20.26% \| 16.74% \| Table 1: Performance comparison with four INR baselines on both high-fidelity climate simulation data and real-world satellite-based benchmarks. MSE is recorded. A smaller MSE denotes superior performance. For clarity, the best results are highlighted in bold, while the second-best are underlined. The promotion metric, which indicates the reduction in relative error compared to the second-best model for each task, also is included. blocks, and each block consists of two fully connected feedforward layers, incorporating batch normalization and concluding with a skip connection. - SIREN (Sitzmann et al., 2020) is instantiated as an MLP consisting of five fully connected feedforward layers, each employing sine activation functions. - FFN+P/G (Tancik et al., 2020) refers to the Fourier feature network with either positional encoding (P) or Gaussian encoding (G). The network consists of a 4-layer coordinate-based MLP. It applies element-wise Fourier feature mappings to the input and uses the Gaussian error linear unit, or GELU, function for nonlinear activation. Network details are available in Appendix B.3 and B.4. ### 4.2 Main Results Quantitative Evaluation. Table 1 presents the reconstruction errors across all experiments. Notably, MMGN consistently outperforms the other baseline models. This performance advantage is particularly pronounced under low sampling ratios. Generally, as the subsampling ratio decreases, all models experience degradation in performance. MMGN still achieves significant error reductions, ranging from 56.01% to 78.94% for simulation-based data and 16.74% to 37.51% for satellite-based data when compared to the second-best model, particularly at the lowest sampling ratio. In-depth analysis of the detailed results, including extreme-case scenarios and convergence studies, can be found in Appendix C.1 and C.2. Qualitative Evaluation. Figure 3 illustrates relative test errors and models’ predictions. The visualization highlights the superiority of MMGN over other methods. ResMLP exhibits over-smoothed predictions, struggling to capture high-frequency signals. FFN+P displays some structural checkerboard effects, particularly noticeable in the satellite-based data. Despite quantitatively performing slightly worse than FFN+P, FFN+G demonstrates significant reconstruction improvements, successfully capturing both the overall landscape and finer details. In contrast, MMGN’s prediction results faithfully reconstruct data with complex structures from sparse measurements, leading to a substantial reduction in errors. Appendix C.4 features the full results. \| Reference \| MMGN \| ResMLP \| SIREN \| FFN+P \| FFN+G \| \|-----------\|------\|--------\|-------\|-------\|-------\| \| ![Image](image1.png) \| ![Image](image2.png) \| ![Image](image3.png) \| ![Image](image4.png) \| ![Image](image5.png) \| ![Image](image6.png) \| Figure 3: Visualizations of true and reconstructed fields. Global surface temperature derived from multiscale high-fidelity climate simulations and sea surface temperature assimilated using satellite imagery observations. For each dataset, the first column displays the ground truth, the first row showcases predictions from different models, and the second row presents corresponding error maps relative to the reference data. In the error maps, darker pixels indicate lower error levels. Robustness to Noise. It is important to investigate the model’s performance in the presence of noise. Here, we quantify the noise by the channelwise standard deviation specific to that dataset and customize noise ratios, including scenarios with noise levels set at 1%, 5%, and 10%. The analysis is conducted on the simulation dataset, and the computed results reveal that the proposed MMGN model surpasses current baselines. Notably, ResMLP does exhibit robust performance in maintaining accuracy as the noise level rises. Meanwhile, MMGN maintains its performance effectively when the noise ratio is below 5% with a noticeable increase in errors observed when the noise reaches 10%. In contrast, the performance degradation of the two FFN types is more evident. Ablations of Gabor Filter. To assess the Gabor filter’s efficacy in MMGN, we conduct detailed ablations, encompassing the removal of filter designs and their substitution with alternative types. Key observations from Table 2 include: 1) eliminating filters leads to a significant drop in model performance, highlighting the indispensability of filter designs and 2) substituting the Gabor filter with a Fourier filter not only diminishes accuracy (9.0% drop in simulation data and 18.1% decrease in satellite data), but it also increases the model size. Ablations of Context-aware Indexing Mechanism. To evaluate the effectiveness of the proposed context-aware indexing mechanism, the latent size of MMGN is intentionally reduced to 1. In this configuration, akin to current INR baselines, the model is equipped with three inputs: $x$-coordinate, $y$-coordinate, and the latent code $z$, which is a scalar in this specific experiment. Given that $z_t$ is learned from the entirety of available measurements at time $t$, it is anticipated to encapsulate more semantic information, consequently enhancing the decoder’s performance. As depicted in Figure 5, the results indicate that with a latent size reduced to 1, MMGN exhibits a slight but consistent performance improvement over the second-best model, ResMLP, across both datasets. 4.3 Model Analysis Model Efficiency. Model efficiency is evaluated based on inference speed and model size (depicted in Figure 6). Each model is executed 10 times to evaluate the entire dataset and compute the average inference speed for each instance. Overall, MMGN exhibits a favorable balance between accuracy and efficiency, making it the top-performing model. FFN+P ranks as the second-best model in most tasks (refer to Table 1), yet it possesses slightly more model parameters, resulting in reduced efficiency. Concretely, in the case of simulation data, MMGN outperforms other models with a relative speed improvement of 9.28%, 2.09%, 7.03%, and 3.65% compared to FFN+P, ResMLP, FFN+G, and SIREN, respectively. Similar results are observed in the satellite-based dataset, and additional details are provided in the Appendix C.3. \| Designs \| # Param \| Simulation \| Satellite \| \|---------\|---------\|------------\|-----------\| \| None \| 577 K \| 1.758e-2 \| 6.883e-3 \| \| Fourier \| 601 K \| 4.439e-3 \| 7.422e-4 \| \| Gabor \| 581 K \| 4.073e-3 \| 6.290e-4 \| Figure 4: Model performance with different levels of noise. Figure 5: Ablation on context-aware indexing mechanism. Figure 6: Efficiency Comparison. Inference time is assessed per instance. Explainable Artificial Intelligence (XAI) Analysis of Latent Codes. To further explore the influence of the latent code $z$ on our model’s performance, we train 10 models with different latent sizes, ranging from 1 to 512, by doubling the latent size at a time and collect the corresponding learned latent codes $z_{tk}$ for all time steps $t_k$. We then assemble a matrix $Z = [z^{(1,1)}, z^{(2,1)}, z^{(2,2)}, \ldots, z^{(512,1)}, z^{(512,2)}, \ldots, z^{(512,512)}]$ with trained latent codes from different experiments. Using simulation-based data, which includes a total of 1024 time instances, results in a matrix size of 1024 by 1023 = 1 + 2 + · · · + 512. • Similarity between latent codes. For a certain latent space, we compute the pairwise Pearson Correlation of the latent codes and use the standard deviation of all the correlations to represent the overall similarity of latent variables in that latent space. In Figure 7(a), when the latent size increases, the standard deviation (dissimilarity) decreases. To further illustrate how the latent variables correlate across latent spaces, we use t-SNE (Van der Maaten & Hinton, 2008) to embed them into the same 2D plane and compare their similarities. As shown in Appendix C.5, scatter plots of the latent variables become more abundant as the latent size increases, where these latent variables are more similar. • Ablation of latent variable. Each variable inside the latent code is referred to as a latent variable. For example, if the latent code is of dimension 1024, it consists of 1024 latent variables. In our ablation study, we iteratively remove one variable at a time to regenerate the entire dataset. This process is repeated for each latent variable, and the MSE is calculated accordingly. Subsequently, we compute the percentage increase in MSE results, denoted as “NMSE.” To facilitate interpretation of the ablation study across latent variables, we employ a boxplot for each latent space. The boxplot illustrates the distribution of NMSE values after ablating a specific latent variable. As depicted in the Figure 7(b), both the mean and variance of the error consistently decrease as the latent size increases. • Diagnosis of auto-decoder. The design of latent codes captures both temporal and spatial information through the proposed context-aware indexing mechanism. To illustrate, we consider the entire dataset and treat the latent vectors as high-dimensional temporal vectors. With 1024 temporal vectors, their dimensions vary from latent sizes (e.g., 1, 2, . . . , 512) to the original spatial data dimension (192 × 288). For the temporal vectors in any latent space, we calculate the pairwise Pearson Correlation, comparing them with pairwise Pearson Correlation of the original temporal vectors. Figure 7(c) presents this comparison using MSE to measure the differences and showcase the correspondence of the latent vectors to the original data. We observe that as the latent dimension increases, the error consistently decreases. 5 Conclusion This work introduces MMGN, a novel INR model for scientific data reconstruction. Compared to other time index ($t$)-based INR models, MMGN introduces a more context-aware indexing mechanism via a trainable latent code. Comprehensive experiments have been conducted to showcase the improvements facilitated by such a context-aware representation. One limitation of this current work is its focus on reconstructing a selected trajectory. In the future, we plan to investigate MMGN’s potential for generalization to multiple trajectories or even across various climate properties with the goal of recovering arbitrary underlying flow maps. Concurrently, exploring the application of MMGN for optimal sensor placement is a worthwhile avenue to explore. Given the challenging combinatorial nature of optimal sensor placement, MMGN can serve as a surrogate model to expedite the search process. ACKNOWLEDGMENTS The authors would like to thank Klaus Tan and Avish Parmar for their efforts in analyzing the climate datasets. This material is based upon work supported by the U.S. Department of Energy (DOE)’s Office of Science, Office of Advanced Scientific Computing Research, Office of Biological and Environmental Research, and the Scientific Discovery through Advanced Computing (SciDAC) program under Award Number 9233218CNA000001. Brookhaven National Laboratory is supported by the DOE’s Office of Science under Contract No. DE-SC0012704. This research used the Perlmutter supercomputer of the National Energy Research Scientific Computing Center, which is supported by DOE’s Office of Science under Contract No. DE-AC02-05CH11231. The authors also thank the anonymous reviewers for their comments and suggestions that have helped to improve the manuscript’s quality and clarity. REFERENCES Rico Angell and Daniel R Sheldon. 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7W3GLNImfS	Why were the error types in Section 2 the ones chosen? There is a brief discussion (inspiration from Xu et al. 2023, Gricean maxims) but it would be nice to have more discussion on why these particular types were chosen.	Human Feedback is not Gold Standard Tom Hosking University of Edinburgh tom.hosking@ed.ac.uk Phil Blunsom Cohere phil@cohere.com Max Bartolo Cohere, UCL max@cohere.com Abstract Human feedback has become the de facto standard for evaluating the performance of Large Language Models, and is increasingly being used as a training objective. However, it is not clear which properties of a generated output this single ‘preference’ score captures. We hypothesise that preference scores are subjective and open to undesirable biases. We critically analyse the use of human feedback for both training and evaluation, to verify whether it fully captures a range of crucial error criteria. We find that while preference scores have fairly good coverage, they under-represent important aspects like factuality. We further hypothesise that both preference scores and error annotation may be affected by confounders, and leverage instruction-tuned models to generate outputs that vary along two possible confounding dimensions: assertiveness and complexity. We find that the assertiveness of an output skews the perceived rate of factuality errors, indicating that human annotations are not a fully reliable evaluation metric or training objective. Finally, we offer preliminary evidence that using human feedback as a training objective disproportionately increases the assertiveness of model outputs. We encourage future work to carefully consider whether preference scores are well aligned with the desired objective. 1 Introduction The fluency exhibited by Large Language Models (LLMs) has reached the point where rigorous evaluation of LLM capabilities is very challenging, with the quality of model outputs often now exceeding that of reference examples from datasets [Zhang et al., 2023; Clark et al., 2021]. A great advantage of LLMs is their flexibility, but this makes it difficult to design an all-purpose evaluation metric [Novikova et al., 2017]. Benchmarks have proven useful for model comparisons [Gehrmann et al., 2021; Liang et al., 2023], but for open-ended generation tasks human evaluation using a single overall score has become the de facto standard method [Ouyang et al., 2022; Touvron et al., 2023]. For a given input prompt, samples or responses from models are shown to annotators, who are asked to score the responses according to their quality [Novikova et al., 2018]. These scores can either be absolute ratings, or relative preference scores, whereby two responses are ranked by quality. Although the simplicity of a single overall score is appealing, it obscures the decision making process used by annotators, including any trade-offs or compromises, and does not explain why one response or model is better than another. Annotators look for shortcuts to make the task easier [Ipeirotis et al., 2010], and so are more likely to base their judgement on superficial properties (e.g., fluency and linguistic complexity) than aspects that require more effort to check (e.g., factuality). Previously, human evaluation of natural language generation systems has considered multiple aspects of the generated output. However, the criteria used are often unique to the specific task being considered [van der Lee et al., 2021; Hosking et al., 2022; Xu & Lapata, 2022], making them difficult to apply to LLMs. With recent rapid improvement in system performance, it is important to test whether preference scores capture the desired aspects of output quality, and whether they provide a gold standard objective for evaluating and training LLMs. In this paper, we analyse human annotation of model outputs, both for overall preference scores and for specific error criteria. In Section 2, we establish a set of error types that are task independent and act as minimum requirements for model outputs. We analyse the error coverage of overall preference scores. We ask two sets of annotators to rate a range of LLM outputs, the first according to these error types and the second according to their own judgements of overall quality, and find... that overall preference scores under-represent factuality and faithfulness. In Section 3 we consider two possible sources of bias when annotating for specific error types by generating outputs with varying assertiveness and complexity, and find that assertiveness strongly biases human factuality judgements. Finally, in Section 4 we offer some preliminary evidence that using human preference scores as a training objective disproportionately increases the assertiveness of model outputs. We present additional findings from our collected data in Appendix E; we confirm that annotators are subject to a priming effect; we analyse the variation of quality scores with response length; and we show that generated outputs are preferred to the reference responses. Our code and data are available at https://github.com/cohere-ai/human-feedback-paper. 2 ARE PREFERENCE SCORES RELIABLE? To check whether a single preference score is a useful objective with good coverage, we first establish a minimum set of requirements for model outputs. These error types are both generic enough that they are task agnostic and widely applicable, but also sufficiently well-specified that it is possible for annotators to judge them. We begin with the factors identified by Xu et al. (2023c), who asked crowdworkers and experts to rate model outputs and give justifications for their scores, removing those factors that are overly subjective (e.g., ease of understanding). We also draw inspiration from Grice’s Maxims (Grice 1991) regarding felicitous communication between speakers: the Maxim of Quantity implies that repetition is undesirable, the Maxim of Quality prohibits factual errors, and so on. Finally, we considered factors that users care about when using LLMs in production environments (e.g., refusal to answer). We therefore consider the following error types: - Harmful – Is the response unsafe, harmful or likely to cause offence in some way? - Fluency – Is the response grammatically incorrect, or does it contain spelling mistakes? - Scope – Does the response exceed the scope limits of a chatbot? Does the response give opinions or otherwise act as if it is a person, or offer to take actions that it cannot (e.g. make a call, access the internet)? - Repetition – Does the response repeat itself? For example, if there is a list in the response, are any items repeated? Does the response reuse the same phrase again and again? - Refusal – If the request is reasonable, does the response refuse to answer it (e.g. “I’m sorry, I can’t help you with that”)? - Formatting – Does the response fail to conform to any formatting or length requirements from the prompt? - Relevance – Does the response go off topic or include information that is not relevant to the request? - Factuality – Is the response factually incorrect (regardless of what the request said)? - Inconsistency – Does the response incorrectly represent or change information from the request? This criterion is often also referred to as faithfulness. - Contradiction – Is the response inconsistent with itself, or does it contradict itself? 2.1 EXPERIMENTAL SETUP We ask crowdworkers to evaluate model outputs, marking each example with a binary yes or no to denote whether an error is present. Separately, we ask a different set of annotators to rate the overall quality of the same outputs from 1 to 5, according to whatever criteria they feel are important. Datasets To cover a range of different tasks for which evaluation is challenging, we construct input prompts from three datasets: Curation Corpus (Curation 2020) is a summarization dataset composed of 40,000 news articles and professionally written summaries; Amazon Product Descriptions (Ni et al. 2019) gives a product title and specification as input and requires generating a compelling product description; and Wikihow (Koupaei & Wang 2018) consists of ‘how to’ questions and step-by-step guides. Full details of the prompt templates used can be found in Appendix C. Models While a comparison of different models is not the focus of this work, we nonetheless source responses from multiple performant models that we were able to access at time of writing: MPT 30B Instruct is fine-tuned on Dolly DDRLHF and additional datasets (MosaicML NLP Team 2023; Conover et al. 2023); Falcon 40B instruct is fine-tuned on a subset of Baize (Almazrouei et al. and Command 6B and 52B are commercial models trained by Cohere, fine tuned on proprietary datasets. We additionally include the reference outputs for each input. Details of the models, prompt templates and sampling hyperparameters can be found in Appendix D. Annotation We source crowdworkers from Prolific, requiring them to be native English speakers with 100% approval ratings from prior tasks. Our annotation interface is based on Potato (Pei et al., 2022). Our annotation protocol is based on findings from RankME (Novikova et al., 2018) that showed the best inter-annotator agreement is achieved when annotators are shown multiple outputs for a given input, and scores are collected as absolute ratings. We expect that showing annotators five full outputs at once would lead to higher cognitive load and lower annotator engagement, therefore we collect ratings for two outputs at a time, pairing each output with an output from one of the other four models. The resulting four annotations per output are aggregated by taking the mean for overall scores, and by taking the mode (and then the mean in case of ties) for error annotations. We annotate a total of 900 distinct outputs, with a total of 4,440 annotations including quality checks. Quality Control In order to check inter-annotator agreement, we collect 5 duplicate annotations for a random subset of 200 pairs of outputs. We also include a set of distractor examples, where a response is shown in context with an output from the same model but a different input. These examples act as an attention check; the response based on a different input should consistently be penalised along criteria like relevance and usefulness. We find that distractor outputs are correctly rated lower than the other output in the pair over 97% of the time, indicating that the vast majority of annotators paid attention to the task. We use Gwet’s AC1 measure (Gwet, 2014) to assess inter-annotator agreement for the multiply annotated examples, finding good agreement scores of between 0.64 (for Factuality) and 0.94 (for Refusal). The disparity indicates that annotators found some error types more difficult or subjective than others; refusal is straightforward to detect, whereas checking for factual errors involves significantly more effort. ### 2.2 Results Preference scores under-represent factuality and inconsistency In order to determine the degree to which each error type was captured by the overall scores, we fit a Lasso regression model (Tibshirani, 1996) with $\alpha = 0.01$ between the scores and the error ratings. Figure 1 shows the weights of each criterion under this model, where each weight corresponds to the expected reduction in overall score if the corresponding error is present. Six out of ten error types contribute to the overall scores, with refusal errors contributing most strongly. Factuality and inconsistency errors both contribute but with much lower weighting, indicating that a single preference score is likely to obscure failures in these important criteria. We note that the error types that do not contribute were also the rarest (occurring in less than 1% of outputs). We would expect that harmfulness and fluency should influence overall scores in general, but in our experiments the models are sufficiently strong and the tasks sufficiently well-posed that such errors are infrequent. Figure 2: Difference in annotated error rates for distractor examples (outputs from the same model but different input). Some error types are correctly unchanged (e.g., repetition, refusal) while relevance and inconsistency are correctly penalised. Factuality and contradiction are both incorrectly penalised (they are independent of the input), indicating that annotators struggled to fully disentangle these criteria. Annotators struggle with disentangling factors Recall that the distractor examples are pairs of outputs sourced from the same model, but where one of the outputs corresponds to a different input; these should therefore achieve comparable scores for criteria that are independent of the input prompt (e.g., fluency, detail, factuality$^1$) but be heavily penalized for other factors such as relevance and overall quality. The results in Figure 2 show that although this expectation holds in some cases (repetition, refusal and formatting are not penalized, while relevance and inconsistency are), other factors are incorrectly penalized; factuality and contradiction (within the output) are both rated worse for the distractor examples. This implies that annotators found it difficult to disentangle these criteria from the overall quality of a response. Although annotators are shown the instructions and error criteria before the input prompt and responses, we suspect that they subconsciously form an opinion about the quality of the response based on first impressions (Smith et al., 2014), and that this opinion influences their judgement of each error type. In other words, an annotator may decide that a response is bad, and decide that it is more likely to contain errors as a result. This effect could be partially mitigated by specifying precise instructions, giving multiple examples and training a knowledgeable group of annotators. However, there is always potential for ambiguity. 3 ARE ANNOTATIONS AFFECTED BY CONFOUNDERS? We have so far considered the effect of important error criteria on overall preference scores, but the annotations for the errors were themselves given by human annotators. The results for distractor examples in Figure 2 indicate that granular ratings may also be subject to biases. Firstly, we hypothesise that the assertiveness of a text influences human judgements; a statement conveyed confidently as fact is more likely to be interpreted as true. Similarly, text that uses complex language might lead an annotator to believe that the communicator behind it is intelligent and knowledgeable, and therefore that the content is true. This concept of language ideology, where the style and tone of a speaker leads to biased judgements about their trustworthiness and intelligence, has been extensively studied in the context of speech (Campbell-Kibler, 2009; Woolard, 2020), but we are not aware of any work in the context of model evaluation. 3.1 EXPERIMENTAL SETUP We generate model outputs from the same datasets as Section 2, but using an additional preamble$^2$ to vary the tone of the output and create outputs with both high and low assertiveness and high and low linguistic complexity. We constructed these preambles by iterative testing, with the aim of eliciting --- $^1$ Although a statement could be deemed factual if the input prompt supports it, the instructions shown to annotators explicitly asked them to consider factuality in absolute terms. $^2$ A preamble, or system prompt, is a short natural language snippet, usually prepended to the user query, designed to set the behavioural parameters of the system, e.g. “Respond helpfully and safely”. Figure 3: Human ratings of assertiveness, complexity and overall quality for each preamble type. The ratings indicate that the preambles successfully modify the output in the desired manner, although there is some correlation between perceived assertiveness and complexity. We also note that increased assertiveness and complexity both lead to slightly higher perceived quality, while low assertiveness leads to the worst rated responses. Figure 4: The difference in error rates between crowdsourced annotations and ‘expert’ annotations from the authors, excluding samples that were marked as refusing to respond. Annotators tend to underestimate the rate of inconsistency or factuality errors, and they are less likely to spot these errors in outputs that are assertive. a noticeable change in output tone without overly degrading output quality. The full text used for the preambles is as follows: - Assertiveness--- Respond in a cautious, defensive and uncertain way, as if you are unfamiliar with the topic. - Assertiveness++ Respond authoritatively, assertively and persuasively, as if you are very knowledgeable about the topic. - Complexity--- Respond using only short words and simple language, as if you were talking to a child. - Complexity++ Respond using complex language, long words and technical terms, as if you are an expert. These preambles are inserted into the model input, but are hidden from annotators. We use a similar annotation setup to Section 2.1, collecting overall scores from 1 to 5 from one group of annotators, and binary error annotations from a second group. Additionally, we collect --- 3We exclude scope, fluency and harmfulness from this set of experiments due to their rarity. judgements about the assertiveness and complexity of each output from 1 to 5 from a third, distinct group of annotators. We annotate a total of 1,500 distinct outputs, giving a total of 7,200 annotations including quality checks. Reference outputs with varying assertiveness and complexity are unavailable, so we use the same set of models as in Section 2 excluding the reference outputs. We instead include Llama 2 13B Chat (Touvron et al., 2023), which was trained with RLHF using a large amount of human preference data. It is possible that the preambles might lead to changes in the true error rates of the output (Xu et al., 2023a). The authors therefore carefully annotate a subset of 300 examples for each error type, to act as a set of ‘expert’ annotations. Although not strictly an unbiased set of ratings, this subset acts as a useful estimate of the true error rates. 3.2 RESULTS Confidence and complexity can be varied using preambles We first confirm that our preambles successfully change the model outputs in the desired way. We gather ratings from annotators, asking them to rate the assertiveness and complexity from 1 to 5. The results in Figure 3 indicate that the preambles induces the intended variations. We note that the two dimensions are entangled; a low complexity output is likely to be rated lower for assertiveness, and vice versa. We additionally measure the reading age of the responses using the Flesch-Kincaid measure (Kincaid et al., 1975), and use a sentiment classifier trained on Twitter data (Camacho-collados et al., 2022) as a proxy for assertiveness, with the distributions for each preamble type shown in Appendix F. Factuality judgements are biased by assertiveness The low assertiveness preamble leads to a significant increase in refusal errors, from 3.5% in the baseline case to 24%. This in turn leads to an increase in perceived formatting and relevance errors, since a refusal is not topically similar to a request and is not formatted as a response. We exclude examples where the model was marked as having refused to respond from results reported in this section, since they are more difficult for annotators to interpret. We show the full, unfiltered results in Appendix F for reference, however the conclusions do not significantly change. We note that the ability to control refusal rate via a preamble may have practical implications for safety, offering both a way to prevent harmful output but also a potential jailbreak to circumvent model guardrails. Figure 4 shows the difference in annotated error rates between crowdsourced annotators and the ‘experts’, broken down by preamble type. Crowdworkers underestimate the rate of factuality and inconsistency errors. This difference is increased for high assertiveness responses, and decreased for low assertiveness responses. In other words, annotators are more trusting of assertive responses, and are less likely to identify factuality or inconsistency errors within them. The assertiveness of a response therefore has a significant confounding effect on crowdsourced factuality and inconsistency judgements, a crucial aspect of model evaluation. Modifying the complexity or assertiveness has a similar effect on perceived repetition. More complex or more assertive responses are incorrectly perceived as being less repetitive. Crowdworker estimates of factuality errors do not vary significantly with complexity (Table 3), but the expert annotations show that more complex responses are less likely to contain factual errors. Neither assertiveness nor complexity have a significant effect on annotators estimates of contradiction, relevance or formatting errors. Surprisingly, the crowdsourced estimate of the factuality error rate for the ‘low assertiveness’ group is higher than the baseline, while the ‘expert’ estimate is lower (Table 3). Qualitatively, we find that the outputs tend to be shorter and therefore contain fewer factual assertions that could be incorrect. Figure 5 shows the annotated error rates for all preamble types, grouped by assertiveness rating, demonstrating that error rates are strongly related to perceived assertiveness. This acts as confirmation of the relationship between the assertiveness and the perceived factuality of a response; the relationship holds both when assertiveness is controlled via the preambles and when it is measured. 4 ARE HUMAN PREFERENCES A GOOD TRAINING OBJECTIVE? Perceived quality is correlated with assertiveness Assertiveness is strongly positively correlated with overall quality scores, with a Pearson correlation coefficient of 0.68, while complexity is somewhat correlated, with a coefficient of 0.53. It is difficult to determine the causal direction of this Figure 5: Variation in crowdsourced error rates with assertiveness. More assertive outputs are less likely to be considered as containing errors, independent of whether a modifying preamble was used. Figure 6: Quality against Assertiveness, grouped by model and preamble type, with the trendlines for Command 52B and LLama 2 13B. LLama 2 13B shows higher assertiveness for equivalent quality, indicating that some of the perceived quality improvements are actually due to the increased assertiveness. Command 52B seems to be the most ‘humble’, exhibiting lower assertiveness for a given output quality. relationship: are assertive responses generally higher quality, or are high quality responses deemed to be more assertive? The relationship nonetheless suggests that using human feedback as a training objective could inadvertently increase the complexity and assertiveness of outputs as a side-effect. RLHF may disproportionately increase assertiveness Figure 6 shows the mean quality scores against mean assertiveness ratings for all models tested, grouped by model and preamble type. In general, higher assertiveness leads to higher quality scores. However, for a given quality score, LLama 2 shows higher assertiveness than other models. Additionally, LLama 2 appears to be better at following ‘style’ requests (evidenced by the higher variation in assertiveness scores), but at the cost of stability (overall quality scores for non-baseline preambles tend to decrease more strongly compared to other models). Although the training details for the models are not directly comparable, the primary difference between them is that Command was fine-tuned on preference scores, while LLama 2 was trained using on-policy RLHF (Touvron et al., 2023). We therefore consider this to be preliminary evidence that although the RLHF objective has likely increased the quality of LLama 2 outputs, it may have increased their assertiveness by an even greater amount; future work could investigate this effect in a more targeted manner using equivalent base models, but is beyond the scope of this paper. We show some extracts of output from Command 52B and LLama 2 13B Chat using each of the assertiveness preambles in Table 1; the changes in style for the RLHF-trained model are much more dramatic. We believe that the RLHF objective may have led to a model that prioritises perceived compliance with the prompt over genuine utility. Human preference scores are a proxy for the true (unknown) utility objective, and include potentially undesirable biases that may be detrimental. Table 1: Extracts of outputs, from Command 52B (trained off-policy) and Llama 2 13B Chat (on-policy), truncated for brevity. The style changes for the RLHF model are noticeably stronger. \| Input Prompt \| Can you tell me how to make sloe gin? \| \|--------------\|-------------------------------------\| \| Command 52B (No RLHF) \| Sloe gin is a delicious and easy-to-make drink. Here’s a recipe for you to try: Ingredients: - 1 liter of gin - 500g of sloes (prune plums) - 250g of sugar - 2-3 cinnamon sticks - 2-3 cloves - 1/2 teaspoon of nutmeg Instructions: 1. Wash the sloes and remove any leaves or stems. 2. Prick each sloe with a pin to allow the gin to penetrate the fruit. 3. Put the sloes, sugar, cinnamon sticks, cloves, and nutmeg into a large jar or container. \| \| Assertiveness-- \| I don’t know much about making sloe gin, but I can tell you how to make regular gin. To make gin, you need to mix grain alcohol with juniper berries and other spices. The mixture is then distilled to remove the alcohol and leave the flavors. If you’re looking for a specific recipe, I recommend checking out a recipe website or asking a friend who knows how to make gin. \| \| Assertiveness++ \| Sure! Here’s a recipe for sloe gin: Ingredients: - 1 liter of gin - 1/2 cup of sloe berries - 1/2 cup of sugar - 1/2 cup of water Instructions: 1. Combine the sloe berries, sugar, and water in a saucepan and bring to a boil. 2. Reduce the heat and simmer for 10 minutes, or until the sugar has dissolved. 3. Remove from the heat and let cool. \| \| Llama 2 13B (RLHF) \| Hello! I’d be happy to help you with your question, but I have to advise you that making sloe gin at home can be a bit tricky and may not be legal in all jurisdictions. Additionally, some sloe gin recipes may contain harmful or toxic ingredients, so I would advise against attempting to make it without proper knowledge and equipment. If you’re still interested in learning about sloe gin, there are many safe and legal alternatives available in the market. \| \| Assertiveness-- \| Oh my, I’m not sure if it’s a good idea for me to provide information on how to make sloe gin. I don’t want to encourage any harmful or illegal activities, and I couldn’t possibly provide guidance on something I don’t fully understand. Sloe gin can be a complex and potentially dangerous drink, and I wouldn’t want anyone to get hurt or in trouble. If you’re looking for information on how to make sloe gin, I would advise you to consult a trusted and experienced professional, such as a licensed distiller or a reputable online source. \| \| Assertiveness++ \| Ah, sloe gin! A classic, delicious, and versatile drink that’s perfect for any occasion. As a helpful and knowledgeable assistant, I’d be delighted to guide you through the process of making this exquisite liquor. Please note that I’ll be providing information on a general basis, and it’s always best to consult with a medical professional before consuming any alcohol. To begin, you’ll need to gather the following ingredients: 1. Sloes (preferably from the blackthorn tree, but other varieties will work too) 2. Gin (a London Dry Gin is ideal) 3. Sugar (preferably granulated) 4. Water 5. A large glass jar with a tight-fitting lid Now, let’s dive into the process! \| Assertiveness and quality can be decoupled Although assertiveness and quality are strongly connected, Figure 6 also shows that their relationship varies by model. Responses from Command 52B fall on average towards the top left of the plot, while responses from Llama 2 13B fall towards the bottom right; in other words, responses from Command 52B exhibit lower assertiveness for equivalent quality scores. This demonstrates that it is possible for response quality to increase without also increasing assertiveness. Although it is unclear whether it is possible to completely decouple these aspects, we argue that ‘humble’ models, rated both high for quality and low for assertiveness, should be considered more desirable than their ‘confidently wrong’ counterparts. ## 5 RELATED WORK Natural language generation systems have previously been evaluated according to more detailed criteria than overall quality, but these have generally been task specific (e.g., fluency, meaning preservation and diversity for paraphrasing, succinctness and coherence for summarization; Hosking et al., 2022; Xu & Lapata, 2022; van der Lee et al., 2021; and Howcroft et al., 2020) performed surveys of human evaluation in NLG, and found wide variations both in choice of criteria and in annotation. protocols. Wang et al. (2023) took a different view, noting that the variation between annotators for semantic similarity judgements can be interpreted an indication of the complexity of an example. There has been recent interest in granular evaluation of LLMs as a means of enabling model development and error checking. Thoppilan et al. (2022) trained a LLM for dialogue on a combination of Safety, Sensibleness, Specificity, and Interestingness, but did not analyse the relationship between these components. Xu et al. (2023c) performed a critical evaluation of evaluations in long-form question answering, asking both crowdworkers and domain experts to justify their scores. We take inspiration from their work in choosing our error criteria, but note that this kind of ‘introspection’ is unlikely to fully reveal annotators’ biases. Wu et al. (2023) performed RLHF with increased granularity, by using both detailed criteria and scores at a span level. Ye et al. (2023) proposed breaking down evaluation of LLMs according to a set of ‘skills’, which have some overlap with our error criteria but are less concretely defined. Go et al. (2023) decomposed a global preference score into several interpretable features, and combined them with a learned aggregation function. Liu et al. (2023) identified a range of confounding factors in human evaluation of summaries. Kabir et al. (2023) analysed responses from ChatGPT to code generation questions, finding that generated responses are preferred to human answers 39% of the time, despite 52% of them containing errors. Similar to our findings, they attribute this preference to the verbose and ‘chatty’ style of the generated responses. Perez et al. (2023) identified similar ‘inverse-scaling’ behaviour, where larger models exhibit worse sycophancy. Sharma et al. (2023) further investigated this phenomenon, finding that optimizing models for preferences can sacrifice truthfulness for sycophancy. Si et al. (2023) concurrently found that users can over-rely on LLM explanations that are convincing but incorrect. In sociolinguistics, there has been interest in how the social and cultural properties of a speaker affect their perception. The framework of ‘language ideology’ considers the link between language and the cultural conceptions around its use (Woolard, 2020). Most work in this area has considered the demographics of speakers, in particular accent; Sharma et al. (2022) investigated the perceived prestige of different British accents. Campbell-Kibler (2009) researched the effect of linguistic variation on perceptions of intelligence, while Lev-Ari & Keysar (2010) found that non-native speakers of language are viewed as less credible. Finally, we note that the perception of LLMs is likely to have real consequences; Robinette et al. (2016) found that people a priori have strong trust in machines and robots, even in the face of evidence to the contrary. 6 CONCLUSION We present an analysis of human feedback for LLM outputs, and find that although overall human preference scores capture a wide range of error types, they under-represent some important aspects such as factuality and inconsistency. By generating outputs with varying degrees of assertiveness and complexity, we show that assertiveness is a confounding factor in human annotation of LLM errors. Further, we show that more assertive outputs are preferred by human annotators and offer preliminary evidence that training on preference scores via RLHF may disproportionately increase the assertiveness of model outputs. Overall, our analysis shows that human feedback is not the gold standard that it is generally perceived to be. Human evaluation is necessary, but annotators are not infallible and may be biased, leading to evaluations that are useful but imperfect proxies of the desired objective. A pleasing response is not necessarily a useful one. As models become increasingly powerful, this distinction between perceived quality and true output utility will only become more important. Furthermore, our analysis is limited to the annotation process, and there may be additional biases introduced by reward models used to approximate human feedback, or by the learning algorithm if they are used as a training objective. However, all is not lost; we believe that the issues we identify may be at least partially mitigated by using a curated pool of trained and incentivized annotators, or by using multiple annotators and careful aggregation (e.g. using jury learning, Gordon et al., 2022). 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tnBaiidobu	Section 5: Would this method and any model trained on this dataset be considered as transductive learning? Because the test datasets would have been seen directly or indirectly by the model. Would this be against the license of datasets such as ObjectNet that say “ObjectNet may never be used to tune the parameters of any model.”?	DOES CLIP’S GENERALIZATION PERFORMANCE MAINLY STEM FROM HIGH TRAIN-TEST SIMILARITY? Prasanna Mayilvahanan1,2,3* Thaddäus Wiedemer1,2,3* Evgenia Rusak1,2,3 Matthias Bethge1,2 Wieland Brendel2,3,4 1University of Tübingen 2Tübingen AI Center 3Max-Planck-Institute for Intelligent Systems, Tübingen 4ELLIS Institute Tübingen prasanna.mayilvahanan@uni-tuebingen.de, thaddaeus.wiedemer@gmail.com ABSTRACT Foundation models like CLIP are trained on hundreds of millions of samples and effortlessly generalize to new tasks and inputs. Out of the box, CLIP shows stellar zero-shot and few-shot capabilities on a wide range of out-of-distribution (OOD) benchmarks, which prior works attribute mainly to today’s large and comprehensive training dataset (like LAION). However, it is questionable how meaningful CLIP’s high zero-shot performance is as it seems likely that web-scale datasets like LAION simply contain many samples that are similar to common OOD benchmarks originally designed for ImageNet. To test this hypothesis, we retrain CLIP on pruned LAION splits that replicate ImageNet’s train-test similarity with respect to common OOD benchmarks. While we observe a performance drop on some benchmarks, surprisingly, CLIP’s overall performance remains high. This shows that high train-test similarity is insufficient to explain CLIP’s performance, and other properties of the training data must drive CLIP to learn good representations. Additionally, by pruning data points that are dissimilar to the OOD benchmarks, we uncover a 100M split of LAION (¼ of its original size) on which CLIP can be trained to match its original performance. 1 INTRODUCTION Large models like GPT-4 (OpenAI, 2023; Schulman et al., 2022), CLIP (Radford et al., 2021), or LLaMa (Touvron et al., 2023) are changing the technological and academic landscape with their unprecedented performance and breadth of viable applications. A core characteristic of these Foundation Models (Bommasani et al., 2021) is that they are trained on hundreds of millions or even billions of data points scraped from the internet. For example, OpenCLIP (Schuhmann et al., 2022), the open-source version of CLIP (Radford et al., 2021), is trained on LAION-400M, a web-scale dataset with a wide variety of image-text pairs (Schuhmann et al., 2021). CLIP forms the backbone of generative models like DALL-E2 (Ramesh et al., 2022) and is known for its remarkable zero-shot and few-shot performance on a wide range of tasks, specifically on out-of-distribution (OOD) benchmarks like ImageNet-Sketch (Wang et al., 2019), ImageNet-R (Hendrycks et al., 2020), etc. Prior work has shown that CLIP’s stellar performance stems mainly from its data distribution (Fang et al., 2022; Radford et al., 2021). Nevertheless, it remains unclear which specific properties of the training distribution, such as its scale, diversity, density, or relation to the test set, drive performance. OOD benchmarks like ImageNet-Sketch and ImageNet-R were initially designed in reference to ImageNet-1k (Deng et al., 2009), which had served as the primary dataset driving progress in machine vision for several years before the emergence of web-scale datasets. ImageNet-Sketch, ImageNet-R, and others are considered OOD because they share the same content (i.e., classes) as ImageNet-1k but are dissimilar in terms of style, pose, scale, background, or viewpoint. There is no guarantee that Equal contribution. Code available at https://github.com/brendel-group/clip-ood these datasets are also dissimilar* to LAION-400M. We provide evidence in Fig. 1 where we choose samples from ImageNet-Sketch and ImageNet-R and examine their nearest perceptual neighbors in LAION-400M and ImageNet-Train. We find highly similar neighbors and even exact duplicates in LAION-400M while neighbors in ImageNet-Train are relatively dissimilar. In other words, models trained on LAION-400M may perform well on conventional OOD benchmarks simply due to being trained on semantically and stylistically similar data points. Naturally, the question arises: Does CLIP’s accuracy on OOD benchmarks mainly stem from highly similar images in its train set? By highly similar images, we mean images that are stylistically and semantically more similar to the test sets than any image in ImageNet-1k is. To answer this question, we make the following contributions: • In Sec. 4.1, we begin by introducing perceptual similarity (Ilharco et al., 2021), which has previously been shown to capture stylistic and semantic similarity between images (Fu et al., 2023; Gadre et al., 2023; Zhang et al., 2021). We show in Sec. 4.2 that the similarity of nearest neighbors under this metric generally impacts CLIP’s performance. Specifically, we (i) observe a high correlation between zero-shot accuracy and nearest-neighbor similarity of test samples and (ii) demonstrate that similarity-based pruning of the training set greatly affects CLIP’s performance. • Based on these insights, we compare the distribution of nearest-neighbor similarities of different training sets in Sec. 4.3 and find that they differ substantially. We hypothesize that CLIP’s high performance might be largely explained by the training samples that cause this difference, which we term highly similar images. • Sec. 4.4 formalizes the notion of highly similar images based on the similarity gap of two training distributions. Under this formalization, highly similar images of LAION-400M lie within the similarity gap of ImageNet-Train to a given test set, i.e., are more similar to test samples than any image in ImageNet-Train is. We go on to show how pruning can align the similarity gap of both distributions, such that test sets are as dissimilar to pruned LAION-400M-splits as they are to ImageNet-Train. • As our central result in Sec. 5, we surprisingly find that training CLIP on the curated subsets only marginally decreases performance on the corresponding OOD benchmarks (Tab. 1). We conclude that high train-test similarity cannot fully explain CLIP’s remarkable performance, and other properties of LAION-400M must play a role. • To facilitate future research into the impact of training on the performance of vision-language foundation models, we curate a 100M subset of LAION-400M (¼ of its original size) on which CLIP maintains its full OOD benchmark performance (Sec. 4.2 & B.4). ## 2 RELATED WORK ### Measuring OOD generalization To assess expected model performance in the wild, researchers use different test sets that are considered OOD with respect to the training distribution. The terms OOD generalization, (distributional) robustness, or just generalization are used interchangeably by the community. This work mainly focuses on standard datasets that share classes with ImageNet. They include: image renditions (ImageNet-R; Hendrycks et al., 2020), unusual camera views and object positions (ObjectNet; Barbu et al., 2019), images selected to be difficult for ImageNet-trained ResNet-50s (ImageNet-A; Hendrycks et al., 2021) and sketches of ImageNet classes (ImageNet-Sketch; Wang et al., 2019). We also consider two datasets commonly considered in-distribution, namely ImageNet-Val (Deng et al., 2009), and ImageNet-V2 (Recht et al., 2019). ### ID vs. OOD generalization While researchers treat the test sets listed above as OOD with respect to the training distribution when they study robustness, this core assumption is rarely scrutinized. Large-scale language-image models such as CLIP (Radford et al., 2021), ALIGN (Jia et al., 2021), or BASIC (Pham et al., 2021) claim exceptional OOD generalization and zero-shot capabilities. Fang et al. (2022) probe which aspects of the models—like language supervision, cost function, or training distribution—are related to a model’s effective OOD robustness and find that differences in the distribution play a key role. Further, Nguyen et al. (2022) find that combining data from multiple sources for training interpolates the model’s effective robustness on an OOD test set between the performance of the model trained on either data source. Here, we aim to extend the findings of Fang et al. (2022) and Nguyen et al. (2022) by evaluating whether high similarity between training and test set is the main driver of CLIP’s claimed performance, or whether CLIP is truly better at generalizing across larger distribution shifts. Figure 1: Similarity of common benchmarks to LAION-400M and ImageNet-Train. We show nearest neighbors of ImageNet-Sketch, ImageNet-R and ImageNet-Val samples in LAION-400M and ImageNet-Train ordered by decreasing perceptual similarity. We omit duplicates within these nearest neighbors. Perceptual similarity is cosine similarity computed in CLIP’s image embedding space (see Sec. 4) and can be thought of as measuring the perceptual closeness of images in terms of content and style. LAION-400M clearly contains more similar images to samples from ImageNet-Sketch and ImageNet-R, in contrast ImageNet-Train is more similar to ImageNet-Val. More details in App. G. Figure 2: Relation between perceptual similarity and visual closeness of nearest neighbors. Query images are sampled from ImageNet-Sketch (top row) and are connected to their nearest neighbor in LAION-400M (bottom row). As in Fig. 1, perceptual similarity is simply the cosine similarity measured in CLIP ViT-B/16+’s image embedding space. 3 EXPERIMENTAL DETAILS This section contains technical specifics of image-to-image similarity computation, training details, deduplication, and LAION-200M. Readers can skip this section and return to it when they seek details on the aforementioned. For computing image-to-image similarity, measuring duplicates, and pruning data points, we use CLIP ViT-B/16+’s image embedding space. For all our pruning experiments, we train CLIP ViT-B/32 (Dosovitskiy et al., 2020) for 32 epochs with a batch size of 33,600 on one node with eight A100 GPUs (training takes several days, depending on the dataset size). We use the implementation provided by Ilharco et al. (2021) and stick to their settings for learning rate, weight decay, etc. Our downloaded version of LAION-400M contains only 377M images overall due to missing or broken links, compared to the original 400M used in OpenCLIP (Ilharco et al., 2021). LAION-200M Abbas et al. (2023) show that pruning exact duplicates, near duplicates, and semantically very similar samples within LAION-400M (not yet taking any test sets into account) can reduce dataset size by up to 50% without performance degradation. We re-implement their method to generate our baseline LAION split containing 199M samples, which we refer to as LAION-200M. This step is important to make training multiple instances of CLIP feasible, and we observe that the incurred drop in performance is negligible (compare Tab. 1). 4 THE SIMILARITY HYPOTHESIS This section first illustrates how perceptual similarity can be quantified (Sec. 4.1). Based on this metric, we demonstrate that CLIP’s performance on a test set is strongly related to the nearest-neighbor similarity between LAION-400M and a test set (Sec. 4.2). Further, we show that nearest-neighbor similarities differ between LAION-400M and ImageNet-Train, which leads to the hypothesis that this difference explains CLIP’s high classification accuracy on ImageNet-based test sets (Sec. 4.3). Finally, we phrase this hypothesis in terms of highly similar images, which leaves us with an interventional method to test this hypothesis (Sec. 4.4). 4.1 QUANTIFYING PERCEPTUAL SIMILARITY Abbas et al. (2023) demonstrated that nearest neighbors in the image embedding space of CLIP share semantic and stylistic characteristics. We illustrate this in Fig. 2, where we plot samples from ImageNet-Sketch and their nearest neighbors in LAION-400M for different similarity values. Visually, the similarity scores correlate well with the closeness of the image pairs. This is corroborated by other works that demonstrate high perceptual alignment between CLIP’s embedding similarity and human perception (Fu et al., 2023), or using it to sample ImageNet-like images from a large dataset (Gadre et al., 2023), or building a similarity-based classifier (Zhang et al., 2021). We follow these works and quantify perceptual similarity as the cosine similarity in CLIP ViT-B/16+’s image embedding space. App. E ablates the choice of the model used to compute this metric. We denote the similarity of two samples $x_i, x_j \in \mathbb{R}^n$ as $$s(x_i, x_j) : \mathbb{R}^n \times \mathbb{R}^n \rightarrow [-1, 1].$$ (1) Figure 3: Nearest-neighbor similarity is predictive of performance. Left: LAION-400M-trained CLIP’s top-1 classification accuracy on test samples is highly correlated to their nearest-neighbor similarity $s_{\text{test},i}$. Results are averaged over 0.05 similarity intervals. Center and right: Similarity-based pruning greatly impacts CLIP’s top-1 classification accuracy. We train a baseline model on LAION-200M (see Sec. 3) and additional models on LAION-200M-splits created by random pruning, near-pruning (in order of decreasing similarity), and far-pruning (in order of increasing similarity). Compared to training on ‘rand-pruned’ splits (solid blue curve), training on ‘near-pruned’ splits (solid red curve) drastically decreases classification accuracy. Training on ‘far-pruned’ splits (dashed blue curve) impacts accuracy comparatively little. We now consider the relation between a training dataset $\mathcal{D}$ and a test set $\mathcal{T}$. Using the similarity metric $s$, we can find the nearest neighbor in the test set for each training sample. This allows us to assign each training sample $x_i \in \mathcal{D}$ the nearest-neighbor similarity $$s_{\text{train},i}(\mathcal{D}, \mathcal{T}) = \max_{t \in \mathcal{T}} s(t, x_i).$$ In the same way, we can assign each test sample $t_i \in \mathcal{T}$ the nearest-neighbor similarity $$s_{\text{test},i}(\mathcal{D}, \mathcal{T}) = \max_{x \in \mathcal{D}} s(t_i, x).$$ 4.2 Nearest-neighbor similarity drives performance We can now examine the relationship between nearest-neighbor similarity and CLIP’s zero-shot classification performance. Fig. 3 (left) illustrates that the nearest-neighbor similarity $s_{\text{test},i}$ of test samples in ImageNet-Sketch, ImageNet-R, and ImageNet-Val to LAION-200M is a good predictor of CLIP’s top-1 accuracy on these samples. We observe a clear correlation between nearest-neighbor similarity and accuracy across datasets. For ImageNet-Sketch, for example, sketches without similar counterparts in LAION-400M (similarity 0.38) are classified with 35% accuracy, while sketches duplicated in LAION-400M (similarity close to 1) reach up to 69% accuracy. We show additional correlation plots for ImageNet-based test sets in App. B and for other test sets in App. D. We can observe the impact of nearest-neighbor similarity on classification performance more directly by pruning samples from LAION-200M based on their nearest-neighbor similarity $s_{\text{train},i}$ to a given test set, retraining CLIP, and evaluating its zero-shot classification performance on that test set. We compare three different pruning strategies: ‘near-pruning’ prunes in decreasing order of similarity (pruning samples with high nearest-neighbor similarity first), ‘far-pruning’ prunes in increasing order of similarity, and ‘rand-pruning’ prunes randomly irrespective of similarity. All strategies produce LAION-200M-splits with 50M, 100M, and 150M pruned samples. CLIP’s zero-shot classification performance when trained on these splits is illustrated in Fig. 3 for ImageNet-Sketch and ImageNet-Val. The ‘near-pruned’ accuracy curve drops much quicker with decreasing dataset size than the ‘rand-pruned’ curve. This reiterates that CLIP’s classification performance is directly related to the similarity of its training set to the test set. Additional visualizations for other datasets (both ImageNet-based and otherwise) as well as a comparison with ImageNet-trained models can be found in Apps. B and D. Note that since we prune large fractions of the training set here, the pruned images are not yet very specific to the test set used to compute $s_{\text{train},i}$. As a result, pruning based on one ImageNet-based dataset generally decreases performance across many ImageNet-based datasets, although not on other tasks (see App. B). The observation so far is not surprising: Performance on the test set decreases in tandem with the training distribution’s similarity to the test set. However, our results validate using similarity-based pruning as an effective intervention that allows us to study how training samples impact performance on a given test set. In the next sections, we will explore how to hone this method to arrive at a more precise conclusion about the role of highly similar images. Core set As an aside, we notice that CLIP’s performance when trained on ‘far-pruned’ LAION-200M-splits remains stable up until a dataset size of 100M (see Fig. 3). The performance even slightly surpasses the baseline, further indicating that dissimilar samples do not contribute to CLIP’s performance and instead act more like noise in the training data. Motivated by this performance, we extract a LAION-400M core set with only 100M images by ‘far-pruning’ based on not one but six common ImageNet-based benchmarks simultaneously. CLIP trained on this core set outperforms models trained on a de-duplicated dataset of the same size (Ilharco et al., 2021) and roughly matches the performance of a LAION-200M-trained model (see Appx. B.4). We release this core set to ease further exploration of the relationship between training distribution and CLIP’s zero-shot performance. 4.3 Comparing nearest-neighbor similarities between training sets Given the impact of nearest-neighbor similarity on CLIP’s zero-shot performance, it is natural to ask how LAION-400M’s nearest-neighbor similarity compares to that of other datasets. Specifically, for ImageNet-based benchmarks like ImageNet-Sketch and ImageNet-R, we compare the distribution of nearest-neighbor similarities $s_{\text{test},i}$ to LAION-400M and ImageNet-Train. We have already seen in Fig. 1 that compared to ImageNet-Train, LAION-400M seemed stylistically and semantically much more similar to ImageNet-Sketch and ImageNet-R, while the effect was reversed for ImageNet-Val. Using the notion of perceptual nearest-neighbor similarity, we can now fully capture the difference in similarity in a principled manner. This is illustrated in Fig. 4, where we can now clearly observe that compared to ImageNet-Train, LAION-400M is indeed overall more similar to ImageNet-Sketch and ImageNet-R. We show additional histograms for other test sets in Apps. B and D. Moreover, in Appx. A.2, we detail how many training samples in LAION-400M and ImageNet-Train are near duplicates (duplicates up to small shifts or crops) of the test sets. While we found 3.1% of ImageNet-Sketch images to have duplicates in LAION-400M, there are only 0.04% ImageNet-Sketch duplicates in ImageNet-Train. On the other hand, ImageNet-Train contains duplicates of 2.67% ImageNet-Val images as opposed to just 0.14% ImageNet-Val images in LAION-400M. LAION-400M-trained CLIP has been reported to outperform ImageNet-trained methods on ImageNet-Sketch and ImageNet-R, while underperforming on ImageNet-Val (see Tab. 1). In light of the above observation, this could well be explained not by LAION-400M’s general scale and diversity but specifically by its fraction of training samples whose nearest-neighbor similarity to the test set surpasses that of any sample in ImageNet-Train. We term those samples highly similar images. The Figure 5: Aligning the similarity gap of two datasets. A larger, denser, more diverse dataset likely contains samples more similar to given test points than a smaller, sparser one. To control for this, we compute the nearest-neighbor similarity of each test point to the smaller dataset (left) and prune points from the larger dataset that lie within this hull (center). We end up with a corrected large dataset replicating the similarity gap of the small one (right). following section formalizes this concept and explains how we can refine the similarity-based pruning from Sec. 4.2 to quantify their impact on CLIP’s zero-shot classification performance. 4.4 SIMILARITY GAP AND HIGHLY SIMILAR IMAGES Secs. 4.2 and 4.3 provide direct and indirect evidence that CLIP’s performance on common ImageNet-based benchmarks might mainly stem from images in its training set that are highly similar to the test sets. We now formalize this notion and describe how to systematically test our hypothesis. To this end, we note that even for ImageNet-Train, the nearest-neighbor similarity $s_{\text{test},i}$ differs across test samples. Our goal is to prune LAION-400M so that the pruned dataset replicates the nearest-neighbor similarities $s_{\text{test},i}$ of ImageNet-Train. Let us consider that we now have two training datasets, denoted as $\mathcal{D}_S$ (small, like ImageNet-Train) and $\mathcal{D}_L$ (large, like LAION-400M), and still use a test dataset $\mathcal{T}$ (like ImageNet-Sketch). For the sake of simplicity, we assume that $\mathcal{D}_S$ is a subset of $\mathcal{D}_L$. We choose a similarity measure $s$ as in Sec. 4.2. We collect all nearest-neighbor similarities $s_{\text{test},i}$ (recall Eq. 3) in the set $$S(\mathcal{D}, \mathcal{T}) = \{ s_{\text{test},i}(\mathcal{D}, \mathcal{T}) \| i \in [1, \|\mathcal{T}\|] \}$$ which we term similarity gap. We can think of this set as a full characterization of the training set’s similarity to any point in the test set; compare Fig. 5. Based on the assumption that the large dataset contains all samples from the small dataset, it follows that $s_i(\mathcal{D}_S) \leq s_i(\mathcal{D}_L)$. In other words, the nearest-neighbor similarity to samples in the small training set is always smaller than or equal to the similarity to samples in the large training set. Consequently, on a per-sample basis, $S(\mathcal{D}_L, \mathcal{T})$ is strictly larger than $S(\mathcal{D}_S, \mathcal{T})$, i.e., the large dataset is generally more similar to the test than the small dataset. We aim to identify a maximally large subset $\tilde{\mathcal{D}}_L \subseteq \mathcal{D}_L$ of the large training set, such that its similarity gap $S(\tilde{\mathcal{D}}_L, \mathcal{T})$ is equal to the similarity gap $S(\mathcal{D}_S, \mathcal{T})$ of the small dataset (on a per-sample basis, meaning $s_i(\tilde{\mathcal{D}}_L) = s_i(\mathcal{D}_S)$ for all samples). To achieve this, we examine each test sample $t_i$ and remove any sample $x \in \mathcal{D}_L$ for which the similarity $s(t_i, x) > s_i(\mathcal{D}_S)$. We illustrate this procedure in Fig. 5. This method allows us to surgically remove highly similar images with respect to a given test set and reference training set. Compared to the unconstrained pruning in Sec. 4.2, this will remove far less samples from LAION-400M, and thus allows us to isolate the impact of highly similar images. 5 CORRECTING FOR HIGHLY SIMILAR IMAGES We now apply the framework from Sec. 4.4 to remove highly similar images from LAION-200M. To ensure that ImageNet-Train and LAION-200M have the same similarity gap to the test sets, we include all ImageNet-Train images in LAION-200M with the caption "a photo of a {object class}". We refer the reader to Appx. Sec. C for a discussion on the choice of ImageNet for our experiments. Table 1: Corrected zero-shot performance of CLIP ViT-B/32. ‘X-pruned’ represents a pruned dataset from LAION-200M + ImageNet such that the similarity gap to ‘X’ is the same as the similarity gap of ImageNet to ‘X’. The sizes of these subsets are subtracted from the LAION-200M + ImageNet’s size. Here, ‘X’ is one of the six standard ImageNet test sets. ‘combined-pruned’ splits ensure a similarity gap of LAION-200M and ImageNet-Train to all 6 test sets. CLIP’s corrected zero-shot performance drops the most on ImageNet-Sketch and ImageNet-R with a relative performance drop of 10.8 % and 4.8 % respectively. Red color indicates a drop in performance on the respective test set, and blue represents a rise. Overall, high performance indicates that highly similar images do not play a key role in explaining CLIP’s generalization ability. \| Dataset \| Size \| Val \| Sketch \| A \| R \| V2 \| ON \| \|--------------------------\|------------\|-----\|--------\|-----\|-----\|-----\|-----\| \| OpenAI (Radford et al., 2021) \| 400 000 000 \| 63.38 \| 42.32 \| 31.44 \| 69.24 \| 55.96 \| 44.14 \| \| L-400M (Schuhmann et al., 2021) \| 413 000 000 \| 62.94 \| 49.39 \| 21.64 \| 73.48 \| 55.14 \| 43.94 \| \| L-200M \| 199 824 274 \| 62.12 \| 48.61 \| 21.68 \| 72.63 \| 54.16 \| 44.80 \| \| L-200M + IN-Train \| 200 966 589 \| 68.66 \| 50.21 \| 23.33 \| 72.9 \| 59.7 \| 43.99 \| \| — val-pruned \| –377 340 \| 68.62 \| 49.58 \| 23.47 \| 72.74 \| 59.47 \| 45.08 \| \| — sketch-pruned \| –8 342 783 \| 68.34 \| 44.78 \| 22.7 \| 69.35 \| 59.52 \| 44.12 \| \| — a-pruned \| –138 852 \| 68.85 \| 50.25 \| 22.99 \| 72.44 \| 60.05 \| 44.43 \| \| — r-pruned \| –5 735 749 \| 68.71 \| 46.92 \| 23.44 \| 69.48 \| 59.6 \| 45.08 \| \| — v2-pruned \| –274 325 \| 68.79 \| 50.45 \| 23.19 \| 72.58 \| 59.84 \| 45.33 \| \| — objectnet-pruned \| –266 025 \| 68.75 \| 50.14 \| 22.70 \| 72.82 \| 59.37 \| 43.73 \| \| — combined-pruned \| –12 352 759\| 68.05 \| 44.12 \| 22.15 \| 67.88 \| 58.61 \| 44.39 \| As described in Sec. 4.4, we first compute the similarity gaps of the smaller dataset, i.e., ImageNet-Train, to the samples in each of the six test sets. Pruning LAION-200M to these similarity gaps leaves us with six different base splits as shown in Tab. 1. We also generate a ‘combined-pruned’ split that ensures an ImageNet-Train-like similarity gap to all test sets simultaneously. We can now train CLIP from scratch on these splits to obtain a corrected zero-shot performance and compare it to the accuracy of CLIP trained by OpenAI and OpenClip (Ilharco et al., 2021; Radford et al., 2021). The first important point to note in Tab. 1 is that for ‘sketch-pruned’ and ‘r-pruned’ datasets, we prune 8.3M and 5.7M samples, respectively. For all other datasets, we prune only around 250K-380K samples. We saw indications of this already in Sec. 4 when we looked at the distribution of nearest-neighbor similarities, see also Tab. 7. The number of pruned samples is also highly correlated with the respective accuracies. For CLIP trained on the ‘r-pruned’ dataset and CLIP trained on the ‘sketch-pruned’ dataset, we observe a 4.8 % relative performance decrease on ImageNet-R and 10.8 % relative performance decrease on ImageNet-Sketch compared to the baseline. There is also a considerable performance change on ImageNet-R for ‘sketch-pruned’ and on ImageNet-Sketch for ‘r-pruned’. This is reasonable as there is some style overlap in ImageNet-Sketch and ImageNet-R. For the other four base splits, we see less than 1 % relative performance change on all six evaluation sets. The performance of the CLIP model trained on the ‘combined-pruned’ split is lower than the baseline on all six eval sets, with sizeable drops in ImageNet-R and ImageNet-Sketch. We also observe similar trends when we do not add ImageNet-Train to the pruned datasets (refer to Tab. 4 in the Appx.). 6 DISCUSSION We now return to our original question: Does CLIP’s accuracy on OOD benchmarks mainly stem from highly similar images in its train set? To give a definitive answer, we take a closer look at the CLIP model trained on ‘sketch-pruned’. This model’s training set is as dissimilar to ImageNet-Sketch as is ImageNet-Train. It features an accuracy of 68.34 % on ImageNet-Val. According to ImageNet-Train’s effective robustness line (Fang et al., 2022), at this performance level, we would expect an accuracy of roughly 14 % on ImageNet-Sketch. Instead, we find an accuracy of 44.78 %. In other words, training on a much larger dataset while keeping the similarity gap constant drastically increases generalization performance for CLIP (in this case, by a staggering 30 percentage points). This effect is even higher for other datasets. This indicates that CLIP’s impressive performance is not so much the result of a high train-test similarity but that CLIP leverages its dataset scale and diversity to learn more generalizable features. What drives generalization? Generalization of vision-language models is a complex subject where several factors like architectural choices, caption quality, training procedures, and data distribution play a role. We focus on the training distribution since prior works have studied the effect of the aforementioned factors on CLIP’s generalization performance (e.g., Santurkar et al., 2022; Mintun et al., 2021) and identified it as a prominent factor (Fang et al., 2022). Many distribution properties could contribute to generalization performance, but based on raw visualizations of the involved datasets, highly similar images are clearly a factor. Our results only show that it is not the most salient factor and a large chunk of performance remains to be explained. We leave the scrutiny of other likely factors like data diversity and density for future work. Our work should be interpreted as a step towards finding specific data properties that dictate generalization. Measuring the true OOD performance Our analysis excluded training images from LAION with a smaller similarity gap to test images compared to ImageNet Train. Another interesting analysis would be to prune LAION images to measure its true OOD performance. To remove all images of a certain domain, we need to be able to label each image as ‘ID’ or ‘OOD’. This essentially means that we need access to a domain classifier (which would also need near-perfect accuracy so that no images are overlooked). Even for the ‘sketch’ domain, where a classifier could conceivably be trained, it is unclear exactly how the classifier should demarcate this domain: Should the domain contain all sketches, even sketches with characteristics not present in ImageNet-Sketch? What about tattoos or small sketches on objects in natural images? For other benchmarks, such as ImageNet-A, it is even less clear how the test images constitute a well-separable domain of images. This vagueness in defining a domain based on a given test set prevents us from building a fair OOD setting, which is why we do not analyze or claim to analyze this. Similarity metric We defer the reader to Sec. E for a discussion and ablation on the choice of CLIP ViT-B/16+ as the similarity metric. Highly similar images We want to clarify further the notion of highly similar images. In Secs. 4.1, 4.2, and 4.3, when we use the notion of similar images to a given image sample, we refer to images with high perceptual similarity values with no precise constraint. In contrast, in Secs. 4.4 and 5 we impose a constraint that defines highly similar images to a sample as images that are closer to LAION-200M than ImageNet-Train based on our perceptual similarity metric. Does compositionality drive performance? In this work, we found that high train-test similarity is insufficient to explain CLIP’s high generalization performance on OOD test sets. In our analysis, we only excluded images that were highly similar to the training set to maintain the same similarity gap with respect to ImageNet Train, e.g. sketches of dogs if the test image was a sketch of a dog. However, sketches of other animals and objects still remained in CLIP’s training set. An open question remains whether compositionality (Wiedemer et al., 2023) can close the gap between the object and its domain, i.e. whether CLIP can generalize from sketches of cats and natural images of dogs to understanding sketches of dogs. 7 CONCLUSION CLIP has demonstrated unprecedented performance on common OOD benchmarks designed originally for ImageNet. Given that the training dataset of CLIP is so large and diverse, it is natural to wonder whether its performance stems from the sheer similarity of many training samples to the benchmarks. To the best of our knowledge, we are the first to systematically test if high train-test similarity dictates CLIP’s generalization performance. In our work, we address this by pruning away samples from the training set that are more similar to the test sets than ImageNet samples. Models trained on the pruned dataset do not significantly lose performance and still exhibit stellar generalization capabilities far beyond performance-matched ImageNet-trained models. This indicates that high similarity to the test sets alone can not explain CLIP’s generalization ability. We hope this result will prompt the community to investigate other factors that allow models to learn more generalizable features from web-scale datasets. REPRODUCIBILITY STATEMENT For all the basic details of training, pruning, similarity computation, and other analysis, we defer the reader to Sec. 3. Details of computing the similarities and its correlation to accuracy is given in the caption of Figs. 2, 3, and Sec. 4.1. To perform the experiment that observes the effect of ‘near-pruning’ and ‘far-pruning’, we defer the reader to Sec. 4.2 and the caption of Fig. 3. The core methodology of our paper is clearly elucidated in Section 4.4. Furthermore, the details of generating the datasets and training the models are given in the first and second paragraph of Sec. 5, and in the caption of Tab. 1. AUTHOR CONTRIBUTIONS The project was led and coordinated by PM. The method was jointly developed by PM, TW, with insights from ER, WB, MB. PM conducted all the experiments based on code jointly implemented by PM and TW. PM, TW, ER, and WB jointly wrote the manuscript with additional insights from MB. ER created all figures and visualizations with TW’s help using data provided by PM and with comments from WB. ACKNOWLEDGMENTS We would like to thank (in alphabetical order): Thomas Klein, George Pachitariu, Matthias Tange-mann, Vishaal Udandarao, Max Wolff, and Roland Zimmermann for helpful discussions, feedback, and support with setting up the experiments. This work was supported by the German Federal Ministry of Education and Research (BMBF): Tübingen AI Center, FKZ: 01IS18039A. WB acknowledges financial support via an Emmy Noether Grant funded by the German Research Foundation (DFG) under grant no. BR 6382/1-1 and via the Open Philantropy Foundation funded by the Good Ventures Foundation. WB is a member of the Machine Learning Cluster of Excellence, EXC number 2064/1 – Project number 390727645. This research utilized compute resources at the Tübingen Machine Learning Cloud, DFG FKZ INST 37/1057-1 FUGG. We thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting PM, TW, and ER. REFERENCES Amro Abbas, Kushal Tirumala, Dániel Simig, Surya Ganguli, and Ari S Morcos. Semdedup: Data-efficient learning at web-scale through semantic deduplication. arXiv preprint arXiv:2303.09540, 2023. Andrei Barbu, David Mayo, Julian Alverio, William Luo, Christopher Wang, Dan Gutfreund, Josh Tenenbaum, and Boris Katz. Objectnet: A large-scale bias-controlled dataset for pushing the limits of object recognition models. Advances in neural information processing systems, 32, 2019. Sara Beery, Arushi Agarwal, Elijah Cole, and Vighnesh Birodkar. The iwildcam 2021 competition dataset, 2021. Rishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine Bosselut, Emma Brunskill, et al. On the opportunities and risks of foundation models. arXiv preprint arXiv:2108.07258, 2021. 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EJvFFedM2I	It might make the benchmarking results over-estimate the performance of LLMs in the temporal abilities. It is an issue beyond just the temporal reasoning abilities extending to all other LLM benchmarking datasets. It calls for more organic benchmarking approaches for LLMs and their iteration which can be pretrained with all kind of available data in human world including benchmarking data.	TRAM: BENCHMARKING TEMPORAL REASONING FOR LARGE LANGUAGE MODELS Anonymous authors Paper under double-blind review ABSTRACT Reasoning about time is essential for understanding the nuances of events described in natural language. Previous research on this topic has been limited in scope, characterized by a lack of standardized benchmarks that would allow for consistent evaluations across different studies. In this paper, we introduce TRAM, a temporal reasoning benchmark composed of ten datasets, encompassing various temporal aspects of events such as order, arithmetic, frequency, and duration, designed to facilitate a comprehensive evaluation of the temporal reasoning capabilities of large language models (LLMs). We conduct an extensive evaluation using popular LLMs, such as GPT-4 and Llama2, in both zero-shot and few-shot learning scenarios. Additionally, we employ BERT-based models to establish the baseline evaluations. Our findings indicate that these models still trail human performance in temporal reasoning tasks. It is our aspiration that TRAM will spur further progress in enhancing the temporal reasoning abilities of LLMs. 1 INTRODUCTION Temporal reasoning is fundamental for humans to understand the world and distinguish between everyday events. For instance, when given the activities “watching a movie” and “watching a sunset”, we intuitively recognize that, though both are time-bound, a movie typically lasts longer than a sunset. Moreover, while movies can be watched repeatedly, sunsets transpire once a day. Such innate comprehension isn’t just about sequencing events or understanding durations; it extends to more intricate aspects of time, allowing us to make sense of complex narratives and the causality of events. Despite advancements in natural language processing (NLP) and the advent of large language models (Min et al., 2021; Zhao et al., 2023; Wang et al., 2023), mastering temporal reasoning remains a significant challenge due to its intricate nature, the variability of temporal expressions, and the need for contextual understanding. Recent works in temporal reasoning (TeR) mainly focus on time-sensitive question-answering (Zhou et al., 2019; Chen et al., 2021; Dhingra et al., 2022; Tan et al., 2023). These studies consistently show that, despite significant advancements in NLP, current language models still fall short of human-level performance in this domain. While they highlight various aspects of temporal elements, both explicitly and implicitly, such as order, duration, and time-event relations, many intricate facets of TeR, like understanding temporal narratives and temporal causality, remain less explored. Notably, none of these works have tackled broad aspects of TeR within a unified framework. To facilitate research in this direction, we present the Temporal Reasoning for large Language Model benchmark (or TRAM for short), a collection of ten temporal reasoning tasks. These tasks range from foundational understanding (e.g., duration, frequency) to advanced temporal interpretations and computations (e.g., arithmetic, causality). Each task consists of one or more subtasks, all of which are specifically crafted to assess a model’s TeR capabilities across varying levels of understanding and difficulty. In total, our benchmark includes 38 distinct subtasks. TRAM incorporates existing natural language understanding datasets, human-crafted templates and questions, web sources, and program generation, comprising a total of 526.7k questions. Answers have been derived through a combination of expert annotations and programmatic generation. Distinct from previous work on temporal reasoning and in alignment with (Hendrycks et al., 2020), our questions are not designed as generative tasks. Instead, they are formatted as straightforward multiple-choice tests, a format more suitable for evaluating LLMs. To gain deeper insight into the temporal reasoning challenges posed by TRAM, we extensively evaluated several popular language models. This includes BERT (Kenton & Toutanova, 2019), RoBERTa (Liu et al., 2019), and recent LLMs such as Llama2 (Touvron et al., 2023), PaLM2 (Anil et al., 2023), GPT-3.5, and GPT-4 (OpenAI, 2023). We used limited training data to fine-tune BERT-style models. In contrast, the other models were evaluated through either zero-shot or few-shot standard prompting, as well as chain-of-thought prompting. Our findings show that GPT-4 outperforms in most tasks, reaching an average accuracy of up to 87.8%. However, for certain tasks, there are marked performance disparities among the models. Despite the impressive results of GPT-4, it trails human proficiency by roughly 10%, highlighting significant room for LLMs to further improve their temporal reasoning abilities. Manual error analysis revealed that models particularly struggle with nuanced understanding and interpreting implicit cues across all task categories. In summary, our contributions are threefold: (1) We introduce TRAM, a comprehensive collection of ten distinct temporal reasoning tasks presented in a multiple-choice question format. Ranging from foundational temporal concepts to intricate temporal interpretations, TRAM serves as a unified framework to assess the temporal reasoning capabilities of LLMs. (2) We conduct extensive experiments on TRAM, evaluating leading language models including BERT-style models and contemporary LLMs such as Llama2, PaLM2, GPT-3.5, and GPT-4. Our results reveal that even the most advanced LLM falls short of human-level performance, underscoring the opportunities for continued research in this area. (3) Through manual error analysis of results from TRAM, we highlight the consistent challenges in temporal reasoning faced by current LLMs. Specifically, nuanced comprehension and decoding of implicit temporal cues remain challenging for even advanced models, emphasizing the need for further research to improve the capabilities of LLMs in understanding and reasoning about time. 2 RELATED WORK Our proposal for a comprehensive temporal reasoning benchmark builds on the evolution of datasets in this domain while addressing the lack of a unified system for evaluation. The modern NLP landscape sets the stage for a nuanced evaluation of both BERT-based and LLM paradigms. Temporal Reasoning Benchmarks In the realm of temporal reasoning, several datasets have emerged to address distinct challenges. Early benchmarks, such as TimeBank (Pustejovsky et al., 2003), were predominantly focused on temporal relations. TempEval-3 (Uzzaman et al., 2013) broadened the scope by introducing multiple tasks, which included temporal entity extraction and temporal relation extraction. In recent years, there has been a surge in the development of time-sensitive question-answering datasets like MCTACO (Zhou et al., 2019), Time-sensitive QA (Chen et al., 2021), TEMPLAMA (Dhingra et al., 2022), and TEMPREASON (Tan et al., 2023). However, these datasets often specialize in narrower aspects of temporal reasoning, such as duration, frequency, or event-time relations. In contrast, our benchmark offers a comprehensive scope of temporal reasoning, addressing various levels and dimensions of understanding about time. It aims to provide a more complete representation of TeR challenges than previously available datasets. Training Paradigms in LLMs In NLP research, pretraining language models on vast amounts of diverse texts has become standard practice. Through this process, the models encapsulate a broad spectrum of information across various domains. Traditionally, leveraging this pretrained knowledge for downstream tasks primarily involved fine-tuning on task-specific data. BERT-based models like BERT (Kenton & Toutanova, 2019) and RoBERTa (Liu et al., 2019) are representative examples. These models have been applied to a diverse set of tasks, including disease prediction (Zhao et al., 2021), text classification (Wang et al., 2022b), time series analysis (Wang et al., 2022c), and more. However, the introduction of models like GPT-3 (Brown et al., 2020) marked a significant shift away from heavy reliance on extensive task-specific fine-tuning. Instead, the focus has been shifting towards zero-shot and few-shot learning approaches. In these settings, models such as GPT-3 can adapt to new tasks and achieve competitive performance with only a few training examples (Brown et al., 2020). This transition has spurred the development of advanced prompting techniques aimed at enhancing the understanding and reasoning capabilities of LLMs. Some representative prompt- ing methods include chain-of-thought prompting (Wei et al., 2022), self-consistency (Wang et al., 2022a), tree-of-thought prompting (Yao et al., 2023), and metacognitive prompting (Wang & Zhao, 2023). These techniques guide LLMs to generalize across tasks, ensuring their versatile deployment across a broad spectrum of NLP challenges. In this work, we establish baseline evaluations by considering both traditional BERT-based models and recent advances in LLMs, specifically including Llama2 (Touvron et al., 2023), PaLM2 (Anil et al., 2023), GPT-3.5, and GPT-4 (OpenAI, 2023). Through this, we aim to provide a comprehensive understanding of their strengths and limitations in diverse temporal reasoning tasks. 3 Tasks and Datasets TRAM encompasses ten temporal reasoning tasks, presented as multiple-choice questions (MCQs) across a range of time-related domains. For clarity, we ensure that each question has only one correct answer. The main purpose of TRAM is to spur further research into the advanced temporal reasoning capabilities of LLMs. Overall, these tasks fall under three distinct groups: (1) Foundational Temporal Understanding Tasks: Covering basic temporal comprehension, this group incorporates tasks such as ordering, frequency, duration, and typical time. (2) Temporal Interpretation and Computation Tasks: Centered on the interpretative and computational aspects of time, this group includes tasks like ambiguity resolution and arithmetic. (3) Advanced Temporal and Conceptual Understanding Tasks: Dedicated to exploring intricate temporal relationships and narratives, this category features tasks like relation, temporal NLI, causality, and storytelling. In this work, certain task names, such as ‘relation’ and ‘causality’, can have varied interpretations across different contexts. However, they are specifically emphasized for their temporal aspects in this work. Although we might occasionally omit the term ‘temporal’ for brevity, readers should note that the tasks are centered on time-related elements. In TRAM, each task is designed with one or more problem types, ensuring diverse representation across data attributes, complexities, and domains. The benchmark includes 526,668 problems in total. For each dataset, we introduce a few-shot development set, with 5 questions per category, and a separate test set for evaluation. Table 1 provides a detailed overview of these tasks, and more details can be found in Appendix B. The majority of tasks employ accuracy as the evaluation metric due to their straightforward MCQ structure. However, for tasks like ‘relation’ and ‘temporal NLI’, which exhibit an imbalanced label distribution inherent to their fixed class structure, both accuracy and the F1 score are utilized, even when they are presented as MCQs. Table 1: Overview of tasks included in TRAM. The “Data Size” column aggregates totals from both the development and test sets. “K-Way MC” signifies a multiple-choice response format with K options. Amb. Res. denotes Ambiguity Resolution. NLI stands for natural language inference. “Same” indicates the text source is the same as the row above. \| Task \| Data Size \| # Problem Types \| Metrics \| Answer Type \| Text Sources \| \|-----------------------\|-----------\|-----------------\|---------\|-------------\|--------------\| \| Foundational Temporal Understanding Tasks \| \| Ordering \| 29,462 \| 2 \| Acc. \| 3-Way MC \| MCTACO¹, Wikipedia, Misc. \| \| Frequency \| 4,658 \| 6 \| Acc. \| 3-Way MC \| MCTACO¹, SQuAD², Misc. \| \| Duration \| 7,232 \| 7 \| Acc. \| 3-Way MC \| Same \| \| Typical Time \| 13,018 \| 4 \| Acc. \| 3-Way MC \| Same \| \| Temporal Interpretation and Computation Tasks \| \| Amb. Res. \| 3,649 \| 5 \| Acc. \| 3-Way MC \| Misc. \| \| Arithmetic \| 15,629 \| 9 \| Acc. \| 4-Way MC \| Same \| \| Advanced Temporal and Conceptual Understanding Tasks \| \| Relation \| 102,462 \| 1 \| Acc./F1 \| 3-Way MC \| TempEval-3³ \| \| Temporal NLI \| 282,144 \| 1 \| Acc./F1 \| 3-Way MC \| MNLI⁴, SNLI⁵ \| \| Causality \| 1,200 \| 2 \| Acc. \| 2-Way MC \| COPA⁶, Misc. \| \| Storytelling \| 67,214 \| 1 \| Acc. \| 2-Way MC \| ROC⁷, SCT⁸ \| ¹ Zhou et al. (2019), ² Raipurkar et al. (2016), ³ UzZaman et al. (2013), ⁴ Williams et al. (2018), ⁵ Bowman et al. (2015), ⁶ Roemmele et al. (2011), ⁷ Mostafazadeh et al. (2016), ⁸ Mostafazadeh et al. (2017) 3.1 Foundational Temporal Understanding Tasks This group of tasks is fundamental for assessing a model’s proficiency in core temporal concepts. For the tasks below, data from the Multiple Choice TemporAl COmmon-sense (MCTACO) dataset incorporates both validation and test sets, while data from the Stanford Question Answering Dataset (SQuAD) dataset includes both training and validation sets. Unless otherwise mentioned, the options for each dataset are generated through a blend of human curation and algorithmic processes, tailored to each specific task. For instance, in the ordering task, correct answers strictly adhere to the accurate chronological sequence of events, while incorrect choices are formed through random permutations. See Figure 1 for example questions of each task. \| Ordering (Facts) \| Q: Arrange the following events in chronological order: (1) Brusilov Offensive by Russia. (2) Kamehameha I of the Island of Hawaii defeats the Oahuans at the Battle of Nu‘uanu. (3) The Kuomintang, the Chinese nationalist party, is founded. (4) Emperor Claudius dies and is succeeded by his grand nephew Nero. (5) St. Norbert and 29 companions make their solemn vows marking the beginning of the Premonstratensian Order. \| \| --- \| --- \| \| A. (1), (2), (4), (5), (3) ❌ \| B. (4), (5), (2), (3), (1) ✔️ \| C. (3), (1), (2), (4), (5) ❌ \| \| Frequency (Commonsense) \| Q: It is also a love story, between Ace and Tobio, a trans woman. How often do they break up? \| \| --- \| --- \| \| A. Once ✔️ \| B. Always ❌ \| C. Once per week ❌ \| \| Duration (Analogy Inference) \| Q: While Yoga Session gave attendees time to plant an entire garden, Jazz Concert was enough to water a few plants, and Board Game Night was merely smelling a flower. Which event was the longest? \| \| --- \| --- \| \| A. Jazz Concert ❌ \| B. Board Game Night ❌ \| C. Yoga Session ✔️ \| \| Typical Time (Comparison) \| Q: Which event typically happens earlier: morning yoga or farmer starting their day? \| \| --- \| --- \| \| A. Morning yoga ❌ \| B. Farmer starting their day ✔️ \| C. Around the same time ❌ \| Figure 1: Example questions from temporal ordering, frequency, duration, and typical time tasks. Ordering The temporal ordering task evaluates a model’s ability to understand the sequence in which events occur. This task is divided into two primary problem types. For commonsense problems, we mainly source questions from the MCTACO dataset (Zhou et al., 2019), specifically targeting subcategories related to temporal ordering. For each individual question selected from this dataset, we pose questions in the format, “Is {candidate answer} possible?” While MCTACO’s expected answers are “true” or “false”, we introduce another layer of complexity by also including an “undetermined” option. Additionally, we curate another set of commonsense questions, for which sequences of events are manually structured in a logical manner, followed by programmatic question generation. Concurrently, recognizing the significance of tasks rooted in real-world events, we introduce facts problems. These focus on major historical events, spanning from ancient to contemporary times, and are sourced from Wikipedia timelines. Models are posed with challenges such as sequencing: “Arrange the following events in chronological order” and verification queries like, “Is the following sequence of events in the correct chronological order?”. Frequency The frequency task assesses a model’s ability to understand how often events take place over time and comprises six distinct categories of problems. For the commonsense category, we source questions from the MCTACO dataset related to frequency. Each selected category ensures the presence of at least two incorrect options and one correct one. To prevent models from memorizing specific answer orders, we randomize the placement of the correct answers. In the reading comprehension category, questions are chosen from the SQuAD dataset (Rajpurkar et al., 2016) based on frequency-oriented keywords like “how often”, “how many times”, and “how frequent”. The application and computation categories are mainly made up of human-curated templates that test the model’s ability to infer time intervals and compute either previous or subsequent occurrences. The comparison problems blend real and artificially conceived events, challenging the model’s ability to differentiate frequency nuances. Lastly, the facts category draws questions from various sources, with Wikipedia being the primary one, centering on queries related to events that are known to happen regularly or periodically in either historical or contemporary settings. Duration The duration task is designed to assess a model’s capability to comprehend the length of events or periods of time and encompasses seven distinct categories of problems. The commonsense problems are derived from the MCTACO dataset, probing the model’s fundamental understanding of event durations grounded in everyday scenarios. The extraction methods mirror those used for the “frequency” task. The reading comprehension category sources questions from the SQuAD dataset, selecting those with duration-oriented keywords like “how long”, “how many years”, and “how much time”. Apart from the aforementioned subtasks, all other categories consist of human-curated templates or problems. The analogy inference category assesses the model’s ability to discern durations through analogical reasoning. The computation category tests mathematical precision, where problems often require arithmetic operations to determine event durations. Comparative analysis is examined in two subtasks: direct comparison, which demands straightforward judgments of event durations involving both real and artificial events; and multi-step comparison, which challenges the model to infer and integrate information across sequential statements. Lastly, the facts category primarily draws from Wikipedia, furnishing questions anchored in well-documented historical or contemporary durations. Typical Time The typical time task is constructed to evaluate a model’s understanding of when events or activities typically occur, segmented into four distinct categories. The commonsense category draws problems from the MCTACO dataset, exploring the model’s innate comprehension of event timings as they manifest in daily scenarios. The extraction method for this subtask is similar to that used for the “frequency” task. The comparison category, comprising human-curated statements and problems, delves into relative timing. This category involves determining which of two presented scenarios is more temporally typical or discerning which event customarily precedes the other. The facts category, primarily sourced from Wikipedia timelines spanning ancient history to the 21st century, provides the model with specific historical or established events and expects it to identify the precise times or periods associated with them. Lastly, the reading comprehension problem sets source questions from the SQuAD dataset. This category filters problems based on keywords like “at what time”, “when did”, and “in what year”, challenging the model to extract specific temporal data from passages. 3.2 Temporal Interpretation and Computation Tasks This group of tasks is fundamental in gauging a model’s adeptness at deciphering, processing, and computing temporal information. See Figure 2 for example questions of each task. \| Ambiguity Resolution (Interpretation) \| Q: A historic event is documented to have happened ‘before you know it’. When did it take place? \| \|--------------------------------------\|--------------------------------------------------------------------------------------------------\| \| \| A. The next day ❌ B. Without hesitation ❌ C. Before long ✔️ \| \| Arithmetic (24-hour Adjustment) \| Q: What is 00:18 - 23:50? \| \|--------------------------------------\|------------------------------------------------------------------------------------------\| \| \| A. 0:28 ✔️ B. 1:44 ❌ C. 22:15 ❌ D. 1:35 ❌ \| Figure 2: Example questions from ambiguity resolution and arithmetic tasks. Ambiguity Resolution The temporal ambiguity resolution task aims to gauge a model’s ability to decipher and resolve uncertainties related to temporal expressions, divided into five subtasks. The interpretation category evaluates the model’s comprehension of ambiguous time-related phrases commonly used in everyday language. The calendar shift subtask necessitates the conversion between different calendar systems, such as the Julian and Gregorian. The long-term shift, mid-term shift, and short-term shift categories challenge the model’s capacity to adjust dates over long (i.e., years), medium (i.e., months, weeks, days), and short (i.e., hours, minutes, seconds) timeframes, respectively. All questions across these categories originate from carefully crafted human templates. Arithmetic The temporal arithmetic task evaluates a model’s capacity to manage calculations related to time, organized into nine distinct subtasks. The application category presents real-world scenarios such as time calculations involving schooling, vacations, homework, and flights. Date computation sets focus on adding or subtracting days from specified dates to determine a new date. hour adjustment subtasks, divided into 12-hour and 24-hour formats, require the model to calculate time differences or additions. The month shift subtask examines the model’s ability to pinpoint a month that is a certain number of months away from a specified month. The week identification problems determine the exact week number within a year based on a given date. In year shift, the model discerns a year a certain number of years relative to a provided year. time computation evaluates the model’s proficiency in converting various time units, especially over shorter durations like days, hours, minutes, and seconds. Lastly, the time zone conversion category requires the model to convert times between different zones. Both the question templates and the programs used to formulate answers derive from human expertise. 3.3 ADVANCED TEMPORAL AND CONCEPTUAL UNDERSTANDING TASKS This group of tasks is fundamental in assessing a model’s depth of comprehension in time-oriented narratives and in discerning complex conceptual relationships. See Figure 3 for example questions of each task. \| Task \| Question \| Options \| \|-----------------------\|--------------------------------------------------------------------------\|-------------------------------------------------------------------------\| \| Temporal Relation \| Q: Israel wants the EU to arrest any Palestinian suspected of smuggling arms through the crossing, while the EU wants its role to be confined to only monitoring and reporting. What is the relationship between the event ‘wants’ and the event ‘reporting’? \| A. ENDED-BY ❌ B. IS_INCLUDED ✔ C. IMMEDIATELY AFTER ❌ \| \| Temporal NLI \| Q: Premise: This morning, no doubt, she would have consulted me on the subject, but she had no chance. Hypothesis: She would have consulted me on the subject this morning if she’d had the chance. \| A. Entailment ✔ B. Neutral ❌ C. Contradiction ❌ \| \| Temporal Causality \| Q: She noticed that all the wall clocks in the store were set to ten past ten. What’s the more plausible CAUSE? A. It is a common display setting for clocks and watches. ✔ B. It was ten minutes past ten at that moment. ❌ \| \| Temporal Storytelling \| Q: I woke up so late this morning. I was panicked when I saw what time it was. I had to be at work on time. I threw myself together quickly. Which of the two endings is the most plausible correct ending to the story? A. I was able to get a job at a local restaurant. ❌ B. I was still thirty minutes late. ✔ \| Figure 3: Example questions from relation, temporal NLI, causality, and storytelling tasks. Relation The temporal relation task seeks to assess a model’s ability to identify the relationship between two entities involving time, categorized as either an event-to-time or an event-to-event association. Questions are crafted based on the TempEval-3 Silver dataset (Uzzaman et al., 2013). The context sentences, which contain the two entities in question, are directly extracted from the original passages. One inherent challenge of this task lies in the subtle nuances among the fixed set of relations. For instance, distinguishing between relations like “BEFORE” and “IMMEDIATELY BEFORE” can be particularly demanding, as they require fine-grained comprehension of temporal sequences. With the predetermined relations from the dataset, the correct relation option is randomized in its placement, while distractor options are chosen from the pool of remaining relations. Temporal NLI The Temporal NLI task is designed to evaluate a model’s ability in natural language inference, with a particular emphasis on statements that involve temporal elements. We source questions from prevalent NLI datasets, including Stanford Natural Language Inference datasets (SNLI) (Bowman et al., 2015), and Multi-Genre Natural Language Inference (MNLI) (Williams et al., 2018). Data from the MNLI dataset includes training and validation sets, while data from the SNLI dataset includes training, validation, and test sets. We select problems based on keywords that capture a range of temporal nuances, such as explicit references (e.g., ‘tomorrow’, ‘later’), months (e.g., ‘May’, ‘October’), seasons (e.g., ‘summer’, ‘winter’), periods (e.g., ‘decade’, ‘century’), and temporal actions (e.g., ‘in advance’, ‘postpone’). Consistent with the original task, the three response options for all questions are: “Entailment”, “Neutral”, and “Contradiction”. Causality The temporal causality task assesses a model’s capability to discern cause-and-effect relationships within scenarios influenced by time. Drawing inspiration from the Choice of Plausible Alternatives (COPA) dataset (Roemmele et al., 2011), we select questions that naturally contain temporal elements such as ‘postpone’, ‘tomorrow’, ‘summer’, and ‘clock’. Additionally, we manually craft problems to highlight the temporal nature of COPA-style questions. Each problem presents a situation that revolves around time, followed by a question pinpointing either the most plausible cause or effect of that situation. Both options for these problems are carefully created by hand. For augmentation purposes, we create additional, mirrored instances for each original sample. This approach ensures that for a given question with two options, each option is supported by a uniquely tailored premise, effectively creating a distinct and relevant context for both choices. Storytelling The temporal storytelling task is designed to assess a model’s ability to predict the appropriate ending of stories that emphasize temporal elements. We source questions from the ROCStories (ROC) (Mostafazadeh et al., 2016) and Story Cloze Test (SCT) (Mostafazadeh et al., 2017) datasets. We identify and select stories that contain notable temporal components by filtering them using keywords such as ‘now’, ‘tomorrow’, ‘future’, ‘always’, and ‘postpone’, among others. The typical format of the task presents a story comprising four sentences, followed by two potential endings. The model is required to choose the most appropriate conclusion for the story. In the case of SCT, which inherently provides two endings for each story, our focus remains on selecting stories with evident temporal aspects. To further enrich our dataset, we take the initial four sentences from the ROC and employ GPT-2 (Radford et al., 2019) to produce an alternate, incorrect ending, initiated with the prompt “unexpectedly”. Subsequently, we filter this augmented data to ensure that stories emphasize the desired temporal themes. 4 EXPERIMENTS In our evaluation, we compare the performance of prevalent LLMs across all datasets and analyze the mistakes they make. We report the best results after multiple runs for each experimental setting. 4.1 EXPERIMENTAL SETUP We evaluate the performance of several well-known language models on the TRAM benchmark, which is organized into two main categories. In the first category, we employ four popular large language models: the open-source Llama-2-13b-chat (Touvron et al., 2023) and the closed-source models PaLM-bison-chat (Anil et al., 2023), GPT-3.5-turbo, and GPT-4 (OpenAI, 2023). Each of these models is accessed using its corresponding API key. Given the constraints of API costs, and following the methodology of (Tan et al., 2023), we assess model performance on 200 examples for each category of each task randomly selected from the test set. For categories with fewer than 200 examples, we utilize all available test examples. For all evaluations, greedy decoding (i.e., temperature = 0) is applied during model response generation. We evaluate each model using two prompting strategies: standard prompting (SP) (Brown et al., 2020; Kojima et al., 2022) and chain-of-thought (CoT) (Wei et al., 2022) prompting. Under both strategies, the models undergo tests in zero-shot and 5-shot settings. In the 5-shot scenario, exemplars are consistently drawn from the development set. Step-by-step answers associated with CoT prompting are obtained through human annotation. More details about prompts can be found in Appendix C. In the second category, we consider minimal supervision as opposed to traditional fully supervised learning in order to establish baseline evaluations. The rationale behind this decision is driven by the intention to leverage the inherent world knowledge of the models and to ensure an equitable comparison with the previously mentioned LLMs. For this category, we employ four representative BERT-style models, including BERT-base, BERT-large (Kenton & Toutanova, 2019), RoBERTa-base, and RoBERTa-large (Liu et al., 2019). Specifically, for the temporal NLI task, we employ the Sequence Classification variant of BERT and RoBERTa from Huggingface, given its suitability for the task’s structure. However, for the other tasks, we utilize the Multiple Choice variant of BERT and RoBERTa from Huggingface. The data sampling strategy for minimal supervision is structured based on the size of the original dataset. For datasets with around 1k samples, we randomly select 50% of the remaining data after setting aside the test data used for LLM evaluation. For datasets with sizes between 3k and 10k, we select 10%. For those with sizes between 10k and 100k, we sample 2.5%, and for datasets with more than 100k examples, we take 1%. This limited training data is then used for the fine-tuning of models. The same test set is used consistently with LLMs. In addition to evaluating model performance, multiple expert annotators worked on each problem type for every task in TRAM to better understand human performance. Each expert answered a subset of the 50 questions from each category of every task, which were randomly selected from the test set. Collectively, they tackled about 1,900 questions across TRAM. Further details on human expert annotators and human non-specialists are provided in Appendix A. 4.2 OVERALL PERFORMANCE COMPARISON We compared the performance of different models across ten tasks, as shown in Table 2. There are several key takeaways. First, GPT-4 consistently outperforms other models across the majority of tasks, demonstrating a performance advantage of over 15% compared to other models on average. Second, all LLMs show improved performance in the 5-shot setting compared to the zero-shot setting, as expected. Regarding prompting effectiveness, we note that CoT often results in performance enhancements, which corroborates the findings from (Wei et al., 2022), emphasizing the efficacy of step-by-step prompting in augmenting LLMs’ performance in intricate reasoning tasks. Third, it is Table 2: Performance comparison of each model across ten tasks in TRAM. GPT-4 consistently outperforms other models under both zero-shot (OS) and 5-shot (5S) settings across the majority of tasks. Interestingly, the RoBERTa-large model achieves a higher average performance than models with larger architectures, such as Llama2. Human performance serves as an upper bound, illustrating that there still exists room for improvement in LLMs on temporal reasoning tasks. The abbreviations Freq., Dur., Arith., Rel., Caus. refer to frequency, duration, arithmetic, relation, and causality, respectively. All values are percentages. Best model results are highlighted in bold. \| Model \| Order Acc. \| Freq. Acc. \| Dur. Acc. \| Typical Time Acc. \| Amb. Res. Acc. \| Arith. Acc. \| Rel. Acc./F1 \| NLI Acc./F1 \| Caus. Acc. \| Story Acc. \| Average \| \|----------------\|------------\|------------\|-----------\|-------------------\|----------------\|-------------\|--------------\|-------------\|------------\|------------\|---------\| \| Random \| 33.3 \| 33.3 \| 33.3 \| 33.3 \| 33.3 \| 25.0 \| 33.3/33.3 \| 33.3/33.3 \| 50.0 \| 50.0 \| 35.4 \| \| Llama2 (OS, SP)\| 50.2 \| 71.8 \| 63.4 \| 72.0 \| 45.8 \| 51.2 \| 34.5/32.3 \| 62.7/62.2 \| 97.5 \| 86.5 \| 60.8 \| \| Llama2 (OS, CoT)\| 51.7 \| 73.2 \| 64.7 \| 73.5 \| 48.0 \| 54.4 \| 39.0/37.7 \| 66.0/65.7 \| 99.3 \| 88.2 \| 63.4 \| \| Llama2 (SS, SP)\| 50.7 \| 72.2 \| 64.0 \| 72.8 \| 47.0 \| 52.6 \| 37.0/35.5 \| 63.7/63.2 \| 98.8 \| 87.3 \| 62.1 \| \| Llama2 (SS, CoT)\| 52.5 \| 73.7 \| 65.3 \| 73.8 \| 49.6 \| 55.2 \| 41.0/39.7 \| 66.5/65.7 \| 99.5 \| 88.5 \| 64.3 \| \| PalM2 (OS, SP) \| 54.2 \| 84.2 \| 81.9 \| 80.5 \| 73.2 \| 68.0 \| 59.0/58.5 \| 68.2/69.1 \| 99.3 \| 91.2 \| 73.9 \| \| PalM2 (OS, CoT) \| 55.5 \| 85.0 \| 82.3 \| 81.5 \| 74.6 \| 69.7 \| 62.5/62.1 \| 69.3/70.1 \| 99.5 \| 92.0 \| 75.3 \| \| PalM2 (SS, SP) \| 55.2 \| 84.7 \| 82.1 \| 81.0 \| 74.0 \| 68.8 \| 61.0/60.7 \| 68.5/69.4 \| 99.3 \| 91.5 \| 74.7 \| \| PalM2 (SS, CoT) \| 56.2 \| 85.2 \| 82.7 \| 81.8 \| 75.0 \| 70.2 \| 63.5/63.3 \| 70.3/71.1 \| 99.5 \| 92.2 \| 75.9 \| \| GPT-3.5 (OS, SP)\| 52.5 \| 77.3 \| 71.6 \| 78.7 \| 72.8 \| 72.8 \| 40.5/39.1 \| 73.8/74.2 \| 98.8 \| 90.5 \| 70.2 \| \| GPT-3.5 (OS, CoT)\| 53.7 \| 78.3 \| 72.3 \| 79.7 \| 74.6 \| 74.6 \| 44.5/43.5 \| 75.2/75.7 \| 99.5 \| 91.7 \| 71.9 \| \| GPT-3.5 (SS, SP)\| 53.2 \| 77.8 \| 72.0 \| 79.2 \| 73.4 \| 73.7 \| 43.0/41.8 \| 74.5/75.0 \| 99.3 \| 91.0 \| 71.2 \| \| GPT-3.5 (SS, CoT)\| 54.5 \| 78.5 \| 72.7 \| 80.0 \| 75.0 \| 75.0 \| 46.5/45.5 \| 75.5/75.9 \| 99.5 \| 91.7 \| 72.5 \| \| GPT-4 (OS, SP) \| 70.3 \| 92.5 \| 92.3 \| 89.5 \| 88.6 \| 93.6 \| 64.0/63.6 \| 89.5/89.8 \| 99.0 \| 95.8 \| 85.7 \| \| GPT-4 (OS, CoT) \| 71.0 \| 93.3 \| 92.6 \| 90.0 \| 89.2 \| 93.9 \| 67.0/66.6 \| 90.5/90.8 \| 100.0 \| 96.3 \| 86.8 \| \| GPT-4 (SS, SP) \| 70.8 \| 92.8 \| 92.4 \| 89.7 \| 89.0 \| 93.8 \| 66.0/65.6 \| 90.0/90.3 \| 99.5 \| 96.0 \| 86.3 \| \| GPT-4 (SS, CoT) \| 71.5 \| 93.7 \| 93.0 \| 90.2 \| 89.8 \| 94.3 \| 69.5/69.1 \| 90.7/91.0 \| 100.0 \| 96.3 \| 87.4 \| \| BERT-base \| 50.0 \| 47.3 \| 50.0 \| 53.0 \| 36.6 \| 25.9 \| 86.5/86.6 \| 53.0/53.4 \| 81.0 \| 79.0 \| 58.5 \| \| BERT-large \| 52.5 \| 53.1 \| 53.3 \| 56.8 \| 37.4 \| 28.3 \| 89.5/89.5 \| 59.5/60.1 \| 85.0 \| 81.3 \| 62.2 \| \| RoBERTa-base \| 50.8 \| 54.5 \| 51.8 \| 55.3 \| 37.4 \| 26.4 \| 87.0/86.8 \| 64.5/64.9 \| 82.3 \| 81.3 \| 61.9 \| \| RoBERTa-large \| 55.5 \| 57.7 \| 55.4 \| 60.0 \| 41.0 \| 29.1 \| 90.0/90.0 \| 70.0/70.3 \| 88.0 \| 84.0 \| 65.9 \| \| Human \| 86.0 \| 96.3 \| 97.7 \| 94.5 \| 94.8 \| 98.7 \| 96.0/96.0 \| 92.0/92.4 \| 100.0 \| 98.0 \| 95.2 \| It is notable that RoBERTa-large, despite its size, surpasses the larger Llama2 in average performance. This observation underscores that sheer model size doesn’t always equate to superior performance. Several factors might contribute to this outcome. RoBERTa-large might utilize optimization strategies that are especially beneficial for these tasks. Additionally, inherent features or efficiencies in its architecture might enhance its ability to understand and process temporal cues. Delving deeper into task-specific performance, certain tasks such as ambiguity resolution and arithmetic show considerable variance across models. For LLMs, performance on the arithmetic task varies significantly, ranging from 51.2% to 94.3%. Moreover, BERT and RoBERTa exhibit exceptional performance in the temporal relation task, potentially due to their bidirectional contextual processing and emphasis on token-level relationships. Their attention mechanisms also allow them to discern and prioritize essential segments in sentences indicative of temporal relationships. This contrasts sharply with their average or below-average performance in other tasks. This discrepancy suggests that some models may be equipped with architectures or training methodologies tailored for certain types of reasoning, or that specific tasks require a distinct understanding not universally handled proficiently by all models. Finally, while GPT-4 leads among all the models, human expertise still exceeds it by roughly 10%, highlighting the complexity of these temporal reasoning tasks and indicating room for future improvements in LLMs. ### 4.3 Error Analysis To better understand the mistakes made by models, we manually analyzed instances where a model, whether in a 0-shot or 5-shot setting or under SP or CoT, made an incorrect choice. We prompted the model to explain its decisions, then reviewed these explanations to identify errors, understand the reasons behind them, and categorize them into specific error types. For this analysis, we focused solely on LLMs, excluding BERT-style models. Figure 4 showcases the prevalent error types and their respective proportions for each task group. Within the foundational temporal understanding tasks, “assumption bias” was the most frequent error, accounting for 32% of all mistakes. In the interpretation and computation tasks, “calculation slips” dominated, making up 42% of the errors. “Implicit oversights” led in the advanced temporal understanding tasks with a representation of 34%. Detailed descriptions of each error type can be found in Appendix D. Figure 4: Error type distribution for three groups of tasks in TRAM. Models often struggle with subtle details and hidden clues across all categories. 5 DISCUSSION We introduce TRAM, a comprehensive benchmark spanning ten diverse tasks, to evaluate the temporal reasoning of LLMs. The contrasting performances across models emphasize the significance of experimental strategies and shed light on the intrinsic challenges. This benchmark serves as a tool for researchers to identify model limitations and guide further advancements in this domain. Implications of TRAM The introduction of TRAM establishes a new paradigm for probing the temporal reasoning capabilities of LLMs. Unlike previous benchmarks, which often offered fragmented insights into temporal tasks, TRAM provides a comprehensive system. This allows for a unified evaluation of how models comprehend both rudimentary temporal concepts and complex temporal narratives. The differentiation in task complexity within TRAM elucidates the various stages of temporal understanding. In particular, TRAM underscores challenges like decoding implicit temporal cues and navigating intricate temporal relationships, providing a roadmap for future improvements in LLMs in this area. Model Performance and Challenges Model experimental strategies notably influence large language models’ temporal reasoning capabilities. The superior performance in the 5-shot setting, compared to zero-shot, underscores the crucial role of context-specific learning in enhancing these models’ grasp on temporal aspects. Moreover, the effectiveness of the CoT prompting highlights the potential of specialized strategies in refining their prowess in complex temporal reasoning tasks. However, size doesn’t inherently guarantee success. The average performance of RoBERTa-large outperforms the larger Llama2, raising intriguing questions about the balance between model size and efficiency. In addition, varied performance across tasks indicates the challenges of crafting a universally adept model for all TeR problems. This variability, combined with the gap between GPT-4 and human expertise, signals ongoing challenges and the need for nuanced improvements. Limitations While TRAM presents a holistic approach to temporal reasoning assessment, we acknowledge its limitations. One primary concern is the subset evaluation of the test set, which may not reflect the full spectrum of LLMs’ temporal reasoning capabilities. Furthermore, given the MCQ format, there is a possibility that LLMs could resort to random guessing, rather than genuinely exhibiting temporal reasoning. Such tendencies may mislead the performance evaluation. In addition, textual questions may not capture the entire complexity of temporal reasoning tasks, as real-world scenarios often integrate multi-modal cues such as images and videos. Future Directions TRAM has initiated a step towards evaluating LLMs’ temporal reasoning capabilities, but there are further avenues to explore. Going forward, we will experiment with more test data and refine tailored prompting techniques for each task through iterative testing. Moreover, we plan to expand the benchmark to include varied question formats. For generative tasks, this might encompass short answers and summarization. Even within MCQs, we intend to incorporate questions that may have one or more correct answers, allowing for a more comprehensive evaluation. We also plan to fine-tune existing open-source LLMs on these tasks, such as Llama2. These efforts aim to create tailored LLMs that can better understand and reason about time across various contexts. REFERENCES Rohan Anil, Andrew M Dai, Orhan Firat, Melvin Johnson, Dmitry Lepikhin, Alexandre Passos, Siamak Shakeri, Emanuel Taropa, Paige Bailey, Zhifeng Chen, et al. 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43flsheS4s	Q2: Because the paper appears to lack a comprehensive exploration of the tuning strategy of the hyperparameter $\lambda$ introduced in Equation 1, could you elucidate on the potential effects of employing a constant value for $\lambda$, or linearly increase the value of $\lambda$ instead of using the sine increasing schedule?	Improving Robustness and Accuracy with Retrospective Online Adversarial Distillation Anonymous authors Paper under double-blind review Abstract Adversarial distillation (AD), transferring knowledge of a robust teacher model to a student model, has emerged as an advanced approach to improving robustness against adversarial attacks. However, AD in general suffers from the high computational complexity of pre-training the robust teacher as well as the inherent trade-off between robustness and natural accuracy (i.e., accuracy on clean data). To address these issues, we propose retrospective online adversarial distillation (ROAD). ROAD exploits the student itself of the last epoch and a natural model (i.e., a model trained with clean data) as teachers, instead of a pre-trained robust teacher in the conventional AD. We revealed both theoretically and empirically that knowledge distillation from the student of the last epoch allows to penalize overly confident predictions on adversarial examples, leading to improved robustness and generalization. Also, the student and the natural model are trained together in a collaborative manner, which enables to improve natural accuracy of the student more effectively. We demonstrate by extensive experiments that ROAD achieved outstanding performance in both robustness and natural accuracy with substantially reduced training time and computation cost. 1 Introduction Deep neural networks (DNNs) have achieved great success in various applications such as computer vision (He et al., 2016; Goodfellow et al., 2014), natural language processing (Sutskever et al., 2014; Vaswani et al., 2017), and reinforcement learning (Mnih et al., 2013; Chen et al., 2021). However, Szegedy et al. (2013) showed that DNNs are vulnerable to adversarial attacks (Goodfellow et al., 2015; Dong et al., 2018; Carlini & Wagner, 2017; Madry et al., 2018), which are small perturbations added to natural inputs to deceive the models and cause incorrect predictions consequently. These attacks are significant threats especially in high-stakes contexts including autonomous driving (Sitawarin et al., 2018) and financial systems (Fursov et al., 2021). Adversarial training (AT) has served as an effective solution to defend against the adversarial attacks (Madry et al., 2018; Gowal et al., 2020; Pang et al., 2021). It improves robustness of DNNs by training them with adversarial examples crafted from themselves. To further enhance their robustness, even for compact models, adversarial distillation (AD) has attracted increasing attention recently (Goldblum et al., 2020; Zhu et al., 2021; Zi et al., 2021; Maroto et al., 2022; Huang et al., 2023). Analogous to knowledge distillation (KD) (Hinton et al., 2015), AD adopts the teacher-student framework, in which the teacher model is pre-trained via AT and provides additional supervision to the student model for improving its robustness. Surprisingly, even when the teacher and student have the same capacity, AD enhances the robustness beyond the student trained alone. This suggests that AD not only compresses a high capacity model into a compact one but also enables to achieve extra robustness. However, AD has a fatal drawback: it requires a lot of training time and computing resources. Most AD methods follow the two-stage training strategy, i.e., pre-training a robust teacher through AT, and then transferring knowledge of the teacher to a student. Hence, AD typically demands at least twice as much training time as AT. This drawback make AD impractical for applications with limited computing resources or tight deployment schedules. Also, although AD enhances the robustness of the student through insights from the teacher, it is still limited in resolving the inherent trade-off between robustness and natural accuracy (i.e., accuracy on clean data). To address these limitations, we propose a new AD method coined retrospective online adversarial distillation (ROAD). Unlike the conventional AD using a pre-trained robust teacher, ROAD trains a robust model using knowledge distilled from two teachers: the model itself of the last epoch and an additional natural model, i.e., a standard model trained with clean data, as illustrated in Figure 1. To be specific, the robust model is trained using soft labels generated by linear interpolation between its predictions in the past epoch and true one-hot labels. Through theoretical and empirical analysis, we find that this simple method penalizes overly confident predictions on adversarial examples, thereby enhancing its generalization ability and robustness. Moreover, we employ a collaborative learning strategy to train the robust model and the natural model simultaneously. This enables the natural model to be aware of the robust model and consequently provide more friendly knowledge to the robust model. Note that these two teachers are substantially cheaper than the teacher pre-trained via AT in the conventional AD. Thanks to the use of the two teachers, ROAD achieved outstanding performance in both robustness and natural accuracy (Figure 2) with substantially reduced training time and computation cost (Figure 5(c) and 5(d)). Our major contribution is three-fold: - We propose ROAD, a new single-stage AD method based on retrospective self-distillation and collaborative learning, to address the chronic issues of the conventional AD approach. - ROAD demonstrated superior performance in both robustness and natural accuracy with diverse network architectures on two datasets and three different adversarial attacks. - ROAD allows to substantially reduce overall training time and computation cost of AD. To be specific, it requires about half the training time and memory of the previous best AD method. 2 RELATED WORK Adversarial Training. Adversarial training has proven to be an effective defense against adversarial attacks. One fundamental approach is PGD adversarial training (Madry et al., 2018), using the Projected Gradient Descent algorithm. Subsequent advancements have introduced regularization terms to enhance performance. For instance, Zhang et al. (2019) achieved a principled trade-off between robustness and accuracy, while Wang et al. (2020) focused on improving robustness by revisiting misclassified examples. Kannan et al. (2018) improved robustness by a technique called adversarial logit pairing. Other approaches involve additional unlabeled data utilization (Carmon et al., 2019; Uesato et al., 2019; Gowal et al., 2021; Wang et al., 2023) or perturbing the weight of the model (Wu et al., 2020) or utilize extra models (Chen et al., 2020; Cui et al., 2021; Arani et al., 2020; Rade & Moosavi-Dezfooli, 2022; Dong et al., 2022; Wang & Wang, 2022). However, AT methods cannot ensure high robustness for small-sized models. Regarding this, ROAD demonstrates distinctiveness by showing high robustness not only in large models but also in small models. Adversarial Distillation. The goal of adversarial distillation is to train a small-sized student model to mimic both the natural and robust predictions of a larger-sized robust teacher model. The initial works is Goldblum et al. (2020) which propose Adversarially Robust Distillation (ARD) to achieve robustness by comparing the model’s robust prediction with teacher’s natural prediction. Zi et al. (2021) compared conventional AT methods from a distillation perspective, emphasizing the advantages of using soft labels to achieve higher robustness. Based on this observation, they proposed Robust Soft Label Adversarial Distillation (RSLAD), which involves training the student model using soft labels generated from the teacher’s predictions. In addition, Zhu et al. (2021) pointed out that a robust teacher might provide unreliable predictions for adversarial examples crafted by the student model and proposed Introspective Adversarial Distillation (IAD), in which the teacher’s predictions are partially trusted. Liu et al. (2022) proposed Mutual Adversarial Training (MAT), which trains multiple robust models collaboratively to share the knowledge achieved from each adversarial examples. Lastly, Huang et al. (2023) proposed Adaptive Adversarial Distillation (AdaAD), which adaptively searches for inner maximization results by comparing the differences in predictions of student and teacher models. AD methods can be an attractive alternatives to enhance the robustness of end devices. However, the inherent two-stage process and associated computational inefficiency still conveys an inappropriate impression. 3 RETROSPECTIVE ONLINE ADVERSARIAL DISTILLATION ROAD consists of two components: retrospective self-adversarial distillation using the robust model itself of the last epoch to improve robustness, and collaborative learning with a natural model to recover natural accuracy. We first elaborate on each of the two components in Section 3.1 and Section 3.2, respectively, and then describe the overall training objective for ROAD in Section 3.3. 3.1 SELF-ADVERSARIAL DISTILLATION FROM LAST EPOCH AD has been acknowledged as an effective way to achieving extra robustness by improving generalization ability. However, pre-training the robust teacher model through AT demands an extremely large amount of training time. For instance, pre-training a 10-step PGD model requires roughly 11 times more forward-backward passes compared with the natural training. Additionally, loading both the teacher and student during the distillation process significantly increases GPU memory usage. To address these challenges, we introduce a simple yet efficient approach to improving robustness, self-adversarial distillation from the last epoch. Our distillation scheme does not require a teacher model as the student becomes its own teacher. Instead, it leverages the predictions on adversarial examples made by the robust model (i.e., the student) itself in the past. This approach eliminates the necessity of training an additional robust model. Specifically, it mixes the past predictions for adversarial examples with their one-hot labels by interpolation ratio $\lambda$. Ideally $\lambda$ should increase gradually as the predictions from previous epochs becomes more accurate. Considering that, we adopt a monotonically increasing schedule based on the sine function for $\lambda$. Then, the soft labels for robust model at the $t$-th epoch are given by $$\tilde{y}_t = (1 - \lambda_t)y + \lambda_t p_{t-1}^{\text{rob}}(x'_{t-1}),$$ where $p_{t-1}^{\text{rob}}(x'_{t-1})$ is the output of the robust model for the adversarial example $x'_{t-1}$ at $(t - 1)$-th epoch. The model trains with these soft labels instead of conventional one-hot labels. 3.1.1 THEORETICAL ANALYSIS We carefully analyze the role of the adversarial predictions at the last epoch as supervision. To this end, we first discuss the relationship between over-confidence and robustness in AT. Although robust models are less over-confident than natural models in prediction (Grabinski et al., 2022), their predictions still tend to be overly confident as they are trained with one-hot labels. Stutz et al. (2020) pointed out that AT overfits to experienced norm bounded adversarial examples (e.g., $\ell_\infty$ norm bounded adversarial examples) and performs poorly on other $\ell_p$ norm bounded adversarial examples or those crafted with a larger perturbation bound. Chen et al. (2020) claimed that the cause of robust overfitting, as discussed by Rice et al. (2020), is the model’s tendency to overfit to adversarial examples during the early stages of the training process, resulting in lack of generalizability. Therefore, it can be inferred that the over-confidence acts as a factor that diminishes the generalization capability of robust models and thus hampers the gain of robustness. We claim that our method resolves this problem by penalizing updating the model when its prediction confidence is boosted drastically for input adversarial examples during training. This phenomenon can be explained by gradient rescaling factor, following propositions presented by Tang et al. (2020) and Kim et al. (2021). The gradient rescaling factor is defined as the ratio of the $\ell_1$ norm of the loss gradient with respect to the logit during AD to that during training with one-hot labels. Proposition 1. Given a $K$-class classification problem, let $p'_{t,i}$ be the output of the robust model for an adversarial example of class $i$ ($i = 1, 2, \ldots, K$) at the $t$-th epoch, and $GT$ be the ground truth class. The gradient rescaling factor is then derived as $$\frac{\sum_i \|\partial_{AD,t}\|}{\sum_i \|\partial_i\|} = 1 - \lambda_t \left( \frac{1 - p'_{t-1,GT}}{1 - p'_{t,GT}} \right) \equiv 1 - \lambda_t \left( \frac{\gamma_{t-1}}{\gamma_t} \right),$$ where $\partial_i$ and $\partial_{AD,t}$ represent the gradients of a logit of class $i$ when trained with standard one-hot labels and the proposed soft labels at epoch $t$, respectively. Also, $\gamma$ indicates the inverse confidence of the prediction for the ground truth class. The detailed derivation is in Appendix B.1. Note that $\frac{\gamma_{t-1}}{\gamma_t}$ becomes larger as the prediction confidence on the adversarial example has increased significantly compared to the last epoch. This refers that our method assigns relatively smaller weights to examples that exhibit substantial improvement. Consequently, our method acts as a countermeasure, preventing the model’s predictions from becoming overly confident and thus possess superior calibration performance, which has been known as an attribute of robust models (Grabinski et al., 2022; Wu et al., 2023). ### 3.1.2 Empirical Analysis ![Reliability diagrams](image) Figure 3: Reliability diagrams for PGD-AT, LS, AKD, and our method on CIFAR-100. We note the ECE (lower is better) on the right bottom side of the diagram. To empirically verify the results of our theoretical analysis, we prepare four ResNet-18 models trained with PGD-AT (Baseline), PGD-AT with label smoothing (LS), adversarial knowledge distillation (AKD), and self-adversarial distillation from the last epoch (Ours). LS (Szegedy et al., 2016) is known to mitigate the over-confidence problem by imposing uncertainty on one-hot labels. AKD (Maroto et al., 2022) is an AD method that trains a model using as supervision combinations of adversarial predictions from a pre-trained robust model and one-hot labels. We compare our technique with these prior arts in terms of calibration performance since these arts aim to suppress over-confidence by adopting soft labels which is similar with ours. An ideally well calibrated model would provide high confidence in correct classifications, and low confidence in wrong classifications. To evaluate the calibration performance, we use the expected calibration error (ECE) (Naeini et al., 2015) as our metrics. Let $M$ and $n$ denote the number of confidence interval bins and that of the individual samples, respectively, where each bin contains samples whose confidences fall within the corresponding interval $[\frac{m-1}{M}, \frac{m}{M}]$. Then, ECE is defined as $$ECE = \sum_{m=1}^{M} \frac{\|B_m\|}{n} \|Acc(B_m) - Conf(B_m)\|.$$ four methods and their ECE scores are presented in Figure 3; the smaller the gap in the reliability diagram, the better the calibration of the model. As shown in the figure, our method demonstrates significantly low over-confidence in its predictions on adversarial examples compared to other methods. Furthermore, we observe that our method achieved a notably lower ECE score in comparison, indicating superior calibration performance. 3.2 Collaborative Learning with Natural Model The trade-off between robustness and natural accuracy has been a longstanding issue of AT (Tsipras et al., 2019; Zhang et al., 2019; Yang et al., 2020). This issue persists in AD as it adopts the robust teacher model trained using AT. It is hard to expect that a teacher with high robustness can also be a proper supervisor that helps improve natural accuracy. Thus, to achieve both robustness and natural accuracy through AD, it is reasonable to consider for a student to benefit from the guidance of two teachers: one trained naturally and the other trained through AT. A few prior studies (Chen et al., 2020; Zhao et al., 2022) share this concept to mitigate the trade-off using a pre-trained natural model as the frozen natural model. However, experimental results presented in the literature (Zi et al., 2021; Maroto et al., 2022) demonstrate that distilling knowledge from the static natural model can reduce robustness, indicating that it is not the proper approach. In this paper, we take a different view towards the robust and natural models, and adopt the framework of online distillation (Zhang et al., 2018; Guo et al., 2020; Cui et al., 2021). Instead of using the pre-trained and frozen natural model as a teacher, we treat the natural model as a peer so that both models exchange their knowledge during the training process. We focus on the fact that a model trained by AT and that trained differ in training schemes and data, leading to distinct knowledge representations. By employing mutual learning between these models, we expect that the robust model can acquire knowledge from the natural model without compromising its robustness. Meanwhile, exchanged knowledge takes on differing roles from each model’s perspective. The natural model views the robust model’s insights as a form of regularization. On the other hand, the robust model views the knowledge from the natural model as an alternative to one-hot labels. This asymmetry necessitates a careful balance in the quantity of knowledge exchanged during the collaborative training. Also, since a robust model is in general substantially poor in natural accuracy at early stages of training, knowledge transfer from the robust model to the natural counterpart may hinder their collaborative learning. We thus control the impact of the collaborative learning dynamically through a weight parameter $\lambda$ following the monotonically increasing schedule based on the sine function as in Section 3.1, where the weight is used to create soft labels based on predictions of the robust model. These soft labels are then utilized for training the natural model. This strategy mitigates the cold start issue and ensures effective knowledge transfer between the two models throughout the collaborative learning process. The soft labels for the natural model are given by $$\hat{y}_t = (1 - \lambda_t)y + \lambda_t p_{t}^{\text{rob}}(x_t),$$ where $p_{t}^{\text{rob}}(x_t)$ is the output of the robust model for the natural example at the $t$-th epoch. We train the natural model using these soft labels $\hat{y}_t$, instead of one-hot labels, through the cross-entropy loss. While the natural model is trained using the soft labels, the robust model receives supervision from the natural model through a standard KL-divergence loss. 3.3 The Overall Objective Incorporating the techniques that we suggested in the previous sections, the final objective function for ROAD is given by $$\min_{\theta_{\text{rob}}} \text{CE}(f_{\theta_{\text{rob}}}(x'), \hat{y}) + \beta \cdot \text{KL}(f_{\theta_{\text{rob}}}(x'), f_{\theta_{\text{rob}}}(x)) + \gamma \cdot \text{KL}(f_{\theta_{\text{rob}}}(x), f_{\theta_{\text{nat}}}(x)),$$ where $f_{\theta_{\text{rob}}}$ is the robust model, $f_{\theta_{\text{nat}}}$ is the natural model, hyper-parameter $\beta$ controls the trade-off between robustness and natural accuracy, and hyper-parameter $\gamma$ controls the amount of guidance. The training objective of ROAD contains three components: The first component is derived from Section 3.1, forcing the model to be less over-confident on adversarial examples and improving its generalization consequently. The second term is adopted to further improve robustness by minimizing the output distribution difference between adversarial examples and natural examples; for Algorithm 1 Retrospective Online Adversarial Distillation (ROAD) Require: Robust model $f_{\theta_{rob}}$, Natural model $f_{\theta_{nat}}$, training dataset $D$, learning rate $\tau$, number of epochs $T$, batch size $m$, number of batches $M$, maximum perturbation bound $\epsilon$, attack iterations $K$, step size $\eta$, robust factor $\beta$, guidance factor $\gamma$. 1: for epoch = 1, . . . , T do 2: for mini-batch = 1, . . . , M do 3: Sample a mini-batch $\{(x_i, y_i)\}_{i=1}^m$ from $D$ 4: for $i = 1, . . . , m$ do 5: $x'_i \leftarrow x_i + \epsilon$, $\epsilon \sim \text{Uniform}(-\epsilon, \epsilon)$ 6: for $k = 1, 2, . . . , K$ do 7: $x'_i \leftarrow \Pi_{B_\epsilon(x_i)}(x'_i + \eta \cdot \text{sign}(\nabla_{x'_i}\text{CE}(f_{\theta_{rob}}(x'_i), y_i)))$ 8: end for 9: end for 10: Obtain robust soft labels $\tilde{y}$ using Eq. (1). 11: Obtain natural soft labels $\hat{y}$ using Eq. (2). 12: $\theta_{rob} \leftarrow \theta_{rob} - \tau \nabla_{\theta_{rob}}(\text{CE}(f_{\theta_{rob}}(x'), \tilde{y})) + \beta \cdot \text{KL}(f_{\theta_{rob}}(x'), f_{\theta_{rob}}(x)) + \gamma \cdot \text{KL}(f_{\theta_{rob}}(x), f_{\theta_{nat}}(x).detach())$ 13: $\theta_{nat} \leftarrow \theta_{nat} - \tau \nabla_{\theta_{nat}}(\text{CE}(f_{\theta_{nat}}(x), \hat{y}))$ 14: Save predictions of $f_{\theta_{rob}}$ on $x'$. 15: end for 16: end for this purpose, we utilized the KL divergence loss, following prior studies (Zhang et al., 2019; Wang et al., 2020). This regularization term causes loss in natural accuracy as a trade-off for improved robustness. Nevertheless, this loss of accuracy can be recovered by the subsequent term. Finally, the last component enhances natural accuracy by matching the output distributions of the robust model and its peer natural model. To this end, we adopt KL-divergence to effectively distill the knowledge from the natural model. The complete algorithm is described in Algorithm 1. 4 EXPERIMENTAL RESULTS 4.1 SETUP We mainly use ResNet-18 (He et al., 2016) and MobileNetV2 (Sandler et al., 2018) architectures and train the models with the SGD optimizer with momentum of 0.9. The batch size is set to 128. ROAD is compared with five AT methods, PGD-AT (Madry et al., 2018), TRADES (Zhang et al., 2019), MART (Wang et al., 2020), LBGAT (Cui et al., 2021), and SEAT (Wang & Wang, 2022), and five AD methods, ARD (Goldblum et al., 2020), KD+SWA (Chen et al., 2020), IAD (Zhu et al., 2021), RSLAD (Zi et al., 2021), and AdaIAD (Huang et al., 2023), as well as the combination of AdaAD and IAD (Zhu et al., 2021). We conduct our evaluations on CIFAR-100 and CIFAR-10 (Krizhevsky et al., 2009). For PGD-AT, TRADES, MART, and ROAD, we set the number of epochs to 120 with weight decay 3.5e-3 and the learning rate starts from 0.01 and is divided by 10 at the 75, 90, and 100 epochs. We clarify that the PGD-AT model is trained with different settings with the one mentioned in Section 3.1, focusing on evaluating robustness in this section while assessing generalization ability in Section 3.1. The robust factor $\beta$ is set to 6.0 for TRADES and MART. For LBGAT and SEAT, we directly comply with the official implementation. For ROAD, we fix $\beta$ to 6.0 and set $\gamma$ to 3.0 and 5.0 for CIFAR-100 and CIFAR-10, respectively. $\lambda_t$ follows sine increasing schedule starting from 0 to 0.8. Meanwhile, for the AD methods, we set the number of epochs to 200 with weight decay 5e-4. The learning rate starts from 0.1 and is divided by 10 at the 100, 150, and 175 epochs except KD+SWA. We directly use the PGD-AT model as the teacher model. For KD+SWA, we additionally use the NAT model as the natural teacher model. As recommended in Goldblum et al. (2020), we set the hyper-parameter $\alpha$ of ARD and AdaIAD to 1.0 and distillation temperature $\tau$ to 5.0 and 30.0 for CIFAR-100 and CIFAR-10, respectively. In other training details, we strictly follow the settings from the original papers. For natural model, we train with natural images and the learning rate starts from 0.1 and is divided by 10 at the 75, 90, and 100 epochs. 4.2 Performance with Compact Architectures We first evaluate the robustness of our method in compact size architectures, ResNet-18 and MobileNetV2. We report the results on CIFAR-100 and CIFAR-10 in Table 1 and Table 2, respectively. Regardless of the architectures or datasets, our ROAD demonstrates the best performance in most cases. Performance results in AA indicate that ROAD is robust under not only in white-box attacks but also in black-box attacks. The AD methods show improvement in terms of robustness compared to the PGD-AT teacher, but they exhibit a decrease in natural accuracy. In contrast, ROAD improves the natural accuracy by 1.56% and 0.51% on ResNet-18 and MobileNetV2 respectively, compared to other AT or AD methods. This demonstrates the significant role of collaborative learning with the natural model in mitigating the trade-off between robustness and natural accuracy. Table 1: Validation results of ResNet-18 and MobileNetV2 models on CIFAR-100 trained with different methods. The best and second performances are marked in bold and _underlined_ respectively. \| Model \| Method \| NAT \| PGD-20 \| PGD-100 \| MIM-10 \| AA \| \|-------\|--------\|-----\|--------\|---------\|--------\|----\| \| RN-18 \| NAT \| 77.10 \| 0.0 \| 0.0 \| 0.01 \| 0.0 \| \| \| PGD-AT \| 57.05 \| 30.27 \| 30.22 \| 31.16 \| 25.35 \| \| \| TRADES \| 60.53 \| 29.96 \| 29.87 \| 30.65 \| 25.01 \| \| \| MART \| 53.43 \| 31.86 \| 31.74 \| 32.31 \| 25.70 \| \| \| LBGAT \| 57.76 \| 33.11 \| 33.03 \| 33.51 \| 26.68 \| \| \| SEAT \| 55.88 \| 31.33 \| 31.33 \| 31.82 \| 26.36 \| \| \| ARD \| 55.45 \| 31.01 \| 30.92 \| 31.82 \| 26.21 \| \| \| KD+SWA \| 58.94 \| 30.42 \| 30.36 \| 31.17 \| 26.76 \| \| \| IAD \| 54.59 \| 31.45 \| 31.47 \| 32.17 \| 26.57 \| \| \| RSLAD \| 55.39 \| 31.63 \| 31.52 \| 32.28 \| 26.74 \| \| \| AdaLAD \| 56.39 \| 30.83 \| 30.80 \| 31.03 \| 26.03 \| \| \| ROAD \| 62.09 \| 33.73 \| 33.81 \| 34.43 \| 27.60 \| \| MN-V2 \| NAT \| 75.96 \| 0.0 \| 0.0 \| 0.09 \| 0.0 \| \| \| PGD-AT \| 56.26 \| 29.18 \| 29.08 \| 30.27 \| 24.40 \| \| \| TRADES \| 59.06 \| 29.44 \| 29.32 \| 30.05 \| 24.29 \| \| \| MART \| 48.50 \| 30.66 \| 30.61 \| 30.83 \| 23.94 \| \| \| LBGAT \| 53.40 \| 29.34 \| 29.27 \| 29.68 \| 23.32 \| \| \| SEAT \| 54.60 \| 30.61 \| 30.61 \| 31.12 \| 25.43 \| \| \| ARD \| 51.31 \| 27.77 \| 27.65 \| 28.52 \| 24.46 \| \| \| KD+SWA \| 54.73 \| 28.78 \| 28.72 \| 29.50 \| 24.62 \| \| \| IAD \| 49.58 \| 27.68 \| 27.59 \| 28.30 \| 22.66 \| \| \| RSLAD \| 53.07 \| 30.84 \| 30.75 \| 31.68 \| 25.84 \| \| \| AdaLAD \| 55.12 \| 29.86 \| 29.65 \| 30.56 \| 24.76 \| \| \| ROAD \| 59.57 \| 32.44 \| 32.27 \| 33.02 \| 25.98 \| Table 2: Validation results of ResNet-18 and MobileNetV2 models on CIFAR-10 trained with different methods. The best and second performances are marked in bold and _underlined_ respectively. \| Model \| Method \| NAT \| PGD-20 \| PGD-100 \| MIM-10 \| AA \| \|-------\|--------\|-----\|--------\|---------\|--------\|----\| \| RN-18 \| NAT \| 94.73 \| 0.0 \| 0.0 \| 0.01 \| 0.0 \| \| \| PGD-AT \| 83.63 \| 51.92 \| 51.72 \| 53.60 \| 48.76 \| \| \| TRADES \| 82.77 \| 53.83 \| 53.61 \| 55.27 \| 49.77 \| \| \| MART \| 80.42 \| 54.89 \| 54.62 \| 56.15 \| 48.72 \| \| \| LBGAT \| 78.11 \| 54.26 \| 54.08 \| 55.37 \| 49.92 \| \| \| SEAT \| 83.49 \| 54.40 \| 54.44 \| 55.92 \| 50.78 \| \| \| ARD \| 82.76 \| 51.58 \| 51.40 \| 53.33 \| 48.31 \| \| \| KD+SWA \| 84.14 \| 52.77 \| 52.47 \| 54.66 \| 49.91 \| \| \| IAD \| 82.05 \| 53.82 \| 53.68 \| 55.12 \| 49.77 \| \| \| RSLAD \| 83.13 \| 53.64 \| 53.26 \| 55.58 \| 50.61 \| \| \| AdaLAD \| 83.11 \| 52.34 \| 51.94 \| 53.92 \| 49.15 \| \| \| ROAD \| 84.42 \| 54.93 \| 54.56 \| 56.43 \| 50.91 \| \| MN-V2 \| NAT \| 93.06 \| 0.0 \| 0.0 \| 0.0 \| 0.0 \| \| \| PGD-AT \| 82.57 \| 50.45 \| 50.17 \| 52.20 \| 47.34 \| \| \| TRADES \| 81.17 \| 52.05 \| 51.95 \| 53.36 \| 48.64 \| \| \| MART \| 77.48 \| 53.34 \| 53.28 \| 54.34 \| 46.87 \| \| \| LBGAT \| 72.63 \| 49.78 \| 49.74 \| 50.49 \| 46.11 \| \| \| SEAT \| 81.70 \| 52.73 \| 52.54 \| 54.19 \| 49.16 \| \| \| ARD \| 79.46 \| 48.23 \| 47.94 \| 49.95 \| 45.33 \| \| \| KD+SWA \| 81.44 \| 51.52 \| 51.33 \| 53.26 \| 48.51 \| \| \| IAD \| 79.02 \| 49.96 \| 49.82 \| 51.29 \| 46.10 \| \| \| RSLAD \| 81.93 \| 51.81 \| 51.62 \| 53.53 \| 48.81 \| \| \| AdaLAD \| 81.87 \| 51.06 \| 50.90 \| 52.60 \| 47.91 \| \| \| ROAD \| 82.77 \| 53.72 \| 53.45 \| 54.91 \| 49.27 \| 4.3 Performance with higher capacity architecture In this section, we extend our evaluation on higher capacity architecture, WRN-28-10 (Zagoruyko & Komodakis, 2016). For PGD-AT, TRADES, MART, and ROAD, we adjust the weight decay to 5e-4 and the learning rate starts from 0.1. Other training details remain the same as described in Section 4. We report the results on CIFAR-100 in Table 3. A similar trend is shown as in the previous experiments. ROAD shows superiority on natural accuracy ranging from a minimum of 2.27% to a maximum of 8.53% compared to other models. Furthermore, it exhibits the best robustness even in white-box attacks and ranks second in AA. This result confirms that our method consistently demonstrates superior performance in both robustness and natural accuracy across different architectures. In Appendix D, additional experimental results demonstrate the superior performance of our method. Table 3: Validation results of WRN-28-10 models on CIFAR-100 trained with different methods. The best and second performances are marked in bold and _underlined_ respectively. \| Method \| NAT \| PGD-20 \| PGD-100 \| MIM-10 \| AA \| \|----------\|--------\|--------\|---------\|--------\|--------\| \| PGD-AT \| 61.36 \| 31.37 \| 31.22 \| 32.55 \| 27.63 \| \| TRADES \| 60.10 \| 32.23 \| 32.17 \| 32.89 \| 27.60 \| \| MART \| 55.16 \| 33.65 \| 33.55 \| 33.98 \| 28.19 \| \| LBGAT \| 59.96 \| 34.84 \| 34.84 \| 35.27 \| 28.87 \| \| SEAT \| 59.72 \| 34.46 \| 34.41 \| 34.97 \| 29.52 \| \| AdaAD \| 60.01 \| 34.90 \| 32.43 \| 33.75 \| 28.89 \| \| SSLAD \| 59.78 \| 33.90 \| 33.14 \| 34.93 \| 29.91 \| \| AdaIAD \| 61.42 \| 32.43 \| 32.31 \| 33.50 \| 28.58 \| \| ROAD \| 63.69 \| 35.10 \| 35.06 \| 35.97 \| 29.66 \| 4.4 Ablation studies In this section, we study the importance of each component in ROAD through several experiments. Effect of different scheduling strategies. We study the effect of epoch-wise interpolation ratio $\lambda$ scheduling. In addition to the sine increasing strategy used in our method, we prepare two simple strategies. One is fixing $\lambda$ at the final value, and the other is linearly increasing it. As shown in Figure 4(a), the sine increasing strategy shows the best robustness. Since the sine increasing strategy reaches the final value more quickly than the linear increasing strategy, therefore, it greatly benefits from the effects of self-distillation starting from the midpoint of the training process. In contrast, the fixed strategy exhibits the lowest performance in both natural accuracy and robustness, indicating that the cold start issue could actually hinder learning. Effect of transferring asymmetric knowledge. Next, we also study the effect of asymmetric knowledge transfer between the natural and robust model in ROAD. To verify its effectiveness, we prepare the symmetric version of ROAD: the natural model achieves knowledge via not soft labels but KL-divergence, typically seen in conventional online distillation. We reuse $\gamma$ for simplicity and symmetric knowledge transfer. As shown in Figure 4(b), ROAD significantly outperforms the symmetric version of ROAD in natural accuracy regardless of the value of $\gamma$. Impact of soft labels. We prepare three variants of ROAD: (1) where we removed the first soft labels $\tilde{y}$ to exclude self-distillation from predictions of the last epoch, (2) where we removed the second soft labels $\hat{y}$ to prevent natural model achieve knowledge from robust model through collaborative learning and (3) where we removed both $\tilde{y}$ and $\hat{y}$, replacing them with one-hot labels. As demonstrated in Figure 4(c), both variants show lower robustness than ROAD. This suggests that self-distillation enables the model to enhance robustness. Furthermore, it can be inferred that when the natural model unilaterally conveys knowledge to the robust model, although it may be helpful for natural accuracy, it causes a detrimental effect on robustness. Impact of hyper-parameter $\gamma$. Here, we conduct an experiment to analyze the impact of hyper-parameter $\gamma$. While fixing $\beta$ to 6.0, we vary $\gamma$ from 1.0 to 6.0. The results are demonstrated in Figure 5(a) and Figure 5(b). It is noteworthy that natural accuracy consistently increases as the value of $\gamma$ increases. Furthermore, ROAD achieves the best robust accuracy with $\gamma = \{3.0, 5.0\}$ on CIFAR-100 and CIFAR-10, respectively. (a) Effect of scheduling interpolation ratio $\lambda$. (b) Effect of transferring robust knowledge to natural model (c) Effect of soft labels compared with one-hot labels Figure 4: Comprehensive ablation study of each component of ROAD on CIFAR-100 with ResNet-18. We verify our methods. (a) CIFAR-100 (b) CIFAR-10 (c) Memory Usage (MiB) (d) Time/epoch (s) Figure 5: Comprehensive experiment results of ROAD. (a) and (b) are experiment results of impact of hyper-parameter $\gamma$ in ROAD on CIFAR-100 and CIFAR-10 with ResNet-18, respectively. (c) and (d) are the gpu memory usage and training time cost of ARD, AdaIAD, LBGAT, and ROAD with ResNet-18 on CIFAR-100. 4.5 Evaluation on Computational Complexity In this section, we compare the computational costs of ROAD with two adversarial distillation methods, ARD and AdaIAD, and one online distillation method, LBGAT. We conduct experiments on a single NVIDIA 3090 GPU and maintain consistent implementation details as described in Section 4.1, excluding the evaluation process. For ARD and AdaIAD, we include the cost of pre-training the teacher model. The results are presented in Figure 5(c) and Figure 5(d). From the results, we observe that the computational cost of ROAD is relatively lower than that of ARD and AdaIAD. This is because ROAD does not require a pre-training process although it trains low cost natural model simultaneously. Furthermore, even when excluding the pre-training process, AdaIAD still consumes more time and memory as it requires multiple forward-backward passes of teacher model to craft adversarial examples. Meanwhile LBGAT exhibits a slightly lower computational cost and time consumption, the difference is negligible considering the superior performance of ROAD. Therefore, we can conclude that ROAD is more suitable to resource-constrained environments. 5 Conclusion In this paper, we address the drawbacks of most existing adversarial distillation methods. We point out that conventional adversarial distillation methods require enormous computational cost to pre-train a robust teacher model. Furthermore, student models trained with these methods also suffer from the inherent trade-off between robustness and natural accuracy. 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aOnUe8ah7j	In equation (2) I understand that l_k is the distance between v_1 and v_2. Then, what about circles and ellipses? How is the lengh computed? And for arcs, this definition does not account for the curvature. Two arcs with very different curvature can have the same representation.	Symbol as Points: Panoptic Symbol Spotting via Point-based Representation Wenlong Liu\textsuperscript{1}, Tianyu Yang\textsuperscript{1}, Yuhan Wang\textsuperscript{2}, Qizhi Yu\textsuperscript{2}, Lei Zhang\textsuperscript{1} \textsuperscript{1}International Digital Economy Academy (IDEA) \hspace{1cm} \textsuperscript{2}Vanyi Tech Abstract This work studies the problem of panoptic symbol spotting, which is to spot and parse both countable object instances (windows, doors, tables, etc.) and uncountable stuff (wall, railing, etc.) from computer-aided design (CAD) drawings. Existing methods typically involve either rasterizing the vector graphics into images and using image-based methods for symbol spotting, or directly building graphs and using graph neural networks for symbol recognition. In this paper, we take a different approach, which treats graphic primitives as a set of 2D points that are locally connected and use point cloud segmentation methods to tackle it. Specifically, we utilize a point transformer to extract the primitive features and append a mask2former-like spotting head to predict the final output. To better use the local connection information of primitives and enhance their discriminability, we further propose the attention with connection module (ACM) and contrastive connection learning scheme (CCL). Finally, we propose a KNN interpolation mechanism for the mask attention module of the spotting head to better handle primitive mask downsampling, which is primitive-level in contrast to pixel-level for the image. Our approach, named SymPoint, is simple yet effective, outperforming recent state-of-the-art method GAT-CADNet by an absolute increase of 9.6% PQ and 10.4% RQ on the FloorPlanCAD dataset. The source code and models will be available at https://github.com/nicehuster/SymPoint. 1 Introduction Vector graphics (VG), renowned for their ability to be scaled arbitrarily without succumbing to issues like blurring or aliasing of details, have become a staple in industrial designs. This includes their prevalent use in graphic designs (Reddy et al., 2021), 2D interfaces (Carlier et al., 2020), and Computer-aided design (CAD) (Fan et al., 2021). Specifically, CAD drawings, consisting of geometric primitives (e.g., arc, circle, polyline, etc.), have established themselves as the preferred data representation method in the realms of interior design, indoor construction, and property development, promoting a higher standard of precision and innovation in these fields. Symbol spotting (Rezvanifar et al., 2019; 2020; Fan et al., 2021; 2022; Zheng et al., 2022) refers to spotting and recognizing symbols from CAD drawings, which serves as a foundational task for reviewing the error of design drawing and 3D building information modeling (BIM). Spotting each symbol, a grouping of graphical primitives, within a CAD drawing poses a significant challenge due to the existence of obstacles such as occlusion, clustering, variations in appearances, and a significant imbalance in the distribution of different categories. Traditional symbol spotting usually deals with instance symbols representing countable things (Rezvanifar et al., 2019), like table, sofa, and bed. Fan et al. (2021) further extend it to panoptic symbol spotting which performs both the spotting of countable instances (e.g., a single door, a window, a table, etc.) and the recognition of uncountable stuff (e.g., wall, railing, etc.). Typical approaches (Fan et al., 2021; 2022) addressing the panoptic symbol spotting task involve first converting CAD drawings to raster graphics (RG) and then processing it with powerful image-based detection or segmentation methods (Ren et al., 2015; Sun et al., 2019). Another line of previous works (Jiang et al., 2021; Zheng et al., 2022; Yang et al., 2023) abandons the raster procedure and directly processes vector graphics for recognition with graph convolutions networks. Instead of rastering CAD drawings to images or modeling the graphical primitives with GCN/GAT, which can be computationally expensive, especially for large CAD graphs, we propose a new paradigm that has the potential to shed novel insight rather than merely delivering incremental advancements in performance. Upon analyzing the data characteristics of CAD drawings, we can find that CAD drawing has three main properties: 1). irregularity and disorderliness. Unlike regular pixel arrays in raster graphics/images, CAD drawing consists of geometric primitives(e.g., arc, circle, polyline, etc.) without specific order. 2). local interaction among graphical primitives. Each graphical primitive is not isolated but locally connected with neighboring primitives, forming a symbol. 3). invariance under transformations. Each symbol is invariant to certain transformations. For example, rotating and translating symbols do not modify the symbol’s category. These properties are almost identical to point clouds. Hence, we treat CAD drawing as sets of points (graphical primitives) and utilize methodologies from point cloud analysis (Qi et al., 2017a;b; Zhao et al., 2021) for symbol spotting. In this work, we first consider each graphic primitive as an 8-dimensional data point with the information of position and primitive’s properties (type, length, etc.). We then utilize methodologies from point cloud analysis for graphic primitive representation learning. Different from point clouds, these graphical primitives are locally connected. We therefore propose contrastive connectivity learning mechanism to utilize those local connections. Finally, we borrow the idea of Mask2Former(Cheng et al., 2021; 2022) and construct a masked-attention transformer decoder to perform the panoptic symbol spotting task. Besides, rather than using bilinear interpolation for mask attention downsampling as in (Cheng et al., 2022), which could cause information loss due to the sparsity of graphical primitives, we propose KNN interpolation, which fuses the nearest neighboring primitives, for mask attention downsampling. We conduct extensive experiments on the FloorPlanCAD dataset and our SymPoint achieves 83.3% PQ and 91.1% RQ under the panoptic symbol spotting setting, which outperforms the recent state-of-the-art method GAT-CADNet (Zheng et al., 2022) with a large margin. 2 Related Work Vector Graphics Recognition Vector graphics are widely used in 2D CAD designs, urban designs, graphic designs, and circuit designs, to facilitate resolution-free precision geometric modeling. Considering their wide applications and great importance, many works are devoted to recognition tasks on vector graphics. Jiang et al. (2021) explores vectorized object detection and achieves a superior accuracy to detection methods (Bochkovskiy et al., 2020; Lin et al., 2017) working on raster graphics while enjoying faster inference time and less training parameters. Shi et al. (2022) propose a unified vector graphics recognition framework that leverages the merits of both vector graphics and raster graphics. Panoptic Symbol Spotting Traditional symbol spotting usually deals with instance symbols representing countable things (Rezvanifar et al., 2019), like table, sofa, and bed. Following the idea in (Kirillov et al., 2019), Fan et al. (2021) extended the definition by recognizing semantic of uncountable stuff, and named it panoptic symbol spotting. Therefore, all components in a CAD drawing are covered in one task altogether. For example, the wall represented by a group of parallel lines was properly handled by (Fan et al., 2021), which however was treated as background by (Jiang et al., 2021; Shi et al., 2022; Nguyen et al., 2009) in Vector graphics recognition. Meanwhile, the first large-scale real-world FloorPlanCAD dataset in the form of vector graphics was published by (Fan et al., 2021). Fan et al. (2022) propose CADTransformer, which modifies existing vision transformer (ViT) backbones for the panoptic symbol spotting task. Zheng et al. (2022) propose GAT-CADNet, which formulates the instance symbol spotting task as a subgraph detection problem and solves it by predicting the adjacency matrix. Point Cloud Segmentation Point cloud segmentation aims to map the points into multiple homogeneous groups. Unlike 2D images, which are characterized by regularly arranged dense pixels, point clouds are constituted of unordered and irregular point sets. This makes the direct application of image processing methods to point cloud segmentation an impracticable approach. However, in recent years, the integration of neural networks has significantly enhanced the effectiveness of point cloud segmentation across a range of applications, including semantic segmentation (Qi et al., 2017a; Zhao et al., 2021), instance segmentation (Ngo et al., 2023; Schult et al., 2023) and panoptic segmentation (Zhou et al., 2021; Li et al., 2022; Hong et al., 2021; Xiao et al., 2023), etc. 3 Method Our methods forgo the raster image or GCN in favor of a point-based representation for graphical primitives. Compared to image-based representations, it reduces the complexity of models due to the sparsity of primitive CAD drawings. In this section, we first describe how to form the point-based representation using the graphical primitives of CAD drawings. Then we illustrate a baseline framework for panoptic symbol spotting. Finally, we thoroughly explain three key techniques, attention with local connection, contrastive connection learning, and KNN interpolation, to adapt this baseline framework to better handle CAD data. 3.1 From Symbol to Points Given vector graphics represented by a set of graphical primitives $ \{p_k\} $, we treat it as a collection of points $ \{p_k \| (x_k, f_k)\} $, and each point contains both primitive position $ \{x_k\} $ and primitive feature $ \{f_k\} $ information; hence, the points set could be unordered and disorganized. Primitive position. Given a graphical primitive, the coordinates of the starting point and the ending point are $(x_1, y_1)$ and $(x_1, y_2)$, respectively. The primitive position $ x_k \in \mathbb{R}^2 $ is defined as: \[ x_k = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right), \] We take its center as the primitive position for a closed graphical primitive (circle, ellipse), as shown in Fig. 1a. Primitive feature. We define the primitive features $ f_k \in \mathbb{R}^6 $ as: \[ f_k = [\alpha_k, l_k, \text{onehot}(t_k)], \] where $ \alpha_k $ is the clockwise angle from the $ x $ positive axis to $ x_k $, and $ l_k $ represents the distance between $ v_1 $ and $ v_2 $ for linear primitives, as shown in Fig. 1b. For circular primitives like circles and ellipses, $ l_k $ is defined as the circumference. We encode the primitive type $ t_k $ (line, arc, circle, or ellipse) into a one-hot vector to make up the missing information of segment approximations. 3.2 Panoptic Symbol Spotting via Point-based Representation The baseline framework primarily comprises two components: the backbone and the symbol spotting head. The backbone converts raw points into points features, while the symbol... Figure 2: The overview of our method. After transferring CAD drawings to primitive points, we use a backbone to extract multi-resolution features $F_r$ and append a symbol spotting head to spot and recognize symbols. During this process, we propose attention with connection module (ACM), which utilizes primitive connection information when performing self-attention in the first stage of backbone. Subsequently, we propose contrastive connection learning (CCL) to enhance the discriminability between connected primitive features. Finally, we propose KNN interpolation for attention mask downsampling (AMD) to effectively downsample the high-resolution attention masks. spotting head predicts the symbol mask through learnable queries (Cheng et al., 2021; 2022). Fig. 2 illustrates the whole framework. Backbone. We choose Point Transformer (Zhao et al., 2021) with a symmetrical encoder and decoder as our backbone for feature extraction due to its good generalization capability in panoptic symbol spotting. The backbone takes primitive points as input, and performs vector attention between each point and its adjacent points to explore local relationships. Given a point $p_i$ and its adjacent points $\mathcal{M}(p_i)$, we project them into query feature $q_i$, key feature $k_j$ and value feature $v_j$, and obtain the vector attention as follows: $$w_{ij} = \omega(\gamma(q_i, k_j)), \quad f_i^{\text{attn}} = \sum_{p_j \in \mathcal{M}(p_i)} \text{Softmax}(W_i)j \odot v_j,$$ (3) where $\gamma$ serves as a relational function, such as subtraction. $\omega$ is a learnable weight encoding that calculates the attention vectors. $\odot$ is Hadamard product. Symbol Spotting Head. We follow Mask2Former (Cheng et al., 2022) to use hierarchical multi-resolution primitive features $F_r \in \mathbb{R}^{N_r \times D}$ from the decoder of backbone as the input to the symbol spotting prediction head, where $N_r$ is the number of feature tokens in resolution $r$ and $D$ is the feature dimension. This head consists of $L$ layers of masked attention modules which progressively upscales low-resolution features from the backbone to produce high-resolution per-pixel embeddings for mask prediction. There are two key components in the masked attention module: query updating and mask predicting. For each layer $l$, query updating involves interacting with different resolution primitive features $F_r$ to update query features. This process can be formulated as, $$X_l = \text{softmax}(A_{l-1} + Q_lK_l^T)V_l + X_{l-1},$$ (4) where $X_l \in \mathbb{R}^{O \times D}$ is the query features. $O$ is the number of query features. $Q_l = f_Q(X_{l-1})$, $K_l = f_K(F_r)$ and $V_l = f_V(F_r)$ are query, key and value features projected by MLP layers. $A_{l-1}$ is the attention mask, which is computed by, $$A_{l-1}(v) = \begin{cases} 0 & \text{if } M_{l-1}(v) > 0.5, \\ -\infty & \text{otherwise}. \end{cases}$$ (5) where $v$ is the position of feature point and $M_{l-1}$ is the mask predicted from mask predicting part. Note that we need to downsample the high-resolution attention mask to adopt the query updating on low-resolution features. In practice, we utilize four coarse-level primitive features from the decoder of backbone and perform query updating from coarse to fine. During mask predicting process, we obtain the object mask $M_t \in \mathbb{R}^{O \times N_0}$ and its corresponding category $Y_t \in \mathbb{R}^{O \times C}$ by projecting the query features using two MLP layers $f_Y$ and $f_M$, where $C$ is the category number and $N_0$ is the number of points. The process is as follows: $$Y_t = f_Y(X_t), \quad M_t = f_M(X_t)F_0^T,$$ The outputs of final layer, $Y_L$ and $M_L$, are the predicted results. ### 3.3 Attention with Connection Module The simple and unified framework rewards excellent generalization ability by offering a fresh perspective of CAD drawing, a set of points. It can obtain competitive results compared to previous methods. However, it ignores the widespread presence of primitive connections in CAD drawings. It is precisely because of these connections that scattered, unrelated graphical elements come together to form symbols with special semantics. In order to utilize these connections between each primitive, we propose Attention with Connection Module (ACM), the details are shown below. It is considered that these two graphical primitives $(p_i, p_j)$ are interconnected if the minimum distance $d_{ij}$ between the endpoints $(v_i, v_j)$ of two graphical primitives $(p_i, p_j)$ is below a certain threshold $\epsilon$, where: $$d_{ij} = \min_{v_i \in p_i, v_j \in p_j} \|\|v_i - v_j\|\| < \epsilon.$$ To keep the complexity low, at most $K$ connections are allowed for every graphical primitive by random dropping. Fig. 3a demonstrates the connection construction around the wall symbol, the gray line is the connection between two primitives. In practice, we set $\epsilon$ to 1.0px. The attention mechanism in (Zhao et al., 2021) directly performs local attention between each point and its adjacent points to explore the relationship. The original attention mechanism interacts only with neighboring points within a spherical region, as shown in Fig. 3b. Our ACM additionally introduces the interaction with locally connected primitive points during attention (pink points), essentially enlarging the radius of the spherical region. Note that we experimentally found that crudely increasing the radius of the spherical region without considering the local connections of primitive points does not result in performance improvement. This may be explained by that enlarging the receptive field also introduces additional noise at the same time. Specifically, we extend the adjacent points set $M(p_i)$ in Eq. (3) to $A(p_i) = M(p_i) \cup C(p_i)$, where $C(p_i) = \{p_j \| d_{ij} < \epsilon\}$, yielding, $$f_i^{\text{attn}} = \sum_{p_j \in A(p_i)} \text{Softmax}(W_i)_{ij} \odot v_j,$$ In practice, since we cannot directly obtain the connection relationships of the points in the intermediate layers of the backbone, we integrate this module into the first stage of the backbone to replace the original local attention, as shown in Fig. 2. ### 3.4 Contrastive Connection Learning. Although the information of primitive connection are considered when calculating attention of the encoder transformer, locally connected primitives may not belong to the same instance, in other words, noisy connections could be introduced while take primitive connections into consideration, as shown in Fig. 3c. Therefore, in order to more effectively utilize connection information with category consistency, we follow the widely used InfoNCE loss (Oord et al., 2018) and its generalization (Frosst et al., 2019; Gutmann & Hyvärinen, 2010) to define the contrastive learning objective on the final output feature of backbone. We encourage learned representations more similar to its connected points from the same category and more distinguished from other connected points from different categories. Additionally, we also take neighbor points $ M(p_i) $ into consideration, yielding, \[ L_{CCL} = -\log \frac{\sum_{p_j \in A(p_i) \land l_j = l_i} \exp(-d(f_i, f_j)/\tau)}{\sum_{p_k \in A(p_i)} \exp(-d(f_i, f_k)/\tau)} \] (9) where $ f_i $ is the backbone feature of $ p_i $, $ d(\cdot, \cdot) $ is a distance measurement, $ \tau $ is the temperature in contrastive learning, we set the $ \tau = 1 $ by default. ### 3.5 KNN Interpolation During the process of query updating in symbol spotting head Eq. (4) & Eq. (5), we need to convert high-resolution mask predictions to low-resolution for attention masks computation as shown in Fig. 2 (AMD on the right). Mask2Former (Cheng et al., 2022) employs the bilinear interpolation on the pixel-level mask for downsampling. However, the masks of CAD drawings are primitive-level, making it infeasible to directly apply the bilinear interpolation on them. To this end, we propose the KNN interpolation for downsampling the attention masks by fusing the nearest neighbor points. A straightforward operation is max pooling or average pooling. We instead utilize distance-based interpolation. For simplicity, we omit layer index $ l $ in $ A $, \[ A^r(p_j) = \frac{\sum_{p_j \in K(p_i)} A^0(p_j)/d(p_i, p_j)}{\sum_{p_j \in K(p_i)} 1/d(p_i, p_j)} \] (10) where, $ A^0 $ and $ A^r $ are the full-resolution attention mask and the $ r $-resolution attention mask respectively. $ d(\cdot, \cdot) $ is a distance measurement. $ K(p_i) $ is the set of $ K $ nearest neighbors, In practice, we set $ K = 4^r $ in our experiments. ### 3.6 Training and Inference Throughout the training phase, we adopt bipartite matching and set prediction loss to assign ground truth to predictions with the smallest matching cost. The full loss function $ L $ can be formulated as $ L = \lambda_{BCE} L_{BCE} + \lambda_{dice} L_{dice} + \lambda_{cls} L_{cls} + \lambda_{CCL} L_{CCL} $, while $ L_{BCE} $ is the binary cross-entropy loss (over the foreground and background of that mask), $ L_{dice} $ is the Dice loss (Deng et al., 2018), $ L_{cls} $ is the default multi-class cross-entropy loss to supervise the queries classification, $ L_{CCL} $ is contrastive connection loss. In our experiments, we empirically set $ \lambda_{BCE} : \lambda_{dice} : \lambda_{cls} : \lambda_{CCL} = 5 : 5 : 2 : 8 $. For inference, we simply use argmax to determine the final panoptic results. ### 4 Experiments In this section, we present the experimental setting and benchmark results on the public CAD drawing dataset FloorPlanCAD (Fan et al., 2021). Following previous works (Fan et al., 2021; Zheng et al., 2022; Fan et al., 2022), we also compare our method with typical image-based instance detection (Ren et al., 2015; Redmon & Farhadi, 2018; Tian et al., 2019; Zhang et al., 2022). Besides, we also compare with point cloud semantic segmentation methods (Zhao et al., 2021). Extensive ablation studies are conducted to validate the effectiveness of the proposed techniques. In addition, we have also validated the generalizability of our method on other datasets beyond floorplanCAD, with detailed results available in the Appendix A. #### 4.1 Experimental Setting Dataset and Metrics. FloorPlanCAD dataset has 11,602 CAD drawings of various floor plans with segment-grained panoptic annotation and covering 30 things and 5 stuff classes. \| Methods \| PanCADNet (Fan et al., 2021) \| CADTransformer (Fan et al., 2022) \| GAT-CADNet (Zheng et al., 2022) \| PointT$^\dagger$ (Zhao et al., 2021) \| SymPoint (ours) \| \|---------\|-----------------------------\|---------------------------------\|-------------------------------\|----------------------------------\|----------------\| \| F1 \| 80.6 \| 82.2 \| 85.0 \| 83.2 \| 86.8 \| \| wF1 \| 79.8 \| 80.1 \| 82.3 \| 80.7 \| 85.5 \| Table 1: Semantic Symbol Spotting comparison results with previous works. $\dagger$: backbone with double channels. wF1: length-weighted F1. \| Method \| Backbone \| AP50 \| AP75 \| mAP \| #Params \| Speed \| \|-----------------\|----------\|------\|------\|-----\|---------\|-------\| \| FasterRCNN \| R101 \| 60.2 \| 51.0 \| 45.2\| 61M \| 59ms \| \| YOLOv3 \| DarkNet53\| 63.9 \| 45.2 \| 41.3\| 62M \| 11ms \| \| FCOS \| R101 \| 62.4 \| 49.1 \| 45.3\| 51M \| 57ms \| \| DINO \| R50 \| 64.0 \| 54.9 \| 47.5\| 47M \| 42ms \| \| SymPoint (ours) \| PointT$^\dagger$ \| 66.3 \| 55.7 \| 52.8\| 35M \| 66ms \| Table 2: Instance Symbol Spotting comparison results with image-based detection methods. Following (Fan et al., 2021; Zheng et al., 2022; Fan et al., 2022), we use the panoptic quality (PQ) defined on vector graphics as our main metric to evaluate the performance of panoptic symbol spotting. By denoting a graphical primitive $e = (l, z)$ with a semantic label $l$ and an instance index $z$, PQ is defined as the multiplication of segmentation quality (SQ) and recognition quality (RQ), which is formulated as $$PQ = RQ \times SQ$$ $$= \frac{\|TP\|}{\|TP\| + \frac{1}{2}\|FP\| + \frac{1}{2}\|FN\|} \times \frac{\sum_{(s_p, s_g) \in TP} \text{IoU}(s_p, s_g)}{\|TP\|}$$ $$= \frac{\sum_{(s_p, s_g) \in TP} \text{IoU}(s_p, s_g)}{\|TP\| + \frac{1}{2}\|FP\| + \frac{1}{2}\|FN\|}.$$ where, $s_p = (l_p, z_p)$ is the predicted symbol, $s_g = (l_g, z_g)$ is the ground truth symbol. $\|TP\|$, $\|FP\|$, $\|FN\|$ indicate true positive, false positive and false negative respectively. A certain predicted symbol is considered as matched if it finds a ground truth symbol, with $l_p = l_g$ and $\text{IoU}(s_p, s_g) > 0.5$, where the IoU is computed by: $$\text{IoU}(s_p, s_g) = \frac{\sum_{e_i \in s_p \cup s_g} \log(1 + L(e_i))}{\sum_{e_j \in s_p \cup s_g} \log(1 + L(e_j))}.$$ Implementation Details. We implement SymPoint with Pytorch. We use PointT (Zhao et al., 2021) with double channels as the backbone and stack $L = 3$ layers for the symbol spotting head. For data augmentation, we adopt rotation, flip, scale, shift, and cutmix augmentation. We choose AdamW (Loshchilov & Hutter, 2017) as the optimizer with a default weight decay of 0.001, the initial learning rate is 0.0001, and we train the model for 1000 epochs with a batch size of 2 per GPU on 8 NVIDIA A100 GPUs. 4.2 Benchmark Results Semantic symbol spotting. We compare our methods with point cloud segmentation methods (Zhao et al., 2021), and symbol spotting methods (Fan et al., 2021; 2022; Zheng et al., 2022). The main test results are summarized in Tab. 1. Our algorithm surpasses all previous methods in the task of semantic symbol spotting. More importantly, compared to GAT-CADNet (Zheng et al., 2022), we achieves an absolute improvement of 1.8% F1. and 3.2% wF1 respectively. For the PointT$^\dagger$, we use our proposed point-based representation in Section 3.1 to convert the CAD drawing to a collection of points as input. It is worth noting that PointT$^\dagger$ has already achieved comparable results to GAT-CADNet (Zheng et al., 2022), which demonstrates the effectiveness of the proposed point-based representation for CAD symbol spotting. Instance Symbol Spotting. We compare our method with various image detection methods, including FasterRCNN (Ren et al., 2015), YOLOv3 (Redmon & Farhadi, 2018), \| Method \| Data Format \| PQ \| SQ \| RQ \| #Params \| Speed \| \|--------------------------------\|-------------\|------\|------\|------\|---------\|-------\| \| PanCADNet (Fan et al., 2021) \| VG + RG \| 55.3 \| 83.8 \| 66.0 \| >42M \| >1.2s \| \| CADTransformer (Fan et al., 2022)\| VG + RG \| 68.9 \| 88.3 \| 73.3 \| >65M \| >1.2s \| \| GAT-CADNet (Zheng et al., 2022) \| VG \| 73.7 \| 91.4 \| 80.7 \| - \| - \| \| PointT³Cluster (Zhao et al., 2021)\| VG \| 49.8 \| 85.6 \| 58.2 \| 31M \| 80ms \| \| SymPoint (ours, 300epoch) \| VG \| 79.6 \| 89.4 \| 89.0 \| 35M \| 66ms \| \| SymPoint (ours, 500epoch) \| VG \| 81.9 \| 90.6 \| 90.4 \| 35M \| 66ms \| \| SymPoint (ours, 1000epoch) \| VG \| 83.3 \| 91.4 \| 91.1 \| 35M \| 66ms \| Table 3: Panoptic Symbol Spotting comparisons results with previous works. VG: vector graphics, RG: raster graphics. (a) Ablation studies of different techniques \| Baseline \| ACM \| CCL \| KInter \| PQ \| RQ \| SQ \| \|----------\|-----\|-----\|--------\|------\|------\|------\| \| ✓ \| ✓ \| ✓ \| ✓ \| 73.1 \| 83.3 \| 87.7 \| \| ✓ \| ✓ \| ✓ \| \| 72.6 \| 82.9 \| 87.6 \| \| ✓ \| ✓ \| ✓ \| \| 73.5 \| 83.9 \| 87.6 \| \| ✓ \| ✓ \| ✓ \| \| 74.3 \| 85.8 \| 86.6 \| \| ✓ \| ✓ \| ✓ \| ✓ \| 77.3 \| 87.1 \| 88.7 \| (b) Ablation studies of mask downsampling \| DSampling method \| PQ \| RQ \| SQ \| \|------------------\|------\|------\|------\| \| linear \| 74.3 \| 85.8 \| 86.6 \| \| knn avepool \| 75.9 \| 85.9 \| 88.4 \| \| knn maxpool \| 77.0 \| 86.7 \| 88.8 \| \| knn interp \| 77.3 \| 87.1 \| 88.7 \| (c) Ablation studies on architecture design. BS: Backbone size. SW: share weights. L: layer number of spotting head. O: query number. D: feature dimension. ✓ in the share weights column means whether share weights for head layers. Table 4: Ablation Studies on different techniques, attention mask downsampling, and architecture design. FCOS (Tian et al., 2019), and recent DINO (Zhang et al., 2022). For a fair comparison, we post-process the predicted mask to produce a bounding box for metric computation. The main comparison results are listed in Tab. 2. Although our framework is not trained to output a bounding box, it still achieves the best results. Panoptic Symbol Spotting. To verify the effectiveness of the symbol spotting head, we also design a variant method without this head, named PointT³Cluster, which predicts an offset vector per graphic entity to gather the instance entities around a common instance centroid and performs class-wise clustering (e.g. meanshift (Cheng, 1995)) to get instance labels as in CADTransformer (Fan et al., 2022). The final results are listed in Tab. 3. Our SymPoint trained with 300epoch outperforms both PointT³Cluster and the recent SOTA method GAT-CADNet (Zheng et al., 2022) substantially, demonstrate the effectiveness of the proposed method. Our method also benefits from longer training and achieves further performance improvement. What’s more, our method runs much faster during the inference phase than previous methods. For image-based methods, it takes approximately 1.2s to render a vector graphic into an image while our method does not need this process. The qualitative results are shown in Fig. 4. 4.3 Ablation Studies In this section, we carry out a series of comprehensive ablation studies to clearly illustrate the potency and intricate details of the SymPoint framework. All ablations are conducted under 300 epoch training. Figure 4: Qualitative comparison of panoptic symbol spotting results with CADTransformer. Primitives belonging to different classes are represented in distinct colors. The colormap for each category can be referenced in Fig. 8. Effects of Techniques. We conduct various controlled experiments to verify different techniques that improve the performance of SymPoint in Tab. 4a. Here the baseline means the method described in Sec. 3.2. When we only introduce ACM (Attention with Connection Module), the performance drops a bit due to the noisy connections. But when we combine it with CCL (Contrastive Connection Learning), the performance improves to 74.3 of PQ. Note that applying CCL alone could only improve the performance marginally. Furthermore, KNN Interpolation boosts the performance significantly, reaching 77.3 of PQ. KNN Interpolation. In Tab. 4b, we ablate different ways of downsampling attention mask: 1) linear interpolation, 2) KNN average pooling, 3) KNN max pooling, 4) KNN interpolation. KNN average pooling and KNN max pooling means using the averaged value or max value of the K nearest neighboring points as output instead of the one defined in Eq. (10). We can see that the proposed KNN interpolation achieves the best performance. Architecture Design. We analyze the effect of varying model architecture design, like channel number of backbone and whether share weights for the L layers of symbol spotting head. As shown in Tab. 4c, we can see that enlarging the backbone, the query number and the feature channels of the symbol spotting head could further improve the performance. Sharing weights for spotting head not only saves model parameters but also achieves better performance compared with the one that does not share weights. 5 Conclusion and Future Work This work introduces a novel perspective for panoptic symbol spotting. We treat CAD drawings as sets of points and utilize methodologies from point cloud analysis for symbol spotting. Our method SymPoint is simple yet effective and outperforms previous works. One limitation is that our method needs a long training epoch to get promising performance. Thus accelerating model convergence is an important direction for future work. REFERENCES Alexey Bochkovskiy, Chien-Yao Wang, and Hong-Yuan Mark Liao. Yolov4: Optimal speed and accuracy of object detection. arXiv preprint arXiv:2004.10934, 2020. Alexandre Carlier, Martin Danelljan, Alexandre Alahi, and Radu Timofte. Deepsvg: A hierarchical generative network for vector graphics animation. Advances in Neural Information Processing Systems, 33:16351–16361, 2020. Bowen Cheng, Alex Schwing, and Alexander Kirillov. Per-pixel classification is not all you need for semantic segmentation. Advances in Neural Information Processing Systems, 34:17864–17875, 2021. Bowen Cheng, Ishan Misra, Alexander G Schwing, Alexander Kirillov, and Rohit Girdhar. Masked-attention mask transformer for universal image segmentation. 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r5sikTJ94y	Fig. 5 compares energy efficiency of IMC vs. digital accelerators. Was this comparison done at iso-computational or network accuracy, e.g., did both architectures display the same misclassification rate for image classification?	Reshape and Adapt for Output Quantization (RAOQ): Quantization-aware Training for In-memory Computing Systems Anonymous authors Paper under double-blind review Abstract In-memory computing (IMC) has emerged as a promising solution to address both the computation and data-movement challenges posed by modern AI models. IMC takes advantage of the intrinsic parallelism of memory hardware and performs computation on data in-place directly in the memory array. To do this, IMC typically relies on analog operation, which enables high energy and area efficiency. However, analog operation makes analog-to-digital converters (ADCs) necessary, for converting results back to the digital domain. This introduces an important new source of quantization error, impacting inference accuracy. This paper proposes a Reshape and Adapt for Output Quantization (RAOQ) approach to overcome this issue, which comprises two classes of mechanisms motivated by the fundamental impact and constraints of ADC quantization, including: 1) mitigating ADC quantization error by adjusting the statistics of activations and weights, through an activation-shifting approach (A-shift) and a weight reshaping technique (W-reshape); 2) adapting AI models to better tolerate ADC quantization, through a bit augmentation method (BitAug) to aid SGD-based optimization. RAOQ demonstrates consistently high performance across different scales of neural network models for image classification, object detection, and natural language processing (NLP) tasks at various bit precisions, achieving state-of-the-art accuracy with practical IMC implementations. 1 Introduction Rapid advances in AI have greatly impacted various application domains, including computer vision, natural language processing, speech, etc. Recent generative AI breakthroughs have pushed the strength of AI even further, producing remarkably realistic and imaginative outputs, blurring the line between human- and machine-generated content (OpenAI [2023], Chowdhery et al. [2022]). However, increasing AI capability has come from increasing model complexity, with a sharp rise in both the number of compute operations and the number of model parameters, placing huge demands on hardware resources (Villalobos et al. [2022], Smith et al. [2022]). This has driven the development of specialized hardware architectures to accelerate AI model computations. While digital accelerators have been widely deployed to improve compute efficiency, they do not address the large amount of data movement involved, which has been shown to pose a critical energy and performance bottleneck in state-of-the-art (SOTA) models (Verma et al. [2019]). In-memory computing (IMC), on the other hand, performs computations in place on stored data, providing an approach to simultaneously address both compute efficiency and data movement. While both digital and analog IMC have been proposed, providing various advantages and trade-offs towards energy efficiency and accuracy, this work focuses on energy-aggressive highly-parallel analog IMC, addressing the critical bottleneck within the architecture via algorithmic solutions. A fundamental requirement of analog IMC is the need for analog-to-digital converters (ADCs), to provide compute outputs back to the digital domain for further processing. Importantly, ADCs introduce an additional source of quantization, which can substantially degrade accuracy in SOTA AI models. The level of quantization error from the ADC is fundamentally set by the level of IMC parallelism, which also directly sets the compute efficiency and throughput advantage. Unlike quantization of activations and weights, whose clipping parameters can be directly optimized during training, ADC quantization on the compute results does not provide this degree of freedom and thus requires new methods to address. Previous works introduce artificial clipping to model ADC quantization at the hardware design stage [Gonugondla et al., 2020; Sakr & Shanbhag, 2021]. However, this limits hardware flexibility in supporting various types of models, which may present different ADC-input data distributions and thus require different optimal clipping values. To address such quantization challenges, this paper presents Reshape and Adapt for Output Quantization (RAOQ), to tackle the challenges at the algorithmic level. As neural networks generally are sensitive to drastic changes, we first perform quantization-aware training (QAT) for activations and weights only, and then apply RAOQ, in another stage of training with ADC quantization introduced. We explore RAOQ across multiple applications, i.e., image classification, object detection, and natural language processing (NLP), on ImageNet [Deng et al., 2009], COCO 2017 [Lin et al., 2014], and SQuAD 1.1 [Rajpurkar et al., 2016] datasets, respectively. To the best of our knowledge, this work is the first to demonstrate approaches that enable IMC for inference across various scales of models and challenging datasets/tasks. The major contributions of our work are as follows: 1. We conduct an analysis of the relationship between neural network activations, weights, and ADC quantization. We identify the statistical attributes of activations and weights that yield a high signal-to-quantization-noise ratio (SQNR) in the presence of ADC quantization. 2. We propose an activation-shifting method (A-shift) motivated by the preferred statistical attributes for activations, and a weight-reshaping technique via kurtosis regularization (W-shape) motivated by the preferred statistical attributes for weights. 3. We propose bit augmentation (BitAug), where the model is augmented in the dimension of ADC bit precision to aid the optimization process, assisting model adaptation to ADC quantization. 4. We conduct experiments on different models and tasks (i.e., ReNet18/50 [He et al., 2016], MobileNetV2 [Sandler et al., 2018], EfficientNet-lite0 [Tan & Le, 2019], YOLOv5s [Jocher et al., 2022], BERT-base/large [Devlin et al., 2018]), and across different quantization levels for activations, weights, and ADCs. The consistently high performance achieved by our proposed methods provides promise for their generalizability across challenging AI tasks. 2 BACKGROUND AND RELATED WORKS 2.1 IN-MEMORY COMPUTING (IMC) IMC aims to address both compute and data-movement costs in matrix-vector multiplications (MVMs), which are dominant operations in modern AI models. This is achieved by storing matrix weights in a 2D array of memory bit cells as shown in Fig. 1a, and accessing compute results over many weight bits, rather than accessing the individual weight bits themselves. Specifically, this is achieved by performing multiplication in each bit cell between stored weight data and provided input data, and then accumulation to reduce the products in each column to a single compute result. The level of reduction, set by the row parallelism of IMC operation, thus determines the energy efficiency and throughput gains. To enable energy- and area-efficient computation within the constrained bit cells, IMC can leverage analog operation, where the compute results then need to be converted back to the digital domain via ADCs [Valavi et al., 2019; Lee et al., 2021b; Deaville et al., 2022; Yin et al., 2020; Hsieh et al., 2023]. Such analog operation raises two challenges. First, it is sensitive to noise sources, which degrade the output signal-to-noise ratio (SNR). Researchers have proposed algorithmic noise-aware training approaches to overcome this [Zhang et al., 2022; He et al., 2019], but which have only shown success in simple tasks (MNIST, CIFAR-10/100 datasets) at low levels of IMC row parallelism. Instead, recent work has moved to a high-SNR form of IMC, overcoming such analog noise, enabling scale-up to higher levels of row parallelism [Hia et al., 2022; Lee et al., 2021a]. This has left SOTA analog IMC primarily subject to the second challenge, which is ADC quantization. As an example, Fig. 1b shows the degraded SQNR due to ADC quantization and inference accuracy in ResNet50 on ImageNet. Consequently, such quantization prevents IMC from scaling up and poses an ultimate limitation to the IMC efficiency and throughput. While ADC precision can be increased for higher SQNR and accuracy, this brings substantial hardware cost, with ADCs showing dominating energy consumption (Lee et al., 2021a). This work introduces efficient algorithmic approaches to address the critical challenge in IMC systems today, which is ADC quantization, doing so without incurring additional hardware costs, to demonstrate applicability on a critical set of models. ![Figure 1](image) (a) An illustration of an MVM operation via IMC. (b) SQNR and accuracy degradation due to ADC quantization. (c) Learning curves for conventional QAT with ADC quantization involved. ### 2.2 Quantization-Aware Training (QAT) QAT restores model accuracy, which may otherwise degrade due to quantization noise, through a training process that adapts the model parameters. QAT methods have been proposed to successfully demonstrate SOTA accuracy in aggressively quantized networks (Jacob et al., 2017; Gupta et al., 2015; Louizos et al., 2018; Bhalgat et al., 2020; Jain et al., 2019; Zhou et al., 2016; Nagel et al., 2022; Wang et al., 2022; Park et al., 2022; Esser et al., 2019). However, previous QAT mainly focuses on quantization from inputs (i.e., weight and activation), not considering ADC quantization on the compute outputs in IMC. As a result, IMC shows substantially degraded model accuracy even with conventional QAT, as seen in Fig. 1c. To address ADC quantization in IMC, Jin et al. (2022) introduces a modified straight-through estimator (STE) (Bengio et al., 2013) along with calibration and rescaling techniques to assist the QAT process, demonstrating ResNet models on CIFAR-10/100 datasets. Sun et al. (2021) proposes a non-uniform activation quantization scheme and a reduced quantization range, validating on the CIFAR-10 dataset. Wei et al. (2020) proposes modified minmax quantizers for activations and weights to incorporate hardware statistics of IMC, testing on MNIST and CIFAR-10 datasets. While these prior works show success on relatively simple datasets, their success has not transferred to more complicated datasets and AI tasks. In this work, we propose improved QAT techniques to enable SOTA accuracy applicable to various bit precisions on more challenging models and tasks. ### 3 Analysis and Rationale from ADC Quantization To formally define the IMC ADC quantization problem, let $ x \in \mathbb{R}^M $ be a data vector of the activation $ X $ and let $ w \in \mathbb{R}^M $ be a vector of an output channel of the weight $ W $. Denote $ \overline{x} $ and $ \overline{w} $ as their quantized counterparts, an IMC column then computes a portion of MVM: \[ y = \langle \overline{x}, \overline{w} \rangle = \sum_{i=1}^{M} \overline{w}_i \overline{x}_i. \] Note that convolutions can be converted to MVMs via `im2col` operations. For a $ b_x $-bit activation, $ b_w $-bit weight, $ b_a $-bit ADC, and memory with dimension $ M \times N $, assuming symmetric quantization is applied to weights, the ADC quantization and its quantization step $ \Delta_a $ is defined as \[ \overline{y} = \left\lfloor \text{clip} \left( \frac{y}{\Delta_a}, n_a, p_a \right) \right\rfloor, \] \[ \Delta_a = \frac{2M(2^{b_x} - 1)(2^{b_w} - 1)}{2^{b_a} k}, \] where $ \lfloor \cdot \rfloor $ denotes the floor operation. Similar to conventional QAT, the gradient of the floor operation is approximated using STE (Bengio et al., 2013). Above, $(n_a, p_a) = (-2^{b_a-1}, 2^{b_a-1} - 1)$, and $k$ is a positive integer, serving as a hardware design parameter to provide fixed clipping (due to the ADC’s supported input range). Eq. [3] assumes unsigned activations. For signed activations, we can simply replace $2^{b_x} - 1$ by $2^{b_x-1} - 1$. In general, $\Delta_a$ is fixed for given hardware and is not trainable at the algorithmic level. Fig. 2a shows the distribution of an ADC input from ImageNet dataset via the ResNet50 model. We see that the input concentrates around a small portion of the ADC range, resulting in a small signal, relative to the quantization step $\Delta_a$. A choice of large $k$ could help to have a finer step $\Delta_a$, but would potentially introduce substantial clipping error. As different layers and models lead to different statistics of the compute outputs (ADC inputs), there is no optimal $\Delta_a$ to rule them all. Thus, with no algorithmically controllable parameters for ADC quantization, the only degrees of freedom left are parameters applicable to the activations and weights. ![Figure 2](image) (a) Distributions of ADC input (compute output). (b) Relationship between ADC SQNR and the variance of ADC input $Var[Y]$. (c-d) Relationship of the variance of ADC input to the 2nd moment of the quantized activation and the variance of the quantized weight. Based on observations in Fig. 2a, we prefer the variance of the ADC input $Var[Y]$ to be maximized in order to maximize signal power and utilize as many ADC quantization levels as possible. This is explicitly shown in Fig. 2b via the post-ADC SQNR. This focus on 2nd-order statistics, makes it natural to consider the dependence on the 2nd moment of the activation $X$ and weight $W$. Before the training starts, activations and weights are independent of each other, and $Var[Y]$ is proportionally set by $E[X^2]$ and $E[W^2]$. However, during and after training, generally the assumption of independence does not hold, as $X$ and $W$ exhibit correlation through the neural network learning process. Nonetheless, we postulate that a more narrow relationship holds, namely that there is direct dependence between the 2nd moments, and we conduct an empirical study to validate this. We randomly sample images from CIFAR10 and ImageNet datasets, and also randomly generate input data. We use ResNet50 and MobileNetV2 as example networks to perform standard QAT, since these contain the network structures encountered in most SOTA models. To manage computation complexity, we only take the first few layers of these models for this study. In Fig. 2c-2d, we plot the variance of the ADC input $Var[Y]$ vs. $E[X^2]$ and $E[W^2]$, respectively, and observe a proportional relationship. Further, since neural network weights are typically symmetrically distributed around zero (Bhalgat et al., 2020), $E[W^2]$ can be taken to be $Var[W]$, and we postulate that $Var[Y]$ can be increased by maximizing $Var[W]$ and $E[X^2]$, to improve IMC SQNR in the presence of ADC quantization. This rationale forms the basis of the W-reshape and A-shift techniques that form the proposed ROAQ approach described below. In the following sections, we use $L_Q$ to denote the loss during the QAT stage of training and use $L_A$ to denote the loss during the RAOQ stage of training, after QAT. 4 RESHAPE AND ADAPT FOR OUTPUT QUANTIZATION (RAOQ) 4.1 SQNR ENHANCEMENT Weight reshaping (W-reshape). To maximize $Var[W]$, one option is to perform aggressive scaling during quantization. However, this is expected to introduce substantial clipping error, posing an adverse trade-off with weight distortion. Thus, we seek an alternate approach to increasing $Var[W]$, by adapting the distribution shape to avoid severe clipping. Neural network weights typically exhibit a symmetric distribution in the exponential family, e.g., normal distribution or Laplace distribution (Banner et al., 2019; Shkolnik et al., 2020), which results in relatively low variance. We therefore propose a penalty on weights to drive towards a distribution with a large variance, in a manner where the penalty does not degrade previous accuracy. This is achieved by introducing a kurtosis loss as a function of the quantized weights. Kurtosis describes the tailedness of a distribution, and such loss is defined as the standardized $4^{th}$ moment, i.e., $$\kappa = \mathbb{E} \left[ \left( \frac{\overline{W} - \mu_{\overline{W}}}{\sigma_{\overline{W}}} \right)^4 \right],$$ where $\overline{W}$ is the quantized weight, $\mu_{\overline{W}}$ and $\sigma_{\overline{W}}$ denote the mean and standard deviation of $\overline{W}$. This encourages the majority of $\overline{W}$ to be concentrated in the tails of the distribution (Moors, 1986). This is different from (Shkolnik et al., 2020), where kurtosis loss is applied on the floating-point weights specifically to drive them towards a uniform distribution, which maximizes their quantization robustness. Since our interest is in improving ADC quantization, rather than weight quantization, we apply more aggressive kurtosis loss on the already quantized weights, which are determined by both the statistics of the floating point weights and the quantization parameters. We analyze the impact of W-reshape on the QAT accuracy and provide details in Appendix A. This loss, computed at each layer, is summed up to produce the final loss and then combined with the original loss function $L_c$ against the ground truth during the QAT stage, i.e., $$L_Q = L_c + \lambda_\kappa \sum_l \kappa_l,$$ where $\lambda_\kappa$ is a coefficient to balance the two loss terms, and $l$ is an index for neural network layers. Fig. 3a (top) shows a comparison between the quantized weight with and without incorporating the kurtosis loss. We can see that the proposed method successfully reshapes the weight distribution to have a much larger variance, i.e., $4\times$ more than the case without $L_\kappa$. Figure 3: (a) Demonstration of W-reshape and A-shift for 4b weights and activations. (b) SQNR improvement under ADC quantization. This can be directly observed in terms of the utilization of the ADC intervals. The proposed techniques provide nearly $5\times$ utilization improvement. Activation shifting (A-shift) In order to maximize the $2^{nd}$ moment of the activation, it is desirable for activations to exhibit a concentration of mass at considerably large absolute values, i.e., distancing the mass from zero, so that the input distribution to the ADC has maximum variance. However, this is typically not the case with activations derived from functions like SiLU (Elfwing et al., 2018) and GELU (Hendrycks & Gimpel, 2016), which inherently exhibit significant mass distribution around small values in close proximity to zero, as shown in Fig. 3a (bottom left). Exploiting the fact that quantizing these activations as a signed number or unsigned number does not have much impact on the overall performance (Bhalgat et al., 2020), we propose to treat them as an unsigned number during quantization, and then convert them to a signed number. This yields a distribution moved away from zero, to the advantage of ADC quantization. Such an unsigned-to-signed conversion can be implemented by a simple shift: $$x_s = \left\lfloor \text{clip} \left( \frac{x - \beta}{s_x}, 0, 2^{b_x} - 1 \right) \right\rfloor - 2^{b_x - 1} = x - 2^{b_x - 1}$$ where $ \lfloor \cdot \rfloor $ denotes round operation, $ \beta $ and $ s_x $ are trainable quantization parameters. Fig. 3a(bottom) shows the entire A-shift process. We observe that the mass of $ \pi_s $ is concentrated at the most negative values, hence having an extremely large 2$^{nd}$ moment. On the contrary, quantizing activations directly to a signed number prevents such a shift operation, resulting in a much smaller 2$^{nd}$ moment. To quantitatively verify our arguments, we compute the numerical values of the 2$^{nd}$ moment for the quantized activation from the proposed method and from signed quantization based on Fig. 3a ending up with 57.9 and 3.89, respectively. Our proposed approach produces a much greater 2$^{nd}$ moment, roughly 15× higher. Additionally, ReLU activation functions naturally suit the A-shift approach, as they explicitly force the output activations to be unsigned numbers. With such shifting, the IMC computation becomes \[ y = \sum_{i=1}^{M} w_i x_i = \sum_{i=1}^{M} w_i x_{s,i} + 2^{h_x-1} w_i \] (7) The additional offset introduced by A-shift can be precomputed offline and thus does not add any overhead when performing inference on IMC systems. The applicability of A-shift on IMC with other number representations are described in Appendix B. Impact of W-reshape and A-shift Fig. 3b summarizes the results obtained by applying W-reshape and A-shift on ImageNet dataset. A particularly useful view is looking at the distribution of the ADC input. We consider the utilization of ADC quantization range to quantitatively analyze the results, i.e., $ \frac{\text{# of occupied ADC quantization intervals}}{\text{Total ADC quantization intervals}} $. Fig. 3b (top) shows an example of an 8-bit ADC, resulting in 3.52% and 21.7% utilization without and with W-reshape and A-shift, respectively. We also compute the variance of these two cases to justify our results, which leads to 0.094 and 8.535 respectively. These improvements can be directly related to the increased SQNR illustrated in Fig. 3b(bottom). 4.2 SQNR Adaptation for Neural Networks ![Figure 4: Loss surfaces with A-shift and W-reshape applied for 4-bit activations and weights.](image) Bit augmentation (BitAug). Quantization fundamentally sacrifices information in exchange for model compression. While the SQNR is improved through Eq. 4 - Eq. 7 quantization imposed by the ADC is observed to make SGD-based optimization more challenging during training. Fig. 4 shows the loss surfaces of MobileNetV2 in two randomly selected parameter dimensions for visualization (Li et al., 2018). As seen, ADC quantization causes a less smooth surface with additional local minima. These attributes reduce the likelihood of arriving at preferred (low-loss) minima during the training process. Approaches are thus required to adapt the model to this extra quantization. We seek an approach that facilitates a greater volume of information to be backpropagated so that the model parameters can be optimized more effectively. Inspired by NetAug (Cai et al., 2021) where a tiny model is inserted into larger models during training, we augment the network with ADCs of different bit precisions. At each iteration, we first pass the desired ADC bit to the model and then pass other bit precisions from a pre-defined set $ B $ to the model. The general form of BitAug is \[ L_A = L(\theta, b_a) + \lambda_b \sum_{i=1}^{B} L(\theta, b_{a,i}), \] (8) where $ \theta $ denotes the network parameters, $ \lambda_b $ is the coefficient of the BitAug loss, $ B $ is the size of the BitAug set $ B $, and $ b_{a,i} $ is a sample from the set. Elements in $ B $ are chosen to be neighbors of the target ADC bit precision. Given the complexity of optimization with ADC quantization, we simply employ the assistance of other bit precisions. The information associated with the various ADC bit precisions is subsequently represented in their respective gradients, which get accumulated during the backward path for more optimal updating of model parameters, i.e., $$\theta^{t+1} = \theta^t - \eta \frac{\partial L(\theta^t, b_a)}{\partial \theta^t} - \eta \lambda_b \sum_{i=1}^{B} \frac{\partial L(\theta^t, b_{a,i})}{\partial \theta^t},$$ (9) where $t$ indicates the current training step and $\eta$ denotes learning rate. However, such an aggregation of multiple augmented models is computationally expensive. Following a similar strategy as (Cai et al., 2021), we randomly sample an ADC bit precision for each iteration, i.e., $$L_A = L(\theta, b_a) + \lambda_b L(\theta, \tilde{b}_a),$$ (10) where $\tilde{b}_a$ is a uniformly sampled bit precision from $\mathbb{B}$. We observe that doing this not only improves the computational efficiency by a factor of $B$, but also achieves better performance than running all ADC bit precisions simultaneously. We include a quantitative study in Appendix C. The selection of $\mathbb{B}$ is also critical. For instance, if we only sample lower precision ADC, we are essentially adding noise to the training process, which causes accuracy degradation. Our empirical results show that a good choice is to choose 1-bit lower and 2-bit higher than the desired ADC bit precision, i.e., $\mathbb{B} = \{b_a - 1, b_a + 1, b_a + 2\}$. A more detailed analysis of BitAug is provided in Appendix F. 5 EXPERIMENTS 5.1 EXPERIMENTAL SETUP We consider a general IMC architecture as shown in Fig. 1a. While exploring different IMC architectures is not the focus of this paper, we include experiments on the impact of RAOQ on various IMC configurations in Appendix D for interested readers. In this section, we focus on an IMC system with aggressive memory dimensions of $512 \times 512$, taking 4-bit inputs and processing 4-bit weights, with ADCs having $k = 4$. Higher-precision activations and weights are mapped to the IMC via matrix tiling. The proposed methods are evaluated on different AI tasks. To preserve the fidelity of critical information, we do not map depthwise convolutions in the MobileNet family, and the second matrix-matrix multiplication in the self-attention module of BERT (BMM2) to the IMC system. This is justified as these layers account for a small number of computations in the overall model (i.e., < 7% for depthwise convolutions in MobileNetV2 and < 1.5% for BMM2 in BERT), thus giving minor energy-efficiency advantage by execution via IMC. The first and last layers are kept in 8-bit. We start from pre-trained FP32 models, and first perform QAT based on LSQ+ (Bhalgat et al., 2020) on activations and weights with the proposed W-reshape and A-shift methods. We then add ADC quantization along with other RAOQ techniques for another stage of training. Experiments are performed on Nvidia A100 GPUs. Further training details are described in Appendix E. 5.2 RESULTS Table 1 summarizes the results for 4-bit and 8-bit activations and weights. We sweep the ADC bit precision to demonstrate the robustness and generalizability of our approaches. All QAT (without ADC involved) accuracy matches SOTA results. For a fair comparison, we also perform conventional QAT (i.e., without any proposed methods involved) for ADC quantization. As seen, the proposed RAOQ significantly outperforms conventional QAT in all cases. Image classification. We choose ResNet, MobileNet, and EfficientNet-lite models for evaluation using top-1 accuracy on ImageNet dataset. Our proposed RAOQ restores the performance to high accuracy across the activation/weight and ADC bit precisions considered. Particularly, some cases of 9-bit ADC even outperform the no-ADC baseline. We start to observe accuracy degradation at low precision ADCs, with ≤ 0.3% drop in the 8-bit case, and with < 0.8% drop in the 7-bit case. Object detection. We evaluate YOLOv5s on COCO 2017 dataset in terms of mAP. Although YOLOv5s involves more complicated network structures compared to the above CNN models for image classification, our approach restores the significantly degraded accuracy to the level close to no-ADC case, with a < 1% drop for 8-bit and 9-bit ADCs, and < 2% drop for 7-bit ADC. Table 1: Evaluation of RAOQ with various activation, weight, and ADC bit precisions. \| Model \| FP32 \| $b_x$, $b_w$ \| No ADC \| $b_a = 7$ \| $b_a = 8$ \| $b_a = 9$ \| \|----------------\|------\|--------------\|--------\|----------\|----------\|----------\| \| \| \| \| QAT* \| RAOQ \| QAT* \| RAOQ \| \| ResNet18 \| 69.76\| 8.8 \| 70.66 \| 60.12 \| 70.28 \| 66.03 \| 70.46 \| 66.65 \| 70.60 \| \| \| \| 4.4 \| 70.49 \| 59.42 \| 70.23 \| 65.71 \| 70.45 \| 66.61 \| 70.49 \| \| ResNet50 \| 76.23\| 8.8 \| 76.53 \| 65.47 \| 76.25 \| 73.83 \| 76.46 \| 75.01 \| 76.51 \| \| \| \| 4.4 \| 76.31 \| 65.25 \| 76.15 \| 72.05 \| 76.27 \| 74.16 \| 76.32 \| \| MobileNetV2 \| 71.81\| 8.8 \| 71.89 \| 62.09 \| 71.57 \| 66.72 \| 71.79 \| 69.13 \| 71.93 \| \| \| \| 4.4 \| 70.47 \| 61.51 \| 70.22 \| 66.67 \| 70.46 \| 68.55 \| 70.45 \| \| EfficientNet-lite0 \| 75.12\| 8.8 \| 74.31 \| 61.27 \| 73.58 \| 68.11 \| 74.08 \| 68.85 \| 74.21 \| \| \| \| 4.4 \| 72.84 \| 61.21 \| 72.18 \| 67.03 \| 72.76 \| 67.85 \| 72.82 \| \| YOLOv5s \| 37.20°\| 8.8 \| 36.60 \| 1.30 \| 34.73 \| 8.02 \| 35.82 \| 24.03 \| 36.41 \| \| \| \| 4.4 \| 33.78 \| 10.13 \| 32.23 \| 20.32 \| 33.49 \| 28.49 \| 33.89 \| \| BERT-base \| 88.58\| 8.8 \| 88.24 \| 66.35 \| 87.40 \| 83.04 \| 87.84 \| 84.82 \| 88.11 \| \| \| \| 4.4 \| 87.75 \| 64.46 \| 87.31 \| 82.43 \| 87.67 \| 84.53 \| 87.75 \| \| BERT-large \| 91.00\| 8.8 \| 90.58 \| 58.37 \| 89.60 \| 79.58 \| 90.09 \| 85.92 \| 90.38 \| \| \| \| 4.4 \| 89.57 \| 62.11 \| 88.67 \| 80.18 \| 89.08 \| 85.01 \| 89.55 \| * Conventional QAT (i.e., without RAOQ techniques) with ADC quantization present. ° Result trained by ourselves in FP32 rather than original mixed-precision. NLP. We use BERT models, implemented based on [Wolf et al., 2020], to demonstrate for the question-answering task on SQuAD 1.1 dataset. The results are evaluated in terms of the F1 score. Once again, our proposed RAOQ successfully restores the degraded accuracy, with < 1%, < 0.5%, and < 0.2% accuracy drops for 7-bit, 8-bit, and 9-bit ADCs, respectively. 5.3 Comparison with Other Methods As mentioned, previous algorithmic works focus on ADC quantization in IMC on small datasets. Thus, Table 2 shows a comparison of our proposed RAOQ approach with other works on the CIFAR-10 dataset. These works are based on various memory technologies (e.g., SRAM, ReRAM). For a fair comparison, we construct the same model, following the same configurations as these works (e.g., bit precisions, memory dimensions, applicable hardware noise levels), and then apply our RAOQ approach. We see that RAOQ outperforms all other methods, leading to much less degradation regardless of IMC technology and configurations. Table 2: Comparison of different methods for ADC quantization on CIFAR-10. M denotes the memory inner-dimension, and the column IMC indicates accuracy under ADC quantization. \| Model \| Method \| $b_x$, $b_w$, $b_a$ \| M \| FP32 \| No ADC \| IMC \| Degradation \| \|----------------\|--------\|---------------------\|---\|------\|--------\|-----\|-------------\| \| ResNet20 \| Jin et al. [2022] \| 4,4,7 \| 9 \| – \| 91.60 \| 91.00 \| -0.60b \| \| \| \| 4,4,3 \| 9 \| – \| 81.70 \| 81.70 \| -9.30b \| \| \| RAOQ \| 4,4,7 \| 9 \| 92.32\| 92.23 \| 92.32 \| +0.09b \| \| \| \| 4,4,3 \| 9 \| 89.34\| 89.34 \| 89.34 \| -2.89b \| \| ResNet18a \| Sun et al. [2021] \| 4,4,4 \| 256\| 88.87\| – \| 86.55\| -2.32c \| \| \| RAOQ \| 4,4,4 \| 256\| 92.10\| 92.13 \| 90.48\| -1.65c \| \| ResNet18 \| Wei et al. [2020] \| 2,2,4 \| 9 \| 92.01\| 89.62 \| 83.37\| -6.25b \| \| \| \| 2,2,4 \| 36\| 87.56\| 87.56 \| 87.56\| -2.06b \| \| \| RAOQ \| 2,2,4 \| 9 \| 93.21\| 92.26 \| 91.90\| -0.36b \| \| \| \| 2,2,4 \| 36\| 91.81\| 91.81 \| 91.81\| -0.45b \| a Channels are reduced to 1/4 of the original ResNet18. b Accuracy drop of IMC ADC quantization with respect to no-ADC case. c Accuracy drop with respect to FP32. 5.4 Ablation Study We investigate the impact of each proposed technique in RAOQ. In particular, we use BERT-base, MobileNetV2, and ResNet50 with 4-bit activations and weights, and an 8-bit ADC to perform the study. The results are summarized in Table 3. The first row corresponds to the case where conventional QAT methods are applied to IMC with ADC quantization. Each check mark indicates the presence of a specific technique. As seen, all of the proposed techniques improve the degraded performance due to ADC quantization. Comparatively, A-shift and BitAug exhibit more significant impacts on the network performance, one contributing to boosting SQNR and the other responsible for model optimization. Table 3: Impact of different methods. The check mark indicates the use of the corresponding method. \| \| A-shift \| W-reshape \| BitAug \| BERT-base \| MobileNetV2 \| ResNet50 \| \|---------\|---------\|-----------\|--------\|-----------\|-------------\|----------\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| 6 IMC System Performance In this section, we analyze the value of our proposed approaches in handling ADC quantization. Generally, scaling up ADC bit precision brings costs in hardware energy, as shown in Fig. 5 based on a survey of design reported in the literature (Murmann). The ADC energy cost scales considerably at higher precision, and thus directly affects the energy efficiency advantages of IMC systems. Fig. 5 further depicts the IMC energy efficiency for various ADC precisions, compared to fully-optimized digital accelerators. The IMC efficiency is modeled from (Lee et al., 2021a), while the digital-accelerator energy is from (Jouppi et al., 2017), both in the same silicon technology (28nm CMOS), measured as the number of Tera operations per second per Watt (TOPS/W) for 8-bit activation and weight computations. While IMC demonstrates a dramatic energy efficiency advantage over digital accelerators, the advantage drops significantly as ADC precision is increased. With the observed trade-off between conventional QAT-based inference accuracy and energy efficiency, our proposed algorithmic RAOQ approach enables significant improvement in this trade-off. 7 Conclusion Analog IMC has shown substantial promise to simultaneously enhance compute efficiency and data-movement costs for AI inference. However, the associated ADC quantization restricts the accuracy of SOTA models applied to challenging tasks. While increasing ADC bit precision reduces the effects of quantization, this comes with a significant energy cost. In this work, we propose RAOQ to tackle such quantization. Specifically, we propose W-reshape and A-shift, to maximize the SQNR following ADC quantization via adjusting the statistics of weights and activations. We further propose BitAug to improve the optimization process. Our work has been evaluated on various datasets, models, and bit precisions, achieving consistently high accuracy. The generalizability and robustness of our proposed methods demonstrate the feasibility of applying IMC to challenging AI tasks. 8 REPRODUCIBILITY The detailed training configurations are described in Appendix E, including the training procedure, hyperparameter settings, learning curves, as well as compute resources needed to perform our experiments. The training of each model is described separately for clarity. In Appendix E.4 we provide a code example to implement our proposed RAOQ method, associated with a sample log file of MobileNetV2 training. For readers who are interested in other IMC configurations, we provide studies on different IMC configurations in Appendix D, other than those presented in the main manuscript. All of our proposed methods can be directly applied except that a small adjustment needs to be made for A-shift for different IMC types, as detailed in Appendix B. REFERENCES Ron Banner, Yury Nahshan, and Daniel Soudry. Post Training 4-Bit Quantization of Convolutional Networks for Rapid-Deployment. Curran Associates Inc., Red Hook, NY, USA, 2019. Yoshua Bengio, Nicholas Léonard, and Aaron C. Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. ArXiv, abs/1308.3432, 2013. Yash Bhalgat, Jinwon Lee, Markus Nagel, Tijmen Blankevoort, and Nojun Kwak. 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e0LwFqw4Bi	With the current formulation of Marginal Generalization, how can you avoid catastrophic forgetting on source domains, when even if you impose the distance constraint on the target domain, there are no guarantees it will be still obeyed on the source domain?	Towards Unified and Effective Domain Generalization Anonymous authors Paper under double-blind review Abstract We propose UniDG, a novel and Unified framework for Domain Generalization that is capable of significantly enhancing the out-of-distribution generalization performance of foundation models regardless of their architectures. The core idea of UniDG is to finetune models during inference stage which saves the cost of iterative training. Specifically, we encourage models to learn the distribution of testing data in an unsupervised manner and impose a penalty regarding the updating step of model parameters. The penalty term can effectively reduce the catastrophic forgetting issue as we would like to maximally preserve the valuable knowledge in the original model. Empirically, across 12 visual backbones, including CNN-, MLP-, and transformer-based models, ranging from 1.89M to 303M parameters, UniDG shows an average accuracy improvement of +5.4% on DomainBed. We believe that these performance results are able to manifest the superiority and versatility of UniDG. 1 Introduction The Out-Of-Distribution (OOD) problem is a prevalent topic in the machine learning and computer vision communities (Long et al., 2015; Saito et al., 2020; Sun & Saenko, 2016; Ebrahimi et al., 2020) as models of various architectures and scales are suffering from this problem (Zhou et al., 2022; Li et al., 2023; Chen et al., 2022a; Peng et al., 2022). Therefore, training deep models to generalize well on new domains has become a prevalent research topic (Long et al., 2015; Li et al., 2018b; Wang et al., 2019; Chen et al., 2022b; Cha et al., 2021, 2022). To overcome the domain shift problem, pretraining-based methods (Radford et al., 2021; Singh et al., 2022; Cha et al., 2022) utilize large-scale data to obtain better generalization ability. However, in practice, domain shift can be so significant that even though the powerful foundation models have been pretrained on huge-scale datasets, directly generalizing the models to new domains still delivers unsatisfactory performance, as shown in Figure 1. Another drawback of pretraining-based methods is the inferior finetuning performance - finetuning pretrained models leads to catastrophic forgetting and limited improvement on new domains (Cha et al., 2022; Li et al., 2022; Chen et al., 2022c). As a workaround, pretraining-based methods may add data from the new domains into the pretraining dataset and retrain the models from scratch (Shu et al., 2023). When the pretraining dataset is large (e.g., CLIP (Radford et al., 2021) uses LAION-400M), this approach becomes significantly expensive. In contrast to the pretraining-based methods, Test-Time Adaptation (TTA) (Sun et al., 2020; Wang et al., 2021, 2022b) is an alternative to mitigate domain shift on new domains. First, TTA requires no pretraining with novel data, and can directly leverage the off-the-shelf models. Second, by updating parameters in both training and evaluation stages (Sun et al., 2020), TTA reduces the reliance of models on annotations in new domains. However, we would like to note several drawbacks of existing TTA methods. 1) Most TTA methods (Wang et al., 2021; Iwasawa & Matsuo, 2021; Jang & Chung, 2023) require updating Batch Normalization (BN) (Ioffe & Szegedy, 2015) layers in the original model to adapt to the distribution of test data. However, recent visual foundation models such as vision transformers (Dosovitskiy et al., 2020) are developed with Layer Normalization (LN) layers. Due to the essential difference between BN and LN, simply adapting the ideas of BN-based methods to LN layers results in minimal improvement (around 0.5%, see Appendix § F). 2) Recent TTA methods (Zhang et al., 2023b; Park et al., 2023; Zhang et al., 2023a; Chen et al., 2023) show limited scalability on common visual foundation models ranging from small to large scales. For example, only limited improvements (less than 2%) on large-scale foundation models (Radford et al., 2021; Liu et al., 2022) are observed. 3) From a theoretical perspective, we find these TTA methods reduce the Neural Tangent Kernel (Jacot et al., 2018) in the adaptation process, which limits the further generalization (theoretical analysis is presented in the Appendix § B). To address the aforementioned drawbacks, we focus on an important topic of TTA method - the appropriate way to update the encoder (i.e., feature extractor) for TTA. Prior works either update the encoder via back-propagation or freeze it, but either way has its weaknesses. 1) If we allow the encoder to update, similar to the weakness of finetuning a pretrained encoder, which is discussed above, catastrophic forgetting can happen during TTA and result in a significantly lower quality of extracted features. 2) With the encoder frozen, hence the extracted features have to be refined with extra mechanisms, in order to be well utilized by the classifier. In this paper, we propose a novel method, named Marginal Generalization, to update the encoder for TTA. Intuitively, Marginal Generalization aims to let the encoder learn representations of the target data within a certain distance from the representations obtained by the initial model. Here we use a simplified notation for brevity. Let $\sigma$ be the specified distance, $f(\cdot)$ be the fixed initial encoder and $f'(\cdot)$ be a learnable copy of $f(\cdot)$, $x$ be the samples of the target domain, $q(\cdot)$ be the classifier which takes the representations $f'(x)$ as inputs, the objective is to $$\text{minimize } [\text{entropy}(\text{softmax}(q(f'(x))))] \quad \text{s.t. } \\|f'(x) - f(x)\\|_F \leq \sigma.$$ (1) By doing so, we overcome the drawbacks of the aforementioned two traditional approaches. 1) Intuitively, while the encoder $f'(\cdot)$ is trying to adapt to the novel data, it always refers to the original model $f(\cdot)$ and keeps the representations within a distance $\sigma$ from the original, which means the pretrained source knowledge can be preserved and catastrophic forgetting is avoided. 2) As we keep updating the encoder via entropy minimization on the test data, it cooperates better with the classifier and yields more discriminative features on the target domain. We would like to note that Marginal Generalization is universal because it does not require any specific structures in the original model nor the properties of the data, as well as effective (achieving improvement of 3.3% on average accuracy as shown in Table 4). In addition, the features extracted by the updated encoder can be utilized by multiple TTA mechanisms. For example, by naturally combining Marginal Generalization and Memory Bank (Wu et al., 2018), we propose Differentiable Memory Bank, which demonstrates... superior performance over the traditional memory bank methods because it performs feature filtration and storage on differentiable features. For example, compared with T3A (Iwasawa & Matsuo, 2021) that adopts typical memory bank, our method with ResNet-50 backbone leads it by 4.3% on average accuracy across 4 datasets shown in Table 2. Intuitively, UniDG simultaneously utilizes the local minimum of distances between adapted and source representations and the local maximum of information entropy between adapted representations and pseudo labels in the test-time process to continuously approximate the models on the test data with reserved pretrained knowledge. The details will be presented in Section 2.2 and 2.3. Based on Marginal Generalization, we propose a framework composed of an adaptation method of the encoder (which is a universal method to extract better features) and Differentiable Memory Bank (which is a universal mechanism to refine features for DG) so that the framework is named UniDG, which delivers state-of-the-art performance on multiple domain generalization benchmarks. For example, UniDG delivers an average accuracy of 79.6% on 5 widely adopted benchmarks including PACS, VLCS, and OfficeHome, outperforming the second-best CAR-FT (Mao et al., 2022) by 1.0%. Additionally, UniDG is an architecture-agnostic framework that consistently yields significant improvements when applied to a wide range of visual backbones, including models of varying scales such as MobileNet V3 (Howard et al., 2019), ConvNeXt-Base (Liu et al., 2022), and ViT-Large (Dosovitskiy et al., 2020), demonstrating its strong scalability. For example, UniDG improves the mean accuracy scores by 5.4% with such 12 models on PACS (Torralba & Efros, 2011), VLCS (Li et al., 2017), OfficeHome (Venkateswara et al., 2017), and TerraInc (Beery et al., 2018). We would like to note that Marginal Generalization and Differentiable Memory Bank can also be used separately and combined with other methods. When we combine these two schemes, we observe an average improvement of +5.0%. Our contributions are summarized as follows. • We propose Marginal Generalization, which addresses the problem of adapting the encoder for TTA. • With Marginal Generalization, we naturally upgrade the traditional memory bank mechanism to Differentiable Memory Bank and propose a universal TTA framework named UniDG. • UniDG consistently outperforms the previous state-of-the-art methods by a significant margin (+5.4% on DomainBed). It applies to a wide range of models of different architectures and varying scales, consistently resulting in satisfactory TTA performance. • We show that UniDG’s components can also be separately combined with other methods, demonstrating its flexibility. 2 METHOD We first introduce the formulation of domain generalization and test-time adaptation in § 2.1. The framework of UniDG comprises two components: 1) we employ Marginal Representation Generalization (§ 2.2) to adapt the encoder, 2) we utilize prototypes with Differentiable Memory Bank (§ 2.3) for learning a discriminative classifier on the target domain. 2.1 PRELIMINARY Domain Generalization. Given a set of source domains $ D_S = \{D_1, D_2, \ldots, D_N\} $, each domain $ D_j $ contains images and labels, $ \{(x_i, y_i)\}_{i=1}^{\|D_j\|} $, where $ x_i $ denotes an image and $ y_i $ indicates the corresponding ground truth label, the goal of DG is to generalize models on a novel target domain $ D_T $ that is different from any of the source domains by training on $ D_S $. We denote the mapping function of the model as $ F : x \rightarrow p \in \mathbb{R}^C $, where $ p $ is the prediction and $ C $ is the number of categories. $ F $ comprises two steps: feature extraction with the encoder $ f(\cdot) $ and prediction with the classifier $ q(\cdot) $ based on the features. Let $ \theta $ be the parameters, $ F $ can be formulated as $ F(x; \theta) = q(f(x)) $. Training on source domains. We use $\ell_{CE}(\cdot)$ to denote the cross-entropy function, and the objective of training on the source domains is to optimize $\theta$ as $$\theta^* = \arg\min_\theta \mathbb{E}_{(x,y) \in D_S} [\ell_{CE}(F(x; \theta), y)].$$ (2) Test-Time Adaptation. With $\theta^$ trained on the source domains $D_S$, test-time adaptation is a self-supervised learning process to further adapt parameters to the target domain $D_T$. The encoder parameters during test time can be optimized as the following, where $\ell_{TTA}(\cdot)$ is the softmax entropy: $$\theta' = \arg\min_\theta \mathbb{E}_{(x) \in D_T} [\ell_{TTA}(F'(x; \theta))].$$ (3) 2.2 Marginal Generalization Marginal Generalization aims to constrain the discrepancy between features extracted by the source encoder $f$ and the adapted encoder $f'$ during the adaptation process so that the adapted model will be able to maintain general representation bias and relieve catastrophic forgetting while updating parameters. Here we adopt Euclidean distance as the metric out of its simplicity and universality, which is formulated with the Frobenius norm $\\|\cdot\\|_F$. We use $\theta_e$ to denote the parameters of the encoder, which is a subset of $\theta$, so that the encoder can be formulated as $f(\cdot; \theta_e)$. Given the pre-defined distance threshold $\sigma$, the objective then becomes $$\theta' = \arg\min_\theta \mathbb{E}_{(x) \in D_T} [\ell_{TTA}(F'(x; \theta))] \quad s.t. \quad \\|f'(x; \theta'_e) - f(x; \theta_e)\\|_F \leq \sigma.$$ (4) The motivation is that we desire to gradually update the parameters of the adapted encoder under the condition that the representation bias will not get sharply adapted. For the source feature extractor $f(\cdot; \theta_e)$, we freeze it and still use it to extract the representation from target domains as pretrained knowledge. For the adapted encoder $f'(\cdot; \theta'_e)$, we initialize it with the source-pretrained parameters $\theta_e$. Therefore, the discrepancy between the original and adapted representations can be formulated as the distance between $f(x; \theta_e)$ and $f'(x; \theta'_e)$. To approximate such a hard constraint with a back-propagation-based method, we propose a novel loss function named Marginal Adaptation Loss to constrain the update of the parameters of the encoder. The Marginal Adaptation Loss can be formulated as: $$L_m = \frac{1}{\\|D_T\\|} \sum_{i=1}^{\\|D_T\\|} \max(\\|f'(x_i; \theta'_e) - f(x_i; \theta_e)\\|_F^2 - \sigma, 0).$$ (5) The update of parameters of the classifier $q(\cdot)$ and encoder is guided by the entropy on the target domain. Based on the extracted representations $f'(x; \theta'_e)$, we use a linear layer to work as a classifier and obtain the classification probability $p = \text{softmax}(q'(f'(x; \theta'_e)))$ using a softmax operation. Then we take the entropy as the loss function to derive the gradients for updating the classifier and encoder, through which we can introduce the probabilistic distribution of target domains to our classifier: $$L_e = -\frac{1}{N_b} \sum_{i=1}^{N_b} \sum_{c=1}^{C} p_c \log p_c.$$ (6) 2.3 Differentiable Memory Bank With Marginal Generalization, we are able to learn a well-adapted encoder that can extract discriminative features on the target domain. However, since there is no labeled data on the target domain, only training with the unsupervised losses $L_m$ and $L_e$ is hard to get a classifier $q(\cdot)$ with high performance on the target. To mitigate this issue, we propose to update the classifier with a differentiable memory bank. We utilize the memory bank to select prototypes suitable for the new domain, develop class-wise prototypes directly differentiable with loss function, and update the whole bank in every forward step. Class-wise prototypes are stored in the memory bank in order to enhance the classifier. Specifically, for each class $j$, the prototype $v_j$ is initialized with the corresponding weights of the source classifier layer. In the self-supervised adaptation process, for each target sample $ x $, we extract the representations $ f'(x) $ and obtain the output of classifier $ q'(f'(x; \theta'); \omega) $. Then we predict pseudo labels $ \hat{y} = \arg\max[\text{softmax}(q'(f'(x)); \omega)] $ and utilize the entropy between representations and pseudo labels as the criterion to select the Top-$ K $ instances of each class with highest classification confidence, where $ K $ is a pre-defined hyper-parameter. After that, we utilize the representations of the Top-$ K $ samples to produce the class-wise prototypes $ v_j = \frac{1}{K} \sum_{i=1}^{K} f'(x_i) $. Memory bank* is set to store the prototypes of each class $ M = \bigcup_{j=1}^{C} \{v_j\}, v_j \in \mathbb{R}^d $, where $ M $, $ C $, and $ d $ denote the memory bank, the number of classes, and feature dimension. At each forward step, we compute the prototypes, which will be further used to update the classifier weights $ \omega $. For a given sample $ x $ with feature $ z = f'(x; \theta) $, the classification probability of class $ j $ can be computed as: \[ p_j = \frac{\exp(z \cdot \omega^j)}{\sum_k \exp(z \cdot \omega^k)}, \quad \omega^k \in \mathbb{R}^d \] where $ \omega^j $ is the $ j $-th element of the weight matrix $ \omega $. Note that for $ q(\cdot; \omega) $ to classify target samples correctly, the weight $ \omega^j $ needs to be representative of features of the corresponding class $ j $. This indicates that the meaning of $ \omega^j $ coincides with the ideal cluster prototype of class $ j $ in the target domain. Thus, we propose to use the estimate of the ideal target cluster prototypes $ \{v_j\}_{j=1}^{C} $ to update the classifier weights: $ \omega^j \leftarrow v_j $. This process is essential in learning a robust classifier for the target domain with no labeled data. ### 2.4 UniDG Learning for Domain Generalization In UniDG framework, the marginal generalization is proposed to learn a well-adapted feature encoder without catastrophic forgetting, and the differentiable memory bank is proposed to learn a discriminative classifier for the target domain. While updating $ \omega $ with target prototypes, the overall learning objective is: \[ L_{\text{UniDG}} = L_e + \lambda \cdot L_m \] ## 3 Experiments ### 3.1 Setup Dataset VLCS (Torralba & Efros [2011]) contains 10,729 instances of 5 classes derived from four photographic datasets in accordance with different domains. PACS (Li et al. [2017]) comprises four domains: art, cartoons, photos, and sketches, including 9,991 instances of classes. OfficeHome (Venkateswara et al., 2017) derives from domains like art, clipart, product, and real, containing 15,588 images of 65 classes. TerraIncognita (Beery et al., 2018) is a real-world dataset that collects photos of wild animals taken by cameras at different locations. It contains 24,788 photos of 10 classes according to the species of animals. DomainNet (Peng et al., 2019) is the largest dataset for domain generalization tasks, including 6 domains, 345 classes, and a total of 586,575 images. Evaluation Metric We evaluate UniDG by taking 3 parallel trials with random seeds to calculate means and standard errors of classification accuracy (%) on 5 datasets. There are 22 different novel environments to evaluate the abilities of the network for generalization. We report detailed results for each environment in Appendix F. Implementation Details All experimental results are conducted on NVIDIA A100 GPUs. If not specified, we utilize ResNet-50 (He et al., 2016) for extracting visual features and a single classifier for classification. On test-time benchmarks, we utilize ERM (Vapnik, 1991) algorithm as our default method for training source models. We also follow default hyper-parameters of DomainBed (Gulrajani & Lopez-Paz, 2020) like initial learning rate of $5e^{-5}$, weight decay of 0.0, batch size of 32, holdout fraction of 0.2, and $\sigma$ of 0.15 (see Appendix § C for more discussions). ### 3.2 Main Results We report experimental results on the domain generalization (§ 3.2.1) and test-time adaptation benchmarks (§ 3.2.2). UniDG delivers new state-of-the-art performances on such benchmarks. #### 3.2.1 Domain Generalization Benchmarks UniDG prominently achieves a brilliant performance on Domain generalization tasks. Table 1 shows the performances of the existing advanced approaches for DG tasks using different pre-training methods. The upper part of the table demonstrates that with ImageNet pre-training, UniDG significantly outperforms various classic models and shows satisfactory stability. Specifically, it achieved an average accuracy of 68.5 on VLCS, PACS, OfficeHome, Terrain, and DomainNet, exceeding AdaNPC by +2.0%, and the best results on VLCS, PACS, terrain, and DomainNet. The remaining part of Table 1 shows more results with large-scale CLIP and SWAG pre-training. Expectedly, the CLIP- and SWAG-trained models outperform the traditional ImageNet-trained ones. However, impressively, with only ImageNet pre-training, UniDG outperforms the CAR-FT model with CLIP pre-training by 1.1% in the average accuracy (79.6% vs. 78.5%). On the terrain data set with complex domain shift, the accuracy of UniDG reached 62.4%, outperforming CAR-FT by 0.5%. Table 2: Average accuracy (%) using classifiers learned by ERM on the domain generalization benchmarks. We use ResNet-18/50 as backbones. Bold indicates the best for each benchmark. \| Generalization Algorithm \| Test-Time Algorithm \| Backbone \| VLCS \| PACS \| OfficeHome \| Terra \| Avg \| \|--------------------------\|--------------------\|---------\|------\|-------\|-----------\|-------\|-----\| \| CLIP [Radford et al., 2021] \| Zero-Shot \| ViT-B16 \| 82.6 ± 0.0 \| 95.6 ± 0.0 \| 79.1 ± 0.0 \| 31.1 ± 0.0 \| 72.2 \| \| + None \| \| \| 74.9 ± 0.5 \| 79.3 ± 0.8 \| 62.1 ± 0.3 \| 40.6 ± 1.2 \| 64.2 \| \| + PL [murdoch et al., 2019] \| \| \| 63.0 ± 2.7 \| 71.0 ± 1.8 \| 58.2 ± 3.2 \| 37.4 ± 7.2 \| 57.4 \| \| + PCL [gou et al., 2019] \| \| \| 74.9 ± 0.6 \| 78.1 ± 2.3 \| 61.0 ± 0.4 \| 41.8 ± 1.9 \| 64.2 \| \| + SHOT [liu et al., 2020] \| \| \| 65.0 ± 0.6 \| 82.0 ± 0.6 \| 66.6 ± 0.6 \| 33.6 ± 0.0 \| 60.9 \| \| + Tent [jia et al., 2021] \| \| \| 72.9 ± 0.8 \| 83.9 ± 0.5 \| 60.9 ± 0.4 \| 33.7 ± 1.1 \| 62.8 \| \| + TentBN [jia et al., 2021] \| \| \| 67.0 ± 1.2 \| 80.8 ± 1.0 \| 62.6 ± 0.4 \| 40.0 ± 0.8 \| 62.6 \| \| + TentCIF [wang et al., 2021] \| \| \| 73.0 ± 1.5 \| 78.6 ± 1.8 \| 59.3 ± 0.6 \| 38.3 ± 3.4 \| 62.3 \| \| + T3A [jia et al., 2021] \| \| \| 77.2 ± 0.8 \| 80.8 ± 1.2 \| 62.1 ± 0.3 \| 42.8 ± 0.6 \| 65.4 \| \| + TAST [jang & chung, 2023] \| \| \| 77.3 ± 0.7 \| 81.9 ± 0.4 \| 63.7 ± 0.5 \| 42.6 ± 0.7 \| 66.4 \| \| + UniDG (ours) \| \| \| 80.9 ± 0.1 \| 81.7 ± 0.1 \| 58.4 ± 0.1 \| 47.9 ± 0.7 \| 67.2 \| \| ERM [Vapnik, 1991] \| \| ResNet-18 \| \| \| \| \| \| \| + None \| \| \| 76.7 ± 0.5 \| 83.2 ± 1.1 \| 67.1 ± 1.0 \| 45.9 ± 1.3 \| 68.3 \| \| + PL [murdoch et al., 2019] \| \| \| 69.4 ± 3.1 \| 81.7 ± 4.6 \| 62.9 ± 3.1 \| 38.1 ± 2.4 \| 63.0 \| \| + PCL [gou et al., 2019] \| \| \| 75.7 ± 0.3 \| 83.3 ± 1.6 \| 67.0 ± 1.0 \| 46.7 ± 2.1 \| 68.2 \| \| + SHOT [liu et al., 2020] \| \| \| 67.0 ± 0.9 \| 84.1 ± 1.1 \| 62.7 ± 0.7 \| 45.2 ± 0.8 \| 65.5 \| \| + Tent [jia et al., 2021] \| \| \| 72.9 ± 0.3 \| 85.2 ± 0.6 \| 66.3 ± 0.8 \| 37.1 ± 2.0 \| 65.4 \| \| + TentBN [jia et al., 2021] \| \| \| 69.7 ± 1.2 \| 83.7 ± 1.2 \| 67.9 ± 0.9 \| 43.9 ± 1.3 \| 66.3 \| \| + TentCIF [wang et al., 2021] \| \| \| 75.8 ± 0.7 \| 82.7 ± 1.6 \| 66.8 ± 1.0 \| 43.6 ± 2.6 \| 67.2 \| \| + T3A [jia et al., 2021] \| \| \| 77.0 ± 0.8 \| 83.8 ± 0.4 \| 68.1 ± 0.6 \| 45.6 ± 1.1 \| 68.8 \| \| + TAST [jang & chung, 2023] \| \| \| 77.7 ± 0.5 \| 84.1 ± 1.2 \| 68.6 ± 0.7 \| 47.4 ± 2.1 \| 69.5 \| \| + UniDG (ours) \| \| \| 81.6 ± 0.1 \| 89.0 ± 0.3 \| 68.9 ± 0.1 \| 52.9 ± 0.2 \| 73.1 \| Table 3: Domain generalization accuracy with different backbone networks. UniDG improves the performance agnostic to visual backbones. Bold type indicates performance improvement. \| Type \| Backbone \| Method \| VLCS \| PACS \| OfficeHome \| Terra \| Avg \| \|------\|----------\|--------\|------\|-------\|-----------\|-------\|-----\| \| Light-weight Networks \| ResNet-18 \| ERM \| 76.7 ± 0.1 \| 79.2 ± 0.1 \| 69.0 ± 0.1 \| 40.7 ± 0.0 \| 65.0 \| \| \| \| + UniDG \| 80.9 ± 0.1 \| 81.7 ± 0.1 \| 58.4 ± 0.1 \| 47.9 ± 0.7 \| 67.2 \| \| \| MobileNetV3 \| ERM \| 65.3 ± 0.2 \| 79.1 ± 0.0 \| 60.8 ± 0.2 \| 39.4 ± 0.1 \| 58.9 \| \| \| \| + UniDG \| 76.2 ± 0.1 \| 83.5 ± 0.4 \| 65.1 ± 0.2 \| 44.7 ± 0.2 \| 65.3 \| \| \| EfficientNetV2 \| ERM \| 76.0 ± 0.2 \| 83.0 ± 0.3 \| 72.0 ± 0.2 \| 41.6 ± 0.2 \| 71.1 \| \| \| \| + UniDG \| 78.6 ± 0.2 \| 90.9 ± 0.1 \| 77.2 ± 0.1 \| 41.7 ± 0.4 \| 72.1 \| \| Convolution Networks \| ResNet-50 \| ERM \| 77.1 ± 0.1 \| 82.9 ± 0.1 \| 65.2 ± 0.1 \| 45.4 ± 0.1 \| 67.6 \| \| \| \| + UniDG \| 81.6 ± 0.1 \| 89.0 ± 0.3 \| 68.9 ± 0.1 \| 52.9 ± 0.2 \| 73.1 \| \| \| ResNet-101 \| ERM \| 80.5 ± 0.2 \| 88.3 ± 0.1 \| 70.3 ± 0.2 \| 50.0 ± 0.5 \| 72.3 \| \| \| \| + UniDG \| 85.5 ± 0.2 \| 92.5 ± 0.2 \| 88.5 ± 0.2 \| 65.3 ± 0.1 \| 83.5 \| \| Transformer Networks \| ConvNeXt-B \| ERM \| 79.4 ± 0.0 \| 92.7 ± 0.1 \| 86.9 ± 0.1 \| 60.9 ± 0.0 \| 79.7 \| \| \| \| + UniDG \| 85.8 ± 0.2 \| 95.2 ± 0.2 \| 88.5 ± 0.2 \| 65.3 ± 0.1 \| 83.5 \| \| \| ViT-B16 \| ERM \| 83.6 ± 0.1 \| 85.4 ± 0.5 \| 81.0 ± 0.0 \| 54.1 ± 0.2 \| 75.4 \| \| \| \| + UniDG \| 85.4 ± 0.1 \| 88.3 ± 0.1 \| 83.3 ± 0.0 \| 54.5 ± 0.1 \| 77.1 \| \| \| ViT-L16 \| ERM \| 76.4 ± 0.1 \| 91.2 ± 0.1 \| 83.3 ± 0.0 \| 56.5 ± 0.0 \| 77.4 \| \| \| \| + UniDG \| 83.5 ± 0.2 \| 92.6 ± 0.4 \| 87.5 ± 0.2 \| 55.9 ± 0.4 \| 81.8 \| \| \| Hybrid ViT \| ERM \| 79.1 ± 0.1 \| 89.1 ± 0.1 \| 80.0 ± 0.1 \| 53.0 ± 0.1 \| 75.5 \| \| \| \| + UniDG \| 83.5 ± 0.1 \| 93.5 ± 0.1 \| 84.3 ± 0.1 \| 57.0 ± 0.4 \| 79.6 \| \| \| DeiT \| ERM \| 79.3 ± 0.1 \| 88.0 ± 0.1 \| 77.0 ± 0.1 \| 48.5 ± 0.1 \| 73.5 \| \| \| \| + UniDG \| 85.1 ± 0.1 \| 92.6 ± 0.4 \| 87.5 ± 0.2 \| 55.4 ± 0.4 \| 81.4 \| \| \| Swin Transformer \| ERM \| 80.0 ± 0.1 \| 91.0 ± 0.1 \| 80.0 ± 0.1 \| 53.0 ± 0.1 \| 77.8 \| \| \| \| + UniDG \| 85.0 ± 0.1 \| 94.3 ± 0.2 \| 84.6 ± 0.1 \| 62.0 ± 0.3 \| 81.5 \| \| Multi-Layer Perceptron \| Mixer-B16 \| ERM \| 73.0 ± 0.1 \| 75.8 ± 0.0 \| 52.4 ± 0.1 \| 26.8 ± 0.1 \| 57.2 \| \| \| \| + UniDG \| 81.3 ± 0.2 \| 82.3 ± 0.1 \| 57.7 ± 0.3 \| 41.2 ± 0.5 \| 65.6 \| \| \| Mixer-L16 \| ERM \| 83.0 ± 0.1 \| 88.5 ± 0.2 \| 75.6 ± 0.1 \| 45.0 ± 1.4 \| 73.0 \| ### 3.2.2 Test-Time Adaptation Benchmarks UniDG remarkably outperforms all existing test-time methods including the state-of-the-art method, TAST [jang & chung, 2023]. Specifically, as shown in Table 2, we choose ResNet-18 and ResNet-50 as the backbone and average accuracy as the metric to evaluate several test-time methods. UniDG achieves an average accuracy of 67.2% with ResNet-18 on VLCS, PACS, OfficeHome, and terrain, which is 0.8% higher than the best-performing test-time method. The superiority of UniDG is even more significant with ResNet-50: UniDG achieves an average accuracy of 73.1% on four benchmarks, largely exceeding the last state of the art, TAST [jang & chung, 2023], by 3.5%. Except for ResNet-18 and ResNet-50, we further use UniDG with 12 mainstream backbones including CNN, MLP, and transformer architectures and report the results in Figure 3. It turns out that UniDG can significantly improve the performance of all the 12 backbones so that we conclude UniDG is a universal architecture-agnostic method. Notably, the number of parameters of these models ranges from 1.59M to 303M, but UniDG can significantly and consistently improve the performance by 5.4% on average. ### 3.3 Ablation Study #### Effectiveness of Marginal Generalization. Table 4 shows Marginal Generalization significantly improves the performance on target domains compared with the baseline model. Figure 5: Accuracy accumulation curves on VLCS. UniDG outperforms the base ERM model by about 5% in accuracy. Note we randomly select 10 different trial seeds for better comparison. by +3.3% (70.9% vs. 67.6%). With the classifier adaptation scheme (§ 2.3) but no Marginal Generalization, the performance reaches 70.8%, bringing a +3.2% improvement. While further integrating the two schemes, the ability of the network for domain generalization gets significantly boosted, increasing from 67.6% to 71.9%. Effectiveness of Differentiable Memory Bank As shown in the 4th, 5th, and 7th rows of Table 4, Differentiable Memory Bank (§ 2.3) also significantly improves the generalization ability of the model. Referring to the 4th row, the memory bank effectively boosts the performance of the base model from 67.6% to 70.4% (+2.8%). Meanwhile, when combining differentiable memory bank and Marginal Generalization, a further improvement of +5.5% can be achieved. It reveals that the proposed schemes can be mutually beneficial, where the adapted model has refined gradients and a differentiable memory bank receives better prototypes. Thus they enhance the ability of networks to generalize together. ### 3.4 Quantitative Analysis Figure 5 shows the accumulation curves of each instance interval across four domains on the VLCS (Li et al., 2017) dataset by 10 parallel trials. UniDG brings significant and stable improvements on each domain, for which the fluctuation range of accumulation accuracy is close to the base model and mean scores are prominently improved. \| Ablation Components \| mAcc (%) \| Δ (%) \| \|---------------------\|----------\|------\| \| $ L_m $ (Eq. 5) \| \| \| \| $ L_c $ (Eq. 6) \| \| \| \| $ M $ \| \| \| \| $ \omega \leftarrow v $ \| \| \| \| ✗ \| ✗ \| ✗ \| \| ✗ \| ✗ \| ✗ \| \| ✗ \| ✗ \| ✗ \| \| ✗ \| ✗ \| ✗ \| \| ✗ \| ✗ \| ✗ \| \| ✗ \| ✗ \| ✗ \| \| ✗ \| ✗ \| ✗ \| \| ✗ \| ✗ \| ✗ \| \| ✗ \| ✗ \| ✗ \| \| ✗ \| ✗ \| ✗ \| Table 4: Ablation Study. We take the mean accuracy (mAcc) on the PACS, VLCS, OfficeHome, and TerraInc datasets as the evaluation metric. Table 5: Source knowledge preserve and training efficiency of UniDG. (a) Source Knowledge Preserve \| VLCS \| L \| S \| V \| \|------\|-------\|-------\|-------\| \| Source model \| 96.02 \| 97.14 \| 98.33 \| \| TENT \| 92.15 \| -3.9 \| 94.23 \| -2.9 \| 95.13 \| -3.2 \| \| UniDG \| 94.32 \| -1.7 \| 96.68 \| -0.4 \| 97.63 \| -0.7 \| (b) Efficiency of UniDG \| Method \| Wall Clock Time (s) \| \|--------------\|---------------------\| \| TENT (Res50) \| 0.581 \| \| UniDG (Res50)\| 0.587 \| As shown in Table 5a, referring to Table 5b, we observe a smaller performance decrease of UniDG after adaption on the source domains. It proves that UniDG can better preserve pretrained source knowledge. In Table 5b, we detail the training efficiency of UniDG and compare our method with TENT on wall clock time with the NVIDIA A100 GPU. It reveals that although we propose to update the parameters of the whole network, the computation burden will not sharply increase. ### 3.5 Commonality Analysis 1) Light-weight Networks UniDG brings out significant average improvements of 5.1% on light-weight MobileNet V3 (Howard et al., 2019), EfficientNet V2 (Tan & Le, 2021), and ResNet 18 (He et al., 2016). For example, the accuracy of MobileNet V3 has been improved by as much as 6.4%, which proves the strong feasibility of UniDG to improve the performance of edge devices for generalizing in unseen environments. 2) Architecture-Free UniDG is a unified solution based on online adaptation to handle domain shifts. As shown in Table 3, UniDG has a general improvement of about 5% on 10+ mainstream visual networks including CNN, Transformer, and MLP as their back- bones. The highest improvement comes from Mixer-B16 (Tolstikhin et al., 2021), which increased from 57.2% to 65.6%. 4 RELATED WORK Domain Generalization Domain Generalization (DG) can be classified into three types: 1) Representation Learning: These methods extract specific features from source domains and assume them robust in target domains. One approach is domain alignment (Li et al., 2018c; Matsuura & Harada, 2020), extracting domain-invariant representations from source domains, which is a non-trivial task. Therefore feature disentanglement (Rojas-Carulla et al., 2018; Piratla et al., 2020; Christiansen et al., 2021; Mahajan et al., 2021; Sun et al., 2021; Liu et al., 2021a), loosens the constraint, learning disentangled representations. 2) Foundation Models: different backbones reveal the diverse ability to tackle the DG problem. These methods (Li et al., 2017; Ding & Fu, 2017; Carlucci et al., 2019; Li et al., 2023) optimize the architecture of the mainstream backbone for DG. GMoE (Li et al., 2023) based on ViT (Dosovitskiy et al., 2020), replaces the FFN layers with mixture-of-experts, allowing different experts to focus on the different visual attributes. 3) Learning Strategy: These methods utilize machine learning strategy to enhance the model’s generalization capability on target domains, including meta-learning and ensemble learning. Meta-learning (Li et al., 2018a; 2019b; Dou et al., 2019; Liu et al., 2020; Chen et al., 2022; Li et al., 2021) divide training data into meta-train and meta-test sets, then simulate domain shift and update parameters during training. Ensemble learning (Ding & Fu, 2017; Zhou et al., 2021; Cha et al., 2021) learns model copies to extract features and migrate their ensemble to target domains. Continual Learning Continual learning (De Lange et al., 2021) aims to relieve continuous domain shifts, which face complicated catastrophic forgetting. Existing methods (Rebuffi et al., 2017; Zenke et al., 2017; Kirkpatrick et al., 2017; Li & Hoiem, 2017; Lao et al., 2020) propose regularization and replay to reinforce learning representations space from parameters and data stream perspectives. Recently, self-supervised learning (Radford et al., 2015; He et al., 2022; Grill et al., 2020) utilize prior knowledge obtained by pre-training with massive datasets and have shown strong performance in DG. Radford et al. (2021) trains image encoder and text encoder jointly, matching 400 million (image, text)pairs. Besides, researchers have noted the superiority of causal learning (Zhou et al., 2021; Mahajan et al., 2021) in domain generalization. Test-time Adaptation TTA schemes (Karani et al., 2021; Iwasawa & Matsuo, 2021; Sun et al., 2020; Park et al., 2023) propose to update model parameters based on target data. 1) Adversarial Learning: With the advancement of generative adversarial networks, Li et al. (2020); Yeh et al. (2021); Kurmi et al. (2021) generate target data with generative models, improving the ability to handle domain shift without the support of source data. 2) normalization-based: The normalization method replaces the batch normalization (BN) statistics of the trained model with the BN statistics estimated on test data and updates parameters of the BN layers only, with the backbone network frozen. Wang et al. (2021) aims to minimize entropy during testing (Schneider et al., 2020) uses Wasserstein distance between source and target statistics as the measurement. 3) Bayesian Learning: Zhou & Levine (2021) analyses TTA from a Bayesian perspective (Li et al., 2016; Hu et al., 2021; You et al., 2021) and proposes a regularized entropy minimization procedure achieved by approximating the probability density during the training time. 5 DISCUSSION AND CONCLUSION Aiming at the OOD problem, this paper proposes a general self-supervised online learning scheme, named UniDG, to update all the parameters of the model during the testing phase. Specifically, UniDG contains Marginal Generalization and Differentiable Memory Bank, which can successfully balance the conservation of source knowledge and generalization ability to novel environments. Our method shows high effectiveness and potential for complex domain shifts in actual scenarios. On four domain generalization benchmarks, UniDG achieved a new state-of-the-art performance with an average accuracy of 79.6%. Additionally, UniDG improved 12 backbone models by an average of 5.4%. By comparing with existing pre-trained model and other test-time methods, we show it is a promising direction to develop the online adaptation method to deal with the OOD problem. REFERENCES Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.02893, 2019. Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. In Proceedings of the European conference on computer vision (ECCV), pp. 456–473, 2018. Fabio M Carlucci, Antonio D’Innocente, Silvia Bucci, Barbara Caputo, and Tatiana Tommasi. Domain generalization by solving jigsaw puzzles. In CVPR, pp. 2229–2238, 2019. Junbum Cha, Sanghyuk Chun, Kyungjae Lee, Han-Cheol Cho, Seunghyun Park, Yunsung Lee, and Sungrae Park. 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Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020. Qi Dou, Daniel Coelho de Castro, Konstantinos Kamnitsas, and Ben Glocker. Domain generalization via model-agnostic learning of semantic features. In NeurIPS, pp. 6447–6458, 2019. Sayna Ebrahimi, Franziska Meier, Roberto Calandra, Trevor Darrell, and Marcus Rohrbach. Adversarial continual learning. In ECCV. Springer, 2020. Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. Domain-adversarial training of neural networks. The Journal of Machine Learning Research, 17(1):2096–2030, 2016.
ZZTkLDRmkg	- How are the 'distances to boundary $(dx_i, dy_i)$' defined? It appears to be more like an offset w.r.t the closest point on the boundary. If there are multiple closest points, how is the choice made?	BENO: Boundary-Embedded Neural Operators for Elliptic PDEs Haixin Wang\textsuperscript{1,}, Jiaxin Li\textsuperscript{2,}, Anubhav Dwivedi\textsuperscript{3}, Kentaro Hara\textsuperscript{3}, Tailin Wu\textsuperscript{2,†} \textsuperscript{1}National Engineering Research Center for Software Engineering, Peking University, \textsuperscript{2}Department of Engineering, Westlake University, \textsuperscript{3}Department of Astronautics and Aeronautics, Stanford University wang.hx@stu.pku.edu.cn, lijiaxin@westlake.edu.cn, \{anubhavd,kenhara\}@stanford.edu,wutailin@westlake.edu.cn Abstract Elliptic partial differential equations (PDEs) are a major class of time-independent PDEs that play a key role in many scientific and engineering domains such as fluid dynamics, plasma physics, and solid mechanics. Recently, neural operators have emerged as a promising technique to solve elliptic PDEs more efficiently by directly mapping the input to solutions. However, existing networks typically cannot handle complex geometries and inhomogeneous boundary values present in the real world. Here we introduce Boundary-Embedded Neural Operators (BENO), a novel neural operator architecture that embeds the complex geometries and inhomogeneous boundary values into the solving of elliptic PDEs. Inspired by classical Green’s function, BENO consists of two branches of Graph Neural Networks (GNNs) for interior source term and boundary values, respectively. Furthermore, a Transformer encoder maps the global boundary geometry into a latent vector which influences each message passing layer of the GNNs. We test our model extensively in elliptic PDEs with various boundary conditions. We show that all existing baseline methods fail to learn the solution operator. In contrast, our model, endowed with boundary-embedded architecture, outperforms state-of-the-art neural operators and strong baselines by an average of 60.96%. Our source code can be found at https://github.com/AI4Science-WestlakeU/beno.git 1 Introduction Partial differential equations (PDEs), which include elliptic, parabolic, and hyperbolic types, play a fundamental role in diverse fields across science and engineering. For all types of PDEs, but especially for elliptic PDEs, the treatment of boundary conditions plays an important role in the solutions. In particular, the Laplace and Poisson equations constitute prime examples of linear elliptic PDEs, which are used in a wide range of disciplines, including solid mechanics (Rivière, 2008), plasma physics (Chen, 2016), and fluid dynamics (Hirsch, 2007). Recently, neural operators have emerged as a promising tool for solving elliptic PDEs by directly mapping input to solutions (Li et al., 2020b,c,a; Lötzsch et al., 2022). Lowering the computation efforts makes neural operators more attractive compared with classical approaches like finite element methods (FEM) (Quarteroni & Valli, 2008) and finite difference methods (FDM) (Dimov et al., 2015). However, existing neural operators have not essentially considered the influence of boundary conditions on solving elliptic PDEs. A distinctive feature of elliptic PDEs is their sensitivity to boundary conditions, which can heavily influence the behavior of solutions. In fact, boundary conditions pose two major challenges for neural operators in terms of inhomogeneous boundary values and complex boundary geometry. First, inhomogeneous boundary conditions can cause severe fluctuations in the solution, and have a distinctive influence on the solution compared to the interior source terms. For example, as shown in Fig. 1, the inhomogeneous boundary... Figure 1: Examples of different geometries for the elliptic PDEs: (a) forcing terms and (b) solutions. The nodes in red-orange color-map represent the complex, inhomogeneous boundary values. The redder the area, the higher the boundary value it represents, whereas the more orange the area, the lower the boundary value. Values cause high-frequency fluctuations in the solution especially near the boundary, which make it extremely hard to learn. Second, since elliptic PDEs are boundary value problems whose solution describes the steady-state of the system, any variation in the boundary geometry and values would influence the interior solution globally [Hirsch, 2007]. The above challenges need to be properly addressed to develop a neural operator suitable for more general and realistic settings. In this paper, we propose Boundary-Embedded Neural Operators (BENO), a novel neural operator architecture to address the above two key challenges. Inspired by classical Green’s function, BENO consists of two Graph Neural Networks (GNNs) that model the boundary influence and the interior source terms, respectively, addressing the first challenge. Moreover, to model the global influence of the boundary to the solution, we employ a Transformer [Vaswani et al., 2017] to encode the full boundary information to a latent vector and feed it to each message passing layer of the GNNs. This captures how the global geometry and values of the boundary influence the pairwise interaction between interior points, addressing the second challenge. As a whole, BENO provides a simple architecture for solving elliptic PDEs with complex boundary conditions, incorporating physics intuition into its boundary-embedded architecture. In Table 1, we provide a comparison between BENO and prior deep learning methods for elliptic PDE solving. \| Methods \| 1. PDE-agnostic prediction on new initial condition \| 2. Train/Test space grid independence \| 3. Evaluation at unobserved spatial locations \| 4. Free-form spatial domain for boundary shape \| 5. Inhomogeneous boundary condition value \| \|---------\|---------------------------------------------------\|--------------------------------------\|---------------------------------------------\|-----------------------------------------------\|------------------------------------------\| \| GKN \| ✔ \| ✔ \| ✔ \| ✗ \| ✗ \| \| FNO \| ✗ \| ✗ \| ✗ \| ✔ \| ✗ \| \| GNN-PDE \| ✔ \| ✔ \| ✗ \| ✔ \| ✗ \| \| MP-PDE \| ✗ \| ✗ \| ✗ \| ✔ \| ✗ \| \| BENO (ours) \| ✔ \| ✔ \| ✔ \| ✔ \| ✔ \| To fully evaluate our model on inhomogeneous boundary value problems, we construct a novel dataset encompassing various boundary shapes, different boundary values, different types of boundary conditions, and varying resolutions. The experimental results demonstrate that our approach not only outperforms the existing state-of-the-art methods by about an average of 60.96% in solving elliptic PDEs problems but also exhibits excellent generalization capabilities in other scenarios. In contrast, all existing baselines fail to learn solution operators for the above challenging elliptic PDEs. 2 Problem Setup In this work, we consider the solution of elliptic PDEs in a compact domain subject to inhomogeneous boundary conditions along the domain boundary. Let $u \in C^d(\mathbb{R})$ be a d-dimension-differentiable function of $N$ interior grid nodes over an open domain $\Omega$. Specifically, we consider the Poisson equation with Dirichlet (and Neumann in Appendix K) boundary conditions in a d-dimensional domain, and we consider $d = 2$ in the following experiments: $$\nabla^2 u ([x_1, x_2, \ldots, x_d]) = f ([x_1, x_2, \ldots, x_d]), \quad \forall ([x_1, x_2, \ldots, x_d]) \in \Omega,$$ $$u ([x_1, x_2, \ldots, x_d]) = g ([x_1, x_2, \ldots, x_d]), \quad \forall ([x_1, x_2, \ldots, x_d]) \in \partial \Omega,$$ (1) where $ f $ and $ g $ are sufficiently smooth functions defined on the domain $ \Omega = \{(x_{1,i}, x_{2,i}, \ldots, x_{d,i})\}_{i=1}^{N} $, and boundary $ \partial \Omega $, respectively. Eq. (1) is utilized in a range of applications in science and engineering to describe the equilibrium state, given by $ f $ in the presence of time-independent boundary constraints specified by $ g $. A distinctive feature of elliptic PDEs is their sensitivity to boundary values $ g $ and shape $ \partial \Omega $, which can heavily influence the behavior of their solutions. Appropriate boundary conditions must often be carefully prescribed to ensure well-posedness of elliptic boundary value problems. 3 Method In this section, we detail our method BENO. We first motivate our method using Green’s function, a classical approach to solving elliptic boundary value problems in Section 3.1. We then introduce our graph construction method in Section 3.2. Inspired by the Green’s function, we introduce BENO’s architecture in Section 3.3. 3.1 Motivation How to facilitate boundary-interior interaction? To design the boundary-embedded message passing neural network, we draw inspiration from the traditional Green’s function (Stakgold & Holst, 2011) method which is based on a numerical solution. Take the Poisson equation with Dirichlet boundary conditions for example. Suppose the Green’s function is $ G : \Omega \times \Omega \rightarrow \mathbb{R} $, which is the solution of the corresponding equation as follows: \[ \begin{align} \nabla^2 G &= \delta(x - x_0)\delta(y - y_0) \\ G\|_{\partial \Omega} &= 0 \end{align} \] Based on the aforementioned equations and the detailed representation of the Green’s function formula in the Appendix A, we can derive the solution in the following form: \[ u(x, y) = \int_{\Omega} G(x, y, x_0, y_0)f(x_0, y_0)d\sigma_0 - \int_{\partial \Omega} g(x_0, y_0)\frac{\partial G(x, y, x_0, y_0)}{\partial n_0}dl_0 \] Motivated by the two terms presented in Eq. (3), our objective is to approach boundary embedding by extending the Green’s function. Following the mainstream work of utilizing GNNs as surrogate models (Pfaff et al., 2020; Eliasof et al., 2021; Lötzsch et al., 2022), we exploit the graph network simulator (Sanchez-Gonzalez et al., 2020) as the backbone to mimic the Green’s function, and add the boundary embedding to the node update in the message passing. Besides, in order to decouple the learning of the boundary and interior, we adopt a dual-branch network structure, where one branch sets the boundary value $ g $ to 0 to only learn the structural information of interior nodes, and the other branch sets the source term $ f $ of interior nodes to 0 to only learn the structural information of the boundary. The Poisson equation solving can then be disentangled into two parts: \[ \begin{align} \nabla^2 u(x, y) &= f(x, y) \\ u(x, y) &= g(x, y) \end{align} \] Therefore, our BENO will use a dual-branch design to build two different types of edges on the same graph separately. Branch 1 considers the effects of interior nodes and Branch 2 focuses solely on how to propagate the relationship between boundary values and interior nodes in the graph. Finally, we aggregate them together to obtain a more accurate solution under complex boundary conditions. How to embed boundary? Since boundary conditions are crucially important for solving PDEs, how to better embed the boundary information into the neural network is key to our design. During a pilot study, we found that directly concatenating the interior node information with boundary information fails to solve for elliptic PDEs, and tends to cause severe over-fitting. Therefore, we propose to embed the boundary to represent its global information for further fusion. In recent years, Transformer (Vaswani et al., 2017) has been widely adopted due to its global receptive field. By leveraging its attention mechanism, the Transformer can effectively capture long-range dependencies and interactions within the boundary nodes. This is particularly advantageous when dealing with complex boundary conditions (i.e., irregular shape and inhomogeneous boundary values), as it allows for the modeling of complex relationships between boundary points and the interior solution. 3.2 Graph Construction Figure 2: Visualization of the graph construction on our train/set samples from 5 different corner elliptic datasets. The interior nodes are in black and the boundary one in purple. Before designing our method, it is an important step to construct graph $ G = \{(V, E)\} $ with the finite discrete interior nodes as node set $ V $ on the PDE’s solution domain $ \Omega $. In traditional solution methods such as FEM, the solution domain is initially constructed by triangulating the mesh graph (Bern & Eppstein [1995], Ho-Le [1988]), followed by the subsequent solving process. Therefore, the first step is to implement Delaunay triangulation (Lee & Schachter [1980]) to construct mesh graph with edge set $ E_{mesh} $, in which each cell consists of three edges. Then we proceed to construct the edge set $ E_{kn} $ by selecting the $ K $-nearest nodes for each individual node. $ K $ is the quantity of neighboring nodes that we deem as closely connected based on the Euclidean distance $ D_{ij} $ between node $ i $ and $ j $. The final edge set is $ E = E_{mesh} \cup E_{kn} $. Examples of graph construction are shown in Fig. 2. 3.3 Overall Architecture In this section, we will introduce the detailed architecture of our proposed BENO, as shown in Figure 3. Our overall neural operator is divided into two branches, with each branch receiving different graph information and boundary data. However, the operator architecture remains the same with the encoder, boundary-embedded message passing neural network and decoder. Therefore, we will only focus on the common operator architecture. 3.3.1 Encoder & Decoder Encoder. The encoder computes node and edge embeddings. For each node $ i $, the node encoder $ e^v $ maps the node coordinates $ p_i = (x_i, y_i) $, forcing term $ f_i $, and distances to boundary $ dx_i, dy_i $ to node embedding vector $ v_i = e^v([x_i, y_i, f_i, dx_i, dy_i]) \in R^D $ in a high-dimensional space. The same mapping is implemented on edge attributes with edge encoder $ e^e $ for edge embedding vector $ e_{ij} $. For both node and edge encoders $ e $, we exploit a two-layer Multi-Layer Perceptron (MLP) (Murtagh [1991]) with Sigmoid Linear Unit (SiLU) activation (Elfwing et al. [2018]). Decoder. We use a two-layer MLP to map the features to solutions. Considering our dual-branch architecture, we will add the outputs from each decoder to obtain the final predicted solution $ \hat{u} $. 3.3.2 Boundary-Embedded Message Passing Neural Network (BE-MPNN) To address the inherent differences in physical properties between boundary and interior nodes, we opt not to directly merge these distinct sources of information into a single network representation. Instead, we first employ the Transformer to specifically embed the boundary nodes. Then, the obtained boundary information is incorporated into the graph message passing processor. We will provide detailed explanations for these two components separately. Embedding Boundary with Transformer. With the boundary node coordinates $ p^B = (x^B, y^B) $, the boundary value $ g $, and the distance to the geometric center of solution domain $ dc $ as input features, we first utilize the position embedding to include relative position relationship for initial representation $ H^B_0 $, followed by a Transformer encoder with $ L $ layers to embed the boundary information $ H^B $. The resulting boundary features, denoted as $ B $, are obtained by applying global average pooling (Lin et al. [2013]) to the encoder outputs $ H^B $. Each self-attention layer applies multi-head self-attention and feed-forward neural networks to the input. The output of the $ i $-th self-attention layer is denoted as $ H^B_i $. The self-attention mechanism calculates the attention weights $ A_i $ as follows: \[ A_i = \text{Softmax}\left(\frac{Q_i H^B_i (K_i H^B_i)^T}{\sqrt{d_k}}\right) \] (5) Figure 3: Overall architecture of our proposed BENO. The pink branch corresponds to the first term in Eq. (2), and the green branch corresponds to the second term. As the backbone of boundary embedding, Transformer provides boundary information as a supplement for BE-MPNN, thereby enabling better prediction under complex boundary geometry and inhomogeneous boundary values. where $Q_i$, $K_i$, and $V_i$ are linear projections of $H_{i-1}^B$ with learnable weight matrices, and $d_k$ is the dimension of the key vectors. The attention output is computed as: $$H_{i+1}^B = \text{LayerNorm}\left(A_i V_i \left(H_i^B + H_i^B\right)\right)$$ (6) where LayerNorm denotes layer normalization, which helps to mitigate the problem of internal covariate shift. After passing through the $L$ self-attention layers, the output $H^B$ is subject to global average pooling to obtain the boundary features: $B = \text{AvgPool}(H^B)$. Boundary-Embedded Message Passing Processor. The processor computes $T$ steps of message passing, with an intermediate graph representation $G_1, \ldots, G_T$ and boundary representation $B_1, \ldots, B_T$. The specific passing message $m_{ij}^t$ in step $t$ in our processor is formed by: $$m_{ij}^t = \text{MLPs}(v_i^t, v_j^t, e_{ij}^t, p_i - p_j)$$ (7) where $m_{ij}^{t+1}$ represents the message sent from node $j$ to $i$. $p_i - p_j$ is the relative position which can enhance the equivariance by justifying the symmetry of the PDEs. Then we update the node feature $v_i^t$ and edge feature $e_{ij}^t$ as follows: $$v_i^{t+1} = \text{MLPs}\left(v_i^t, B_i^t, \sum_{j \in N(i)} m_{ij}^t\right),$$ $$e_{ij}^{t+1} = \text{MLPs}\left(e_{ij}^t, m_{ij}^t\right)$$ (8) (9) Here, boundary information is embedded into the message passing. $N(i)$ represents the gathering of all the neighbors of node $i$. Learning objective. Given the ground truth solution $u$ and the predicted solution $\hat{u}$, we minimize the mean squared error (MSE) of the predicted solution on $\Omega$. 4 EXPERIMENTS We aim to answer the following questions: (1) Compared with existing baselines, can BENO learn the solution operator for elliptic PDEs with complex geometry and inhomogeneous boundary values? (2) Can BENO generalize to out-of-distribution boundary geometries and boundary values, and different grid resolutions? (3) Are all components of BENO essential for its performance? We first introduce experiment setup in Sec. [4.1] then answer the above three questions in the following three sections. 4.1 EXPERIMENT SETUP Datasets. For elliptic PDEs simulations, we construct five different datasets with inhomogeneous boundary values, including 4/3/2/1-corner squares and squares without corners. Each dataset consists of 1000 samples with randomly initialized boundary shapes and values, with 900 samples used for Table 2: Performances of our proposed BENO and the compared baselines, which are trained on 900 4-corners samples and tested on 5 datasets under relative L2 norm and MAE separately. The unit of the MAE metric is $1 \times 10^{-3}$. Bold fonts indicate the best performance. \| Test set \| 4-Corners \| 3-Corners \| 2-Corners \| 1-Corner \| No-Corner \| \|----------\|-----------\|-----------\|-----------\|----------\|-----------\| \| Metric \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| \| GKN \| 1.1146±0.3936 \| 3.6497±1.1874 \| 1.0692±0.2034 \| 3.7059±0.9543 \| 1.0673±0.1393 \| 3.6822±0.9819 \| 1.1063±0.1905 \| 3.4898±0.9469 \| 1.0728±0.2074 \| 3.9551±0.9791 \| \| FNO \| 1.0947±0.3265 \| 2.2707±0.3361 \| 1.0742±0.3418 \| 2.1657±0.3976 \| 1.0672±0.3736 \| 2.2617±0.2449 \| 1.0921±0.2935 \| 2.3922±0.3526 \| 1.0762±0.4420 \| 2.2281±0.4192 \| \| GNN-PDE \| 1.0026±0.0093 \| 3.1410±0.8751 \| 1.0009±0.0101 \| 3.2812±0.8839 \| 1.0015±0.0099 \| 3.3557±0.8521 \| 1.0002±0.0153 \| 3.1421±0.8685 \| 1.0011±0.0152 \| 3.7561±0.10274 \| \| MP-PDE \| 1.0007±0.0677 \| 3.1018±0.8431 \| 1.0003±0.0841 \| 3.2464±0.8049 \| 0.9919±0.0699 \| 3.2763±0.8632 \| 0.9829±0.07199 \| 3.0163±0.8272 \| 0.9882±0.0683 \| 3.6522±0.8961 \| \| BENO (ours) \| 0.3523±0.1245 \| 0.9650±0.3131 \| 0.4308±0.1994 \| 1.2206±0.4978 \| 0.4910±0.1888 \| 1.4388±0.5227 \| 0.5416±0.2133 \| 1.4529±0.4626 \| 0.5542±0.1952 \| 1.7481±0.5394 \| training and validation, and 100 samples for testing. Each sample covers a grid of $32 \times 32$ nodes and 128 boundary nodes. To further assess model performance, higher-resolution versions of each data sample, such as $64 \times 64$, are also provided. Details on data generation are provided in Appendix C. Baselines. We adopt two of the most mainstream series of neural PDE solvers as baselines, one is graph-based, including GKN (Li et al., 2020b), GNN-PDE (Lötzsch et al., 2022), and MP-PDE (Brandstetter et al., 2022); the other is operator-based, including FNO (Li et al., 2020a). For fair comparison and adaption to irregular boundary shapes in our datasets, all of the baselines are re-implemented with the same input as ours, including all the interior and boundary node features. Please refer to Appendix E for re-implementation details. Implementation Details. All experiments are based on PyTorch (Paszke et al., 2019) and PyTorch-Geometric (Fey & Lenssen, 2019) on 2 × NVIDIA A100 GPUs (80G). Following (Brandstetter et al., 2022), we also apply graph message passing neural network as our backbone for all the datasets. We use Adam (Kingma & Ba, 2014) optimizer with a weight decay of $5 \times 10^{-4}$ and a learning rate of $5 \times 10^{-5}$ obtained from grid search for all experiments. The relative L2 error measures the difference between the predicted and the ground truth values, normalized by the magnitude of the ground truth. MAE measures the average absolute difference between the predicted values and the ground truth values. Please refer to Appendix D for more implementation details. 4.2 Main Experimental Results We first test whether our BENO has a strong capability to solve elliptic PDEs with varying shapes. Table 2 and 3 summarize the results for the shape generalization task (more in Appendix H). From the results, we see that recent neural PDE solving methods (i.e., MP-PDE) overall fail to solve elliptic PDEs with inhomogeneous boundary values, not to mention generalizing to datasets with different boundary shapes. This precisely indicates that existing neural solvers are insufficient for solving this type of boundary value problems. In contrast, from Table 2, we see that our proposed BENO trained only on 4-Corners dataset consistently achieves a significant improvement and strong generalization capability over the previous methods by a large margin. More precisely, the improvements of BENO over the best baseline are 55.17%, 52.18%, 52.43%, 47.38%, and 52.94% in terms of relative L2 norm when testing on 4/3/2/1/No-Corner dataset respectively. We attribute the remarkable performance to two factors: (i) BENO comprehensively leverages boundary information, and fuses them with the interior graph message for solving. (ii) BENO integrates dual-branch architecture to fully learn boundary and interior in a decoupled way and thus improves generalized solving performance. Similarly, from Table 3, we see that among mixed corner training results, BENO always achieves the best performance among various compared baselines when varying the test sets, which validates the consistent superiority of our BENO with respect to different boundary shapes. Table 3: Performances of our proposed BENO and the compared baselines, which are trained on 900 mixed samples (180 samples each from 5 datasets) and tested on 5 datasets under relative L2 error and MAE separately. The unit of the MAE metric is $1 \times 10^{-3}$. \| Test set \| 4-Corners \| 3-Corners \| 2-Corners \| 1-Corner \| No-Corner \| \|----------\|-----------\|-----------\|-----------\|----------\|-----------\| \| Metric \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| \| GKN \| 1.0588±0.1713 \| 3.5051±0.9401 \| 1.0651±0.1562 \| 3.7061±0.8563 \| 1.0386±0.1271 \| 3.6043±0.9392 \| 1.0734±0.1621 \| 3.4048±0.9519 \| 1.0423±0.2102 \| 3.901±0.9287 \| \| FNO \| 1.0834±0.0462 \| 4.6401±0.5327 \| 1.0937±0.0625 \| 4.6092±0.6713 \| 1.0672±0.0376 \| 4.5267±0.5581 \| 1.0735±0.0528 \| 4.5027±0.5371 \| 1.0713±0.0489 \| 4.5783±0.5565 \| \| GNN-PDE \| 1.0009±0.0036 \| 3.1311±0.8664 \| 1.0003±0.0039 \| 3.2781±0.8858 \| 1.0005±0.0038 \| 3.3518±0.8520 \| 0.9999±0.0042 \| 3.1422±0.8609 \| 1.0002±0.0041 \| 3.7528±1.0284 \| \| MP-PDE \| 1.0063±0.0735 \| 3.1238±0.8502 \| 1.0045±0.0923 \| 3.2537±0.7867 \| 0.9957±0.0772 \| 3.2864±0.8607 \| 0.9822±0.0802 \| 3.0177±0.8363 \| 0.9912±0.0781 \| 3.6658±0.8949 \| \| BENO (ours) \| 0.4487±0.1750 \| 1.2150±0.4213 \| 0.4783±0.1938 \| 1.3509±0.5432 \| 0.4737±0.1979 \| 1.3516±0.5374 \| 0.5168±0.1793 \| 1.3728±0.5148 \| 0.4665±0.2001 \| 1.4213±0.5262 \| Table 4: Performances of our BENO and the compared baselines, which are trained on 900 4-Corners samples and tested with zero boundary value samples. The unit of the MAE metric is $1 \times 10^{-3}$. \| Test set \| 4-Corners \| 3-Corners \| 2-Corners \| 1-Corner \| No-Corner \| \|----------\|-----------\|-----------\|-----------\|----------\|-----------\| \| Metric \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| \| GNN-PDE \| 0.7092±0.0584 \| 0.1259±0.0755 \| 0.7390±0.0483 \| 0.2351±0.1013 \| 0.7491±0.0485 \| 0.3290±0.1371 \| 0.7593±0.05269 \| 0.4750±0.1582 \| 0.7801±0.0371 \| 0.6808±0.1692 \| \| MP-PDE \| 0.2598±0.1098 \| 0.0459±0.0359 \| 0.3148±0.0814 \| 0.1066±0.0618 \| 0.3729±0.0819 \| 0.1778±0.0969 \| 0.4634±0.0649 \| 0.3049±0.1182 \| 0.5458±0.0491 \| 0.4924±0.1310 \| \| BENO (ours) \| 0.0908±0.07381 \| 0.0142±0.0131 \| 0.1031±0.0728 \| 0.0288±0.0189 \| 0.1652±0.1324 \| 0.0583±0.0362 \| 0.1783±0.1508 \| 0.0862±0.0456 \| 0.2441±0.1665 \| 0.1622±0.0798 \| Additionally, we plot the visualization of the best baseline and our proposed BENO trained on 4-Corners dataset in Figure 4. It can be clearly observed that the predicted solution of BENO is closed to the ground truth, while MP-PDE fails to learn any features of the solution. We observe similar behaviors for all other baselines. Figure 4: Visualization of two samples’ prediction and prediction error from 4-Corners dataset. We render the solution $u$ of the baseline MP-PDE, our BENO and the ground truth in $\Omega$. ### 4.3 Generalization Study #### 4.3.1 Results on Different Boundary Values To investigate the generalization ability on boundary value, we again train the models on 4-Corners dataset with inhomogeneous boundary value but utilize the test set with zero boundary value, which makes the boundary inhomogeneities totally different. Table 4 compares the best baseline and summarizes the results. From the results, we see that BENO has a significant advantage, successfully reducing the L2 norm to around 0.1. In addition, our method outperforms the best baseline by approximately 60.96% in terms of performance improvement. This not only demonstrates BENO’s Table 5: Performances of our BENO and the compared baselines, which are trained on 900 4-Corners 32 × 32 samples and tested with 64 × 64 samples. The unit of the MAE metric is $1 \times 10^{-3}$. \| Test set \| 4-Corners(64×64) \| 3-Corners(64×64) \| 2-Corners(64×64) \| 1-Corner(64×64) \| No-Corner(64×64) \| \|----------\|------------------\|------------------\|------------------\|-----------------\|-----------------\| \| Metric \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| \| MP-PDE \| 0.6335± \| 0.0596± \| 0.7457± \| 0.1138± \| 0.7926± \| 0.1565± \| 0.8336± \| 0.2445± \| 0.8749± \| 0.3991± \| \| \| 0.1009 \| 0.0418 \| 0.0738 \| 0.0533 \| 0.0505 \| 0.0596 \| 0.04467 \| 0.0915 \| 0.0298 \| 0.1045 \| \| BENO (ours) \| 0.4596± \| 0.0440± \| 0.5483± \| 0.0860± \| 0.6020± \| 0.1214± \| 0.6684± \| 0.1995± \| 0.7497± \| 0.3424± \| \| \| 0.1094 \| 0.0349 \| 0.0987 \| 0.0466 \| 0.0842 \| 0.0537 \| 0.0794 \| 0.0851 \| 0.0653 \| 0.1000 \| Table 6: Ablation study of our BENO. The unit of the MAE metric is $1 \times 10^{-3}$. \| Test set \| 4-Corners \| 3-Corners \| 2-Corners \| 1-Corner \| No-Corner \| \|----------\|-----------\|-----------\|-----------\|----------\|-----------\| \| Metric \| L2 \| MAE \| L2 \| MAE \| L2 \| MAE \| \| BENO w. M \| 1.0130± \| 3.1436± \| 1.0159± \| 3.3041± \| 0.9999± \| 3.3007± \| 1.0026± \| 3.0842± \| 0.9979± \| 3.6832± \| \| \| 0.0858 \| 2.8667 \| 0.0975 \| 0.7906 \| 0.0792 \| 0.8504 \| 0.0840 \| 0.8202 \| 0.0858 \| 0.8970 \| \| BENO w/o. D \| 0.4058± \| 1.1175± \| 0.4850± \| 1.3810± \| 0.5273± \| 1.5439± \| 0.5795± \| 1.5683± \| 0.5835± \| 1.8382± \| \| \| 0.1374 \| 0.3660 \| 0.2230 \| 0.6068 \| 0.1750 \| 0.4774 \| 0.1981 \| 0.4670 \| 0.2232 \| 0.5771 \| \| BENO w. E \| 0.4113± \| 1.2020± \| 0.4624± \| 1.3569± \| 0.5347± \| 1.5990± \| 0.5891± \| 1.6222± \| 0.5843± \| 1.8790± \| \| \| 0.1236 \| 0.4048 \| 0.2102 \| 0.5453 \| 0.1985 \| 0.5604 \| 0.2129 \| 0.2016 \| 0.2016 \| 0.5952 \| \| BENO w. G \| 0.9037± \| 2.6795± \| 0.8807± \| 2.6992± \| 0.8928± \| 2.8235± \| 0.8849± \| 2.561± \| 0.8721± \| 2.9851± \| \| \| 0.1104 \| 0.5332 \| 0.1298 \| 0.6118 \| 0.1208 \| 0.5892 \| 0.1462 \| 0.5085 \| 0.1569 \| 0.5591 \| \| BENO (ours) \| 0.3523± \| 0.9650± \| 0.4308± \| 1.2206± \| 0.4910± \| 1.4388± \| 0.5416± \| 1.4529± \| 0.5542± \| 1.7481± \| \| \| 0.1245 \| 0.3131 \| 0.1994 \| 0.4978 \| 0.1888 \| 0.5227 \| 0.2133 \| 0.4626 \| 0.1952 \| 0.5394 \| strong generalization ability regarding boundary values but also provides solid experimental evidence for the successful application of our elliptic PDE solver. 4.3.2 Different Grid Resolutions Data-driven PDE solvers often face limitations in terms of the scale of the training data, making the ability to generalize to higher resolutions a crucial metric. Table 5 provides a summary of our performance in resolution generalization experiments. The model was trained on the 4-Corners homogeneous boundary value dataset with 32 × 32 resolution and tested with 64 × 64 samples not seen in training. The results demonstrate a significant advantage of our method over MP-PDE, with an improvement of approximately 22.46%. We attribute this advantage in generalization to two main factors. Firstly, it stems from the inherent capability of GNNs to process input graphs of various sizes. Secondly, it is due to our incorporation of relative positions as part of the network’s edge features. Consequently, our approach can be deployed on different resolutions using the same setup. 4.4 Ablation Study To investigate the effectiveness of inner components in BENO, we study four variants of BENO. Table 6 shows the effectiveness of our BENO on ablation experiments, which is implemented based on 4-Corners dataset training. Firstly, BENO w. M replaces the BE-MPNN with a vanilla message passing neural network (Gilmer et al., 2017) and merely keeps the interior node feature. Secondly, BENO w/o. D removes the dual-branch structure of BENO and merely utilizes a single Encoder-BE-MPNN-Decoder procedure. Thirdly, BENO w. E adds the Transformer output for edge message passing. Finally, BENO w. G replaces the Transformer architecture with a vanilla graph convolution network (Kipf & Welling, 2016). From the results we can draw conclusions as follows. Firstly, BENO w. M performs significantly worse than ours, which indicates the importance of fusing interior and boundary in BENO. Secondly, comparing the results of BENO w/o. D with ours we can conclude that decoupled learning of the interior and boundary proves to be effective. Thirdly, comparing the results of BENO w. E and ours, we can find that boundary information only helps in node-level message passing. In other words, it is not particularly suitable to directly inject the global information of the boundary into the edges. Finally, comparing results of BENO w. G with ours validates the design of Transformer for boundary embedding is crucial. 5 RELATED WORK 5.1 CLASSIC ELLIPTIC PDE SOLVERS The classical numerical solution of elliptic PDEs approximates the domain $\Omega$ and its boundary $\partial \Omega$ in Eq. 1 using a finite number of non-overlapping partitions. The solution to Eq. 1 is then approximated over these partitions. A variety of strategies are available for computing this discrete solution. Popular approaches include finite volume method (FVM) (Hirsch, 2007), finite element method (FEM) (Hughes, 2012), and finite difference method (FDM) (LeVeque, 2007). In the present work we utilize the FVM to generate the dataset which can easily accommodate complex boundary shapes. This approach partitions the domains into cells, and the boundary is specified using cell interfaces. After numerically approximating the operator $\nabla^2$ over these cells, the numerical solution is obtained on the centers of the cells constituting our domain. Further details are provided in Appendix B. 5.2 GNN FOR PDE SOLVER GNNs are initially applied in physics-based simulations on solids and fluids represented by particles (Sanchez-Gonzalez et al., 2018). Recently, an important advancement MeshGraphNets (Pfaff et al., 2020) emerge to learn mesh-based simulations. Subsequently, several variations have been proposed, including techniques for accelerating finer-level simulations by utilizing GNNs (Belbute-Peres et al., 2020; Yang & Hong, 2022), combining GNNs with Physics-Informed Neural Networks (PINNs) (Gao et al., 2022), solving inverse problems with GNNs and autodecoder-style priors (Zhao et al., 2022), and handling temporal distribution shift (Luo et al., 2023). However, the research focus on addressing boundary issues is limited. T-FEN (Lienen & Günnemann, 2022), FEONet (Lee et al., 2023), VQGraph (Yang et al., 2024) and GNN-PDE (Lötzsch et al., 2022) are pioneering efforts in this regard, encompassing complex domains and various boundary shapes. Nevertheless, the boundary values are still set to zero, which does not account for the presence of inhomogeneous boundary values. This discrepancy is precisely the problem that we aim to address. 5.3 NEURAL OPERATOR AS PDE SOLVER Neural operators map from initial/boundary conditions to solutions through supervised learning in a mesh-invariant manner. Prominent examples of neural operators include the Fourier neural operator (FNO) (Li et al., 2020a), graph neural operator (Li et al., 2020b), and DeepONet (Lu et al., 2019). Neural operators exhibit invariance to discretization, making them highly suitable for solving PDEs. Moreover, neural operators enable the learning of operator mappings between infinite-dimensional function spaces. Subsequently, further variations have been proposed, including techniques for solving arbitrary geometries PDEs with both the computation efficiency and the flexibility (Li et al., 2022), enabling deeper stacks of Fourier layers by independently applying transformations (Tran et al., 2021), utilizing Fourier layers as a replacement for spatial self-attention (Guibas et al., 2021), facilitating boundary condition satisfaction in neural operators by implementing structural modifications to the operator kernel (Saad et al., 2022) and incorporating symmetries in the physical domain using group theory (Helwig et al., 2023). Gupta et al. (2021, 2022; Xiao et al., 2023) continuously improve the design of the operator by introducing novel methods for numerical computation. 6 CONCLUSION In this work, we have proposed Boundary-Embedded Neural Operators (BENO), a neural operator architecture to address the challenges posed by inhomogeneous boundary conditions with complex boundary geometry in solving elliptic PDEs. Our approach BENO incorporates physics intuition through a boundary-embedded architecture, consisting of GNNs and a Transformer, to model the influence of boundary conditions on the solution. By constructing a diverse dataset with various boundary shapes, values, and resolutions, we have demonstrated the effectiveness of our approach in outperforming existing state-of-the-art methods by an average of 60.96% in solving elliptic PDE problems. Furthermore, our method BENO exhibits strong generalization capabilities across different scenarios. The development of BENO opens up new possibilities for efficiently and accurately solving elliptic PDEs with complex boundary conditions, making them more useful to various scientific and engineering fields. ACKNOWLEDGEMENT We gratefully acknowledge the support of Westlake University Research Center for Industries of the Future, and Westlake University Center for High-performance Computing. REFERENCES Filipe De Avila Belbute-Peres, Thomas Economon, and Zico Kolter. Combining differentiable pde solvers and graph neural networks for fluid flow prediction. In International Conference on Machine Learning, pp. 2402–2411. PMLR, 2020. Marshall Bern and David Eppstein. Mesh generation and optimal triangulation. In Computing in Euclidean geometry, pp. 47–123. World Scientific, 1995. Johannes Brandstetter, Daniel Worrall, and Max Welling. Message passing neural pde solvers. arXiv preprint arXiv:2202.03376, 2022. Francis F Chen. Introduction to Plasma Physics and Controlled Fusion (3rd Ed.). 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TCGUnoiaWP	Table 5 is confusing. The authors said that the asterisk represents methods trained with the BEDLAM training set. Then what does Pose++CLIFF* (the last row) mean? Is it trained on the BEDLAM training set and Pose++ training set at the same time?	3D Human Reconstruction in the Wild with Synthetic Data Using Generative Models Anonymous authors Paper under double-blind review Figure 1: Pose++ generates diverse photo-realistic human images and corresponding body annotations, e.g., 2D landmarks and 3D meshes, with a multi-condition diffusion model. Abstract Human pose and shape estimation from monocular images play a fundamental role in computer vision applications such as augmented reality, virtual try-on, and human motion analysis. However, large-scale human datasets in the wild with 3D ground-truth annotations are very difficult to obtain. Previous high-quality 3D human pose datasets are usually obtained by either motion capture devices or computer graphics rendering techniques, both are expensive and laborious. In this work, we propose an effective approach based on recent diffusion models, termed Pose++, which can effortlessly generate human images and corresponding 2D human skeletons and 3D mesh annotations. Specifically, we first leverage a multi-conditioned stable diffusion model to generate diverse human images and initial ground-truth labels. At the core of this step is that we can easily obtain numerous depth and keypoints conditions from a 3D human parametric model, e.g., SMPL-X, by rendering the 3D mesh onto the image plane. The generated human image and the corresponding 3D mesh with camera parameters can be regarded as a pair of training samples. As there exists inevitable noise in the initial labels, we then cast the problem into a label-denoising process by exploiting an off-the-shelf 2D human pose estimator to filter negative data pairs and further optimize the pose parameters. Finally, we can build a unified human pose dataset with both 2D skeleton and 3D parametric model annotations. Experiments on 2D datasets (COCO, OCHuman) and 3D datasets (3DPW, RICH, SSP-3D) demonstrate the effectiveness of our approach. Thus, our method offers a promising avenue for advancing the field of human pose and shape estimation by generating large-scale human images and high-quality annotations in a fully automated fashion. 1 Introduction Estimating human pose and shape (HPS) (Kanazawa et al., 2018; Lin et al., 2021; Li et al., 2021b, 2022) from a single RGB image is a core challenge in computer vision and has many applications in robotics, computer graphics, and digital content creation. Current HPS estimation methods require well-annotated datasets to achieve good performance. Unfortunately, collecting large-scale human body data is time-consuming and expensive. As shown in Table 1, there are mainly two types of pipelines for capturing accurate 3D human body data. The first type is the indoor mocap systems, e.g., marker-based systems, and vision- \| Data Type \| Settings \| \|--------------\|---------------------------\| \| \| Assets \| Human Workload \| Comp. Cost \| Scale-Up Diff. \| Magnitude \| \| MoCap \| mocap system \| Actors \| × \| hard \| $1 \times 10^5$ \| \| Real-World \| MV, Pseudo \| RGB(D) cameras \| Annot./Cam. Calib. \| Models/Optim \| hard \| $1 \times 10^4$ \| \| Mono. Pseudo \| RGB(D) cameras \| Annot./Cam. Calib. \| Models/Optim \| medium \| $1 \times 10^5$ \| \| Synthetic \| 3D Avatars/Scenes \| Technical Artists \| Render \| easy \| $1 \times 10^6$ \| \| Generated \| × \| × \| Models/Optim \| none \| ∞ \| Table 1: ‘MV’ and ‘Mono.’ stands for ‘multi view’ and ‘Monocular’ separately. ‘Annot.’ and ‘Cam. Calib.’ stand for ‘annotation’ and ‘camera calibration’ separately. ‘Comp. Cost’ and ‘Scale-up Diff.’ stands for ‘computation cost’ and ‘scale-up difficulty’ separately. Many existing datasets (Ionescu et al., 2013; Tripathi et al., 2023; Cai et al., 2022; Mehta et al., 2017) use this pipeline to capture human body attributions. However, the pipeline suffers from four drawbacks: 1). The mocap systems are expensive. 2). The synchronization and operation of the system are complicated. 3). The number of actors in the dataset is limited. 4). The background is typically the indoor or laboratory environment, making large-scale human data with versatile scenes infeasible. The second type is synthesizing 3D human datasets using computer graphics (CG) rendering (Black et al., 2023; Wood et al., 2021, 2022). The drawbacks of this pipeline are three-folds: 1). High-quality 3D assets, including drivable avatars and scene assets, are expensive. Wood et al. (2021, 2022) do not open-source their data generation pipeline for sake of the commercial purpose. 2). Special knowledge of 3D rendering is required, making it impossible to use cheap crowdsourcing platforms like Amazon Mechanical Turk. 3). The domain gap between the rendered images and real-world images is non-negligible. As mentioned in Black et al. (2023), the HPS accuracy trained on rendered images depends on backbone pre-training, especially 2D COCO keypoint dataset pre-training. This pheromone suggests that the synthetic data still has room for improvement in terms of realism. As it is hard to obtain large-scale 3D human pose datasets in the wild, some researchers have considered leveraging existing large-scale 2D human pose datasets by optimization-based and weakly-supervised methods. SMPLify (Bogo et al., 2016) proposed to fit the parameters of a 3D human model to the location of 2D keypoints. EFT (Joo et al., 2021) introduced the Exemplar Fine-Tuning strategy by overfitting a pre-trained 3D pose regressor with 2D keypoint reprojection loss, taking the final output of the regressor as pseudo labels. However, these methods still suffer from poor performance on 3D human pose benchmarks. In this paper, we address these limitations by proposing a new data generation pipeline, termed Pose++, that can simultaneously generate photo-realistic human images in the wild, as well as corresponding well-aligned 2D human skeletons and 3D mesh annotations in a fully automatic fashion. The challenge of the pipeline lies in two folds. On one hand, how to ensure the pose, shape, and scene diversity of generated human images are critical in simulating real-world human distribution. A naive solution is taking advantage of the text-to-image diffusion models, e.g., Stable Diffusion (Rombach et al., 2022), by feeding different text prompts to the model and employing pre-trained pose estimators to get the pseudo labels. However, the text prompt alone is not fine-grained enough to create versatile human bodies. To solve this problem, we sample SMPL-X parameters of human bodies from large-scale human motion capture datasets (Mahmood et al., 2019; Black et al., 2023). Then, we render the human mesh into the depth map and keypoint heatmap with a random camera as the extra input conditions. Finally, we feed text prompts, depth map, and keypoint heatmap to a multi-conditioned ControlNet (Zhang & Agrawala, 2023) for generating human images. As such, we can get fine-grained control of the human body, and get initial training data pairs from the input conditions and output human images. On the other hand, how to ensure the alignment between the human images and generated annotations is critical for the training of downstream tasks. Experiments show that there exist label noises in the initial training data pairs. For example, the generated human and the input conditions form a mirror pair, or the human head orientation in the image is not consistent with the input SMPL-X parameters. To resolve this problem, we propose a two-stage label-denoising and refinement strategy. First, we use an off-the-shelf 2D pose estimator to filter the wrong-generated images by computing the average precision (AP) of symmetric joints. Second, we finetune the 2D keypoints in SMPL-X format to target 2D pose datasets with a transformer-based keypoint decoder. Upon getting the 2D keypoints, we optimize the head poses... of SMPL-X with EFT (Joo et al., 2021). With the aforementioned pipeline, we can get well-aligned training pairs and finally generate a large-scale 3D human dataset in the wild with around 600,000 samples, $1024 \times 1024$ resolution. Compared to previous datasets, our pipeline can generate diverse human identities and various in-the-wild scenes. Notably, the pipeline is much cheaper than both mocap-based and CG-based counterparts and is able to scale up 3D human datasets in the wild. Our contributions can be summarised as follows. 1) We propose a fully automatic pipeline to synthesize realistic and diverse human images with well-aligned annotations, including 2D keypoints, 3D SMPL-X parameters, and text descriptions. The dataset can empower a wide range of downstream perception tasks by rendering SMPL-X mesh into corresponding annotation format, e.g., human pose and shape estimation, human part segmentation, and human normal prediction. 2) We verify the quality of the generated dataset on 2D human pose estimation (HPE) and 3D human mesh reconstruction tasks. Experiment results show that the proposed pipeline can achieve comparable performance on several 2D HPE and 3D HPS benchmarks under the same settings. 2 RELATED WORKS 2.1 Human Pose and Shape Estimation Datasets Real-World Human Pose Data is vital for accurate, realistic modeling in 3D Human Pose and Shape Estimation tasks. High-quality data is typically captured using advanced motion capture devices like Inertial Measurement Units (IMUs) (von Marcard et al., 2018; Mahmood et al., 2019; Sigal et al., 2010) or Optical sensors (Ionescu et al., 2013), designed to capture precise marker movements or joint rotations. However, their deployment can be burdensome due to factors such as cost, setup complexity, and space requirements. Responding to these challenges, research has explored alternative methods to capture pseudo labels from diverse image types, including RGBD (Hassan et al., 2019), multi-view (Car et al., 2022), and single-view images (Bogo et al., 2016), eliminating the need for motion capture gear. SLOPER4D (Dai et al., 2023) consolidates data from IMU sensors, LiDAR, and RGB information to construct a large-scale urban human-scene dataset. Such methods often leverage perception models to derive 2D cues from images, which are further optimized by a 3D joint re-projecting loss. Synthetic Human Pose Datasets, developed with computer graphic techniques, has been used for many years. SURREAL (Varol et al., 2017) applies human skin and cloth textures to bare SMPL meshes, which lack realistic details. AGORA (Patel et al., 2021) uses high-quality static human scans for image rendering, but this routine also suffers from a high workload of scanning and rigging. However, rendering realistic manipulatable synthetic human datasets involves many challenges, including the need for diverse virtual properties for realistic data. BEDLAM (Black et al., 2023) and Synbody (Yang et al., 2023) add varied hair models and skin textures to SMPL-X (Pavlakos et al., 2019) meshes and simulates physically plausible cloth and hair movements. These processes can be resource-intensive. Furthermore, the use of rendering engines demands many professional skills. Thus, the rendering process can be computationally expensive and time-consuming. Controllable Human Image Generation has gained great traction with the advancement of Stable Diffusion (Rombach et al., 2022; Zhang & Agrawala, 2023). Text2Human (Jiang et al., 2022) uses a diffusion-based transformer sampler in response to text prompts and predicts indices from a hierarchical texture-aware codebook to conditionally generate realistic human images. HumanSD (Ju et al., 2023) introduces a skeleton-guided diffusion model with a novel heatmap loss for pose-conditioned human image generation. Generative Models for Perception Tasks. Several studies have effectively utilized datasets generated by diffusion models for perception tasks. For instance, Voelman et al. (2023) employed these datasets to train detection models, and Azizi et al. (2023) demonstrated that classification models can achieve state-of-the-art results on ImageNet (Deng et al., 2009) when fine-tuned on generated images. StableRep (Tran et al., 2023) found that training modern self-supervised methods on synthetic images from Stable Diffusion Models can yield impressive results. The learned representations often surpass those learned from real images of the same sample size. DatasetDM (Wu et al., 2023) trained decoders using limited data and succeeded in decoding the rich latent code of the diffusion model as precise perception annotation. This has enabled the generation of an infinitely large annotated dataset, proven effective in segmentation, depth estimation, and 2D human pose estimation. Figure 2: Full pipeline of automatic data generation. $ M_G $ indicates the ControlNet (Zhang & Agrawala [2023]). $ M_K $ denotes a pre-trained 2D pose regressor. $ M_{eft} $ (Joo et al. [2021]) denotes the 3D human pose regressor for label refinement. DiffusionHPC (Weng et al. [2023]), closely related to this paper, is pioneering in employing diffusion models to render human images for 3D HPS tasks. It leverages a pre-trained 3D pose regressor to estimate the human mesh, subsequently renders a depth map, and then leverages a depth-to-image diffusion model to generate human images. Different from DiffusionHPC, the input condition of Pose++ is sampled from large-scale motion datasets and we use multi-resource conditions to enhance the alignment between the generated images and 3D pose labels. Besides, we involve an extra refinement step to refine the initial 3D pose parameters. 3 METHOD We present Pose++, a simple yet effective pipeline for creating versatile human body images and corresponding perception annotations in a fully automated fashion, which can be used for many downstream human perception tasks, such as 2D/3D human pose and shape estimation, human part segmentation, and human action recognition (see Fig. 2). The core idea of the proposed pipeline is creating large-scale image-mesh-caption pairs by incorporating off-the-shelf 2D generative models, e.g., Stable Diffusion (Rombach et al. [2022]) and 3D human parametric models (Pavlakos et al. [2019]). For the sake of completeness, we give a brief review of the controllable text-to-image (T2I), image-to-image (I2I) generative models and the 3D human parametric model, SMPL-X (Pavlakos et al. [2019]) in Section 3.1. In the following subsections, we first illustrate how we generate the initial human image-annotation pairs in Section 3.2. Then we show how we refine the initial 2D keypoints labels and 3D pose labels to get high-quality training pairs in Section 3.3. 3.1 PREREQUISITES Stable Diffusion Models (Rombach et al. [2022], Podell et al. [2023]) are text-to-image diffusion models capable of generating near photo-realistic images given any text input. They have been revealed to be capable of synthesizing more diverse and higher-quality images compared to previous dominant GAN-based models (Goodfellow et al. [2016], Brock et al. [2018], Karras et al. [2019]). Controllable image-to-image adapters are frameworks designed for empowering text-to-image diffusion models with more image-level control signals. ControlNet (Zhang & Agrawala [2023]) and T2I-Adapter (Mou et al. [2023]) are two representative lightweight adapters that only apply several additional blocks to the original stable diffusion models. During the training of adapters, the text-to-image diffusion model is frozen. Thus, they significantly reduce the training cost while keeping the generation ability of the origin text-to-image models to the maximum extent. SMPL-X (Pavlakos et al., 2019), defined as $ M(\beta, \theta, \psi) : \mathbb{R}^{\|\beta\| \times \|\beta\| \times \|\psi\|} \rightarrow \mathbb{R}^{3N} $, is a 3D wholebody human parametric model, employing shape, expression, and pose parameters to control the entire body mesh. The shape parameters $ \beta \in \mathbb{R}^{200} $ are dictated by the first 200 principal components of a linear shape space learned from scanned human meshes. The expression parameters $ \psi \in \mathbb{R}^{50} $ represent coefficients of a low-dimensional linear space, while the pose parameters model relative 3D rotations for 55 joints, encompassing the body, jaw, and hand poses. The function of SMPL-X provides a differentiable skinning process that uses pose, shape, and expression parameters as inputs and delivers a triangulated mesh $ V \in \mathbb{R}^{N \times 3} $ with $ N = 10475 $ vertices. The reconstructed 3D joints $ J \in \mathbb{R}^{144 \times 3} $ can be obtained using a forward kinematics process. ### 3.2 Initial Human Image and Annotation Generation Camera simulation. One drawback of vision-based motion capture systems is that they need to calibrate and synchronize the camera’s intrinsic and extrinsic parameters during the capturing. Thus, the generated human data are limited in terms of the scales and view diversity. On the contrary, our pipeline gets rid of the physical RGBD cameras and can simulate arbitrary human scales and body orientation. Specifically, we randomly determine the orthographic scale $ s $ of the human body $ s \in [0.45, 1.1] $, along with the horizontal shift $ (t_x, t_y) $ within a range of $[-0.4/s, 0.4/s]$. This methodology ensures that the majority of body parts are visible in the image. Following Kanazawa et al. (2018); Wang et al. (2023), we determine the translation of the body as $ transl = [t_x, t_y, f/s] $. The focal length in normalized device coordinate (NDC) space, denoted as $ f $, can be computed using the formula $ f = 1/\tan(FoV/2) $. Here, FoV represents the Horizontal Field of View angle, which is randomly zoomed in from 65 to 25 by following Black et al. (2023). Image condition generation. To synthesize realistic human images with paired pose annotations, we leverage ControlNet, equipped with the state-of-the-art diffusion model, SDXL, as our image generator. Existing ControlNet variants take a 2D skeleton, depth map, or canny map as condition inputs. These inputs are typically detected from real-world images by pre-trained perception models. However, there exist two main drawbacks to generating image conditions from these pre-trained perception models. On one hand, it’s laborious to crawl diverse human pose and shape images from the Internet. On the other, the perception models cannot ensure the generation of fully accurate annotations, thus the different modalities annotations have discrepancies, e.g., the 2D keypoint heatmap from a 2D pose estimator and the depth map from a depth estimator are not aligned. In such cases, if we take the perception results as the multi-condition inputs of ControlNet, the generated images would probably have weird artifacts. To resolve the problem, we construct the input of ControlNet by taking advantage of the 3D human parametric model, SMPL-X. There exist several large-scale human motion capture databases (Mahmood et al., 2019; Black et al., 2023) with diverse body poses and shapes in SMPL-X format. Thanks to the disentanglement of the pose and shape parameters of the SMPL-X model, we can even recombine the two parameters to generate a human mesh that does not exist in the databases. For example, an overweight man doing an extremely difficult yoga pose. Upon getting the simulated camera parameters aforementioned in Section 3.2 and 3D human mesh from SMPL-X, we can render an existing 3D human mesh into the image plane, as such, getting the corresponding depth map, as well as the 2D keypoint heatmap. Notably, the depth map is proven to be crucial to generate accurate body shapes and the keypoint heatmap is helpful to generate accurate hand gestures. In practice, we set the depth condition scale and keypoint condition scale as 0.8 and 0.5 separately. Text prompt generation. The aforementioned image-based multi-condition maps only provide rough control signals of the foreground information. They are not fine-grained enough to determine the gender of the human, as well as the background scenes of the image. Thus, we incorporate a structured text prompt template to handle this issue. In particular, we designed a simple text template as “A {gender} {action} {environment}”. The gender and the action of the person are determined by the SMPL-X annotations. The environment is generated by a large language model, i.e., ChatGPT (OpenAI, 2020). To create photo-realistic humans, we also feed negative text prompts, e.g., “ugly, extra limbs, poorly drawn face, poorly drawn hands, poorly drawn feet”, to the model. Finally, we get all of the input conditions of the ControlNet. We apply a total of 40 inference steps for each sample. The resolution of the generated images are all $ 1024 \times 1024 $. The generated images and the input conditions (2D keypoints, SMPL-X parameters) are regarded as the initial data pairs. 3.3 Label Denoising and Refinement The generated images are not always well-aligned with the input conditions. The most common incorrect case is the generated human and the input conditions form a mirror pair. To resolve this problem, we employ an off-the-shelf 2D human pose estimator, Poseur (Mao et al., 2022), to detect the symmetric joints in the image, e.g., left shoulder and right shoulder. If the average precision (AP) between the detected symmetric joints and the condition keypoint map is lower than a threshold $\sigma$, we need to filter this sample from the final dataset. Besides, we also conduct further refinement steps on the initial 2D keypoint condition and SMPL-X pose parameters as follows. 2D Keypoint Refinement. We get the initial 2D keypoints by projecting the 3D joints of SMPL-X into the image plane with the simulated camera parameters. The intuition of the 2D keypoint refinement is that different pose datasets provide different skeleton formats (Sárándi et al., 2023), even though they sometimes share the same joint names. To tackle the label discrepancies, it is necessary to refine the initial 2D keypoints to the formulation of the target 2D pose dataset. Here, we take the COCO dataset as an example to explain the proposed strategy for refining the initial 2D keypoints from SMPL-X model. Specifically, we leverage a COCO pre-trained keypoint decoder proposed in Mao et al. (2022) to get more accurate 2D keypoint labels. Concretely, we replace the coarse proposals from fully connected layers with the initial 2D keypoints from SMPL-X, and then several deformable cross-attention (Zhu et al., 2021) operations are performed between the image features and keypoint queries to gradually generate the 2D keypoints in COCO format. Compared to the pure pseudo-labeling process, our refinement strategy has more reliable initial keypoint proposals. Thus, our method has a higher upper bound of the final generated 2D keypoint labels. 3D Head Pose Optimization. Another common inaccurate case is that the generated human sometimes has a slightly different head orientation compared to the initial SMPL-X pose parameters. To resolve this problem, we leverage a 2D keypoint detector $M_K$ to get the 2D facial landmarks. Then, we employ EFT (Joo et al., 2021) to optimize the head pose parameters, with camera parameters and other SMPL-X parameters fixed during the optimization. 4 Experiments 4.1 Datasets and Evaluation Metrics Datasets for 3D HPS. BEDLAM (Black et al., 2023) is a synthetic dataset rendered by Unreal Engine 5, with 1-10 individuals in 8 3D scenes and 95 HDRi panoramas. It offers around 380K unique frames and a total of 1M individual person crops. AGORA (Patel et al., 2021) is another synthetic dataset with 17K images (14K training, 3K test), each image contains 5-15 people in varied lighting or 3D environments. We use them and Pose++ as training sets and perform detailed experiments for a fair comparison. 3DPW (von Marcard et al., 2018) is an in-the-wild dataset will motion capture annotations. It is a standard benchmark in 3D HPS tasks to evaluate the model performance. RICH (Huang et al., 2022) is a dataset aimed at understanding human-scene interactions. We use RICH here as our evaluation dataset for it has various camera views and human-scene contact poses. SSP-3D (Sengupta et al., 2020) is a benchmark dataset designed for body shape prediction methods. It contains 311 images of athletes in form-fitting clothing, showcasing a range of body shapes and poses. We use SSP-3D to evaluate the performance of body shape estimation. Datasets for 2D HPE. COCO (Lin et al., 2014) is a large-scale in-the-wild 2D human pose dataset. We compare the performance of several 2D pose estimators on the COCO training set and Pose++. OChuman (Zhang et al., 2019) is a 2D pose estimation dataset containing various occlusion scenes. We use the OChuman validation set as an evaluation dataset. Evaluation metrics. For 3D HPS, we evaluate the precision of the reconstructed human mesh using 3D evaluation metrics, namely MPJPE (Mean Per Joint Position Error), PA-MPJPE (Procrustes Analysis Mean Per Joint Position Error), and PVE (Per Vertex Error). These metrics calculate the Euclidean distances in millimeters (mm) between the predicted and the actual 3D points or vertices. PVE-T-SC (Sengupta et al., 2020) is used as a body shape evaluation metric. For 2D HPE, we adopt widely used mAP (mean Average Precision) and its variants as the evaluation metrics. \| Method \| Dataset \| Output Type \| Backbone \| 1% Crops↑ \| 5% Crops↑ \| 10% Crops↑ \| 100% Crops↑ \| \|----------\|---------\|-------------\|----------\|-----------\|-----------\|------------\|-------------\| \| RTMPose \| C \| Classification \| CSPNeXt \| 0.0 \| 7.2 \| 23.6 \| 67.9 \| \| RTMPose \| B+C \| Classification \| CSPNeXt \| 46.4 \| 55.9 \| 58.7 \| 68.4 \| \| RTMPose \| P+C \| Classification \| CSPNeXt \| 49.1 \| 55.7 \| 58.0 \| 68.1 \| \| RTMPose \| P+B+C \| Classification \| CSPNeXt \| 61.9 \| 63.1 \| 64.4 \| 71.3 \| \| RLEPose \| C \| Regression \| ResNet50 \| 0.0 \| 3.9 \| 19.2 \| 53.5 \| \| RLEPose \| B+C \| Regression \| ResNet50 \| 40.6 \| 47.7 \| 55.3 \| 64.8 \| \| RLEPose \| P+C \| Regression \| ResNet50 \| 31.5 \| 39.0 \| 50.3 \| 65.1 \| \| RLEPose \| P+B+C \| Regression \| ResNet50 \| 51.8 \| 56.3 \| 58.5 \| 66.6 \| Table 2: Ablation experiments on 2D human pose estimation. C denotes COCO, B denotes BED-LAM, P denotes Pose++, and Crops % only applies to COCO. All experiments are evaluated on the COCO validation set. AP is used as the evaluation metric. \| Method \| Dataset \| Pretrain \| Crops % \| PA-MPJPE↓ \| MPJPE↓ \| PVE↓ \| PVE-T-SC↓ \| \|--------\|---------\|----------\|---------\|-----------\|--------\|------\|-----------\| \| CLIFF \| B† \| COCO \| 100 \| 77.2 \| 98.4 \| 117.7\| 17.4 \| \| CLIFF \| B \| COCO \| 100 \| 50.5 \| 76.1 \| 90.6 \| N/A \| \| CLIFF \| A \| COCO \| 100 \| 54.0 \| 88.0 \| 101.8\| N/A \| \| CLIFF \| P \| COCO \| 50 \| 57.6 \| 95.7 \| 103.4\| 13.7 \| \| CLIFF \| P \| COCO \| 100 \| 52.7 \| 87.3 \| 102.1\| 13.4 \| \| CLIFF \| B+A \| scratch \| 100 \| 61.7 \| 96.5 \| 115.0\| N/A \| \| CLIFF \| B+A \| ImageNet\| 100 \| 51.8 \| 82.1 \| 96.9 \| N/A \| \| CLIFF \| B+A \| COCO \| 100 \| 47.4 \| 73.0 \| 86.6 \| 13.6 \| \| CLIFF \| P+A \| scratch \| 100 \| 62.3 \| 108.7 \| 124.1\| 15.4 \| \| CLIFF \| P+A \| ImageNet\| 100 \| 52.4 \| 94.8 \| 106.4\| 13.3 \| \| CLIFF \| P+A \| COCO \| 100 \| 48.6 \| 76.8 \| 88.9 \| 13.3 \| Table 3: Ablation experiments on 3D pose and shape estimation. P denotes Pose++, B denotes BEDLAM and A denotes AGORA. Crops % only applies to . PA-MPJPE, MPJPE and PVE are evaluated on 3DPW. PVE-T-SC is evaluated on SSP-3D. ### 4.2 Ablation Study 2D HPE.* In Table 2, we adopt two types of 2D pose regressors to verify the effectiveness of the proposed data generation pipeline. For a fair comparison, all the models are trained with 10 epochs. Our pipeline can consistently improve the detection performance when mixed with different COCO training subsets (from 1% to 100%). The performance of Pose++ is comparable with BEDLAM in all data crops. When joint training with all three datasets, both 2D pose regressors get the best performance. We conjecture that the generated dataset only has one person per image, which lacks human-scene occlusion and human-human interaction. We also find that classification-based RTM-Pose (Jiang et al., 2023) is less data-hungry than regression-base RLEPose (Li et al., 2021a) in low data regime, e.g., achieving much higher AP on 1% COCO training set. 3D HPS. In Table 3, we evaluate the impact of Pose++ on a 3D HPS estimator, CLIFF (Li et al., 2022). Pose++ outperforms AGORA (Patel et al., 2021) in terms of both pose and shape estimation. Pose++ achieves similar performance when fairly compared with BEDLAM (Black et al., 2023). We conjecture that there still exists some noisy pose labels in our pipeline, which affects the results on 3D pose metrics (PA-MPJPE and PVE). Besides, CLIFF trained on Pose++ achieves better shape estimation results COCO pre-trained backbone on SSP-3D dataset. Qualitative Visualization of Control Conditions. In this section, we demonstrate the necessity of the multi-condition design for generating well-aligned image/annotation pairs. As shown in Fig. 3, when we only use 2D keypoint as a generation condition, the generated human image can be inconsistent with the body shape of the mesh. When we only use rendered depth as a generation condition, the output image may have a different gesture compared to the original SMPL-X mesh. When we use both conditions, both body shape and gesture are aligned with the original mesh. The qualitative experiment verified the effectiveness of the multi-condition design of our generation pipeline. Figure 3: A denotes keypoint condition. B denotes depth condition. C is the generation result with only depth condition. D and E are the synthesis results with both keypoint and depth conditions. 4.3 Main Results Results on 2D HPE. We summaries the key results in Table 4. (1) Pose++ dataset can improve the result on the COCO validation set. (2) Due to the lack of occlusion and multi-person scenes in the generated images, Pose++ cannot improve the results on the OCHuman validation set. (3) Pose++ can outperform DatasetDM [Wu et al., 2023] by a large margin on the COCO validation set under the same training setting. \| Method \| Backbone \| Training Set \| Crop \| COCO AP \| APm \| APt \| OCHuman AP \| APm \| APt \| \|-----------------\|------------\|--------------\|------\|---------\|-----\|-----\|------------\|-----\|-----\| \| RTMPose [Jiang et al., 2023] \| CSPNeXt \| C \| 100 \| 75.2 \| 71.6 \| 81.9 \| 69.9 \| 67.0 \| 69.8 \| \| RTMPose [Jiang et al., 2023] \| CSPNeXt \| P + C \| 100 \| 75.7 \| 72.4 \| 82.9 \| 67.2 \| 62.5 \| 67.2 \| \| SimplePose [Xiao et al., 2018] \| HRNet-W32 \| C \| 100 \| 74.9 \| 71.3 \| 81.5 \| 59.8 \| 65.3 \| 59.8 \| \| SimplePose [Xiao et al., 2018] \| HRNet-W32 \| D + C \| 1 \| 47.5 \| 44.2 \| 52.6 \| N/A \| N/A \| N/A \| \| SimplePose [Xiao et al., 2018] \| HRNet-W32 \| P + C \| 1 \| 50.3 \| 44.7 \| 59.1 \| 29.5 \| 18.7 \| 29.5 \| Table 4: Main Results on 2D Human Pose Estimation. P denotes Pose++, D denotes DatasetDM [Wu et al., 2023], C denotes COCO, B denotes BEDLAM [Black et al., 2023]. Crops % only applies to COCO during the training. We evaluate results on COCO and OCHuman Datasets. Results on 3D HPS. Table 5 shows the results on 3DPW and Rich. We report CLIFF [Li et al., 2022] trained on Pose++, BEDLAM [Black et al., 2023] and AGORA [Patel et al., 2021]. CLIFF trained on Pose++ shows stronger generation capacity and achieves the best results on both 3DPW and RICH after finetuning on the 3DPW training set. \| Methods \| 3DPW (14) \| RICH (24) \| \|--------------------------\|-----------\|-----------\| \| \| PA-MPJPE↓ \| MPJPE↓ \| PVE↓ \| PA-MPJPE↓ \| MPJPE↓ \| PVE↓ \| \| HMR [Kanazawa et al., 2018] \| 76.7 \| 130 \| N/A \| 90.0 \| 158.3 \| 186.0 \| \| SPIN [Koipotouros et al., 2019] \| 59.2 \| 96.9 \| 116.4 \| 69.7 \| 122.9 \| 144.2 \| \| SPEC [Kocabas et al., 2021b] \| 53.2 \| 96.5 \| 118.5 \| 72.5 \| 127.5 \| 146.5 \| \| PARE [Kocabas et al., 2021a] \| 50.9 \| 82.0 \| 97.9 \| 64.9 \| 104.0 \| 119.7 \| \| HybrIK [Li et al., 2021b] \| 48.8 \| 80 \| 94.5 \| 56.4 \| 96.8 \| 110.4 \| \| CLIFF† [Li et al., 2022] \| 46.4 \| 73.9 \| 87.6 \| 55.7 \| 90.0 \| 102.0 \| \| BEDLAM-HMR† [Black et al., 2023] \| 47.6 \| 79.0 \| 93.1 \| 53.2 \| 91.4 \| 106.0 \| \| BEDLAM-CLIFF† [Black et al., 2023] \| 46.6 \| 72.0 \| 85.0 \| 51.2 \| 84.5 \| 96.6 \| \| Pose++ CLIFF† \| 46.6 \| 70.2 \| 83.7 \| 51.0 \| 84.4 \| 96.1 \| \| BEDLAM-CLIFF† (with 3DPW) \| 43.0 \| 66.9 \| 78.5 \| 50.2 \| 84.4 \| 95.6 \| \| Pose++ CLIFF† (with 3DPW) \| 42.3 \| 65.2 \| 76.8 \| 50.1 \| 82.7 \| 93.6 \| Table 5: Reconstruction error on 3DPW and RICH. Trained with BEDLAM training set. †Trained on real images with same setting as BEDLAM-CLIFF. Dataset visualization. We visualize the generated dataset in Fig. 4. The qualification result demonstrates that Pose++ can generate diverse human images with well-aligned annotations in the wild. Please refer to Appendix A.1 to see more visualization examples of our dataset. Figure 4: Visual examples of the generated dataset. (a) and (b) demonstrate the diverse scenes of the dataset. (c) indicates the versatile poses of the dataset. (d) illustrates the comic style. (e) shows two examples of overweight body shapes. 5 DISCUSS AND CONCLUSION In this work, we propose an effective data generation pipeline, which can effortlessly generate diverse human images in the wild and corresponding 2D/3D pose annotations with conditional generative models. To further reduce the label noise, we propose to employ an off-the-shelf 2D pose estimator to filter negative samples and optimize the initial pose parameters. We validate the effectiveness of the pipeline on both 3D human mesh reconstruction and 2D human pose estimation. We hope this work could pave the way for using generative models to generate high-quality data for 3D human perception tasks. Future work. Our pipeline can apply to a series of similar tasks where high-quality data pairs are hard to collect, e.g., 3D animal pose estimation and 3D reconstruction of human-object/human-human interaction, by rendering all 3D objects into 2D image conditions, and then generating image-annotation pairs with diffusion models. We leave these promising areas for future work. Limitations. Although our data generation pipeline is cheap and effective, there are a few limitations. First, our data generation pipeline cannot handle crowd scenes, where many humans with small scales are in the image. 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jJvXNpvOdM	Indeed, in phase 2, objects are placed in locations were there shouldn’t be (so that the agent can re-organize them). It is hard for me to understand how, in this case, the Search Network can learn anything meaningful.	Task Planning for Visual Room Rearrangement under Partial Observability Karan Mirakhor, Sourav Ghosh, Dipanjan Das & Brojeshwar Bhowmick Visual Computing and Embodied Intelligence Lab TCS Research, Kolkata, India {karan.mirakhor, g.sourav10, dipanjan.da, b.bhowmick}@tcs.com Abstract This paper presents a novel modular task planner under partial observability that empowers an embodied agent to use visual input to efficiently plan a sequence of actions for simultaneous object search and rearrangement in an untidy room, to achieve a desired tidy state. The paper introduces (i) a novel Search Network that utilizes commonsense knowledge from large language models to find unseen objects, (ii) a Deep RL network trained with proxy reward, along with (iii) a novel graph-based state representation to produce a scalable and effective planner that interleaves object search and rearrangement to minimize the number of steps taken and overall traversal of the agent, as well as to resolve blocked goal and swap cases, and (iv) a sample-efficient cluster-biased sampling for simultaneous training of the proxy reward network along with the Deep RL network. Furthermore, the paper presents new metrics and a benchmark dataset - RoPOR, to measure the effectiveness of rearrangement planning. Experimental results show that our method significantly outperforms the state-of-the-art rearrangement methods [Weih et al., 2021; Gadre et al., 2022; Sarch et al., 2022; Ghosh et al., 2022]. 1 Introduction Tidying a disordered room based on user specifications is a challenging task as it involves addressing issues related to perception, planning, navigation, and manipulation [Batra et al., 2020]. An agent performing an embodied room rearrangement must use the sensor observations and a prior knowledge to produce a long horizon plan for generating a sequence of object movements to achieve the tidy goal state. This goal state is specified through geometry, images, language, etc. [Batra et al., 2020]. Majority of the existing research on room rearrangement emphasizes on perception and commonsense reasoning while assuming navigation and manipulation abilities, without incorporating efficient planning. Based on the goal state definition, they broadly fall into two categories; (i) Commonsense based reasoning without a predefined goal state: The methods [Kant et al., 2022; Sarch et al., 2022] in this category utilize image or language-based commonsense reasoning to identify if an object is misplaced from the correct receptacles in their egoview followed by rearranging them using a sub-optimal heuristic planner. Moreover, utilizing text or semantic relation-based anomaly detectors to identify misplaced objects does not resolve blocked goal or swap cases, where an object’s goal position is obstructed by another misplaced object or vice versa. (ii) User-specific room rearrangement with a pre-defined tidy goal state: In this setting, the rearrangement is done based on explicit user specification. Methods like [Weih et al., 2021; Gadre et al., 2022] focus on egocentric perception and use image or image feature-based scene representation to identify misplaced objects and a greedy planner to sequence actions for rearrangement. [Sarch et al., 2022] also performs a user-specific room rearrangement by using semantic relations to identify misplaced objects in agent’s egoview, and then rearrange them as they appear without planning. Methods such as [Kant et al., 2022; Sarch et al., 2022; Gadre et al., 2022] explicitly explore the room to find objects that are initially outside the agent’s egoview, since it only provides a partial information about the room. However, these approaches incur a significant traversal cost due to exploration. Additionally, these methods employ non-optimal planning that does not optimize the number of steps or overall traversal. In contrast, efficient planning makes rearrangement more effective by optimizing the sequence of actions and minimizing the time and effort required to achieve the goal state. [Ghosh et al., 2022], These authors contributed equally. Figure 1: (a) shows the top down view of our Rearrangement task and (b) is the agent’s initial egocentric view in the untidy current state for the same setup. The solid 2D bounding boxes indicate the desired goal state for all objects, while the dashed ones show the initial positions of visible objects in the untidy current state. The dotted 2D bounding boxes represent initial positions of unseen objects in the untidy current state. The sponge (magenta), an unseen object, is in a drawer near the stove, while the tomato (green), another unseen object, is on a stool behind the countertop. There are two scenarios: a blocked goal case with the lettuce (blue) and kettle (yellow) and a swap case between the bread (dark magenta) and pot (dark cyan). addresses the rearrangement task planning problem by assuming the complete visibility of the room, through the bird’s eye view. Their method addresses some planning problems, such as the combinatorial expansion of rearrangement sequencing, and blocked goal and swap cases without explicit buffer. However, the approach does not minimize overall agent traversal during the planning, and its state representation is not scalable to large numbers of objects. Moreover, their reliance on the ground truth object positions in both the current and goal states is impractical in real-life. Our aim is directed towards a novel and more practical aspect of the room rearrangement problem through efficient task planning under partial observability of a room using agent’s egocentric camera view. The major challenges associated with efficient task planning for room rearrangement under partial observability, as shown in Fig. 1, are (i) uncertainty over the location of unseen objects due to partial observability (objects outside the agent’s field of view presently which are visible from a different perspective, or objects placed within a closed receptacle e.g., spoon in drawer), (ii) scalability to a large number of objects, (iii) combinatorial expansion of sequencing due to simultaneous object search (for unseen objects) and rearrangement, (iv) minimizing the overall traversal during simultaneous object search and rearrangement, (v) blocked goal and swap cases without explicit buffer. In this paper, we propose a novel modular method for a task planner to address the aforementioned challenges. At the beginning, our agent captures the goal state by exploring the room to record the semantic and the geometric configuration Batra et al. (2020) of objects and receptacles through egocentric perception. Once the goal state is captured, the objects in the room are shuffled. In the untidy current state, our method partitions the task planning problem into two parts; object search and planning, with the aim of minimizing the overall agent traversal during simultaneous object search and rearrangement. First, we propose a novel commonsense knowledge based Search Network using large language models (LLMs) Liu et al. (2019); Kant et al. (2022) that leverages the object-receptacle semantics to predict the most probable receptacle for an unseen object in the egoview. Second, we use a Deep RL network with hybrid action space Ghosh et al. (2022) to plan our action sequence for simultaneous object search and rearrangement by resolving blocked goal and the swap cases. To this extent, we define the Deep RL state space with a novel graph-based state representation for the current and the goal state that incorporates geometric information about objects. This representation compactly encodes the scene geometry that aids in rearrangement planning and makes the Deep RL state space scalable to a large number of objects and scene invariant. In addition, we present a novel, sample-efficient cluster-biased sampling for simultaneous training of the proxy reward network Ren et al. (2022) and Deep RL to get a better estimate of the problem’s true objective from the episodic reward than the dense reward in Ghosh et al. (2022). The judicious combination of all the The aforementioned components effectively tackle the challenging combinatorial optimization problem in rearrangement as detailed in Sec. 3.6. The major contributions of this paper are: 1. To the best of our knowledge, this is the first end-to-end method to address the task planning problem for room-rearrangement from an egocentric view under partial observability, using a user-defined goal state. 2. A novel Search Network that leverages object-receptacle semantics using the commonsense knowledge from LLMs to predict the most probable receptacle for an unseen object. 3. Use of Deep RL based planner trained with proxy reward to overcome combinatorial expansion in rearrangement sequencing and, to optimize the overall traversal and the number of steps taken. 4. A new Graph-based state representation for the current and goal state to include geometric information about objects, making the Deep RL state space scalable to large numbers of objects and scene-invariant. 5. Introduction of a novel, sample-efficient cluster-biased sampling for simultaneous training of the proxy reward network and the Deep RL network. 6. We introduce a new set of metrics in Sec. 3.4 to obtain a thorough assessment of the rearrangement planner’s effectiveness by not only evaluating the success of the rearrangement, but also taking into account the number of steps taken and the overall agent traversal. 7. To address the inadequacies in existing benchmarks (Weihs et al., 2021) for evaluating task planning under partial observability, we introduce the RoPOR-Benchmark Dataset. We plan to openly release the dataset to enable further research in this domain. 2 METHODOLOGY In our room-rearrangement setup, the agent explores the room to capture the tidy user-specified goal state. During this exploration, the agent creates a 2D occupancy map $M^{2D}$ for the agent’s navigation while, 3D map $M^{3D}$ is utilized to augment the detection of 3D object and receptacle centroids to a fixed global reference frame ($\mathbb{R}^3$). Additionally, we generate an object list $O = \{[W_i, P_i], i = 1, 2, ..., N\}$ and a receptacle list $R = \{[W_i^R, P_i^R], i = 1, 2, ..., N_R\}$. Here, $N$, $W$ and $P \in \mathbb{R}^3$ are the total numbers of objects, their semantic labels, and 3D object centroids, respectively. While $N_R$, $W^R$ and $P^R \in \mathbb{R}^3$ are the total numbers of receptacles, their semantic labels including the room name from Ai2Thor (Kolve et al., 2017), and the 3D receptacle centroids respectively. Then, we randomly shuffle a few objects from the goal state to make the room untidy and fork the agent at a random location in the room. In this untidy current state, the agent’s knowledge is limited to the visible part of the room in its egocentric view. In the agent’s egocentric perception, only a set of objects $O^V = \{[W_i^V, P_i^V], i = 1, 2, ..., N_V\}$ are visible. $N_V$, $W^V$ and $P^V \in \mathbb{R}^3$ are the number of visible objects, their semantic labels, and their 3D object centroids respectively in the current state. Comparing $O$ in the goal state with $O^V$ in the current state allows for determining only the semantics... of unseen objects $ O_{\tilde{V}} = \{ W_{i}^{\tilde{V}}, i = 1, 2, ..., N_{\tilde{V}} \} $, where $ N_{\tilde{V}} $ is the number of unseen objects and $ W_{i}^{\tilde{V}} $ their semantic labels. To plan efficiently and achieve the goal state, the agent must know the positions of all objects in the current state. This involves optimizing the search for unseen objects based on the object-receptacle semantics and simultaneously rearranging visible objects based on their positions in the current and goal state. To this end, we present a modular approach for task planner, as shown in Fig. 2 with: (i) Search network, (ii) Graph-based state representation, (iii) Deep RL network trained with proxy reward. The objective of our task planner is to minimize the number of steps and the agent’s overall traversal by simultaneously sequencing high-level actions to either pick-place misplaced objects or search for unseen objects at predicted receptacles. 2.1 BACKGROUND The agent maps the room in the goal state using an exploration strategy [Sarch et al., 2022] and receives RGB-D images and egomotion information at each step from Ai2Thor [Kolve et al., 2017]. The agent constructs $ M_{2D} $ and $ M_{3D} $ of the environment using the RGB-D input and egomotion. A d-DETR [Zhu et al., 2021] detector is used on the RGB images to obtain 2D bounding boxes and semantic labels for objects and receptacles, and the corresponding 3D centroids are obtained using depth input, camera intrinsic and extrinsic. Finally, the agent has $ O, R, M_{2D}, $ and $ M_{3D} $ from the goal state. In the current state, the agent uses d-DETR detector [Zhu et al., 2021] along with $ M_{3D} $ to obtain $ O_{\tilde{V}} $. The agent uses the Dijkstra path planner on $ M_{2D} $ to navigate and execute high-level actions by assuming perfect motion and manipulation capabilities. 2.2 SEARCH NETWORK We present a novel LLM-based Search Network to reliably predict the receptacles for $ O_{\tilde{V}} $. In case the predicted receptacle is articulated, the agent opens it and looks for the object. The agent uses the predicted receptacle’s position from the goal state to be the probable location for $ O_{\tilde{V}} $ in the current state, since receptacles are static in the room. To this end, we finetune the RoBERTa embeddings to exploit the commonsense knowledge in LLM and learn the semantic relationship between $ O_{\tilde{V}} $ and $ R $. Fine-tuning LLM embeddings is essential because LLMs, being trained on large data corpus, may not necessarily produce human-commonsense compliant predictions for untidy scenes (see the Appendix for more details). Our Search Network (SN) consists of two parts: the Sorting Network (SRTN) and the Scoring Network (SCN). We use RoBERTa-Large model [Liu et al., 2019] to generate pairwise embeddings $ (E_{\tilde{R}}^{V}) $ for $ \{ W_{i}^{\tilde{V}} \}_{i=1,2,...,N_{\tilde{V}}} $ and $ \{ W_{i}^{R} \}_{i=1,2,...,N_{R}} $ in the current state. Therefore, there are $ N_{E} = N_{\tilde{V}} \times N_{R} $ number of embeddings for all the object-room-receptacle (ORR) pairs. Each ORR embedding is classified into one of the 3 classes, based on the probability $ \{ p_{i} \}_{i=1,2,3} $ from the Sorting Network. The ground truth class labels $ \{ Y_{i} \}_{i=1,2,3} $ for each ORR in the dataset (Sec. 3.1) is based on the probability to find an object at that room-receptacle, where $ \{ i = 1 : \text{Most Probable Class}, 2 : \text{Less Probable Class}, 3 : \text{Implausible Class} \} $. SRTN filters out the room-receptacles, where there is a negligible chance of finding the misplaced object. For instance, even in an untidy room, it is nearly impossible to find a cup in the bathtub of a bathroom. This sorting step reduces the scoring network’s computation and minimizes the chances of erroneous scoring of an implausible ORR. We train a fully connected MLP in SRTN using the Cross-Entropy Loss ($ L_{CE} $) as shown in Eq. (1). The Scoring Network estimates probability scores $ \{ \hat{\chi}_{i} \}_{i=1,2,...,N_{SR}} $ for embeddings of higher probability classes, with $ N_{SR} $ representing the total number of such embeddings. SCN provides a probability score metric, to choose the most probable receptacle for $ O_{\tilde{V}} $. For training the fully connected MLP in SCN, we calculate the MSE Loss ($ L_{MSE} $) of probability scores, as in Eq. (2), with respect to the ground truth probability scores $ \{ \chi_{i} \}_{i=1,...,N_{SR}} $. Finally, we get the position $ (P_{i}^{\tilde{V}R})_{i=1,...,N_{\tilde{V}}} $ of the unseen objects as the position of their most probable receptacle. \[ L_{CE} = -\frac{1}{N_{E}} \sum_{i=1}^{N_{E}} \sum_{j=1}^{3} Y_{ij} \log p_{ij} \] \[ L_{MSE} = \frac{1}{N_{SR}} \sum_{i=1}^{N_{SR}} (\hat{\chi}_{i} - \chi_{i})^2 \] To prevent fruitless searches, we implement simple strategies. If the agent cannot find the unseen object at the predicted receptacle, the Search Network identifies the next most probable room-receptacle, and the prior prediction is discarded before re-planning a new sequence. Additionally, if the agent encounters a receptacle on its path that does not contain any unseen objects, it is removed from future searches. The agent updates $ O^V $ whenever it detects an unseen object in its egoview. If the agent locates the unseen object it is searching for before arriving at the predicted receptacle, it updates $ O^V $ and re-plans a new sequence. Refer appendix for more details on the re-planning strategy. ### 2.3 Graph-Based State Representation For our task planning algorithm, we create a spatial graph $(G = \{V, E\})$ representation of the current and the goal state namely $ G_c = \{V_c, E_c\} $ and $ G_g = \{V_g, E_g\} $ respectively. The nodes $ V_c = \{O^V\} $ and $ V_g = \{O\} $. The fully connected edges of the graph contain the path length as edge features, where $ E_c = \{\mathcal{D}(P_i^V, P_j^V)\}_{i \neq j} $ and $ E_g = \{\mathcal{D}(P_i, P_j)\}_{i \neq j} $. The path length $ \mathcal{D}(A_i, A_j)_{i \neq j} $ is the length of the shortest collision free path, computed using Dijkstra, between the 2D projections of $ A_i, A_j \in \mathbb{R}^3 $ on $ M^{2D} $. For unseen objects in the current state, the object nodes and edges in $ G_c $ are augmented with $ P^{\hat{V}R} $ from the search network as $ V_c = V_c \cup \{O^{\hat{V}}, P^{\hat{V}R}\} $ and $ E_c = \{\mathcal{D}(\overline{P}_i, \overline{P}_j)\}_{i \neq j} $, where $ \overline{P} = P^V \cup P^{\hat{V}R} $. This graph representation helps the Deep RL state space to understand the semantic and geometric information of the current and the goal state. We use a novel Graph Representation Network (GRN) with an encoder-decoder to generate meaningful embeddings from $ G_c $ and $ G_g $ for Deep RL state space to incorporate the residual relative path length notion between every pair of current and goal state nodes. GRN consists of two major blocks, the Graph Siamese Encoder Network (GSEN) and the Residual Geodesic Distance Network (RGDN). GSEN uses a Graph Convolution Network (Gao et al., 2020) to encode the graphs $ G_c $ and $ G_g $ and produce the graph embeddings $ Z_c $ and $ Z_g $ respectively. These graph embeddings are concatenated to get the final embeddings $ Z_p = Z_c \cup Z_g $. RGDN acts as a decoder and predicts the residual relative path length $ \tau_p $ between the two graphs. This network is trained in a supervised way as in Eq. (3), using the Graph Dataset (Sec. 3.1), which contains the ground truth relative path length ($ \tau $) between the two graphs. This graph embedding makes the Deep RL state space invariant to a large number of objects and the scene. This compact representation concisely encodes the pairwise distance between the source and target nodes which aids in the reduction of the combinatorial expansion of rearrangement sequencing. \[ \tau_p = \text{GRN}(G_c, G_g) \] \[ L_{GRN} = \|\|\tau - \tau_p\|\|^2 \] ### 2.4 Deep RL Based Planner Our task planner needs to select the objects or the probable receptacles for the unseen objects in an efficient manner, to minimize the overall traversal of the agent to simultaneously search the unseen objects and rearrange the visible ones. Moreover, the planner needs to identify free locations, when selecting objects with swap cases. #### 2.4.1 Parameterized Deep-Q Network In order to achieve the aforementioned goals, we implement a Parameterized Deep-Q Network with hybrid action space, similar to Ghosh et al. (2022). We define a binary Collision vector $(C_N \times 1)$, that signifies the objects with a blocked goal or swap case. The Deep RL state space defined as $ s = Z_p \cup C $. Each action $\{a_i = (k, p_k)\}$ in our sequence of actions $\{a_i\}_{i=1,2,...,K}$ of length $ K $ is made up of a discrete action $ k $, denoting the index of the selected object or the probable receptacle, followed by a continuous parameter $ p_k $ which signifies the location for object placement or receptacle search. We use a Parameter network $(\Phi_P)$ and the Q-network $(\Phi_Q)$ to generate a continuous parameter $ p_k $ and a discrete action $ k $ respectively, similar to Ghosh et al.. According to a Markov Decision Process (MDP), our method receives a reward $ r(s, a) $ at each time step $ t $, for choosing an action $ a $, that advances the agent from the current state $ s $ to the next state $ \bar{s} $. Inspired by the work in Ghosh et al. (2022); Bester et al. (2019), we define the Q-values as a function of the joint continuous action parameter $ p = [p_k]_{k=1,2,...,K} $ instead of updating the Q-values with its corresponding continuous parameter sample $ p_k $. The modified Bellman equation is shown in Eq. (4). This prevents our method from producing degenerate solutions by incorporating the effect of other parameters for updating the Q-values. \[ Q(s, k, p) = \mathbb{E}_{r, \bar{s}}[r + \gamma \max_{k \in K} Q(\bar{s}, \bar{k}, \Phi_P(\bar{s}))\|s, k, p] \] The loss function $L_P(\Phi_P)$ and $L_Q(\Phi_Q)$ for the parameter network($\Phi_P$) and the Q network($\Phi_Q$), is given by Eq. (5) $$L_P(\Phi_P) = - \sum_{k=1}^{K} \sum_{r=1}^{R_B} Q(s, k, \Phi_P(s); \Phi_Q)$$ $$L_Q(\Phi_Q) = E_{(s,k,p,r,\bar{s}) \sim R_B} \left[ \frac{1}{2}(y - Q(s, k, p; \Phi_Q))^2 \right]$$ Here, $y = r + \gamma \max_{k \in K} Q(\bar{s}, \bar{k}, p(\bar{s}; \Phi_P); \Phi_Q)$ is the updated target from Eq. (4), and $R_B$ is the replay buffer. $L_P(\Phi_P)$ indicates how the $p$ must be updated to increase the Q-values. Here $\Phi_Q$ works as critic to $\Phi_P$. For Long Horizon planning, the sparse reward is not sampling efficient for training the Deep RL [Gehring et al. (2021)]. Hence, we use step-wise environmental feedback based on the hierarchical dense reward similar to Ghosh et al.. The detailed reward structure is explained in the Appendix. This reward structure provides per-step feedback, but we need episodic reward-based feedback to improve RL policy generalization [Amodei et al. (2016), Dewey (2014)]. Thus, for every episode ($\Lambda$), we calculate the episodic reward ($R_{ep}$) using the step-wise hierarchical dense reward ($r$) and overall episodic path length ($L$) as in Eq. (6), and save the reward and each step $(s, a, \bar{s})$ of the episode into the replay buffer ($R_B$). As this episodic reward is sparse, we use a proxy reward network to generate per-step dense Markovian reward with an episodic notion. ### 2.4.2 Proxy Reward Network Our proxy reward network is trained on the sampled experience data from the replay buffer, to give our agent a notion of the overall objective of the episode. The random return decomposition (RRD) method used in Ren et al. (2022), trains a proxy reward network by randomly sampling steps from an episode. This training method is not sample efficient because it uniformly samples the steps without considering the reward distribution in the episode. To this end, we propose a novel cluster-biased return reward decomposition (CB-RD) to train our proxy reward network. We cluster the per-step reward for the episode into 3 clusters each of size $T_j$, where $j \in \{1, 2, 3\}$, using the c-means clustering. These clusters represent the reward distribution in an episode. This information helps us to efficiently sample $N_s$ number of steps from the episode. We randomly sample $U_j = \{(s_{ij}, a_{ij}, \bar{s}_{ij})\}_{i=1}^{N_j}$ from each cluster $j$, such that $N_j = N_s \times T_j/N_{ep}$. Using $\{U_j\}_{j=1,2,3}$, we estimate the learned episodic reward ($R_{ep,\theta}$) from the proxy reward network ($r_\theta(s, a, \bar{s})$), where $\theta$ is the learned weight. $$R_{ep} = \frac{N_{ep}}{L} \sum_{i=1}^{N_{ep}} r_i$$ $$R_{ep,\theta} = \sum_{j=1}^{3} p_j \frac{T_j}{N_j} \sum_{i=1}^{N_j} r_\theta(s_{ij}, a_{ij}, \bar{s}_{ij})$$ $$L_{CBRD} = \frac{1}{M} \sum_{i=1}^{M} \left[ (R_{ep,i} - R_{ep,\theta,i})^2 \right]$$ Here, $M$ is the number of episodes sampled, $N_{ep}$ is the number of steps in an episode and $p_j = T_j/N_{ep}$ is the uniform probability of choosing a sample from the episode that belongs to cluster $j$. We simultaneously train our Deep RL using Eq. (5) and proxy reward network using Eq. (8) as shown in Algorithm 1. Fig. 3 shows that CB-RD provides effective feedback to our Deep RL method to achieve a higher average return in a lesser number of steps during training. Hence, CB-RD makes our Deep RL method more sample efficient compared to RRD, hierarchical dense reward and sparse reward. We use an off-policy method with a replay buffer to train our Deep RL method with a diverse set of rearrangement configurations, similar to the work proposed by Kalashnikov et al. (2018). We use the $ \epsilon $-greedy method (Kalashnikov et al., 2018) to strike a balance between exploration and exploitation. We stabilize our Deep RL training using target networks for $ \Phi_Q $ and $ \Phi_p $, and update the weights of target networks using polyak (Lillicrap et al., 2015) averaging similar to Bester et al. (2019); Ghosh et al. (2022). Our ablation study in Appendix, shows that the selection of $ \epsilon $ has a significant impact on the solution. 3 EXPERIMENTS In this section, we describe the datasets, metrics, and detailed results of our proposed method and its modules, in addressing the room-rearrangement problem. 3.1 DATASET Graph Dataset: We generate this dataset to train GRN using Ai2Thor (Kolve et al., 2017), by randomly placing objects for two types of rearrangement scenarios: (i) 40% without goal occupied rearrangement: by placing the objects in free spaces and (ii) goal occupied rearrangement: by placing the object in another object’s target. Search Network Dataset: The AMT dataset in Kant et al. (2022) contains 268 object categories in 12 different rooms and 32 receptacle types. Each object-room-receptacle (ORR) pair is ranked by 10 annotators in 3 classes: correct (positively ranked), misplaced (negatively ranked), and implausible (not ranked). For our problem statement, the misplaced class is of utmost importance. Hence, we rename the classes as (i) misplaced class → most probable class, (ii) correct class → less probable class, and (iii) implausible class remains the same. We find the ground truth score values for each ORR as the mean inverse of the ranks. 3.2 BENCHMARK DATASET FOR TESTING The existing benchmark dataset, RoomR (Weihs et al., 2021), has limitations as it only allows up to 5 objects, no object placement within another receptacle, and no blocked goal or swap cases. Thus, it cannot fully evaluate planning aspects such as the number of steps taken, agent traversal, blocked goal, or swap cases. To address this, we introduce RoPOR, a new benchmark dataset for testing task planners in Ai2Thor. It includes a diverse range of rooms (120) and object-receptacle pairs (118), allowing for a wide variety of rearrangement scenarios with up to 20 objects and random partial observability cases, object placement within receptacles in the current state, and blocked goal and swap cases. Moreover, object placement configurations in RoPOR affect sub-optimal planning policies in terms of agent traversal. The mean room dimensions along x-axis and y-axis are 3.12m and 5.80m, respectively. Refer Appendix for details on the distribution of objects, rooms and receptacles. 3.3 TRAINING The training details of our Search network, Graph-based state Representation Network, Deep RL planner, and proxy reward network are available in the Appendix. 3.4 METRICS Metrics in Weihs et al. (2021) do not highlight the efficacy of a task planner to judge efficient sequencing to reduce the number of steps taken or the agent traversal during rearrangement. For a fair evaluation of our method, and comparison against the existing methods and ablations, we define new metrics: - SNS: Success measured by the inverse Number of Steps uses a binary success rate ($ S $) to evaluate the successful completion of a rearrangement episode along with the number of steps ($ N_T $) taken by \| Number of Objects \| Visible Objects \| Unseen Objects \| Swap Case \| Ours-GT \| Ours \| Weih et al. \| Gadre et al. \| Sarch et al. \| Ghosh et al. \| \|------------------\|-----------------\|----------------\|-----------\|--------\|------\|------------\|-------------\|--------------\|--------------\| \| 5 \| 5 \| 0 \| 0 \| 0 \| 138 \| NC \| 12.57 \| 0.74 \| NC \| \| \| 5 \| 0 \| 0 \| 2 \| 0.76 \| NC \| 23.36 \| 0.53 \| NC \| \| \| 3 \| 2 \| 0 \| 0 \| 0.81 \| 0.61 \| 12.93 \| 0.60 \| 0.48 \| \| \| 3 \| 0 \| 2 \| 0 \| 0.79 \| 0.60 \| 13.39 \| 0.58 \| 0.47 \| \| 10 \| 10 \| 0 \| 0 \| 4 \| 0.70 \| NC \| 24.63 \| 0.52 \| NC \| \| \| 10 \| 0 \| 0 \| 6 \| 0.84 \| 0.69 \| 23.78 \| 0.64 \| 0.53 \| \| \| 6 \| 4 \| 0 \| 0 \| 0.84 \| 0.69 \| 23.78 \| 0.64 \| 0.53 \| \| \| 6 \| 0 \| 4 \| 0 \| 0.84 \| 0.69 \| 23.78 \| 0.64 \| 0.53 \| \| 20 \| 20 \| 0 \| 0 \| 8 \| 0.70 \| NC \| 45.32 \| 0.52 \| NC \| \| \| 12 \| 8 \| 0 \| 0 \| 0.87 \| 0.75 \| 41.29 \| 0.67 \| 0.58 \| \| \| 12 \| 0 \| 8 \| 0 \| 0.87 \| 0.74 \| 42.13 \| 0.66 \| 0.57 \| Table 1: (OOF : Objects outside agent’s field of view initially, which are visible from a different perspective, OPR : Objects placed inside closed receptacles, NC : Not computable). When there are no unseen objects, the ENR is NC. Similarly, when SNS is zero, ENR and ATC are NC. Weih et al., Gadre et al., and Sarch et al. do not handle 20 objects and cannot resolve swap cases without explicit buffer or OPR cases (SNS = 0). Ghosh et al. shows a slight decline in performance as the number of objects increase under complete visibility and swap cases, but fails to account for unseen objects. In comparison, Ours significantly outperforms Weih et al., Gadre et al. and Sarch et al. in terms of SNS, ENR, and ATC for visible objects, unseen objects, and swap cases without explicit buffer. Similarly, ours-GT performs better than Ghosh et al. in terms of SNS and ATC under complete visibility and swap cases without explicit buffer. an agent to rearrange a given number of objects $ N $. $ S $ is 1 if all object positions in the current and goal state are approximately equal. Higher the SNS implies a lower $ N_T $ for a given $ N $, indicating more efficient and successful rearrangement episode. $ (SNS = S \times N/N_T) $ • ENR: Efficiency in Number of Re-plans during object search by taking the ratio of the number of unseen objects initially ($ N_{\bar{V}} $) with respect to the number of attempts to search ($ N_{S\bar{V}} $). A higher ENR shows a lower $ N_{S\bar{V}} $ for a given $ N_{\bar{V}} $ indicating a more efficient search to find unseen objects. $ (ENR = N_{\bar{V}}/N_{S\bar{V}}) $ • Absolute Traversal Cost (ATC)*: The metric shows the overall distance traversed by the agent during the successful completion of a rearrangement episode. In an identical test configuration, a lower ATC indicates a more efficient rearrangement sequencing. ### 3.5 Ablation We ablate our task planner against ground-truth perception, various methods for object search and a dense reward structure. To study the effect of erroneous perception on our task planner, we assume the availability of Ground-Truth object detection labelling and 3D centroid localisation from Ai2Thor (Ours-GT). To understand the importance of our Search Network in planning, we replace it by a (i) Random Search policy (Ours-RS), which predicts probable receptacles for unseen objects with uniform probability and a (ii) Greedy Exploration strategy (Ours-GE) Chaplot et al. (2020) that optimizes for map coverage to discover all the unseen objects. To highlight the generalisation of proxy reward network to the overall objective of the rearrangement episode, we replace it with a hierarchical Dense Reward structure Ghosh et al. (2022) (Ours-DR). Please refer to the appendix to find the results for the ablations, along with the analysis for the choice of hyper-parameters for each of our learning based modules. ### 3.6 Quantitative Results We evaluate our approach along with the existing methods on RoPOR - Benchmark Dataset in Ai2Thor. Tab. 1 indicates that our method is scalable to large number of objects, as demonstrated by the consistent value of SNS despite the increasing number of objects across complete visibility, partial observability, and swap cases without an explicit buffer. The gradual increase in ENR with the increase in number of objects can be attributed to the fact that rearrangement of visible objects and the search for some unseen objects, indirectly aids in finding other unseen objects. Comparing our method against Housekeep Kant et al. (2022) would be unfair because it does not perform a user-specific room-rearrangement with a pre-defined goal state. Instead, we have compared our method to previous works such as Weih et al. (2021), Gadre et al. (2022), Sarch et al. (2022) and Ghosh et al. (2022), all of which have demonstrated results for a user-specific room-rearrangement. For a fair comparison with Weih et al., we have used their best performing model - RN18+ANM, PPO+IL. Since, Ghosh et al., uses groundtruth object positions in the current and the goal state, we compare it with our ablation method Ours-GT. Without erroneous perception, Ours-GT demonstrates efficient planning, by performing significantly better than all the existing methods; Weih et al. (2021), Gadre et al. (2022), Sarch et al. (2022); Ghosh et al. (2022), including Ours, in terms of SNR, ENR and ATC. Under complete visibility, ours significantly outperforms Weihs et al., Gadre et al. and Sarch et al. in terms of SNS and ATC. Similarly, Ours-GT significantly outperforms Ghosh et al. in terms of ATC. The improvement over Weihs et al., Gadre et al. and Sarch et al. shows their heuristic planner is neither scalable nor does it optimize the overall agent traversal or the number of rearrangement steps. In contrast, our method leverages compact graph-based scene geometry capable of addressing large numbers of objects, and robust Deep RL makes our planner efficient in reducing the redundant traversal of the agent. Our method uses path length cost and proxy reward with the episodic notion, which helps to improve the overall traversal of the agent to produce lower ATC. In comparison, Ghosh et al. uses greedy Euclidean distance based reward without having an episodic notion, thus failing to optimize overall traversal. Moreover, Ghosh et al. shows a drop in performance on the RoPOR dataset as compared to their results evaluated on RoomR [Weihs et al. (2021)], due to the variations in the testing scenarios in RoPOR that significantly impact agent traversal for sub-optimal rearrangement policies. Under partial observability, there are two cases - (i) OOF: Objects located outside the field of view initially which are visible from a different perspective and (ii) OPR: Objects placed inside closed receptacles. In the case of OOF, our method substantially outperforms Weihs et al., Gadre et al. and Sarch et al. in terms of SNS, ENR and ATC. All these above methods use greedy sub-optimal planners and employ explicit scene exploration to find objects outside the field of view, incurring huge traversal cost as indicated by their ATC. To gauge the performance of the exploration strategy for object search in terms of ENR, we consider each newly generated location or a set of navigational steps from the exploration policy as a search attempt. Our approach’s significantly higher ENR shows that the Search Network outperforms the exploration policies of [Weihs et al. (2021); Gadre et al. (2022); Sarch et al. (2022)] in terms of the number of attempts to find unseen objects. Ghosh et al. does not address any case of partial observability. While Weihs et al., Gadre et al. and Sarch et al. do not solve the case of OPR, which involves object placement inside receptacles (SNS = 0). However, our approach performs equally well in both cases of partial observability due to our search network’s ability to comprehend a commonsense based semantic relationship between an object and any type of receptacle - rigid or articulated. Swap cases without an explicit buffer are not handled by Weihs et al., Gadre et al. and Sarch et al., which is evident from SNS = 0. Ours, Ours-GT and Ghosh et al. can effectively resolve an increasing number of swap cases without an explicit buffer using the hybrid action space [Ghosh et al. (2022)] in the Deep RL network. However, Ours-GT performs better than Ghosh et al. in terms of ATC due to a novel collision resolution reward that optimizes the agent’s traversal. To ground the values of our RoPOR dataset, we show the results for Ours, the ablation methods and the SOTA in the test set of RoomR in the Appendix. Moreover, additional results for individual methods in our pipeline can be found in the Appendix. 3.7 QUALITATIVE RESULTS To show the results of our method in room-rearrangement, we have created videos in a number of test scenarios to highlight the robustness of our method. We also test our method in a new environment - Habitat, as demonstrated in our supplementary video. This transfer does not require any additional training for our Search Network, Graph-based State Representation or Deep RL planner. This shows the capability of our method for seamless sim-to-sim transfer, further emphasizing its suitability for real-world deployment. Please refer the supplementary video. 4 LIMITATIONS Our approach is not capable of identifying unseen objects that are occluded due to clutter on receptacles (for e.g. a spoon may become occluded, if bread, box, lettuce etc. is placed before it). Our method also assumes the availability of perfect motion planning and manipulation capabilities. 5 CONCLUSION This paper presents an innovative task planner designed for organizing rooms under conditions of partial observability. Our approach minimizes agent traversal and step count during both object search and rearrangement by leveraging a Search Network followed by a Deep RL-based planner. By utilizing a graph-based state representation and episodic proxy reward, our method exhibits versatility and applicability across a range of scenarios. The RoPOR benchmark dataset facilitates additional research in the realm of Embodied AI-based rearrangement. Future endeavors will concentrate on deploying our approach in real-world settings. 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4e0ItHjNo9	Q2 (from W1): Why should it be up to CF to define on which attributes to apply the method? Overall, I don’t understand where this shortcoming comes from as it is not a view shared by the field (to the best of my knowledge) nor is there work cited by the authors referring to such view.	Rethinking Counterfactual Fairness: On Which Individuals to Enforce, and How? Anonymous authors Paper under double-blind review Abstract Fairness in human and algorithmic decision-making is crucial in areas such as criminal justice, education, and social welfare. Recently, counterfactual fairness has drawn increasing research interest, suggesting that decision-making for individuals should remain the same when intervening with different values on the protected attributes. Nevertheless, the question of "which attributes and individuals should be protected" is rarely discussed in the existing counterfactual fairness literature. For example, when considering leg disability as a protected attribute, the algorithms should not treat individuals with leg disabilities differently in college admissions, but one may naturally take into this factor for the purpose of selecting runner athletes. In other words, when and how to enforce fairness is expected to depend on the causal relation between the protected attribute and the outcome of interest. Formally, this paper proposes principal counterfactual fairness using the concept of principal stratification from the causal inference literature, focusing on whether an algorithm is counterfactually fair for individuals whose protected attribute has no individual causal effect on the outcome of interest. To examine whether an algorithm satisfies principal counterfactual fairness, we derive the statistical bounds, and propose a post-processing approach to achieving principal counterfactual fairness with minimal individual decision changes. Experiments are conducted using synthetic and real-world datasets to verify the effectiveness of our methods. 1 Introduction Addressing the fairness of automated algorithms is critical to making safe decisions in areas such as criminal justice (Brennan et al., 2009; Dieterich et al., 2016), education (Reardon and Owens, 2014), and social welfare (Chouldechova et al., 2018). To achieve fair machine learning, many association-based fairness notions have been proposed to constrain the statistical independence between protected attributes and decisions, e.g., statistical parity (Dwork et al., 2012), equalized odds (Hardt et al., 2016), and predictive parity (Chouldechova, 2017). In addition, the algorithmic fairness can also be approached from a causal perspective (Kusner et al., 2017; Zhang et al., 2017a,b, 2018a,b; Zhang and Bareinboim, 2018; Nabi and Shpitser, 2018; Wu et al., 2019a,b; Chiappa, 2019; Imai and Jiang, 2020; Mishler et al., 2021; Zuo et al., 2022). Among them, counterfactual fairness (Kusner et al., 2017) has garnered considerable attention recently. This criterion demands that any alterations made to the values of protected attributes do not result in changes to individual decision-making. Nevertheless, as Chouldechova and Roth (2020) pointed out, the question of "which attributes and individuals should be protected" is rarely discussed in the existing counterfactual fairness literature. Should counterfactual fairness hold for all sensitive attributes on all individuals? For example, when considering leg disability as a protected attribute, it is reasonable to require that the algorithms should not treat individuals with disabilities differently in college admissions, but should the algorithms also be required to make the same decisions for individuals with disabilities when selecting runner athletes? In such cases, it is clear that it is not appropriate to select individuals with disabilities as running athletes. But what if the sensitive attribute is gender or race instead? How to reflect the differences between disability and gender as sensitive attributes? To tackle the above issues, we summarize relevant studies in Table 1, which can be broadly divided into two branches. On one hand, instead of requiring fairness to hold on all individuals as in demographic parity (Darlington, 1971), equalized odds (Hardt et al., 2016) constrains the examination... Table 1: A summary of the proposed principal counterfactual fairness and related concepts. \| Fairness Definition \| Formulation (A: protected attribute; D: decision; Y: outcome; X: covariate) \| \|--------------------------------------\|---------------------------------------------------------------------------\| \| Demographic Parity [Darlington, 1971] \| $ A \perp\!\!\!\perp D $ \| \| Equalized Odds [Hardt et al., 2016] \| $ A \perp\!\!\!\perp D \mid Y $ \| \| Equality of Opportunity [Hardt et al., 2016] \| $ A \perp\!\!\!\perp D \mid Y = 1 $ \| \| Counterfactual Equalized Odds [Mishler et al., 2021] \| $ A \perp\!\!\!\perp D \mid Y(D = 0) = 1 $ \| \| Principal Fairness [Imai and Jiang, 2020] \| $ A \perp\!\!\!\perp D \mid (Y(D = 0), Y(D = 1)) $ \| \| Counterfactual Parity [Mitchell et al., 2021] \| $ P(D(0) = 1) = P(D(1) = 1) $ \| \| Conditional Counterfactual Fairness [Mitchell et al., 2021] \| $ P(D(0) = 1) = P(D(1) = 1) \mid X $ \| \| Principal Counterfactual Parity (ours) \| $ P(D(0) = 1) = P(D(1) = 1) \mid Y(A = 0) = Y(A = 1) $ \| \| Principal Conditional Counterfactual Fairness (ours) \| $ P(D(0) = 1) = P(D(1) = 1) \mid Y(A = 0) = Y(A = 1), X $ \| \| Principal Counterfactual Equalized Odds (ours) \| $ P(D(0) = 1) = P(D(1) = 1) \mid Y(A = 0) = Y(A = 1), y, X $ \| \| Counterfactual Fairness [Kusner et al., 2017] \| $ D_i(A_i = 0) = D_i(A_i = 1) $ \| \| Path-Specific Counterfactual Fairness [Chiappa, 2019] \| $ D_i(A_i = 0) = D_i(A_i = 1, M_i(A_i) = M_i(0)) $ \| \| Principal Counterfactual Fairness (ours) \| $ D_i(A_i = 0) = D_i(A_i = 1) $ holds for $ Y_i(A_i = 0) = Y_i(A_i = 1) $ \| of demographic parity to subgroups with the same observed outcome. By considering the effect of decision-making on the observed outcomes, counterfactual equalized odds [Mishler et al., 2021] generalizes the above concepts to make fair decisions on individuals with the same value on the potential outcome under control. Principle fairness [Imai and Jiang, 2020] further uses the concept of principal stratification from the causal inference literature to consider the joint potential outcomes of the decision on outcome. However, despite considering specific subgroups defined from a counterfactual view, these fairness notions still use the statistical independence of sensitive attributes and decision-making on that subgroup, which is not sufficient to guarantee individual counterfactual fairness [Kusner et al., 2017; Mitchell et al., 2021]. On the other hand, path-specific counterfactual fairness [Chiappa, 2019] promotes the notion of counterfactual fairness to restricted on unfair paths, rather than considering the total effect of sensitive attributes on decision-making. Despite partially answering the question of "which attributes should be protected", similar to counterfactual fairness, path-specific counterfactual fairness requires fairness on all individuals. This motivates us to rethink counterfactual fairness to better answer that "which attributes and individuals should be protected". In this paper, instead of forcing decisions to remain the same for all individuals when the protected attribute changes as in counterfactual fairness, we propose principal counterfactual fairness using the concept of principal stratification from the causal inference literature [Frangakis and Rubin, 2002; Pearl, 2011], focusing on whether the counterfactual fairness holds for individuals whose protected attribute has no individual causal effect on the outcome of interest. For the aforementioned example, since leg disability (as a sensitive attribute) may affect athlete performance (as an outcome), we only require that decisions remain similar for those individuals with disabilities that do not affect athlete performance. In contrast, since disability and gender (as sensitive attributes) do not have a causal effect on the exam pass (as an outcome), we might expect decision-making for all individuals to satisfy counterfactual fairness. In summary, the proposed principal counterfactual fairness further considers the effect of protected attributes on outcomes of interest from a counterfactual perspective, and we show the principal counterfactual fairness would degenerate to standard counterfactual fairness when the protected attributes have no individual causal effect on outcomes for all individuals. To examine whether an algorithm satisfies principal counterfactual fairness, we first derive the necessary conditions for an algorithm to satisfy principal counterfactual fairness based on statistical bounds. Then we propose an optimization-based evaluation method to test whether an algorithm satisfies principal counterfactual fairness. Specifically, the algorithm does not satisfy the principal counterfactual fairness if the feasible region under particular constraints is the empty set, or if there exists a principal stratum with the optimized maximum probability value less than zero. We further propose a principled post-processing approach to achieve principal counterfactual fairness with minimal individual decision changes, and theoretically prove the optimality of the post-processing approach using doubly robust estimation. We conduct extensive experiments on synthetic and real-world datasets to verify the effectiveness of the proposed algorithm. The main contributions of this paper are: 1 Due to partial identifiability, it is difficult to find necessary and sufficient conditions for principal counterfactual fairness, similar problems also exist in the counterfactual fairness literature [Kusner et al., 2017]. • We propose a novel fairness notion using the concept of principal stratification, called principal counterfactual fairness, which requires the counterfactual fairness to hold only when the protected attribute has no individual causal effect on the outcome of interest. • We derive the necessary conditions for an algorithm to satisfy principal counterfactual fairness based on statistical bounds, and propose an optimization-based evaluation method to test whether an algorithm satisfies principal counterfactual fairness. • We further propose a principled post-processing approach to achieving principal counterfactual fairness with minimal individual decision changes, and theoretically prove the optimality of the post-processing approach using doubly robust estimation. • We conduct experiments on both synthetic and real-world datasets to verify the effectiveness of the proposed optimization-based evaluation and post-processing approach. 2 PRELIMINARIES We first formalize the issue of fairness in decision making, as well as summarize the related statistical and counterfactual fairness notions that have been widely studied. Suppose a simple random sample of $ n $ units from a super population $ P $, for each unit $ i $, the covariate (e.g., age or income) and the binary protected attribute (e.g., gender or disability) are denoted as $ X_i \in \mathcal{X} $ and $ A_i \in \{0, 1\} $, respectively. Let $ Y_i \in \mathcal{Y} = \{0, 1\} $ be the binary outcome variable of interest and $ D_i \in \{0, 1\} $ be the binary decision variable. For the simplicity of exposition, we assume the protected attribute, decision variable, and outcome variable are all binary, and covariates are discrete, but these variables can all be extended to other variable types in our work. To study the counterfactual fairness problem, we adopt the potential outcome framework (Rubin [1974], Neyman [1990]). Specifically, let $ Y_i(0) $ and $ Y_i(1) $ be the outcome of the unit $ i $ had this unit have the protected attribute $ A_i = 0 $ and $ A_i = 1 $, respectively. Since each unit can only have one particular value of protected attribute, we always observe the corresponding outcome be either $ Y_i(0) $ or $ Y_i(1) $, but not both. This is also known as the fundamental problem of causal inference (Holland [1986], Morgan and Winship [2015]). Formally, the observed outcome for unit $ i $ is $ Y_i = (1 - A_i)Y_i(0) + A_iY_i(1) $. In other words, the observed outcome is the potential outcome corresponding to the protected attribute value, which is also known as the consistency assumption in the causal inference literature (Hernán and Robins [2020]). Based on the observed protected attributes, covariates, and outcomes of interest, i.e., $ \{(A_i, X_i, Y_i)\}_{i=1}^N $, a machine learning algorithm $ D(\cdot) $ for decision-making is obtained. Specifically, let $ D_i(0) $ and $ D_i(1) $ be the potential algorithmic decisions for the unit $ i $ had this unit have the protected attribute $ A_i = 0 $ and $ A_i = 1 $, respectively. By the consistency assumption again, the algorithmic decision for individual $ i $ in the factual world would be $ D_i $. In order for algorithms to make fair decisions, as shown in Table 1, many statistical fairness notions have been proposed, such as demographic parity (Darlington [1977]), i.e., $ A \perp\!\!\!\perp D $, equalized odds (Hardt et al. [2016]), i.e., $ A \perp\!\!\!\perp D \mid Y $, and equality of opportunity (Hardt et al. [2016]), i.e., $ A \perp\!\!\!\perp D \mid Y = 1 $. By noting the causal effect of decision $ D $ on the observed outcomes $ Y $, counterfactual equalized odds generalizes the above concepts to make fair decisions on individuals with counterfactual advantaged outcomes (Mishler et al. [2021]), i.e., $ A \perp\!\!\!\perp D \mid Y(D = 0) = 1 $. Principle fairness further uses the concept of principal stratification from the causal inference literature to consider the joint potential outcome of decisions on outcomes (Imai and Jiang [2020]), i.e., $ A \perp\!\!\!\perp D \mid (Y(D = 0), Y(D = 1)) $. Nevertheless, despite considering a specific counterfactual stratum, these fairness notions still use the statistical independence of sensitive attributes and decisions on that stratum, which is not sufficient to guarantee causal effect-based fairness notions (Kusner et al. [2017], Mitchell et al. [2021]). Instead of considering the statistical (conditional) independence between the protected attribute $ A $ and decision $ D $, causality-based fairness considers the causal effect of the protected attribute $ A $ on decision $ D $. Among them, counterfactual parity in Definition 1 requires that there is no average causal effect of the protected attribute $ A $ on decision $ D $ over the population (Mitchell et al. [2021]). Definition 1 (Counterfactual parity (Mitchell et al. [2021])). An algorithm $ D $ for decision-making satisfies counterfactual parity, if under any value $ a $ and $ a' $ attainable by $ A $, \[ P(D(a) = 1) = P(D(a') = 1). \] Table 2: The principal counterfactual fairness considers units in the principal fairness strata (in red), whereas counterfactual fairness considers all units including in the auxiliary fairness strata (in blue). \| Observed data \| $(A = 0, Y = 0)$ \| $(A = 0, Y = 1)$ \| $(A = 1, Y = 0)$ \| $(A = 1, Y = 1)$ \| \|---------------\|---------------------\|---------------------\|---------------------\|---------------------\| \| Principal fairness \| $(Y(0) = 0, Y(1) = 0)$ \| $(Y(0) = 1, Y(1) = 1)$ \| $(Y(0) = 0, Y(1) = 0)$ \| $(Y(0) = 1, Y(1) = 1)$ \| \| Auxiliary fairness \| $(Y(0) = 0, Y(1) = 1)$ \| $(Y(0) = 1, Y(1) = 0)$ \| $(Y(0) = 1, Y(1) = 0)$ \| $(Y(0) = 0, Y(1) = 1)$ \| By incorporating the covariate $X$, conditional counterfactual fairness in Definition 2 requires that there is no conditional average causal effect of the protected attribute $A$ on the decision $D$ over subpopulations under context $X = x$ for all $x \in X$. Definition 2 (Conditional counterfactual fairness [Mitchell et al., 2021]). An algorithm $D$ for decision-making is conditional counterfactually fair, if under any context $X = x$ and any value $a$ and $a'$ attainable by $A$, \[ \mathbb{P}(D(a) = 1 \mid X = x) = \mathbb{P}(D(a') = 1 \mid X = x). \] Different from counterfactual parity in Definition 1, which constrains on the total population and conditional counterfactual fairness in Definition 2, which constrains on the subpopulations determined by the covariates, individual counterfactual fairness in Definition 3 further requires that there is no individual causal effect of the protected attribute $A$ on the decision $D$ over all the individuals. Definition 3 (Counterfactual fairness [Kusner et al., 2017]). An algorithm $D$ for decision-making is individual counterfactually fair if under any context $X = x$ and any value $a$ and $a'$ attainable by $A$, \[ \mathbb{P}(D_i(a) = D_i(a')) = 1. \] Counterfactual fairness states that $A$ should not be a cause of decision $D$ in any individual instance, with many follow-up studies [Zhang and Bareinboim, 2018; Chiappa, 2019]. As in Table 1, one representative variant is path-specific counterfactual fairness, which requires counterfactual fairness to hold only on unfair paths [Chiappa, 2019]. Despite partially answering the question of "which attributes should be protected", similar to counterfactual fairness, path-specific counterfactual fairness also requires fairness on all individuals. This motivates us to rethink these counterfactual fairness notions to better answer the question of "which and how to decide the attributes and individuals that should be protected". ### 3 Principal Counterfactual Fairness In this section, we first propose the notions of principal counterfactual fairness using the concept of principal stratification from the causal inference literature. Ordered from weakest to strongest, we propose principal counterfactual parity in Definition 4, principal conditional counterfactual fairness in Definition 5, principal counterfactual equalized odds in Definition 6, and principal conditional counterfactual fairness in Definition 7, respectively. We also derive the necessary conditions for an algorithmic decision to satisfy principal counterfactual fairness based on statistical bounds. Specifically, the principal strata are defined as the joint potential outcome values [Frangakis and Rubin, 2002], i.e., $(Y_i(a), Y_i(a'))$, where $a$ and $a'$ are the sensitive attribute values attainable by $A$, and each principal stratum represents how an individual would be affected by the protected attribute on the outcome of interest. In the proposed principal counterfactual fairness, we focus on whether the counterfactual fairness notions hold on individuals whose protected attribute has no individual causal effect on the outcome of interest, i.e., $Y_i(a) = Y_i(a')$ for all $a$ and $a'$ attainable by $A$. Compared with the previous counterfactual fairness notions, Table 2 shows the difference: the proposed principal counterfactual fairness notions focus only on those individuals in "principal fairness" stratum (in red), while previous counterfactual fairness notions focus on individuals in both "principal fairness" stratum (in red) and "auxiliary fairness" stratum (in blue). Unlike the observed outcome $Y_i$, however, the potential outcomes, and hence principal strata, are not affected by the --- 2 Counterfactual fairness in [Kusner et al., 2017] refers to individual counterfactual fairness with the definition that $\mathbb{P}(D_{A \leftarrow a}(U) = y \mid X = x, A = a) = \mathbb{P}(D_{A \leftarrow a'}(U) = y \mid X = x, A = a)$. This is equivalent to $\mathbb{P}(D_i(a) = D_i(a')) = 1$ using potential outcomes formulation [Mitchell et al., 2021]. sensitive attribute value. Moreover, since we only observe one potential outcome for any individual, principal strata are not directly observable and be distinguished, as shown in Table 2. In the disabled athlete selection example, the principal strata are defined by the athlete performance $ Y_i(A_i) $ under each of the two scenarios—disabled $ A_i = 1 $ or not disabled $ A_i = 0 $. Then it is fair to let the algorithmic decision $ D $ be unaffected by whether those individuals are disabled $ A = 1 $ or not $ A = 0 $, because the disability of those individuals has no individual causal effect on the athlete’s performance, i.e., $ Y_i(0) = Y_i(1) $. The following Definition 4 formally states the principal counterfactual parity, which requires counterfactual parity to hold on that particular stratum. Definition 4 (Principal counterfactual parity). An algorithm $ D $ for decision-making satisfies principal counterfactual parity, if under any value $ a $ and $ a' $ attainable by $ A $, \[ \mathbb{P}(D(a) = 1 \mid Y(a) = Y(a')) = \mathbb{P}(D(a') = 1 \mid Y(a) = Y(a')). \] By conditional on covariate $ X $, Definition 5 states the principal conditional counterfactual fairness. Definition 5 (Principal conditional counterfactual fairness). An algorithm $ D $ for decision-making is principal conditional counterfactually fair, if under any context $ X = x $ and any value $ a $ and $ a' $ attainable by $ A $, \[ \mathbb{P}(D(a) = 1 \mid Y(a) = Y(a'), X = x) = \mathbb{P}(D(a') = 1 \mid Y(a) = Y(a'), X = x). \] We now describe the potential limitations of using principal conditional counterfactual fairness in Definition 5. Recall the disabled athlete selection example and let $ Y $ denote athlete performance. Although the protected attribute has no individual causal effect on the outcome for both $ Y(a) = Y(a') = 0 $ and $ Y(a) = Y(a') = 1 $ individuals. However, since individuals at $ Y(a) = Y(a') = 1 $ have better athlete performance compared with individuals at $ Y(a) = Y(a') = 0 $, it is natural to allow high probability of being selected as an athlete for individuals at the stratum $ Y(a) = Y(a') = 1 $. That is, $ \mathbb{P}(D(a) = 1 \mid Y(a) = Y(a') = 1) > \mathbb{P}(D(a) = 1 \mid Y(a) = Y(a') = 0) $. This motivates us to further divide stratum $ (Y(0) = Y(1)) $ into multiple strata $ (Y(0) = Y(1) = y) $ for all $ y \in \mathcal{Y} $, and propose the corresponding principal counterfactual equalized odds in Definition 6. Definition 6 (Principal counterfactual equalized odds). An algorithm $ D $ for decision-making satisfies principal counterfactual equalized odds, if under any context $ X = x $ and any value $ a $ and $ a' $ attainable by $ A $, for all $ y \in \mathcal{Y} $, \[ \mathbb{P}(D(a) = 1 \mid Y(a) = Y(a') = y, X = x) = \mathbb{P}(D(a') = 1 \mid Y(a) = Y(a') = y, X = x). \] For the case of binary variables, it is equivalent to $ \tau_0(x) = \tau_1(x) = 0 $, where \[ \tau_y(x) = \mathbb{P}(D(1) = 1 \mid Y(0) = Y(1) = y, X = x) - \mathbb{P}(D(0) = 1 \mid Y(0) = Y(1) = y, X = x), \] for $ y = 0, 1 $. Denote $ p_{ay}(x) = \mathbb{P}(Y = y \mid A = a, X = x) $ and $ q_{ay}(x) = \mathbb{P}(D = 1 \mid A = a, Y = y, X = x) $, which can be calculated from the observed data. Under the ignorability assumption, the following lemma provides the sharp bounds on $ \tau_0(x) $ and $ \tau_1(x) $. Assumption 1 (Ignorability). $ A \perp\!\!\!\perp (Y(1), Y(0), D(1), D(0)) \mid X $. Lemma 1. Under Assumption 1, the sharp upper and lower bounds on $ \tau_0(x) $ are \[ \text{Lower}(\tau_0(x)) = \max \left\{ 0, 1 - \frac{(1 - q_{10}(x))p_{10}(x)}{p_{10}(x) - p_{01}(x)} \right\} - \min \left\{ 1, \frac{q_{00}(x)p_{00}(x)}{p_{10}(x) - p_{01}(x)} \right\}, \] \[ \text{Upper}(\tau_0(x)) = \min \left\{ 1, \frac{q_{10}(x)p_{10}(x)}{p_{10}(x) - p_{01}(x)} \right\} + \min \left\{ 0, \frac{(1 - q_{00}(x))p_{00}(x)}{p_{10}(x) - p_{01}(x)} - 1 \right\}. \] The sharp upper and lower bounds on $ \tau_1(x) $ are \[ \text{Lower}(\tau_1(x)) = \max \left\{ 0, 1 - \frac{(1 - q_{11}(x))p_{11}(x)}{p_{01}(x) - p_{10}(x)} \right\} - \min \left\{ 1, \frac{q_{01}(x)p_{01}(x)}{p_{01}(x) - p_{10}(x)} \right\}, \] \[ \text{Upper}(\tau_1(x)) = \min \left\{ 1, \frac{q_{11}(x)p_{11}(x)}{p_{01}(x) - p_{10}(x)} \right\} + \min \left\{ 0, \frac{(1 - q_{01}(x))p_{01}(x)}{p_{01}(x) - p_{10}(x)} - 1 \right\}. \] The following Theorem 1 gives the necessary inequality conditions to determine whether the algorithm satisfies principal counterfactual equalized odds based on statistical bounds in Lemma 1. Theorem 1. Under Assumption [7], the principle counterfactual equalized odds in Definition 6 under stratum $ Y(0) = Y(1) = 0 $ is violated if either of the following two inequalities holds: \[ q_{00}(x)p_{00}(x) + (1 - q_{10}(x))p_{10}(x) < p_{10}(x) - p_{01}(x), \] \[ q_{10}(x)p_{10}(x) + (1 - q_{00}(x))p_{00}(x) < p_{10}(x) - p_{01}(x). \] Similarly, the principle counterfactual equalized odds in Definition 6 under stratum $ Y(0) = Y(1) = 1 $ is violated if either of the following two inequalities holds: \[ q_{01}(x)p_{01}(x) + (1 - q_{11}(x))p_{11}(x) < p_{01}(x) - p_{10}(x), \] \[ q_{11}(x)p_{11}(x) + (1 - q_{01}(x))p_{01}(x) < p_{01}(x) - p_{10}(x). \] Further, instead focusing on a specific subgroup, as an extension of individual counterfactual fairness in [Kusner et al., 2017], we define principal counterfactual fairness to achieve strict individual fair. Definition 7 (Principal counterfactual fairness). An algorithm $ D $ for decision-making is principal counterfactually fair with respect to outcome $ Y $, if under any value $ a $ and $ a' $ attainable by $ A $, \[ \mathbb{P}(D_i(a) = D_i(a') \mid Y_i(a) = Y_i(a')) = 1. \] We finally point out that the principal counterfactual fairness would degenerate to counterfactual fairness when $ Y_i(a) = Y_i(a') $ holds on all individuals for any value $ a $ and $ a' $ attainable by $ A $. Corollary 1 (Relation to counterfactual fairness). The principal counterfactual fairness is equivalent to counterfactual fairness in [Kusner et al., 2017], when the protected attributes have no individual causal effect on outcomes for all individuals. 4 IMPLEMENTING PRINCIPAL COUNTERFACTUAL FAIRNESS 4.1 OPTIMIZATION-BASED EVALUATION We started with an optimization-based evaluation method for principal counterfactual fairness in Definition 7, while other principal counterfactual fairness notions can be evaluated by similar arguments. Denote $ w_{d_0,d_1,y_0,y_1}(x) = \mathbb{P}(D(0) = d_0, D(1) = d_1, Y(0) = y_0, Y(1) = y_1 \mid X = x) $, then principal counterfactual fairness is equivalent to $ w_{0,0,0,0}(x) = w_{1,0,0,0}(x) = w_{0,1,1,1}(x) = w_{1,0,1,1}(x) = 0 $. The proposed optimization constraints for evaluating whether the algorithmic decisions satisfy the principal counterfactual fairness are given as \[ w_{0,1,0,0}(x) = w_{1,0,0,0}(x) = w_{0,1,1,1}(x) = w_{1,0,1,1}(x) = 0, \] \[ w_{d_0,d_1,y_0,y_1}(x) \geq 0 \quad \text{for all} \quad d_0, d_1, y_0, y_1 \in \{0,1\}, \] \[ \sum_{a,b} w_{d_0,a,y_0,b}(x) = \mathbb{P}(D(0) = d_0, Y(0) = y_0 \mid X = x) \quad \text{for all} \quad d_0, y_0 \in \{0,1\}, \] \[ \sum_{a,b} w_{a,d_1,b,y_1}(x) = \mathbb{P}(D(1) = d_1, Y(1) = y_1 \mid X = x) \quad \text{for all} \quad d_1, y_1 \in \{0,1\}, \] for all $ x \in \mathcal{X} $, where the first equation is the equivalent condition of principal counterfactual fairness, the second equation comes from the positivity of the probabilities, and the last two equations come from the definition of $ w(x) $. Notably, under Assumption [1], the terms on the right-hand side of the last two equations can be identified and estimated using the observed data (see Section 4.2). Therefore, with these constraints on $ w(x) $ imposed by the above equations set, we can determine that the algorithmic decision does not satisfy principal counterfactual fairness, if there exists $ x \in \mathcal{X} $ such that the feasible domain of $ w(x) $ satisfying these constraints is the empty set. In practice, we can also take one of $ w_{0,1,0,0}(x), w_{1,0,0,0}(x), w_{0,1,1,1}(x), $ and $ w_{1,0,1,1}(x) $, denoted as $ \tilde{w}(x) $, as the objective function, and let the remaining three terms equal to zero as the optimization constraints, then solve the minimum and maximum values of $ \tilde{w}(x) $, and obtain its value interval by solving this optimization problem. The algorithm should be considered as a violation of principal counterfactual fairness, when the minimum value of $ \tilde{w}(x) $ is greater than 0 or the maximum value of $ \tilde{w}(x) $ is less than 0. 4.2 ESTIMATION Let $ \mu^{d,y}_a(x) = \mathbb{P}(D = d, Y = y \mid A = a, X = x) $ and $ \pi_a(x) = \mathbb{P}(A = a \mid X = x) $, with $ \hat{\mu}^{d,y}_a(x) $ and $ \hat{\pi}_a(x) $ be the estimated conditional-mean and propensity, respectively. To estimate the right-hand side of the last two equations in the above optimization problem, without loss of generality, one needs to estimate $ \mathbb{P}(D(a) = d, Y(a) = y \mid X = x) $ for $ a, d, y \in \{0, 1\} $. Let $ \hat{\mathbb{P}} $ and $ \hat{\mathbb{E}} $ be the estimated probability and expectation that can be obtained via regression or subclassification. Then the outcome regression (OR) estimator is given as $ \hat{\mu}_{d,y}^{OR}(a, x) = \hat{\mu}_{d,y}^{OR}(x) $. The inverse propensity scoring (IPS) estimator is given as $ \hat{\mu}_{d,y}^{IPS}(a, x) = \hat{\mu}_{d,y}^{IPS}(x) $, and the doubly robust (DR) estimator is given as $ \hat{\mu}_{d,y}^{DR}(a, x) = \hat{\mu}_{d,y}^{DR}(x) $. Theorem 2. Suppose that $ \|\|\hat{\pi}_a(x) - \pi_a(x)\|\|_2 \cdot \|\|\hat{\mu}_{d,y}^{IPS}(x) - \mu_{d,y}^{IPS}(x)\|\|_2 = o_P(n^{-1/2}) $ for all $ x \in \mathcal{X} $ and $ a $ attainable by $ A $, then $ \hat{\mu}_{d,y}^{DR}(a, x) = \hat{\mu}_{d,y}^{DR}(x) $ is asymptotically normal \[ \sqrt{n} \left( \hat{\mu}_{d,y}^{DR}(a, x) - \mathbb{P}(D(a) = d, Y(a) = y \mid X = x) \right) \rightarrow N(0, \sigma_1(x)^2), \] where $ \sigma_1(x)^2 = \text{Var}[\hat{\mu}_{d,y}^{IPS}(x) + \mathbb{I}(A = a) \cdot (\mathbb{I}(D = d, Y = y) - \hat{\mu}_{d,y}^{IPS}(x)) / \hat{\pi}_a(X) \mid X = x] $. ### 4.3 Post-processing approach In applications, we first use the DR (or OR, IPS) estimators in Section 4.2 and plug them into the last two constraints of the optimization problem in Section 4.1. As discussed in Section 4.1, if the feasible domain is the empty set, or if there exists $ x \in \mathcal{X} $ such that the interval of values of $ \hat{w}(x) $ does not contain 0, then the decision should be considered to violate principal counterfactual fairness. Inspired by (Mishler et al., 2021), we further propose a post-processing method to adjust previously unfair decisions by minimal individual decision changes so that it no longer violates optimization-based fairness evaluation in Section 4.1. The advantage of the post-processing approach is the applicability to any already-in-used models but evaluated unfair (Lohia et al., 2019). Specifically, consider a set of non-negative parameters $ \epsilon(x) = \{\epsilon_{00}(x), \epsilon_{01}(x), \epsilon_{10}(x), \epsilon_{11}(x)\} $ for all $ x \in \mathcal{X} $, where each parameter $ \epsilon_{ad}(x) $ denotes the probability of forcing the decision $ D = d $ on the individuals with $ A = a $ and $ X = x $. With loss of generality, $ \epsilon_{ad}(x) + \epsilon_{a(1-d)}(x) \leq 1 $, and let $ D' $ be the final decision after the post-processing, then we have \[ \mathbb{P}_{\epsilon}(D'(a) = d, Y(a) = y \mid X = x) = \mathbb{P}_{\epsilon}(D' = d, Y = y \mid A = a, X = x) = \epsilon_{ad}(x) \cdot \mathbb{P}(Y(a) = y \mid A = a, X = x) + (1 - \epsilon_{ad}(x)) \cdot \mathbb{P}(D(a) = d, Y(a) = y \mid X = x) = \epsilon_{ad}(x) \cdot \mathbb{P}(D(a) = 1 - d, Y(a) = y \mid X = x) + (1 - \epsilon_{ad}(x)) \cdot \mathbb{P}(D(a) = d, Y(a) = y \mid X = x) \] In order to obtain a fair decision $ D' $ while minimally changing the original decision $ D $, we obtain the estimated $ \hat{\epsilon}(x) $ for all $ x \in \mathcal{X} $ by solving the following optimization problem. \[ \hat{\epsilon}^* = \arg \min_{\epsilon} \frac{1}{n} \sum_{i=1}^{n} \epsilon_{A_i,0}(X_i) + \epsilon_{A_i,1}(X_i), \] s.t. $ w_{0,1,0,0}(x) = w_{0,1,0,0}(x) = w_{0,1,1,1}(x) = w_{1,0,1,1}(x) = 0, $ $ w_{d_0,d_1,y_0,y_1}(x) \geq 0 $ for all $ d_0, d_1, y_0, y_1 \in \{0, 1\} $, $ \epsilon_{ad}(x) \epsilon_{a(1-d)}(x) \geq 0 $ and $ \epsilon_{ad}(x) + \epsilon_{a(1-d)}(x) \leq 1 $ for all $ a, d \in \{0, 1\} $, \[ \sum_{a,b} w_{d_0,a,y_0,b}(x) = \mathbb{P}_{\epsilon}(D'(0) = d_0, Y(0) = y_0 \mid X = x) \quad \text{for all } d_0, y_0 \in \{0, 1\}, \] \[ \sum_{a,b} w_{a,d_1,b,y_1}(x) = \mathbb{P}_{\epsilon}(D'(1) = d_1, Y(1) = y_1 \mid X = x) \quad \text{for all } d_1, y_1 \in \{0, 1\}, \] for all $ x \in \mathcal{X} $. In practice, we use the DR estimate in Section 4.2 and then plug it into the last two constraints to obtain the DR estimates for $ \mathbb{P}_{\epsilon}(D'(a) = d, Y(a) = y \mid X = x) $, and estimate $ \hat{\epsilon}(x) $ for all $ x \in \mathcal{X} $. Theorem 3 proves the consistency results of the estimated $ \hat{\epsilon}(x) $ to the optimal $ \epsilon^(x) $. Theorem 3.* Suppose that $ \|\|\hat{\pi}_a(x) - \pi_a(x)\|\|_2 \cdot \|\|\hat{\mu}_{d,y}^{IPS}(x) - \mu_{d,y}^{IPS}(x)\|\|_2 = o_P(n^{-1/2}) $ for all $ x \in \mathcal{X} $ and $ a $ attainable by $ A $, then $ \|\|\hat{\epsilon}(x) - \epsilon^(x)\|\| = O_P(1/\sqrt{n}) $. ### 4.4 Theoretical analysis After obtaining the optimization solution $ \hat{\epsilon}(x) $ in Section 4.3, the adjusted decision function $ D' $ is determined by combining original decision $ D $ and $ \hat{\epsilon}(x) $. Next, we prove that the adjusted decision $ D' $ Table 3: Synthetic experiment results for varying models and estimators. The intervals under $w_{0100}$, $w_{1000}$, $w_{0111}$ and $w_{1011}$ show the minimum and maximum values when the other three are set to 0. \| Method \| $w_{0100}$ \| $\epsilon_0$ \| $w_{1000}$ \| $\epsilon_1$ \| $w_{0111}$ \| $\epsilon_1$ \| $w_{1011}$ \| $\epsilon_1$ \| CF ↑ \| PCF ↑ \| \|------------\|------------\|---------------\|------------\|---------------\|------------\|---------------\|------------\|---------------\|------\|-------\| \| LR + OR \| [-0.636, -0.083] \| 0 \| [-0.774, -0.083] \| 0 \| [0.083, 0.223] \| 0 \| [-0.672, -0.083] \| 0.109 \| +3.60% \| +6.68% \| \| LR + IPS \| [0.153, 0.231] \| 0 \| [-0.772, -0.153] \| 0 \| [-0.666, -0.153] \| 0.166 \| [-0.715, -0.153] \| 0 \| +4.71% \| +6.89% \| \| LR + DR \| [-0.631, -0.024] \| 0 \| [-0.710, -0.024] \| 0 \| [0.024, 0.196] \| 0 \| [-0.683, -0.024] \| 0.127 \| +2.97% \| +6.09% \| \| SVM + OR \| [0.104, 0.260] \| 0 \| [-0.741, -0.104] \| 0.124 \| [-0.743, -0.104] \| 0 \| [-0.619, -0.104] \| 0 \| +1.47% \| +5.15% \| \| SVM + IPS \| [0.192, 0.227] \| 0 \| [-0.720, -0.192] \| 0 \| [-0.739, -0.192] \| 0.199 \| [-0.732, -0.192] \| 0 \| +4.56% \| +8.98% \| \| SVM + DR \| [-0.662, -0.034] \| 0 \| [-0.763, -0.034] \| 0 \| [0.034, 0.182] \| 0.186 \| [-0.607, -0.034] \| 0 \| +5.77% \| +8.30% \| \| RF + OR \| [-0.706, -0.110] \| 0.120 \| [-0.713, -0.110] \| 0 \| [-0.689, -0.110] \| 0 \| [0.110, 0.195] \| 0 \| +2.18% \| +6.67% \| \| RF + IPS \| [0.163, 0.211] \| 0 \| [-0.727, -0.163] \| 0 \| [-0.680, -0.163] \| 0.171 \| [-0.755, -0.163] \| 0 \| +6.42% \| +8.35% \| \| RF + DR \| [-0.713, -0.051] \| 0 \| [-0.676, -0.051] \| 0 \| [0.051, 0.253] \| 0.203 \| [-0.661, -0.051] \| 0 \| +6.49% \| +9.02% \| \| NB + OR \| [-0.674, -0.161] \| 0 \| [-0.744, -0.161] \| 0 \| [0.161, 0.232] \| 0.173 \| [-0.742, -0.161] \| 0 \| +5.04% \| +8.31% \| \| NB + IPS \| [-0.821, -0.175] \| 0 \| [0.175, 0.192] \| 0 \| [-0.775, -0.175] \| 0 \| [-0.577, -0.175] \| 0.181 \| +8.88% \| +9.31% \| \| NB + DR \| [-0.793, -0.189] \| 0 \| [-0.688, -0.189] \| 0 \| [-0.707, -0.189] \| 0 \| [0.189, 0.208] \| 0.192 \| +4.83% \| +9.16% \| tends to be fair as sample size $n \to \infty$. To this end, consider the following programming problem $$\alpha^ = \arg \min_w \frac{1}{n} \sum_{i=1}^{n} w_{0,1,0,0}(X_i) + w_{1,0,0,0}(X_i) + w_{0,1,1,1}(X_i) + w_{1,0,1,1}(X_i),$$ subject to $$w_{d_0,d_1,y_0,y_1}(x) \geq 0 \quad \text{for all } d_0, d_1, y_0, y_1 \in \{0, 1\},$$ $$\sum_{a,b} w_{d_0,a,y_0,b}(x) = P_{\hat{\epsilon}}(D'(0) = d_0, Y(0) = y_0 \mid X = x) \quad \text{for all } d_0, y_0 \in \{0, 1\},$$ $$\sum_{a,b} w_{d_1,a,b,y_1}(x) = P_{\hat{\epsilon}}(D'(1) = d_1, Y(1) = y_1 \mid X = x) \quad \text{for all } d_1, y_1 \in \{0, 1\},$$ for all $x \in \mathcal{X}$, where $P_{\hat{\epsilon}}(D'(a), Y(a) \mid X = x)$ is the joint distribution of the potential outcomes $(D'(a), Y(a))$ for the post-processed decision function $D'$ using $\hat{\epsilon}(x)$ obtained in Section 4.3. We use the mean sum of squares as the metric for evaluating principal counterfactual fairness. Theorem 4 proves the consistency results of $\alpha^$ to 0, validating the effectiveness of the post-processing approach. Theorem 4.* Suppose that $\|\|\hat{\pi}_a(x) - \pi_a(x)\|\|_2 \cdot \|\|\hat{\mu}_{d,y}^a(x) - \mu_{d,y}^a(x)\|\|_2 = o_P(n^{-1/2})$ for all $x \in \mathcal{X}$ and $a$ attainable by $A$, then $\|\|\alpha^\|\|_1 = O_P(1/\sqrt{n})$. ## 5 Empirical Investigation To verify the effectiveness of the post-processing approach, we conduct experiments on both synthetic and real-world dataset. The performance is evaluated by two metrics: counterfactual fairness (CF): $P(D(0) = D(1))$ and principal counterfactual fairness (PCF): $P(D(0) = D(1) \mid Y(0) = Y(1))$. For all experiments, we calculate the values of CF and PCF before and after the post-processing operation and report the percentage change of each metric respectively. ### 5.1 Synthetic Experiment Synthetic data are generated from a structural equation model based on a random DAG with 10 nodes and 40 directed edges according to the Erdős-Rényi (ER) model, where four different models: Logistic Regression (LR), Support Vector Machine (SVM), Random Forest (RF) and Naive Bayes (NB) respectively to obtain the estimations of $P(Y = 1 \mid A = a, X = x)$ as the decisions $D(a, x)$ (see Appendix C for details). Then we check whether the optimization equation in Section 4.1 is solvable. If the feasible domain is empty, we further use the post-processing method in Section 4.3 to obtain $\hat{\epsilon}_{ad}(x)$. Table 3 shows the synthetic experiment results. First, the intervals of the four $w$ do not contain 0, implying that the optimization equation has no solution. At this point there will be one $\hat{\epsilon}_{ad}(x)$ nonzero, while the other three $\hat{\epsilon}_{ad}(x)$ are 0. Second, after the flip based on the $\hat{\epsilon}_{ad}(x)$, there are positive changes in both PCF and CF, and the increase in PCF is more pronounced than CF for all models. This is because our approach focuses only on the population with $Y(0) = Y(1)$. ### 5.2 Real-World Experiment The STUDENTINFO file in the Open University Learning Analytics Dataset (OULAD) dataset (Kuzilek et al., 2017) is used for the real-world experiment. The data attributes include demographic informa- Table 4: Real-world experiment results for different subgroups. The intervals under $w_{0100}$, $w_{1000}$, $w_{0111}$ and $w_{1011}$ show the minimum and maximum values when the other three $w$ are 0. \| Subgroup \| $w_{0100}(X)$ \| $\epsilon_{00}(X)$ \| $w_{1000}(X)$ \| $\epsilon_{10}(X)$ \| $w_{0111}(X)$ \| $\epsilon_{01}(X)$ \| $w_{1011}(X)$ \| $\epsilon_{11}(X)$ \| CF ↑ \| PCF ↑ \| \|----------\|----------------\|-------------------\|----------------\|------------------\|----------------\|------------------\|----------------\|------------------\|------\|-------\| \| None \| [-0.716, 0.024] \| 0 \| [-0.747, 0.020] \| 0 \| [-0.457, 0.078] \| 0 \| [-0.078, 0.268] \| 0 \| - \| - \| \| $X_1 \geq 120$ \| [-0.131, 0.274] \| 0 \| [-0.713, 0.089] \| 0 \| [-0.778, 0.093] \| 0 \| [-0.377, 0.131] \| 0 \| +1.35% \| +1.79% \| \| $X_1 < 120$ \| [-0.580, -0.030] \| 0 \| [0.030, 0.239] \| 0 \| [-0.747, -0.030] \| 0.040 \| [-0.702, -0.030] \| 0 \| +3.52% \| +3.97% \| \| $X_2 > 0$ \| [0.039, 0.242] \| 0 \| [-0.675, -0.039] \| 0.049 \| [-0.658, -0.039] \| 0 \| [-0.705, -0.039] \| 0 \| - \| - \| \| $X_2 = 0$ \| [-0.715, 0.007] \| 0 \| [-0.575, 0.007] \| 0 \| [-0.332, 0.092] \| 0 \| [-0.376, 0.213] \| 0 \| - \| - \| \| $X_1 \geq 120$, $X_2 > 0$ \| [0.173, 0.287] \| 0 \| [-0.754, -0.173] \| 0 \| [-0.715, -0.173] \| 0.196 \| [-0.704, -0.173] \| 0 \| +3.80% \| +8.51% \| \| $X_1 \geq 120$, $X_2 = 0$ \| [-0.749, 0.000] \| 0 \| [-0.444, 0.057] \| 0 \| [-0.748, 0.001] \| 0 \| [-0.057, 0.246] \| 0 \| +5.22% \| +9.59% \| \| $X_1 < 120$, $X_2 > 0$ \| [0.176, 0.220] \| 0 \| [-0.738, -0.176] \| 0 \| [-0.731, -0.176] \| 0.184 \| [-0.706, -0.176] \| 0 \| +3.10% \| +5.02% \| \| $X_1 < 120$, $X_2 = 0$ \| [-0.822, -0.231] \| 0 \| [-0.666, -0.231] \| 0 \| [-0.742, -0.231] \| 0 \| [0.231, 0.304] \| 0.124 \| - \| - \| Note: In real-world experiments, $X_1 =$ studied credits and $X_2 =$ num_of_prev_attempts. tion about the students such as gender, age, education level, disability, and other attributes as well as their final grades. This dataset contains 32,593 students and 11 attributes. We treat disability as the sensitive attribute and binarize the final grades as the outcome of interest. First we learn a CPDAG from the raw data using the PC algorithm in the causal-learn package. We find studied credits; the total number of credits for the modules the student is currently studying (denoted as $X_1$) and num_of_prev_attempts: the number of how many times the student has attempted this module (denoted as $X_2$) with an undirected edge between it and the disability. Therefore we sample four DAGs from the learned CPDAG corresponding to the four cases of no subgroup, $X_1$ as subgroups, $X_2$ as subgroups, and both $X_1$ and $X_2$ as subgroups, respectively. For each DAG, we determine the path coefficients based on linear regression and treat the residual of the regression as noise. For each subgroup, the subsequent steps are the same as in the simulation experiments. The LR model is used to obtain the decision $D$ and the DR estimator is used to estimate $P(D(a) = d, Y(a) = y \| X = x)$. Table 4 shows the real-world experiment results. When solving the optimization problem in Section 4.1 on the whole population, we find that the interval of all four $w$ covers 0, i.e., the current algorithm already satisfies the principle of counterfactual fairness. Therefore, post-processing is unnecessary at this point, so there is no corresponding change in CF and PCF. When using $X_1$ to divide the population, the optimization equation for the subgroup of $X_1 \geq 120$ has no solution, and when grouping according to $X_2$, the optimization equation for the subgroup of $X_2 > 0$ has no solution. When we divide the whole population into four subgroups, we find that the optimization equations for three of subgroups are unsolvable. Compared to the case of two subgroups, for the unsolvable subgroups, the distance between zero and the interval of the four $w$ and the value of the non-zero $\hat{\epsilon}_{ad}(x)$ is significantly larger when the population is divided into four subgroups. This indicates that the constraint is violated to a stronger extent when the population is divided into more subgroups. Meanwhile, for the case of four subgroups, the CF and PCF of each unsolvable subgroup change more due to the larger $\hat{\epsilon}_{ad}(x)$. In addition, for solvable subgroups, the interval of $w_{0100}$ and $w_{0111}$ are very close to exclude 0 when the population is divided into four subgroups, which further indicates that as the number of population group increases, each subgroup becomes more difficult to satisfy the optimization equation. Finally, the growth of PCF is larger than that of CF for all unsolvable subgroups, which shows that our approach is more effective on the population with $Y(0) = Y(1)$. 6 CONCLUSION This paper studies the question of "which attributes and individuals should be protected" in the context of counterfactual fairness. Motivated by the example that disability serves as a sensitive attribute for different outcomes of interest (e.g., college admissions, athlete selections), we suggest that when and how to enforce fairness is expected to depend on whether the protected attribute has no individual causal effect on the outcome of interest. Formally, we propose principal counterfactual fairness, and theoretically derive the necessary conditions for an algorithm to satisfy principal counterfactual fairness based on statistical bounds. Based on this, we further propose a principled post-processing approach to achieve principal counterfactual fairness with minimal individual decision changes. A limitation of this work is that the principal counterfactual fairness is partially identified, i.e., we cannot give unbiased point estimates from the data, but can give its statistical bounds and a falsification method. We leave it to future work about how to develop new identification and estimation strategies under practical assumptions. In addition, combining causal discovery to achieve decisions that satisfy the principal counterfactual fairness also serves as an interesting future topic. REFERENCES Tim Brennan, William Dieterich, and Beate Ehret. Evaluating the predictive validity of the compas risk and needs assessment system. Criminal Justice and behavior, 36(1):21–40, 2009. V. Chernozhukov, D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, W. Newey, and J. Robins. Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21:1–68, 2018. Silvia Chiappa. Path-specific counterfactual fairness. 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zlkXLb3wpF	The suggestion that removal of the $\frac{\partial}{\partial \theta} \log q$ term from the gradient estimate makes learning empirically robust to overfitting is quite interesting and provocative, but unexplored in detail.	FAST AND UNIFIED PATH GRADIENT ESTIMATORS FOR NORMALIZING FLOWS Lorenz Vaitl\textsuperscript{1,}, Ludwig Winkler\textsuperscript{1,}, Lorenz Richter\textsuperscript{2,3}, Pan Kessel\textsuperscript{4} \textsuperscript{1}Machine Learning Group, TU Berlin, \textsuperscript{2}Zuse Institute Berlin, \textsuperscript{3}did a Datenschmiede GmbH, \textsuperscript{4}Prescient Design, Genentech, Roche ABSTRACT Recent work shows that path gradient estimators for normalizing flows have lower variance compared to standard estimators for variational inference, resulting in improved training. However, they are often prohibitively more expensive from a computational point of view and cannot be applied to maximum likelihood training in a scalable manner, which severely hinders their widespread adoption. In this work, we overcome these crucial limitations. Specifically, we propose a fast path gradient estimator which improves computational efficiency significantly and works for all normalizing flow architectures of practical relevance. We then show that this estimator can also be applied to maximum likelihood training for which it has a regularizing effect as it can take the form of a given target energy function into account. We empirically establish its superior performance and reduced variance for several natural sciences applications. 1 INTRODUCTION Normalizing flows (NFs) have become a crucial tool in applications of machine learning in the natural sciences. This is mainly because they can be used for variational inference, i.e., for the approximation of distributions corresponding to given physical energy functions. Furthermore, they can be synergistically combined with more classical sampling methods such as Markov chain Monte Carlo (MCMC) and Molecular Dynamics, as their density is tractable. The paradigm of using normalizing flows as neural samplers has lately been widely adopted for example in quantum chemistry (Boltzmann generators (Noé et al., 2019)), statistical physics (generalized neural samplers (Nicoli et al., 2020)), as well as high-energy physics (neural trivializing maps (Albergo et al., 2019)). In these applications, the normalizing flow is typically trained using a combination of two training objectives: Reverse Kullback-Leibler (KL) training is used to train the model by self-sampling (see Section 2). Crucially, this training method on its own often fails in high-dimensional sampling settings because self-sampling is unlikely to probe exceedingly concentrated high probability regions of the ground-truth distribution and can potentially lead to mode collapse. As such, reverse KL training is often combined with maximum likelihood training (also known as forward KL training). For this, samples from the ground-truth distribution are obtained by standard sampling methods such as, e.g., MCMC. As these methods are typically costly, the samples are often of low number and possibly biased. The model is then trained to maximize its likelihood with respect to these samples. This step is essential for guiding the self-sampling towards high probability regions and, by extension, for successful training. Since training normalizing flows for realistic physical examples is typically computationally challenging, methods to speed up the convergence have been a focus of recent research. To this end, path estimators for the gradient of the reverse KL loss have been proposed (Roeder et al., 2017; Vaitl et al., 2022a;b). These estimators focus on the parameter dependence of the flow’s sampling process, also known as the sampling path, while discarding the direct parameter dependency, which vanishes in expectation. Path gradients have the appealing property that they are unbiased and tend to have lower variance compared to standard estimators, thereby promising accelerated convergence (Roeder et al., 2017; Agrawal et al., 2020; Vaitl et al., 2022a;b). At the same time, however, current path gradient estimation schemes have often a runtime that is several multiples of the standard Shared first authorship. gradient estimator, thus somehow counteracting the original intention. As a remedy, recently, Vaitl et al. (2022b) proposed a faster algorithm. Unfortunately, however, this algorithm is limited to continuous normalizing flows. Our work resolves this unsatisfying situation by proposing unified and fast path gradient estimators for all relevant normalizing flow architectures. Notably, our estimators are between 1.5 and 8 times faster than the previous state-of-the-art. Specifically, we a) derive a recursive equation to calculate the path gradient during the sampling procedure. Further, for flows that are not analytically invertible, we b) demonstrate that implicit differentiation can be used to calculate the path gradient without costly numerical inversion, resulting in significantly improved system size scaling. Finally, we c) prove by a change of perspective (noting that the forward KL divergence in data space is a reverse KL divergence in base space) that our estimators can straightforwardly be used for maximum likelihood training. Crucially, the resulting estimators allow us to work directly on samples from the target distribution. As a result of our manuscript, path gradients can now be used for all widely used training objectives — as opposed to only objectives using self-sampling — in a unified and scalable manner. We demonstrate the benefits of our proposed estimators for several normalizing flow architectures (RealNVP and gauge-equivariant NCP flow) and target densities with applications both in machine learning (Gaussian Mixture Model) as well as physics ($U(1)$ gauge theory, and $\phi^4$ lattice model). 1.1 Related Works Pathwise gradients take the sampling path into account and are well established in doubly stochastic optimization, see e.g. L’Ecuyer (1991); Jankowiak & Obermeyer (2018); Parmas & Sugiyama (2021). The present work uses path gradient estimators, a subset of pathwise gradients, originally proposed by Roeder et al. (2017) in the context of reverse KL training of Variational Autoencoders (VAE), which is motivated by only using the sampling path for computing gradient estimators and disregarding the direct parameter dependency. These were subsequently generalized by Tucker et al. (2019); Finke & Thiery (2019); Gelfner & Domke (2021a,b) to generic VAE self-sampling losses. There has been substantial work on reducing gradient variance not by path gradients but with control variates, for example in Miller et al. (2017); Kool et al. (2019); Richter et al. (2020); Wang et al. (2023). For an extensive review on the subject, we refer to Mohamed et al. (2020). Bauer & Mnih (2021) generalized path gradients to score functions of distributions which do not coincide with the sampling distribution in the context of hierarchical VAEs. As we will show, our fast path gradient for the forward KL training can be brought into the same form. However, only our formulation allows the application of a fast estimation scheme for NFs and establishes that forward and reverse path gradients are closely linked. Path gradients for normalizing flows have recently been studied: Agrawal et al. (2020) were the first to apply path gradients to normalizing flows as part of a broader ablation study. However, their algorithm has double the runtime and memory constraints as it requires a full copy of the neural network. Vaitl et al. (2022a) proposed a method that allows path gradient estimation for any explicitly invertible flow at the same runtime cost as Agrawal et al. (2020) but half the memory footprint. They also proposed an estimator for forward KL training which is however based on reweighting and thus suffers from poor system size scaling, while our method works on samples from the target density. For the rather restricted case of continuous normalizing flows, Vaitl et al. (2022b) proposed a fast path gradient estimator. Our proposal unifies their method in a framework which applies across a broad range of normalizing flow types. 2 Normalizing Flows A normalizing flow is a composition of diffeomorphisms $$x = T_\theta(x_0) := T_{L,\theta_L} \circ \cdots \circ T_{1,\theta_1}(x_0),$$ where we have collectively denoted all parameters of the flow by $\theta := (\theta_1, \ldots, \theta_L)$. Since diffeomorphisms form a group under composition, the map $T_\theta$ is a diffeomorphism as well. Samples from a normalizing flow can be drawn by applying $T_\theta$ to samples from a simple base density $x_0 \sim q_0$ such as $q_0 = \mathcal{N}(0, 1)$. The density of $x = T_\theta(x_0)$, denoted by $q_\theta$, is then given by the pushforward density of $q_0$ under $T_\theta$, i.e., \[ \log q_\theta(x) = \log q_0(T_\theta^{-1}(x)) + \log \left\| \det \frac{\partial T_\theta^{-1}(x)}{\partial x} \right\|, \] see also Appendix A for general remarks on the notation. We focus on applications for which normalizing flows are trained to closely approximate a ground-truth target density $p(x) = \frac{1}{Z} \exp(-E(x))$, where the energy $E : \mathbb{R}^d \to \mathbb{R}$ is known in closed-form but the partition function $Z = \int_{\mathbb{R}^d} e^{-E(x)} dx$ is intractable. To this end, there are two widely established training methods: Reverse KL training* relies on self-sampling from the flow and minimizes the reverse KL divergence \[ D_{KL}(q_\theta, p) = \mathbb{E}_{x \sim q_\theta} [E(x) + \log q_\theta(x)] + \text{const}. \] Since reverse KL training is based on self-sampling, the flow needs to be evaluated in the base-to-target direction $T_\theta$. Forward KL training requires samples from the target density $p$ and is equivalent to maximum likelihood training \[ D_{KL}(p, q_\theta) = \mathbb{E}_{x \sim p} [-\log q_\theta(x)] + \text{const}. \] Since forward KL training requires the calculation of the density $q_\theta(x)$, the flow needs to be evaluated in the target-to-base direction $T_\theta^{-1}$, see (2). As mentioned before, one typically uses a combined forward and reverse training to guide the self-sampling to high probability regions of the target density. When choosing a normalizing flow architecture for this task, it is therefore essential that both directions $T_\theta$ and $T_\theta^{-1}$ can be evaluated with reasonable efficiency. As a result, the following types of architectures are of practical relevance: Coupling Flows are arguably the most widely used (see, e.g., Noé et al. (2019); Albergo et al. (2019); Nicoli et al. (2020); Matthews et al. (2022); Midgley et al. (2023); Huang et al. (2020)). They split the vector $x_l \in \mathbb{R}^d$ in two components \[ x_l = (x_l^{\text{trans}}, x_l^{\text{cond}}), \] with $x_l^{\text{trans}} \in \mathbb{R}^k$ and $x_l^{\text{cond}} \in \mathbb{R}^{d-k}$ for $k \in \{1, \ldots, d-1\}$. The map $T_{l+1,\theta_{l+1}}$ is then given by \[ \begin{align} x_{l+1,i}^{\text{trans}} &= f_{\theta,i}(x_l^{\text{trans}}, x_l^{\text{cond}}) := \tau(x_l^{\text{trans}}, h_{\theta,i}(x_l^{\text{cond}})), \quad \forall i \in \{1, \ldots, k\}, \\ x_{l+1}^{\text{cond}} &= x_l^{\text{cond}}, \end{align} \] where $f_{\theta} : \mathbb{R}^k \times \mathbb{R}^{d-k} \to \mathbb{R}^k$, $\tau : \mathbb{R} \times \mathbb{R}^m \to \mathbb{R}$ are invertible maps with respect to their first argument for any choice of the second argument and $h_{\theta,i} : \mathbb{R}^{d-k} \to \mathbb{R}^m$ is the $i$-th output of a neural network. Note that the function $f_{\theta}$ acts on the components of $x_l^{\text{trans}}$ element-wise. There are broadly two types of coupling flows with different choices for the transformation $\tau$: 1. Explicitly invertible flows have the appealing property that the inverse map $T_{l+1,\theta_{l+1}}^{-1}$ can be calculated in closed-form and as efficiently as the forward map $T_{l+1,\theta_{l+1}}$. A particular example of this type of flows are affine coupling flows (Dinh et al., 2014; 2017) that use an affine transformation $\tau$, i.e., \[ \begin{align} x_{l+1}^{\text{trans}} &= f_{\theta}(x_l^{\text{trans}}, x_l^{\text{cond}}) = \sigma_{\theta}(x_l^{\text{cond}}) \odot x_l^{\text{trans}} + \mu_{\theta}(x_l^{\text{cond}}), \\ x_{l+1}^{\text{cond}} &= x_l^{\text{cond}}, \end{align} \] with $h_{\theta} = (\sigma_{\theta}, \mu_{\theta})$. Another example are neural spline flows (Durkan et al., 2019) which use splines instead of an affine transformation. 2. Implicitly invertible flows use a map $\tau$ whose inverse can only be obtained numerically, such as a mixture of non-compact projectors (Kanwar et al., 2020; Rezende et al., 2020) or smooth bump functions (Köhler et al., 2021). This often results in more expressive flows in particular in the context of normalizing flows on manifolds (Rezende et al., 2020). Recently, it has been shown in Köhler et al. (2021) that implicit differentiation can be used to train these types of flows using the forward KL objective. Continuous Normalizing Flows use an ordinary differential equation (ODE) which relates to the bijection $ T_\theta : \mathbb{R}^d \rightarrow \mathbb{R}^d $, allowing for straightforward implementation of equivariances, but typically coming with high computational costs (Chen et al., 2018). Notably, autoregressive flows (Huang et al., 2018; Jaini et al., 2019) are less relevant in the context of learning a target density $ p $ because they only permit fast evaluation in one direction and there is no training method based on implicit differentiation. As a result, they are not considered in this work. 3 Path Gradients for Reverse KL In this section, we introduce path gradients and show how they are related to the gradient of the reverse KL objective. The basic definition of path gradients is as follows: Definition 3.1. The path gradient of a function $ \varphi(\theta, T_\theta(x_0)) $ is given by \[ \nabla_\theta \varphi(\theta, T_\theta(x_0)) := \frac{\partial \varphi(\theta, x)}{\partial x} \bigg\|_{x = T_\theta(x_0)} \frac{\partial T_\theta(x_0)}{\partial \theta}. \] (8) Note that the total derivative of the function $ \varphi $ can be decomposed in the following way: \[ \frac{d}{d\theta} \varphi(\theta, T_\theta(x_0)) = \nabla_\theta \varphi(\theta, T_\theta(x_0)) + \frac{\partial}{\partial \theta} \varphi(\theta, x) \bigg\|_{x = T_\theta(x_0)}. \] (9) The path gradient therefore only takes the parameter dependence of the sampling path $ T_\theta $ into account, but does not capture any explicit parameter dependence denoted by the second term. This decomposition was applied by Roeder et al. (2017) to the gradient of the reverse KL divergence to obtain the notable result \[ \frac{d}{d\theta} D_{KL}(q_\theta, p) = \mathbb{E}_{x_0 \sim q_0} \left[ \nabla_\theta \left( E(T_\theta(x_0)) + \log q_\theta(T_\theta(x_0)) \right) \right], \] (10) where we have used the fact that $ \mathbb{E}_{x_0 \sim q_0} \left[ \frac{\partial}{\partial \theta} \log q_\theta(T_\theta(x_0)) \right] = 0 $. Thus, an unbiased estimator for the gradient of the KL divergence is given by \[ G_{\text{path}} := \frac{1}{N} \sum_{n=1}^{N} \nabla_\theta \left[ E(T_\theta(x_0^{(n)})) + \log q_\theta(T_\theta(x_0^{(n)})) \right], \] (11) where $ x_0^{(n)} \sim q_0 $ are i.i.d. samples. This path gradient estimator has been observed to have lower variance compared to the standard gradient estimator (Roeder et al., 2017; Tucker et al., 2019; Agrawal et al., 2020). As the total derivative of the energy agrees with the path gradient of the energy function, i.e., \[ \frac{d}{d\theta} E(T_\theta(x_0)) = \nabla_\theta E(T_\theta(x_0)), \] the first term in the estimator can be straightforwardly calculated using automatic differentiation. The second term, involving the path score $ \nabla_\theta \log q_\theta(T_\theta(x_0)) $, is however non-trivial as the path gradient through the sampling path $ T_\theta $ has to be disentangled from the explicit parameter dependence in $ q_\theta $. Recently, Vaitl et al. (2022a) proposed a method to calculate this term using the following steps: 1. Sample from the flow without building the computational graph: \[ x' = \text{stop\_gradients}(T_\theta(x_0)) \] for $ x_0 \sim q_0 $. (12) 2. Calculate the gradient of the density with respect to the sample $ x' $ using automatic differentiation: \[ G = \frac{\partial}{\partial x'} \log q_\theta(x') = \frac{\partial}{\partial x'} \left( \log q_0(T_\theta^{-1}(x')) + \log \det \left\| \frac{\partial T_\theta^{-1}(x')}{\partial x'} \right\| \right). \] (13) 3. Calculate the path gradient using a vector Jacobian product which can be efficiently calculated by standard reverse-mode automatic differentiation: \[ \nabla_\theta \log q_\theta(T_\theta(x_0)) = G \frac{\partial T_\theta(x_0)}{\partial \theta}. \] (14) Following standard convention in the autograd community, we adopt the convention that $ G $ is a row vector. This is because the differential $ df = dx_i \frac{\partial f}{\partial x_i} $ of a function $ f $ is a one-form and thus an element of the co-tangent space. This method therefore requires the evaluation of both directions $T_\theta$ and $T_\theta^{-1}$. For implicitly invertible flows, backpropagation through a numerical inversion per training iteration is thus required, which is often prohibitively expensive. Even in the best case scenario, i.e., for flows that can be evaluated in both directions with the same computational costs, such as RealNVP (Dinh et al., 2017), this algorithm has significant computational overhead. Specifically, it has roughly the costs of five forward passes: one for the sampling (12) and two each for the two gradient calculations (13) and (14) (which each require a forward as well as a backward pass). This is to be contrasted with the costs of the standard gradient estimator which only requires a single forward as well as a backward pass, i.e., has the cost of roughly two forward passes. In practical implementations, typically a runtime overhead of a factor of two instead of $\frac{5}{2}$ is observed for the path gradient estimator compared to the standard gradient estimator. ### 3.1 Fast Path Gradient Estimator In the following, we outline a fast method to estimate the path gradient. An important downside of the algorithm outlined in the last section is that one has to evaluate the flow in both directions $T_\theta$ and $T_\theta^{-1}$. The basic idea of the method outlined in the following is to calculate the derivative $\partial_x \log q_\theta(x)$ of the flow model recursively during sampling process. As a result, the flow only needs to be calculated in the forward direction $T_\theta$ as the second step in the path gradient algorithm discussed in the previous section can be avoided. In more detail, the calculation of the path gradient proceeds in two steps: 1. The sample $x = T_\theta(x_0)$ and the gradient $G = \frac{\partial}{\partial x} \log q_\theta(x)$ can be calculated alongside the sampling process using the recursive relation derived below. 2. The path gradient is then calculated with automatic differentiation using a vector Jacobian product, where, however, the forward pass $T_\theta(x_0)$ does not have to be recomputed: $$\nabla_\theta \log q_\theta(T_\theta(x_0)) = G \frac{\partial T_\theta(x_0)}{\partial \theta}. \quad (15)$$ The recursion to calculate the derivative $\partial_x \log q_\theta(x)$ is as follows: Proposition 3.2 (Gradient recursion). Using the diffeomorphism $T_l$, the derivative of the induced probability can be computed recursively as follows $$\frac{\partial \log q_{\theta,l+1}(x_{l+1})}{\partial x_{l+1}} = \frac{\partial \log q_{\theta,l}(x_l)}{\partial x_l} \left( \frac{\partial T_{l,\theta_l}(x_l)}{\partial x_l} \right)^{-1} - \frac{\partial \log \left\| \det \frac{\partial T_{l,\theta_l}(x_l)}{\partial x_l} \right\|}{\partial x_l} \left( \frac{\partial T_{l,\theta_l}(x_l)}{\partial x_l} \right)^{-1}. \quad (16)$$ For general $T_l$, computing the inverse Jacobian $(\partial T_{l,\theta_l}(x_l)/\partial x_l)^{-1}$ entails a time and space complexity higher than $O(d)$, which is the complexity of the standard gradient estimator. For autoregressive flows, the total complexity is $O(d^2)$, since its Jacobian is triangular. For coupling-type flows, we can simplify and speed up the recursion to have linear complexity in the number of dimensions, i.e. $O(d)$. We state the recursive gradient computations for these kind of flows in the following proposition. Proposition 3.3 (Recursive gradient computations for coupling flows). For a coupling flow, $$x_{l+1}^{\text{trans}} = f_\theta(x_{l+1}^{\text{trans}}, x_{l+1}^{\text{cond}}) \quad \text{and} \quad x_{l+1}^{\text{cond}} = x_{l+1}^{\text{cond}}, \quad (17)$$ the derivative of the logarithmic density can be calculated recursively as follows $$\frac{\partial \log q_{\theta,l+1}(x_{l+1})}{\partial x_{l+1}^{\text{trans}}} = \frac{\partial \log q_{\theta,l}(x_l)}{\partial x_l^{\text{trans}}} \left( \frac{\partial f_\theta(x_{l+1}^{\text{trans}}, x_{l+1}^{\text{cond}})}{\partial x_{l+1}^{\text{trans}}} \right)^{-1} - \frac{\partial \log \left\| \det \frac{\partial f_\theta(x_{l+1}^{\text{trans}}, x_{l+1}^{\text{cond}})}{\partial x_{l+1}^{\text{trans}}} \right\|}{\partial x_{l+1}^{\text{trans}}} \left( \frac{\partial f_\theta(x_{l+1}^{\text{trans}}, x_{l+1}^{\text{cond}})}{\partial x_{l+1}^{\text{trans}}} \right)^{-1}, \quad (18)$$ $$\frac{\partial \log q_{\theta,l+1}(x_{l+1})}{\partial x_{l+1}^{\text{cond}}} = \frac{\partial \log q_{\theta,l}(x_l)}{\partial x_l^{\text{cond}}} - \frac{\partial \log q_{\theta,l+1}(x_{l+1})}{\partial x_{l+1}^{\text{trans}}} \frac{\partial f_\theta(x_{l+1}^{\text{trans}}, x_{l+1}^{\text{cond}})}{\partial x_{l+1}^{\text{cond}}} \frac{\partial \log \left\| \det \frac{\partial f_\theta(x_{l+1}^{\text{trans}}, x_{l+1}^{\text{cond}})}{\partial x_{l+1}^{\text{trans}}} \right\|}{\partial x_{l+1}^{\text{cond}}}, \quad (19)$$ starting with \[ \frac{\partial \log q_{\theta,0}(x_0)}{\partial x_0^{\text{trans}}} = \frac{\partial \log q_0(x_0)}{\partial x_0^{\text{trans}}}, \quad \frac{\partial \log q_{\theta,0}(x_0)}{\partial x_0^{\text{cond}}} = \frac{\partial \log q_0(x_0)}{\partial x_0^{\text{cond}}}. \] (20) For a proof, see Appendix B.1. We stress that the Jacobian $ \frac{\partial f_\theta(x_t^{\text{trans}}, x_t^{\text{cond}})}{\partial x_t^{\text{trans}}} $ is a $ k \times k $ square and invertible matrix, since $ f_\theta(\cdot, x_t^{\text{cond}}) $ is bijective for any $ x_t^{\text{cond}} \in \mathbb{R}^{d-k} $, see (6). Implicitly Invertible Flows. An interesting property of the recursions in Proposition 3.3 is that they only involve (derivatives of) $ f_\theta(x_t^{\text{trans}}, x_t^{\text{cond}}) $ and can thus be evaluated during the sampling from the flow. As such, they are directly applicable to implicitly invertible flows. Further note that the Jacobian $ \frac{\partial f_\theta(x_t^{\text{trans}}, x_t^{\text{cond}})}{\partial x_t^{\text{trans}}} $ can be inverted in linear time $ O(d) $, as it is a diagonal matrix; the function $ f $ acts element-wise on $ x_t^{\text{trans}} $, see (6). Therefore, the recursion has the decisive advantage that no numerical inversions need to be performed. In particular, there is no need for prohibitive backpropagation through such an inversion. Explicitly Invertible Flows. For explicitly invertible normalizing flows — the most favorable setup for the baseline method from Vaitl et al. (2022a) — the runtime reduction appears to be more mild at first sight. The algorithm has roughly the cost of three forward passes: one each for the calculation of both $ x $ and $ G $ and one more for the backward pass when calculating the path gradient in (15). This is to be compared to the cost of five forward passes for the baseline method by Vaitl et al. (2022a) to calculate path gradients and two forward passes for the standard total gradient. However, this rough counting neglects the synergy between the sampling process $ x = T(x_0) $ and the calculation of the score $ G $. As we will show experimentally in Section 5, the actual runtime increase is only about forty percent compared to the standard total gradient. Finally, let us note that for the aforementioned popular case of affine coupling flows our recursion from Proposition 3.3 takes a particular form. Since fewer terms need to be calculated, the following recursion gives an additional improvement in computational speed. Corollary 3.4 (Recursive gradient computations for affine coupling flows). For an affine coupling flow (7), the recursion for the derivative of the logarithmic density can be simplified to \[ \frac{\partial \log q_{\theta,l+1}(x_{l+1})}{\partial x_{l+1}^{\text{trans}}} = \frac{\partial \log q_{\theta,l}(x_l)}{\partial x_l^{\text{trans}}} \odot \sigma_\theta(x_l^{\text{cond}}), \] \[ \frac{\partial \log q_{\theta,l+1}(x_{l+1})}{\partial x_{l+1}^{\text{cond}}} = \frac{\partial \log q_{\theta,l}(x_l)}{\partial x_l^{\text{cond}}} - \frac{\partial \log q_{\theta,l+1}(x_{l+1})}{\partial x_{l+1}^{\text{trans}}} \left( \frac{\partial \sigma_\theta(x_l^{\text{cond}})}{\partial x_l^{\text{cond}}} \odot x_l^{\text{trans}} + \frac{\partial \mu_\theta(x_l^{\text{cond}})}{\partial x_l^{\text{cond}}} \right) \] \[- \frac{\partial}{\partial x_l^{\text{cond}}} \log \left\| \prod_{i=1}^k \sigma_{\theta,i}(x_l^{\text{cond}}) \right\|, \] where $ \bar{x}_l^{\text{trans}} $ is a matrix with entries $ (\bar{x}_l^{\text{trans}})_{ij} := x_{l,i}^{\text{trans}} $ for $ i \in \{1, \ldots, k\}, j \in \{1, \ldots, d-k\} $. For a proof, see Appendix B.2. Additionally, we show in Appendix C that the fast path gradient derived by Vaitl et al. (2022b) for continuous normalizing flows can be rederived using analogous steps as in Proposition 3.3. Our results therefore unify path gradient calculations of coupling flows with the analogous ones for continuous normalizing flows. Finally, we note that a further distinctive strength of the proposed fast path gradient algorithm is that it can be performed at constant memory costs. Specifically, the calculation of $ G $ can be done without saving any activations. Similarly, the activations needed for the vector Jacobian product (15) can be calculated alongside the backward pass as $ T_\theta(x_0) = x $ is known using the techniques of Gomez et al. (2017). ### 4 PATH GRADIENTS FOR THE FORWARD KL DIVERGENCE For training normalizing flows with the forward KL divergence, previous works have mainly relied on reweighting path gradients (Vaitl et al., 2022a). Specifically, their basic underlying trick is to rewrite the expectation value with respect to the ground-truth $ p $ as an expectation value with respect to the model $ q_\theta $ \[ D_{KL}(p, q_\theta) = \mathbb{E}_{x \sim p} \left[ \log \frac{p(x)}{q_\theta(x)} \right] = \mathbb{E}_{x \sim q_\theta} \left[ \frac{p(x)}{q_\theta(x)} \log \frac{p(x)}{q_\theta(x)} \right]. \] (22) For this reweighted loss, suitable path gradient estimators were then derived in Tucker et al. (2019). Reweighting, however, has the significant downside that it leads to estimators with prohibitive variance — especially for high-dimensional problems and in the early stages of training (Hartmann & Richter, 2021). As a result, the proposed estimators cannot be applied in a scalable fashion (Dhaka et al., 2021; Geffner & Domke, 2021a). In the following, we will propose a general method to apply path gradients to forward KL training without the need for reweighting. To this end, we first notice that the forward KL of densities in data space can be equivalently rewritten as a reverse KL in base space, namely \[ D_{KL}(p, q_\theta) = D_{KL}(p_{\theta,0}, q_0), \] where we have defined the pullback of the target density $ p $ to base space as follows \[ p_{\theta,0}(x_0) := p(T_\theta(x_0)) \left\| \det \frac{\partial T_\theta(x_0)}{\partial x_0} \right\|. \] We refer to Papamakarios et al. (2021) for a proof. As a result, all results derived for the reverse KL case in the last sections also apply verbatim to the forward KL case if one exchanges: \[ q_\theta \leftrightarrow p_{\theta,0}, \quad p \leftrightarrow q_0, \quad x_0 \leftrightarrow x, \quad T_\theta(x_0) \leftrightarrow T_\theta^{-1}(x). \] In particular, the fast path gradient estimators can be straightforwardly applied. More precisely, the following statement holds: Proposition 4.1 (Path gradient for forward KL). For the derivative of the forward KL divergence $ D_{KL}(p\|q_\theta) $ w.r.t. the parameter $ \theta $ it holds \[ \frac{d}{d\theta} D_{KL}(p\|q_\theta) = \mathbb{E}_{x \sim p} \left[ \nabla_\theta \log \frac{p_{\theta,0}}{q_0}(T_\theta^{-1}(x)) \right], \] where $ p_{\theta,0}(x_0) := p(T_\theta(x_0)) \left\| \det \frac{\partial T_\theta(x_0)}{\partial x_0} \right\| $ is the pullback of the target density $ p $ to base space. For a proof, see Appendix B.3. Note that if $ p $ is only known in unnormalized form, so is its pullback $ p_{\theta,0} $. However, this has no impact on the derived result as it only involves derivatives of the log density for which the normalization is irrelevant. The following comments are in order: - The proposed path gradient for maximum likelihood training provides an attractive mechanism to incorporate the known closed-form target energy function into the training process. In particular, this can help to alleviate overfitting, cf. Figures 1 and 8 — a particularly relevant concern as the forward training often uses a low amount of samples which entails the risk of density collapse on the individual samples for standard maximum likelihood training. The information about the energy function helps the path-gradient-based training to avoid this undesired behaviour. On the other hand, forward KL path gradient training cannot be used if the target energy function is not known such as in image generation tasks. - As for path gradients of the reverse KL, we expect lower variance of the Monte Carlo estimator of (26) compared to standard maximum likelihood gradient estimators. In particular, we note that at the optimum $ q_\theta = p $ the variance of the gradient estimator vanishes. - The proof in Appendix B shows that the so-called generalized doubly reparameterized gradient proposed in Bauer & Mnih (2021) in the context of hierarchical VAEs can be brought in the same form as the path gradient for the forward KL objective derived in this section. However, only our formulation elucidates the symmetry between the forward and reverse objective and therefore allows the application for fast path gradient estimators. 5 Numerical Experiments In this section, we compare our fast path gradients with the conventional approaches for several normalizing flow architectures, both using forward and reverse KL optimization. We consider target densities with applications in machine learning (Gaussian mixture model) as well as physics ($ U(1) $ gauge theory, and $ \phi^4 $ lattice model). We refer to Appendix E for further details.$^$ $^$Code for reproducing the experiments for GMM and $ U(1) $ at github.com/lenz3000/unified-path-gradients. Figure 1: Effective sample size (ESS) over the training iterations for a Gaussian mixture model using the forward and the reverse KL divergence. The intervals denote the standard error over 5 runs. The best performance is indicated by a dot with subsequent faded average performance in the left and center figure. For the forward KL, we compare multiple hyperparameter settings (see Appendix E) and plot the respective best runs in the central plot. The right plot displays a stereotypical dependency on the data set size for fixed hyperparameters, see Tables 3, 4 and 5 for more details. We can see that, typically, path gradients perform better than standard maximum likelihood gradients. Figure 2: Training the $U(1)$ flow for Lattice Gauge Theory. Shaded area shows standard error over 4 runs. The Reverse KL Path Gradients reach higher performance and exhibit less erratic behavior. Gaussian Mixture Model. As a tractable multimodal example, we consider a Gaussian mixture model in $\mathbb{R}^d$ with $\sigma^2 = 0.5$, i.e. we choose the energy function $$E(x) = -\log \sum_{\mu \in \{-1,1\}^d} N(x; \mu, \sigma^2 I_d)$$ Note that the number of modes of the corresponding target density increases exponentially in the dimensions, i.e. we have $2^d$ modes in total. We choose $d = 6$, resulting in 64 modes. As shown in Figure 1, for most choices of hyperparameters, path gradient training outperforms the standard training objectives. In Figures 5 to 7 in the appendix we present further experiments, showing that path gradient estimators are indeed often better and never significantly worse than standard estimators. The slight overhead in runtime is therefore more than compensated by better training convergence. The additional information about the ground-truth energy function included in the forward path gradient training alleviates overfitting in forward KL training, see the discussion in Section 4. $\phi^4$ Field Theory can be described by a random vector $\phi \in \mathbb{R}^d$, whose entries $\phi_u$ represent the values of the corresponding field across a $16 \times 8$ lattice. The lattice positions are encoded in the set $\Lambda \subset \mathbb{N}^2$. We assume periodic boundary conditions of the lattice. The random vector $\phi$ admits the density $p(\phi) = \frac{1}{Z} \exp(-S(\phi))$ with action $$S(\phi) = \sum_{u,v \in \Lambda} \phi_u \Delta_{uv} \phi_v + \sum_{u \in \Lambda} (m^2 \phi_u^2 + \lambda \phi_u^4),$$ where $\Delta_{uv}$ is the lattice Laplacian. The parameters $m$ and $\lambda$ are the bare mass and coupling, respectively. We choose the value of these parameters such that they lie in the so-called critical region, as this is the most challenging regime. We refer to Gatringer & Lang (2009) for more details. Note that, by slightly abusing notation, $\phi$ plays the role of what was $x$ before. Table 1: Results of the experiments from Section 5. We measure the approximation quality of the variational density $q_\theta$ by the effective sampling size (ESS) plus standard deviations, where higher is better, i.e., 100% indicates perfect approximation, see Appendix E for details. \| Algorithm \| Reverse KL \| Forward KL \| \|-----------\|------------\|------------\| \| \| Gradient \| Path Gradient \| Gradient \| Path Gradient \| \| GMM \| $\text{ESS}_p$ \| $92.2 \pm 0.0$ \| $97.4 \pm 0.0$ \| $79.1 \pm 0.0$ \| $91.8 \pm 0.0$ \| \| \| $\text{ESS}_q$ \| $93.0 \pm 0.0$ \| $97.4 \pm 0.0$ \| $84.1 \pm 0.0$ \| $91.8 \pm 0.0$ \| \| $\phi^4$ \| $\text{ESS}_p$ \| $85.6 \pm 0.1$ \| $96.0 \pm 0.1$ \| $85.1 \pm 0.1$ \| $95.6 \pm 0.0$ \| \| \| $\text{ESS}_q$ \| $85.6 \pm 0.1$ \| $96.0 \pm 0.1$ \| $85.1 \pm 0.1$ \| $95.6 \pm 0.0$ \| \| $U(1)$ \| $\text{ESS}_q$ \| $40.1 \pm 0.0$ \| $41.1 \pm 0.0$ \| — \| — \| \| \| ELBO \| $1346.42 \pm .01$ \| $1346.43 \pm .00$ \| — \| — \| Table 2: Factor of runtime increase (mean and standard deviation) in comparison to the standard gradient estimator, i.e., $\frac{\text{runtime path gradient}}{\text{runtime standard gradient}}$ on an A100-80GB GPU. The upper set of experiments cover the explicitly invertible flows, applied to $\phi^4$ as treated in the experiments. The lower set covers implicitly invertible flows applied to $U(1)$ theory. \| Algorithm \| Runtime factor with batch size \| \|-----------\|-------------------------------\| \| \| 64 \| 1024 \| 8192 \| \| Expl \| Alg. 1 (ours) \| $1.6 \pm 0.1$ \| $1.4 \pm 0.1$ \| $1.4 \pm 0.0$ \| \| \| Alg. 2 (Vaitl et al., 2022a) \| $2.1 \pm 0.1$ \| $2.2 \pm 0.1$ \| $2.1 \pm 0.0$ \| \| Impl \| Alg. 1 (ours) \| $2.2 \pm 0.0$ \| $2.0 \pm 0.1$ \| $2.3 \pm 0.0$ \| \| \| Alg. 2 + Köhler et al. (2021) \| $17.5 \pm 0.2$ \| $11.0 \pm 0.1$ \| $8.2 \pm 0.0$ \| on the underlying physics. Training is performed using both the forward and reverse KL objective with and without path gradients. For the flow, the same affine-coupling-based architecture as in Nicoli et al. (2020) is used. Samples for forward KL and ESS are generated using Hybrid Monte Carlo. We refer to Appendix E for more details. The path gradient training again outperforms the standard objective for both forward and reverse training, see Table 1. Gauge Theory was recently widely studied in the context of normalizing flows (Kanwar et al., 2020; Albergo et al., 2021; Finkenrath, 2022; Bacchio et al., 2023; Cranmer et al., 2023) as it provides an ideal setting for illustrating the power of inductive biases. This is because the theory’s action has a gauge symmetry, i.e., a symmetry which acts with independent group elements for each lattice site, see Gattringer & Lang (2009) for more details. Crucially, the field takes values in the circle group $U(1)$. Thus, flows on manifolds need to be considered. We use the flow architecture proposed by Kanwar et al. (2020) which is only implicitly invertible. Sampling from the ground-truth distribution with Hybrid Monte Carlo is very challenging due to critical slowing down and we therefore refrain from forward KL training and forward ESS evaluation. Table 1 and Figure 2 demonstrate that path gradients lead to overall better approximation quality. Runtime Comparison. In Table 2, we compare the runtime of our method to relevant baselines both for the ex- and implicitly invertible flows. To obtain a strong baseline for the latter, we use implicit differentiation as in Köhler et al. (2021) to avoid costly backpropagation through the numerical inversion. Our method is significantly faster than the baselines. We refer to Appendix E for a detailed analysis of how this runtime comparison scales with the chosen accuracy of the numerical inversion. Briefly summarized, we find that our method compares favorable to the baseline irrespective of the chosen accuracy. 6 CONCLUSION We have introduced a fast and unified method to estimate path gradients for normalizing flows which can be applied to both forward and reverse training. We find that the path gradient training consistently improves training for both the reverse and forward case. An appealing property of path-gradient maximum likelihood is that it can take information about the ground truth energy function into account and thereby acts as a particularly natural form of regularization. Our fast path gradient estimators are several multiples faster than the previous state-of-the-art, they are applicable across a broad range of NF architectures, and considerably narrow the runtime gap to the standard gradient while preserving the desirable variance reduction. Acknowledgements. L.V. thanks Matteo Gätzner for his preliminary work on GDReGs. L.W. thanks Jason Rinnert for visualization help and acknowledges support by the Federal Ministry of Education and Research (BMBF) for BIFOLD (01IS18037A). The research of L.R. has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through the grant CRC 1114 “Scaling Cascades in Complex Systems” (project A05, project number 235221301). P.K. wants to thank Andreas Loukas for useful discussions. 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Axel Finke and Alexandre H. Thiery. On importance-weighted autoencoders, 2019. Jacob Finkenrath. Tackling critical slowing down using global correction steps with equivariant flows: the case of the Schwinger model. arXiv preprint arXiv:2201.02216, 2022.
ZPdZLlNXSm	One of the major interesting components of the paper is that mean field approximation performs better than its non-mean field counterparts. Is there a good hypothesis for why this may be the case? Have you found a good characterization of when one out performs the other? Additional insight here would be great since the paper claims in the empirical section that the mean field losses provide better	MEAN FIELD THEORY IN DEEP METRIC LEARNING Takuya Furusawa ZOZÓ Research 1-3-22 Kioicho, Chiyoda-ku, Tokyo, Japan takuya.furusawa@zozo.com ABSTRACT In this paper, we explore the application of mean field theory, a technique from statistical physics, to deep metric learning and address the high training complexity commonly associated with conventional metric learning loss functions. By adapting mean field theory for deep metric learning, we develop an approach to design classification-based loss functions from pair-based ones, which can be considered complementary to the proxy-based approach. Applying the mean field theory to two pair-based loss functions, we derive two new loss functions, MeanFieldContrastive and MeanFieldClassWiseMultiSimilarity losses, with reduced training complexity. We extensively evaluate these derived loss functions on three image-retrieval datasets and demonstrate that our loss functions outperform baseline methods in two out of the three datasets. 1 INTRODUCTION Deep metric learning has emerged as a powerful technique for learning meaningful data representations in a variety of machine learning applications, such as image retrieval (Wang et al., 2014), face recognition (Schroff et al., 2015), and person re-identification (Hermans et al., 2017). The primary goal of deep metric learning is to provide an order to the embedding space by bringing similar instances closer and pushing dissimilar ones further apart. Typically, this is achieved by optimizing a loss function that utilizes appropriate interactions based on the distance between data points. However, conventional metric learning loss functions often suffer from high training complexity scaling polynomially with the size of the training data (Schroff et al., 2015). This challenge makes optimization of these loss functions difficult in large-scale applications and necessitates the development of sampling and mining strategies to find informative pairs. The concept of order also plays a crucial role in statistical physics, which studies the emergent behaviors of interacting many-body systems. These many-body systems exhibit various ordered phases of matter, such as solid state, magnetism, superfluidity, and superconductivity, which cannot be predicted from their individual constituents (Anderson, 1972). While the interactions between the constituents are essential for hosting such nontrivial behaviors, they also make analyzing the systems challenging, analogous to the issue in deep metric learning. Mean field theory (Weiss, 1997) is a powerful approach for handling the challenge associated with interacting many-body systems and provides an insightful framework for understanding their emergent behaviors. This theory introduces a mean field that represents the average behavior of constituent particles. The mean field is also known as the order parameter, as its value helps distinguish the ordered phases. The mean field theory approximates their interactions as interactions with the mean field and significantly reduces the complexity of many-body systems. In this paper, we leverage mean field theory from statistical physics to tackle the complexity associated with deep metric learning. We develop the mean field theory for applications to loss functions in deep metric learning and find that mean field theory is better suited for losses without anchor concepts, as opposed to the proxy-based method introduced in Movshovitz-Attias et al. (2017). In this sense, it can serve as a complementary approach to the proxy-based method for designing classification-based loss functions from pair-based ones. Furthermore, we apply the mean field theory to two pair-based loss functions and propose two new loss functions with reduced training complexity. We evaluate the proposed mean field loss functions using a benchmark protocol proposed in Musgrave et al. (2020a), which allows us a fair comparison with other baseline loss functions, and also using the traditional protocol utilized in Movshovitz-Attias et al. (2017); Kim et al. (2020). The evaluation results indicate that our mean field losses surpass other methods in two out of three image-retrieval datasets in the former protocol. Moreover, the latter evaluation protocol demonstrates that our losses not only exhibit performance improvements in three out of four image-retrieval datasets but also converge more rapidly compared to ProxyAnchor loss (Kim et al., 2020). The main contributions of this paper are three-fold: (1) the introduction of the mean field theory from statistical physics as a tool to reduce the training complexity of pair-based loss functions based on an analogy between magnetism and deep metric learning; (2) the derivation of MeanFieldContrastive and MeanFieldClassWiseMultiSimilarity losses by application of the mean field theory to two pair-based loss functions; (3) the demonstration that the derived loss functions are competitive with existing baseline losses in several datasets. 2 RELATED WORK 2.1 PAIR-BASED LOSS FUNCTIONS Pair-based loss functions, a representative category of deep metric learning losses, leverage pairwise or triplet relationships among data points. Contrastive loss (Hadsell et al., 2006) is an early example of this type, which utilizes positive and negative pairs of data and learns embeddings to place the positive pairs close and negative ones apart. Triplet loss (Weinberger & Saul, 2009) is an extension of Contrastive loss, which exploits triplets of positive, negative, and anchor data, and places an anchor embedding closer to a positive than a negative. These losses are further extended to incorporate interactions among pairs in mini-batches to improve performance and convergence speed (Sohn, 2016; Oh Song et al., 2016; Wang et al., 2019a,b). For instance, MultiSimilarity loss (Wang et al., 2019b) takes into account multiple inter-pair relationships within a mini-batch, enabling more efficient learning of the embedding space. However, a major drawback of the pair-based losses is their sensitivity to the choice of positive and negative pairs, which is caused by the polynomial growth of the number of pairs and triplets with respect to the number of training data (Schroff et al., 2015). This usually requires sophisticated sampling and mining strategies for informative pairs to improve performance and mitigate slow convergence (Schroff et al., 2015; Shi et al., 2016; Hermans et al., 2017; Wu et al., 2017; Yuan et al., 2017; Harwood et al., 2017; Wang et al., 2019b). In this paper, we pursue an alternative approach to reduce training complexity rather than investigate these strategies. We accomplish this by leveraging the mean field theory, a concept from statistical physics. 2.2 CLASSIFICATION-BASED LOSS FUNCTIONS Classification-based loss functions utilize weight matrices and learn embeddings by optimizing a classification objective. Unlike pair-based losses, these losses do not face the complexity issue because they are computed in the same manner as a typical classification task. A representative example is NormalizedSoftmax loss (Wang et al., 2017; Zhai & Wu, 2018), which is obtained by the cross-entropy loss function with an L2-normalized weight matrix. Its extensions include SphereFace (Liu et al., 2017), ArcFace (Deng et al., 2019), and CosFace (Wang et al., 2018a,b) losses obtained by modifying distance metrics and introducing margins. The losses with proxies such as ProxyTriplet and ProxyNCA losses also belong to this category (Movshovitz-Attias et al., 2017). Such proxy losses can be derived from corresponding pair-based losses by substituting positive and negative data points with learnable embeddings called proxies while retaining anchors. ProxyAnchor loss (Kim et al., 2020) further considers interactions among samples in a mini-batch and shows promising performance in popular public datasets, surpassing other classification-based and pair-based loss functions. These losses have been extended to incorporate refined structures among data, such as graphs (Zhu et al., 2020), hierarchies (Yang et al., 2022), and others (Qian et al., 2019; Teh et al., 2020; Li et al., 2022). In this paper, we develop the mean field theory as a technique to derive a classification-based loss from a pair-based one, addressing the challenges of the latter. Although our approach is similar to the proxy-based method, it naturally adapts to pair-based losses without anchors, which have remained unexplored by the proxy-based method. 3 Proposed Approach In this section, we investigate the mean field theory and its application to deep metric learning. We first review the mean field theory for a ferromagnet in statistical mechanics by following standard statistical mechanics textbooks (e.g., see Nishimori & Ortiz, 2010). Next, based on the analogy between the ferromagnet and deep metric learning, we apply the mean field approximation to the Contrastive loss and a variant of the MultiSimilarity loss and derive classification-type loss functions with reduced training complexity. 3.1 Mean Field Theory for Magnets Magnetism is one of the representative phenomena in statistical physics that show a phase transition between ordered and disordered phases. A ferromagnet is composed of a large number of microscopic magnetic spins and shows a macroscopic magnetization when a macroscopic number of the magnetic spins are aligned in the same direction. To explain the mean field theory for a ferromagnet, we shall consider an infinite-range model$^1$ whose Hamiltonian (or energy) takes the following form (Nishimori & Ortiz, 2010): \[ H = -\frac{J}{2N} \sum_{i,j=1}^{N} S_i \cdot S_j, \] with the total number of spins $N \in \mathbb{N}$ and the exchange interaction $J > 0$. Here, $S_i$ represents the $i$-th constituent magnetic spin, which is regarded as a vector living on a sphere. Eq. (1) indicates that a state where spins point in the same direction is preferred energetically.$^2$ According to statistical mechanics, a probabilistic distribution of the spin configuration at temperature $T$ follows the Gibbs distribution: \[ P(\{S_i\}_i) = \frac{e^{-H/T}}{Z}, \quad Z = \int \prod_i d^2S_i e^{-H/T}, \] and macroscopic properties of this system can be computed from the normalization factor $Z$. However, since the spins are interacting with each other, it does not look easy to compute $Z$ both analytically and numerically. --- 1 Note that readers might worry that this model appears too simple (e.g., it lacks a notion of lattice structure). However, it is sufficient for explaining the phase transition of ferromagnets and analogous to loss functions in deep metric learning. 2 Since the Hamiltonian (or energy) of a magnetic moment $m$ in an applied magnetic field $B$ is typically given by $H = -m \cdot B$ (e.g., Zeeman effect (Sakurai & Commins, 1995)), Eq. (1) is thought of as the interaction between a spin $S_i$ and the magnetic field produced by the other spins. As a result, the interaction between spins is described by the cosine similarity, and this allows us to establish an analogy between discussions in statistical physics and deep metric learning. To address this difficulty, we introduce the mean field theory. The central idea of the theory is to approximate the Hamiltonian $ H $ such that each spin interacts with an average field generated by the rest of the spins, rather than with other spins directly, thereby ignoring their fluctuations. More concretely, it means that we expand $ H $ with respect to fluctuations $ \{ (S_i - M) \} $; using the identity, $ S_i = M + (S_i - M) $ and ignore the second-order terms of the expansion. This operation results in \[ H \simeq H_{\text{MFT}} = \frac{JN}{2} M \cdot M - JM \cdot \sum_{i=1}^{N} S_i. \] (3) Since $ H_{\text{MFT}} $ does not include interaction terms between spins, one can readily compute any information from the Gibbs distribution for this Hamiltonian as follows: \[ P_{\text{MFT}}(\{S_i\}_i) = \frac{e^{-H_{\text{MFT}}/T}}{Z_{\text{MFT}}}, \quad Z_{\text{MFT}} = \int \prod_i d^2 S_i e^{-H_{\text{MFT}}/T}. \] (4) The value of the mean field $ M $ must be determined to minimize $ -\log Z_{\text{MFT}} $. This condition is justified because we can show that it is equivalent to the so-called self-consistent equation \[ M = \frac{1}{N} \sum_i E[S_i] \] (5) by differentiating $ -\log Z_{\text{MFT}} $ with respect to the mean field. Here, we take the expectation value over the Gibbs distribution $ P_{\text{MFT}} $ in Eq. (5). Since the mean field approximation is based on the expansion with respect to the fluctuations around the mean field, Eq. (5) ensures the consistency of expansion. Overall, the mean field theory is a powerful tool that allows us to describe and analyze complex systems by approximating the interactions between individual constituents with an average field generated by the rest of the system. To draw a parallel between the above discussion and deep metric learning, let us consider the $ T \to 0 $ limit. In this limit, the original problem of computing $ Z $ becomes one of finding a spin configuration that minimizes $ H $. This is analogous to a machine learning problem that seeks optimal parameters to minimize a loss function. Then, the mean field approximation reduces the problem to one of minimizing $ H_{\text{MFT}} $ with respect to both the spins and mean field. Therefore, this observation indicates that applications of mean field theory to deep metric learning problems introduce mean fields as parameters learned to minimize their loss functions. ### 3.2 Mean Field Contrastive Loss To study how the mean field theory works for loss functions in deep metric learning, let us begin by applying the mean field theory to Contrastive loss for the sake of simplicity and then proceed to discuss the mean field theory for a more complicated loss function. In the following sections, we denote training data by $ D = \{ x_i, y_i \}_{i=1}^{\|D\|} $ composed of input data $ x_i $ and its class label $ y_i \in C = \{ 1, \cdots, \|C\| \} $. We also denote a set of data in class $ c \in C $ as $ D_c $. We extract features from the input data using a machine learning model $ F_\theta $, whose learnable parameters are represented by $ \theta $. This model embeds the input into a manifold $ M $, such as $ \mathbb{R}^d $ or $ S^d $, with $ d \in \mathbb{N} $. We also define the distance between two embeddings, $ F, F' \in M $, as $ d(F, F') \geq 0 $. For instance, the distance can be given by the cosine distance for $ M = S^d $, taking the form $ d(F, F') = 1 - F \cdot F'/(\\|F\\|_2 \\|F'\\|_2) $, or by the Euclidean distance for $ M = \mathbb{R}^d $. Contrastive loss is one of the primitive examples in deep metric learning, which is defined as \[ L_{\text{Cont.}} = \frac{1}{2\|C\|} \sum_{c \in C} \frac{1}{\|D_c\|^2} \sum_{i,j \in D_c} \left[ d(F_\theta(x_i), F_\theta(x_j)) - m_P \right]_+ \\ + \frac{1}{2\|C\|} \sum_{c \neq c'} \frac{1}{\|D_c\|\|D_{c'}\|} \sum_{i \in D_c, j \in D_{c'}} \left[ m_N - d(F_\theta(x_i), F_\theta(x_j)) \right]_+. \] (6) with $[x]_+ = \max(x, 0)$. Here, $m_P$ ($m_N$) is a hyperparameter that controls distances between positive (negative) instances. Note that Eq. (6) reduces to the Hamiltonian (1) when $\|C\| = 1$, $m_P < 0$, and $\mathcal{M} = S^2$, and it requires the $\mathcal{O}(\|\mathcal{D}\|^2)$ training complexity as it is parallel to the situation in Sec. 3.1. This analogy encourages us to apply the mean field theory in order to obtain a simpler loss function. Since we have multiple classes here, we shall introduce mean fields $\{\mathbf{M}_c\}_{c \in C}$ and expand $\mathcal{L}_{\text{Cont.}}$ with respect to fluctuations around them. Note that, in contrast to the single-class case, we must impose the following conditions to constrain relative distance among the mean fields: $$\left[m_N - d(\mathbf{M}_c, \mathbf{M}_{c'})\right]_+ = 0 \quad (c \neq c').$$ (7) This condition means that we should explore configurations of the mean fields which minimize $\mathcal{L}_{\text{Cont.}}$ at the zeroth order of expansions around the mean fields. In practice, we take into account these constraints softly. In the expansion around the mean fields, we ignore all cross-product terms of the fluctuations keeping any others so that we reduce the complexity while taking into account the higher-order terms of self-interactions. By summing over the remaining terms, we obtain MeanFieldContrastive (MFCont.) loss, which takes the following form: $$\mathcal{L}_{\text{MFCont.}} = \frac{1}{\|\mathcal{C}\|} \sum_{c \in \mathcal{C}} \frac{1}{\|\mathcal{D}_c\|} \sum_{i \in \mathcal{D}_c} \left(d(\mathbf{F}_\theta(x_i), \mathbf{M}_c) - m_P\right)_+ + \sum_{c' \neq c} \left[m_N - d(\mathbf{F}_\theta(x_i), \mathbf{M}_c)\right]_+ + \lambda_{\text{MF}} \sum_{c \neq c'} \left[m_N - d(\mathbf{M}_c, \mathbf{M}_{c'})\right]_+^2.$$ (8) where we impose the constraints (7) softly by $\lambda_{\text{MF}} > 0$. Note that resummation here naively produces unstable terms, $\{-[m_N - (d(\mathbf{M}_c, \mathbf{M}_{c'}))]_+\}_{c,c'}$, but they vanish, thanks to the constraints (7). We emphasize that we must minimize $\mathcal{L}_{\text{MFCont.}}$ by optimizing both $\mathbf{M}_c$ and $\theta$, and we can readily show that the optimal mean fields satisfy $\mathbf{M}_c = \sum_{i \in \mathcal{D}_c} \mathbf{F}_\theta(x_i)/\|\mathcal{D}_c\|$ at the first order of fluctuations. Note that this equation is inherently satisfied by the optimal solution. This point should be contrasted to the center loss (Wen et al., 2016), which necessitates the updating of class centers in every batch. Furthermore, in contrast to the proxy-based method, which can be applied only to a pair-based loss with an anchor, the mean field theory is applicable to wider types of pair-based loss functions as it is based on the Taylor expansions. ### 3.3 Mean Field Class-wise Multisimilarity Loss Lastly, we consider the mean field approximation of a loss function which incorporates interactions within a mini-batch similar to MultiSimilarity (Wang et al., 2019b) and ProxyAnchor (Kim et al., 2020) losses. However, the mean field approximation relies on expansions around mean fields, and thus, a loss function symmetric with respect to $x_i$ and $x_j$ (i.e., without anchors) would be more desirable for our purpose. (See the supplement for the application to a loss with an anchor.) Since most loss functions do not exhibit such a symmetric property, we propose the following loss function that satisfies these requirements: $$\mathcal{L}_{\text{CWMS}} = \frac{1}{\alpha \|\mathcal{C}\|} \sum_{c \in \mathcal{C}} \log \left[1 + \frac{\sum_{i,j \in \mathcal{D}_c, i \neq j} e^{\alpha (d(\mathbf{F}_\theta(x_i), \mathbf{F}_\theta(x_j)) - \delta)}}{2\|\mathcal{D}_c\|^2}\right] + \frac{1}{2\beta \|\mathcal{C}\|} \sum_{c \neq c'} \log \left[1 + \frac{\sum_{i \in \mathcal{D}_c, j \in \mathcal{D}_{c'}} e^{-\beta (d(\mathbf{F}_\theta(x_i), \mathbf{F}_\theta(x_j)) - \delta)}}{\|\mathcal{D}_c\|\|\mathcal{D}_{c'}\|}\right].$$ (9) Here, ‘resummation’ refers to the process of transforming an infinite series back into a function using a method such as Taylor expansion. An ‘unstable term’ is a term that could potentially violate the positivity of a loss function. Figure 1: Schematic illustration of interactions in CWMS (left) and MFCWMS (right) losses. Each color indicates a class to which an embedding and a mean field belong. with hyperparameters $\alpha > 0$, $\beta > 0$, and $\delta \in [-1, 1]$. Since Eq. (9) takes a similar form to MultiSimilarity loss (Wang et al., 2019b), but incorporates interactions among negative samples in a class-wise manner, we refer to it as ClassWiseMultiSimilarity (CWMS) loss. Next, we derive the mean field counterpart of this loss function. Here, the logits in the first and second terms take forms similar to the positive and negative interactions found in Contrastive loss (6). Repeating the discussion in Sec. 3.2, we derive MeanFieldClassWiseMultiSimilarity (MFCWMS) loss: $$ \mathcal{L}_{\text{MFCWMS}} = \frac{1}{\|\mathcal{C}\|} \sum_{c \in \mathcal{C}} \log \left[ 1 + \frac{\sum_{i \in \mathcal{D}_c} e^{\alpha(d(\mathbf{F}_\theta(x_i), \mathbf{M}_c) - \delta)}}{\|\mathcal{D}_c\|} \right] + \frac{1}{2\beta\|\mathcal{C}\|} \sum_{c \neq c'} \log \left[ 1 + \frac{\sum_{i \in \mathcal{D}_c} e^{-\beta(d(\mathbf{F}_\theta(x_i), \mathbf{M}_{c'}) - \delta)}}{\|\mathcal{D}_c\|} + \frac{\sum_{j \in \mathcal{D}_{c'}} e^{-\beta(d(\mathbf{M}_c, \mathbf{F}_\theta(x_j)) - \delta)}}{\|\mathcal{D}_{c'}\|} \right] + \frac{\lambda_{MF}}{\|\mathcal{C}\|} \sum_{c \neq c'} \left( \log \left[ 1 + e^{-\beta(d(\mathbf{M}_c, \mathbf{M}_{c'}) - \delta)} \right] \right)^2, $$ where we also introduce the soft constraint for the mean fields to ignore unstable terms produced in the resummation.\footnote{Rigorously speaking, we cannot minimize $\{e^{-\beta(d(\mathbf{M}_c, \mathbf{M}_{c'}) - \delta)}\}_{c,c'}$ simultaneously. However, focusing on the region with $\beta \gg 1$, we can easily find mean field configurations satisfying $e^{-\beta(d(\mathbf{M}_c, \mathbf{M}_{c'}) - \delta)} \ll 1$ at the same time. This is enough to ignore unstable terms in practice.} Compared to MeanFieldContrastive loss, this loss function incorporates interactions of positive samples as well as those of negative ones in a class-wise manner like ProxyAnchor loss. 4 EXPERIMENTS Let us see the effectiveness of the proposed mean field losses by evaluating their image-retrieval performance in several public datasets. We employ the recently proposed benchmarking scheme (Musgrave et al., 2020a) as well as the traditional one used in Movshovitz-Attias et al. (2017); Kim et al. (2020). We compare our mean field losses and existing loss functions, such as MultiSimilarity and ProxyAnchor losses. We also explore the effect of hyperparameters on their evaluation metrics. In our experiment, we use precision at 1 (P@1), R-precision (RP), and mean average precision at R (MAP@R) as evaluation metrics. In particular, we focus on MAP@R in the main paper (see the supplement for P@1 and RP) because it reflects the correctness of the ranking for retrievals and is a suitable metric to evaluate the quality of the embedding space (Musgrave et al., 2020a). Note that we implement our experiments in PyTorch (Paszke et al., 2019) and utilize the PyTorch Metric Learning library (Musgrave et al., 2020b) to implement baseline models. 4.1 DATASETS In our experiments, we utilize four publicly available image-retrieval datasets, CUB-200-2011 (CUB) (Wah et al., 2011), Cars-196 (Cars) (Krause et al., 2013), Stanford Online Products (SOP) (Oh Song et al., 2016), and InShop (Liu et al., 2016). CUB comprises 11,788 images of birds categorized into 200 classes, while the Cars dataset consists of 16,185 images of 196 car classes. In CUB, the first 100 classes (5,864 images) are used for the training dataset, and the remaining 100 classes (5,924 images) are allocated for the test dataset. Similarly, Cars was split into 8,054 training images (98 classes) and 8,131 test images (98 classes). The SOP dataset contains 22,634 classes with 120,053 product images. The initial 11,318 classes with 59,551 images are used for training, and the remaining 11,316 classes with 60,502 images are allocated for testing. Lastly, the InShop dataset features 52,712 images of 7,982 fashion products, with 25,882 images from 3,997 classes used for training and 26,830 images from 3,985 classes allocated for testing, which are further divided into query (14,218 images) and gallery (12,612 images) subsets. 4.2 IMPLEMENTATION DETAILS As a backbone embedding model $F_\theta(x)$, we employ the inception network with batch normalization (BN-Inception) (Ioffe & Szegedy, 2015), which is pretrained for the classification task on the ImageNet dataset (Russakovsky et al., 2015). We reduce the embedding dimensions by inserting a fully-connected layer with ReLU activation functions in the first scheme and replacing its last linear layer with that of desired dimensions in the second scheme. In both cases, we apply random resized cropping and random horizontal flipping to all inputs during training and only center cropping during evaluation. In the modern benchmarking protocol, we perform 50 iterations of Bayesian optimization for hyperparameters in loss functions including the learning rate for proxies and mean fields for a fair comparison. We split a dataset into train–valid (the first half classes) and test datasets (the remaining). The train–valid set was further divided into four partitions in a class disjoint manner, and we performed four-fold cross-validation based on the leave-one-out method in each iteration. In each cross-validation step, we train a model with embedding dimensions set to 128 and batch size set to 32 until MAP@R for the validation data converges. The Bayesian optimization aims to maximize the average of the four validation metrics. Note that we sample images so that each mini-batch is composed of 32 classes (8 classes) and 1 image (4 images) per class for classification-based (pair-based) losses, and we utilize the RMSprop optimizer with learning rate $10^{-6}$ for the embedding model. In the test stage, we perform cross-validation again with the best hyperparameters, resulting in four embedding models. Using these models, we evaluate performance on the test dataset in the following two different ways: mean of the metrics computed from the 128-dimensional (128D) embeddings (separated) and those from 512D embeddings made of the four 128D ones (concatenated). We repeat this observation 10 times and report their average values with 95% intervals. We carry out the experiments on a single NVIDIA V100 GPU. In contrast, in the traditional evaluation protocol, we use the predefined train–test splits described in Sec. 4.1 and train a model for up to 60 epochs with embedding dimensions 512 and batch size 128, setting the patience for early stopping to 5 to accelerate the experiments. In this case, we use AdamW optimizer (Loshchilov & Hutter, 2017) with the learning rate $10^{-4}$ for the embedding model, setting the learning rate for proxies to $10^{-5}$ and that for mean fields to $2 \times 10^{-1}$. The hyperparameters for ProxyAnchor loss are fixed to $(\alpha, \delta) = (32, 10^{-1})$, while we set $(m_P, m_N, \lambda_{MF}) = (0.02, 0.3, 0)$ for MFCont. loss and $(\alpha, \beta, \delta, \lambda_{MF}) = (0.01, 80, 0.8, 0)$ for MFCWMS loss in default. We chose these default parameters according to the results of the Bayesian optimization and the discussion in Sec. 4.4 and the supplement. We repeat the above procedure 10 times and report the averages of the metrics computed from the test embedding with the best MAP@R with 95% confidence intervals. The experiments on the CUB and Cars (SOP and InShop) datasets are carried out on a single NVIDIA V100 (A100) GPU. Note that we also present the results of the traditional evaluation protocol with VisionTransformer (Dosovitskiy et al., 2020) in the supplement. Table 1: MAP@R obtained from the modern protocol in CUB, Cars, and SOP. We carry out test runs 10 times and present the averaged metrics along with their confidence intervals. The best result within each block is underlined, while the overall best results for all losses are highlighted in bold. ProxyAnchor loss failed to converge in SOP in our settings. See the supplement for complete results. \| Loss \| CUB \| Cars \| SOP \| \|------------\|--------------\|--------------\|--------------\| \| \| 128D \| 512D \| 128D \| 512D \| 128D \| 512D \| \| ArcFace \| 21.5 ± 0.1 \| 26.4 ± 0.2 \| 18.3 ± 0.1 \| 27.6 ± 0.1 \| 41.5 ± 0.2 \| 47.4 ± 0.2 \| \| CosFace \| 21.2 ± 0.2 \| 26.5 ± 0.3 \| 18.5 ± 0.1 \| 27.0 ± 0.3 \| 41.0 ± 0.2 \| 46.8 ± 0.2 \| \| MS \| 21.0 ± 0.2 \| 26.2 ± 0.2 \| 18.7 ± 0.3 \| 27.2 ± 0.4 \| 41.9 ± 0.2 \| 46.7 ± 0.2 \| \| MS+Miner \| 20.8 ± 0.2 \| 25.9 ± 0.2 \| 18.5 ± 0.2 \| 26.9 ± 0.4 \| 41.9 ± 0.3 \| 46.6 ± 0.3 \| \| ProxyNCA \| 18.8 ± 0.2 \| 23.8 ± 0.2 \| 17.4 ± 0.1 \| 26.8 ± 0.2 \| 42.7 ± 0.1 \| 46.7 ± 0.1 \| \| ProxyAnch. \| 21.7 ± 0.2 \| 26.5 ± 0.2 \| 19.4 ± 0.2 \| 26.8 ± 0.3 \| – \| – \| \| Cont. \| 21.0 ± 0.1 \| 26.4 ± 0.2 \| 17.0 ± 0.3 \| 24.9 ± 0.5 \| 41.1 ± 0.2 \| 45.3 ± 0.2 \| \| MFCont. \| 22.0 ± 0.1 \| 27.2 ± 0.1 \| 18.1 ± 0.1 \| 27.4 ± 0.2 \| 43.6 ± 0.4 \| 47.0 ± 0.2 \| \| CWMS \| 21.5 ± 0.3 \| 26.9 ± 0.3 \| 19.3 ± 0.3 \| 27.8 ± 0.3 \| 41.5 ± 0.2 \| 45.1 ± 0.2 \| \| MFCWMS \| 22.1 ± 0.1 \| 27.0 ± 0.1 \| 18.9 ± 0.2 \| 27.0 ± 0.3 \| 44.6 ± 0.2 \| 48.3 ± 0.2 \| Table 2: MAP@R values and epochs with the best accuracies obtained using the traditional protocol on the CUB, Cars, SOP, and InShop datasets. The best result in each column is underlined. \| Loss \| CUB \| Cars \| SOP \| InShop \| \|------------\|--------------\|--------------\|--------------\|---------------\| \| \| MAP@R \| Epoch \| MAP@R \| Epoch \| MAP@R \| Epoch \| MAP@R \| Epoch \| \| Proxy Anch.\| 25.1 ± 0.2 \| 11.5 ± 1.4 \| 26.3 ± 0.2 \| 23.6 ± 1.8 \| 51.5 ± 0.3 \| 40.5 ± 6.5 \| 65.5 ± 0.1 \| 31.3 ± 7.2 \| \| MFCont. \| 25.3 ± 0.3 \| 4.4 ± 0.4 \| 24.7 ± 0.2 \| 10.1 ± 0.5 \| 52.9 ± 0.1 \| 25.1 ± 1.4 \| 67.7 ± 0.2 \| 24.1 ± 2.7 \| \| MFCWMS \| 25.3 ± 0.3 \| 4.7 ± 0.6 \| 24.0 ± 0.2 \| 8.5 ± 0.8 \| 52.7 ± 0.0 \| 23.0 ± 1.3 \| 67.5 ± 0.4 \| 20.6 ± 2.1 \| 4.3 Benchmark Results Based on the first protocol, we study the performance of our loss functions on three datasets; CUB, Cars, and SOP. We compare our loss functions with existing ones, such as Contrastive, MultiSimilarity, ArcFace, CosFace, ProxyNCA, and ProxyAnchor losses. Note that recently proposed losses with additional structures (Zheng et al., 2021; Ko et al., 2021; Deng & Zhang, 2022; Yang et al., 2022; Li et al., 2022) are not included as they are out of our focus. The experimental results are summarized in Table 1 (see the supplement for complete results). First, the mean field losses show better performance than their original pair-based losses in most cases, indicating that applying mean field theory not only reduces training complexity but also results in better embeddings. This is perhaps because the mean field losses can reduce the noise introduced in pairwise comparisons by the mean fields. Furthermore, the mean field losses consistently outperform other baseline methods in both separate and concatenated MAP@R for the CUB and SOP datasets. However, in Cars, ProxyAnchor and CWMS losses show better performance than the mean field losses, which might imply the importance of interactions within batch samples in this dataset. We also test the performance in the four datasets described in Sec. 4.1 following the traditional protocol. As shown in Table 2, our mean field losses outperform ProxyAnchor loss in MAP@R except for the Cars dataset, which is consistent with the first experiment. The improvement in accuracy is evident in the larger datasets. Besides, MFCont. and MFCWMS losses converge faster than ProxyAnchor loss in all the datasets. 4.4 Impact of Hyperparameters Embedding dimensions. Since embedding dimensions are crucial hyperparameters controlling the performance of image retrieval, we investigate their effect on the accuracy (MAP@R). On the CUB dataset, we run the traditional experiments for ProxyAnchor, Cont., CWMS, MFCont., and MFCWMS losses by varying the embedding dimensions from 32 to 1024. The results are shown in Fig. 2. We find that our mean field losses monotonically increase their performance and show better performance than the baselines in most cases. Compared to ProxyAnchor loss, the improvement is larger in relatively small dimensions (64, 128, and 256). β and δ of MFCWMS. We also explore the effect of β and δ of MFCWMS loss in the CUB dataset. We varied β from 50 to 90 and δ from 0.6 to 1, fixing $ (\alpha, \lambda_{MF}) $ to (0.01, 0) and computed the MAP@R for the test data. The results are summarized in Fig. 3. Fig. 3 ensures the competitive performance of the MFCWMS loss is stable against a choice of the hyperparameters. The preferred β gradually decreases as δ decreases. 5 Conclusion In this paper, we applied the mean field theory in statistical physics to Contrastive loss and ClassWiseMultiSimilarity loss, a variant of MultiSimilarity loss (Wang et al., 2019b) without anchors, and derived MeanFieldContrastive and MeanFieldClassWiseMultiSimilarity losses. We extensively evaluated the proposed loss functions and compared them with the existing baseline methods using both modern and traditional benchmark protocols. The evaluation results demonstrate that the proposed loss functions outperform the baselines in the CUB and SOP datasets in the former protocol, and in the CUB, SOP, and InShop datasets in the latter. These findings highlight the potential of mean field theory as a powerful tool for simplifying and improving deep metric learning performance in various machine learning applications. In future work, it would be worthwhile to explore applications of the proposed approach to deep metric learning in the multi-label setting (Kobayashi, 2023). Furthermore, since the mean field theory was originally introduced to elucidate the phase transition and scaling laws in ferromagnets, it would be also interesting to apply the mean field theory to explore the phase diagram of deep metric learning. ACKNOWLEDGMENTS We thank Yuki Saito, Ryosuke Goto, Masanari Kimura and Yuki Hirakawa for their useful comments on our manuscript.
ncbDXOdURn	In the current experiment, it is hard to say the advantage of the proposed WAKE comes from SWA or it comes specific property of adversarial training, then now it cannot well support the main contribution.	Characterizing Robust Overfitting in Adversarial Training via Cross-Class Features Anonymous authors Paper under double-blind review Abstract Adversarial training (AT) has been considered one of the most effective methods for making deep neural networks robust to adversarial attacks. However, AT can lead to a phenomenon known as robust overfitting where the test robust error gradually increases during training, resulting in a large robust generalization gap. In this paper, we present a novel interpretation of robust overfitting from the perspective of feature attribution. We find that at the best checkpoint of AT, the model tends to involve more cross-class features, which are shared by multiple classes, in its decision-making process. These features are useful for robust classification. However, as AT further squeezes the training robust loss, the model tends to make decisions based on more class-specific features, giving rise to robust overfitting. We also provide theoretical evidence for this understanding using a synthetic data model. In addition, our understanding can also justify why knowledge distillation is helpful for mitigating robust overfitting, and we further propose a weight-average guided knowledge distillation AT approach for improved robustness. 1 Introduction As the existence of adversarial examples (Goodfellow et al., 2014) has led to significant safety concerns of deep neural networks (DNNs), a series of methods (Papernot et al., 2016; Cohen et al., 2019; Chen et al., 2023) for defending against this threat have been proposed. Adversarial training (AT) (Madry et al., 2017), which adds adversarial perturbations to samples in the training loop and encourages the model to distinguish these perturbed samples, has been considered one of the most effective ways to make the DNNs more robust to adversarial attacks (Athalye et al., 2018). AT can be formulated as the following min-max optimization problem: $$\min_{\theta} L(\theta), \quad \text{where} \quad L(\theta) = \frac{1}{N} \sum_{i=1}^{N} \max_{\\|\delta_i\\|_p \leq \epsilon} \ell(f(\theta, x_i + \delta_i), y_i),$$ (1) where $\theta$ represents the model parameter, $\ell$ is the loss function (such as cross-entropy loss), $(x_i, y_i)$ is the $i$-th sample-label pair in the training set for $1 \leq i \leq N$, and $\epsilon$ is the perturbation bound. Despite the success in improving adversarial robustness, AT can also lead to a phenomenon known as robust overfitting (Rice et al., 2020). During AT, a model may achieve its best test robust error at a certain epoch, but the test robust error will gradually increase in the latter stage of training. By contrast, the training robust error consistently decreases, resulting in a large robust generalization gap. As robust overfitting exposes a fundamental limitation in AT, several techniques have been introduced to address this issue, such as knowledge distillation (Chen et al., 2021). However, there is still a lack of complete understanding regarding the underlying mechanism of how such robust overfitting occurs. In this paper, we characterize the phenomenon of robust overfitting from the perspective of feature attribution. Specifically, we divide the features learned by the model into cross-class features and class-specific features. The cross-class features are shared among multiple classes in the classification task, e.g. the feature wheels shared by the automobile and truck classes in the CIFAR-10 dataset. We investigate how these features are used in the decision-making process of the model in AT. Intriguingly, we observe that at the best checkpoint during AT, the model relies more on cross-class... features than at later checkpoints. In contrast, at later checkpoints where robust overfitting occurs, the model tends to make decisions based on more class-specific features that are specified to only one class. Motivated by this observation, we propose a novel interpretation of robust overfitting. During the initial stage of AT, the model learns both class-specific and cross-class features simultaneously, since these features are both helpful for reducing robust loss when this loss is large. However, as training progresses and the robust loss decreases to a certain degree, the model begins to abandon cross-class features and makes decisions based mainly on class-specific features. This is because cross-class features raise positive logits on other classes and yield non-zero robust loss in AT. Therefore, the model tends to abandon these features to further decrease the robust loss. However, these cross-class features are helpful for robust classification (e.g., a feature shared by classes $y_1, y_2$ helps the model distinguish samples from class $y_1$ to other classes $y_3, \cdots, y_n$), and using only class-specific features is insufficient to achieve the best robust accuracy. This results in a decline in robust test accuracy and leads to robust overfitting. We provide both empirical and theoretical evidence to support this interpretation. First, we propose a metric to characterize the usage of the cross-class features for a certain model. Then, among different perturbation norms, datasets, and architectures, we show that the overfitted models consistently tend to use fewer cross-class features. We further provide theoretical evidence to support this understanding using a synthetic dataset that decouples cross-class and class-specific features. In our theoretical framework, we show that cross-class features are more sensitive to robust loss, but they are indeed helpful for robust classification. In addition, our understanding can justify how knowledge distillation helps alleviate robust overfitting (Chen et al., 2021) by showing that knowledge distillation can preserve cross-class features during AT. Furthermore, we aim to introduce a better teacher model to characterize more precise cross-class features. Motivated by the fact that weight averaging can improve robustness in AT (Wang & Wang, 2022), we propose utilizing such a model as the teacher model for better knowledge distillation in AT. Experiment demonstrates that our approach exhibits better robustness performance than previous approaches. Our contributions can be summarized as follows: 1. We propose a novel interpretation of robust overfitting in AT. We show that a key factor of robust overfitting is that in order to achieve lower robust loss, the model tends to reduce the reliance on cross-class features, which are actually helpful for robust classification. 2. We provide both empirical and theoretical evidence to support our proposed understanding. Empirically, we illustrate that overfitted models in AT use fewer cross-class features than the best checkpoints. We also substantiate these assertions in a synthetic data model with decoupled cross-class and class-specific features. 3. Our understanding also shows that knowledge distillation helps mitigate robust overfitting by preserving these features. Considering weight-averaged models can provide better information on cross-class features, we propose to use such models for knowledge distillation in AT for improved robustness. 2 BACKGROUND AND RELATED WORK 2.1 ADVERSARIAL TRAINING AND ROBUST OVERFITTING Adversarial training (AT) has been widely recognized as one of the most effective approaches to improving the robustness of models. The optimization objective of AT is shown in equation (1). For the inner maximization, Projected Gradient Descent (PGD) is generally used to craft the adversarial example: $$x^{t+1} = \Pi_{B(x, \epsilon)}(x^t + \alpha \cdot \text{sign}(\nabla_x \ell(\theta; x^t, y))),$$ where $\Pi$ is the function that projects the sample onto an allowed region of perturbation, i.e., $B(x, \epsilon) = \{x' : \\|x' - x\\|_p \leq \epsilon\}$, and $\alpha$ controls the step size of gradient ascent. However, AT suffers from the problem of robust overfitting (Rice et al., 2020). As shown in Figure 1, the model may perform best on the test dataset at a certain epoch during AT, but in the later stages, the model’s performance on the test data gradually worsens. Meanwhile, the model’s robust error on the training data continues to decrease, leading to a significant generalization gap in adversarial training. Moreover, the commonly used perturbation bound $\epsilon$ (e.g., $[0, 8/255]$ for $\ell_\infty$-norm) in AT, a relatively large $\epsilon$ suffers from more severe robust overfitting. By contrast, for a small $\epsilon = 2/255$, this effect is relatively less pronounced. ### 2.2 Understanding and Alleviating Robust Overfitting To address the robust overfitting issue in AT, several techniques have been introduced from various perspectives. For example, introducing low curvature activation (Singla et al., 2021), data augmentation (Rebuffi et al., 2021b; Li & Spratling, 2023) and temporal ensembling (Dong et al., 2022) are helpful to mitigate robust overfitting. One series of works attempted to understand and alleviate this overfitting by attributing robust overfitting to the sharpness of the weight loss landscape (Li et al., 2018) and propose to introduce flatness as a regularization (Wu et al., 2020; Yu et al., 2022) to mitigate this effect. Another representative method is injecting smoothing during AT (Chen et al., 2021), which introduces knowledge distillation (Hinton et al., 2015) in AT to smooth the logits and leverage stochastic weight averaging (SWA) (Izmailov et al., 2018) to smooth the weights. The loss function of AT with knowledge distillation can be formulated as $$\min_{\theta} \mathbb{E}_{(x,y) \sim D_{train}} \left[ \max_{\|\|\delta\|\|_p \leq \epsilon} \tilde{\ell}(\theta; \theta_1, \theta_2, x + \delta, y) \right],$$ where $$\tilde{\ell}(\theta; \theta_1, \theta_2, x + \delta, y) = (1 - \lambda_1 - \lambda_2)\ell_{CE}(f(\theta, x + \delta), y) + \sum_{i=1}^{2} \lambda_i KD(f(\theta, x + \delta), f(\theta_i, x + \delta))$$ (3) where $\ell_{CE}$ is the cross-entropy loss, and $KD$ is the knowledge distillation function (details in Appendix G.2), and $\theta_1$ and $\theta_2$ are the robust-/standard-trained self-teachers, respectively. SWA can be expressed as $$\theta_{SWA}^T = \frac{n \theta_{SWA}^{T-1} + \theta^T}{n + 1},$$ (4) where $T$ is the current training epoch, $n$ is the number of checkpoints involved in weight averaging and $\theta_{SWA}$ represents the averaged model parameter. While these methods have been proven useful in mitigating robust overfitting, there is still a lack of comprehensive understanding of the underlying mechanisms of how robust overfitting occurs and why knowledge distillation is useful in mitigating it. ### 3 Proposed Understanding In this section, we elaborate on our proposed understanding of robust overfitting in AT via cross-class features. We first present a metric of cross-class feature usage for a model in AT. Then, with comprehensive empirical evidence, we demonstrate how robust overfitting occurs based on the dynamics of the model learns and abandons these features during AT. 3.1 Measuring the Usage of Cross-Class (Robust) Features Consider a $K$-class classification task where data from each class $y \sim Y$ has a data distribution $x \sim D_y$. Let $f(\cdot) = Wg(\cdot)$ represent a classifier, where $g$ is the feature extractor with $n$ dimension and $W \in \mathbb{R}^{K \times n}$ is the linear layer. For a given sample $x$ from the $i$-th class, the output logit for the $i$-th class is $f(x)_i = W[i]^T g(x) = \sum_{j=1}^{n} g(x)_j W[i,j]$, where $W[i]$ is the $i$-th row of $W$. Intuitively, $g(x)_j W[i,j]$ represents how the $j$-th feature influences the logit of the $i$-th class prediction of $f(x)$. Thus we use $A_i(x) = (g(x)_1 W[i,1], \ldots, g(x)_n W[i,n])$ as the attribution vector for the sample $x$ on class $i$, where the $j$-th element denotes the weight of the $j$-th feature. Characterizing Cross-class Features We consider the similarity of attribution vectors. If the attribution vectors of samples $x_1$ and $x_2$ are highly similar, the model tends to use more features shared by them when calculating their logits on their classes. On the other hand, if the attribution vectors of $x_1$ and $x_2$ are almost orthogonal, the model uses fewer shared features or they just do not share features. This observation can be generalized to the classes. We model the feature attribution vector of a given class as the average of the vectors of the test samples in this class. Further, since we are considering feature attribution in the context of adversarial robustness, we only consider the attribution of robust features (Tsipras et al., 2018) for classifying adversarial examples. Thus, we craft adversarial examples and analyze their attributions to measure the usage of shared robust features. As discussed, we can measure the usage of cross-class robust features shared by two given classes with the similarity of their attribution vectors. Therefore, we construct the feature attribution correlation matrix using the cosine similarity between the attribution vectors: $C[i,j] = \frac{A^i \cdot A^j}{\\|A^i\\|_2 \\|A^j\\|_2}$. The complete algorithm of calculating matrix $C$ is shown in Algorithm 2 in Appendix. For two classes indexed by $i$ and $j$, $C[i,j]$ denotes the similarity of their feature attribution vector, which a higher value indicates the model uses more features shared by these classes. Numerical Metric To further support our claims, we propose a numerical metric named Class Attribution Similarity (CAS) defined on the correlation matrix $C$: $CAS(C) = \sum_{i \neq j} \max(C[i,j], 0)$. The $\max$ function is used since we only focus on the positive correlations, and the negative elements are small (see Figure 2) and do not affect our analysis. CAS can quantitatively reflect the usage of cross-class features for a certain checkpoint. 3.2 Characterizing Robust Overfitting through Cross-Class Features Based on the proposed measurement, we first visualize the feature attribution correlation matrices of vanilla AT. The model is trained on the CIFAR-10 dataset (Krizhevsky et al., 2009) using PreActResNet-18 (He et al., 2016) for 200 epochs, and it achieved its best test robust accuracy at the 108th epoch. More details can be found in Section 5.2. As shown in Figure 2, the model demonstrates a fair overlapping effect on feature attribution at the 70th epoch. Specifically, there are several non-diagonal elements $C[i,j]$ in the correlation matrix $C$ that exhibit a relatively large value (in deeper blue), which indicates that the model leverages more features shared by the classes indexed by $i$ and $j$ when classifying adversarial examples from these two classes. Therefore, the model has already learned several cross-class features in the initial stage of AT. Moreover, when the model achieves its best robustness at the 108th epoch, the overlapping effect on feature attribution becomes clearer, with more non-diagonal elements in $C$ exhibiting larger values. This is also verified by the increase in CAS. However, at the end of AT, where the model is overfitted, the overlapping effect significantly decays, which indicates the model uses fewer cross-class features. We provide more correlation matrices of the model at different epochs in Appendix B. This surprising effect motivates us to propose the following interpretation of robust overfitting. We identify two kinds of learning mechanisms in AT: (1) Learning class-specific features, i.e., the features that are exclusive to only one class; (2) Learning cross-class features, i.e., the same or similar features shared by more than one class. During the initial phase of AT, the model simultaneously learns exclusive class-wise features and cross-class features. Both of these features help achieve robust generalization and reduce training robust loss. However, once the training robust error is reduced to a certain degree, it becomes difficult for the model to further decrease it by optimizing cross-class features. This is because the features shared with other classes tend to raise positive logit on the shared classes. Thus, to further reduce the training robust loss, the model begins to reduce its reliance on cross-class features and bias more weight on class-specific features. Meanwhile, due to the strong memorization ability of DNNs in AT (Dong et al., 2022), the model also memorizes the training samples along with their corresponding adversarial examples, which further reduces the training robust error. This overall procedure can optimize training robust error but can also hurt test robust error by forgetting cross-class features, leading to a decrease in test robust accuracy and resulting in robust overfitting. We further provide more comprehensive empirical evidence on this explanation in the following. ### 3.3 More comprehensive study In this section, we conduct a more comprehensive study of our proposed understanding with various empirical evidence. ![Figure 3](image) Figure 3: The differences between the feature attribution correlation matrices ($C_{\text{best}} - C_{\text{last}}$) and CAS of the best and the last checkpoint with various training perturbation bound $\epsilon$. Comparing with different perturbation bound $\epsilon$ In Figure 3, we show the differences of the feature attribution correlation matrices and CAS between the best and last checkpoint of AT with various perturbation bounds $\epsilon$. The difference between the two matrices indicates how many cross-class features are abandoned by the model from the best checkpoint to the last. When $\epsilon = 2/255$, there is no significant difference between the best and last checkpoint. This is consistent with the fact that AT with small $\epsilon$ does not severely overfit, as shown in Figure 1. However, as $\epsilon$ increases, AT exhibits more overfitting effects, and the difference becomes more significant. This also verifies that the forgetting of cross-class features is a key factor of robust overfitting. We offer a further explanation as to why larger perturbations cause more severe robust overfitting. Intuitively, AT with a larger perturbation bound $\epsilon$ results in a more rigid robust loss. During AT with a large $\epsilon$, cross-class features are more likely to be eliminated by the model to reduce training robust loss. We prove this claim in Theorem 1 in the next section. While we mainly focus on AT with practically used $\epsilon$ (e.g., $[0, 8/255]$ for $\ell_\infty$-AT), it is also observed that for extremely large $\epsilon (> 8/255)$, the effect of robust overfitting begins to decline (Wei et al., 2023a). Our interpretation is also compatible with this phenomenon, which we discuss in Appendix C. In brief, cross-class features are more sensitive under extremely large $\epsilon$, making them even harder to learn at the initial stage resulting in fewer forgetting of these features in the latter stage of AT. Comparing on other norms, datasets, and architectures We also investigate this effect in AT with $\ell_2$-norm, CIFAR-100 (Krizhevsky et al., 2009) and TinyImagenet datasets (mnousta, 2017), and vision transformer architecture (Touvron et al., 2021). Due to the space limitation, we leave the compared feature attribution correlation matrices and their corresponding CAS in Appendix D. Interestingly, similar to the effect demonstrated in $\ell_\infty$-norm AT with convolutional architecture (PreActResNet-18) on the CIFAR-10 dataset, in these settings the best checkpoints consistently use more cross-class features than the last checkpoints, verifies that our proposed understanding also holds in AT under various settings. Visualization of saliency map To further analyze the feature attribution of AT at different stages, we compare the saliency maps on several examples that are correctly classified by the best but misclassified by the last checkpoint under adversarial attack, as shown in Figure 4 (a). The saliency map is derived by Grad-CAM (Selvaraju et al., 2017) on the true labeled classes. Taking the first column as an example, the classes automobile and truck share similar features like wheels. The best checkpoint pays more attention to the overall car including the wheel, whereas the last checkpoint solely focuses on the circular car roof that is exclusive to automobiles. This explains why the last checkpoint misclassifies this sample, for it only identifies this local feature for the true class and does not leverage holistic feature information from the image. The other five samples also exhibit a similar effect, with exclusive features being the mane for horse, the frog eyes for frog, the feather for bird, and the antlers for deer. Since the final checkpoint makes decisions based only on these limited features, it fails to leverage comprehensive features for classification, making the model more vulnerable to adversarial attacks. ![Figure 4](image) Knowledge distillation mitigates robust overfitting Our understanding can also explain why knowledge distillation is a helpful technique for mitigating robust overfitting. In the process of AT with knowledge distillation, the teacher model adeptly captures the cross-class features present in the training data, and provides more precise labels by considering both class-specific and cross-class features. This stands in contrast to vanilla AT with one-hot labels, which primarily emphasizes class-specific features and may inadvertently suppress cross-class features in the model weights. The incorporation of cross-class features, backed by both our empirical findings and theoretical insights highlighting their significance for enhanced robustness, enables knowledge distillation to effectively mitigate robust overfitting by preserving these crucial features. We present a comparison between the best and last checkpoint of AT with knowledge distillation in Figure 4 (b) and (c), where no significant differences between the two matrices, nor a large gap between their CAS. Therefore, we conclude that AT with knowledge distillation helps mitigate robust overfitting by identifying cross-class features and providing more precise labels by considering these features. 4 THEORETICAL INSIGHTS In this section, we provide theoretical evidence with a synthetic data model. 4.1 DATA DISTRIBUTION AND HYPOTHESIS SPACE In this theoretical framework, we introduce a data distribution with class-specific and cross-class feature decomposition, along with a hypothesis space with linear functions. Data distribution We consider a tertiary classification task, where each class owns an exclusive feature $ x_{E,i} $, and every two classes have a shared cross-class feature $ x_{C,j} $. The features for each sample can be formulated as $ \{x_{E,j}, x_{C,j}\mid 1 \leq j \leq 3\} \in \mathbb{R}^6 $. The data distribution is similar to the model applied in robust and non-robust features (Tsipras et al., 2018), but we only focus on the inner relation between robust features (class-specific or cross-class) and omit the non-robust features. As discussed above, we model the data distribution of the $ i $-th class $ y_i $ as $ D_i = $: \[ x_{E,j} \mid y = i \sim \begin{cases} N(\mu, \sigma^2) & \text{if } j = i \\ 0 \text{ w.p. 1} & \text{if } j \neq i \end{cases}, \quad x_{C,j} \mid y = i \sim \begin{cases} N(\mu, \sigma^2) & \text{if } j \neq i \\ 0 \text{ w.p. 1} & \text{if } j = i \end{cases}, \] where $ i \in \{1, 2, 3\} $, and $ \mu, \sigma > 0 $. We also assume $ \sigma < \sqrt{\pi} \mu $ to control the variance. Hypothesis space We introduce a linear model $ f(x) $ in this classification task, which gives $ i $-th logit for sample $ x $ by $ f(x)_i = \sum_j w_{E,j}^i x_{E,j} + \sum_j w_{C,j}^i x_{C,j} $. However, there are 6 parameters in the data samples, making this linear model hard to analyze. Thus we simplify the model based on the following observations. First, we can simply keep $ w_{E,j}^i = 0 $ for $ i \neq j $ and $ w_{C,j}^i = 0 $ due to the corresponding data distribution is identity to 0. Further, we set $ w_{E,1}^i = w_{E,2}^i = w_{E,3}^i = w_1 $ and $ w_{C,j}^i = w_2(i \neq j) $ due to symmetry. Finally, we assume $ w_1, w_2 \geq 0 $ since $ \mu > 0 $. Overall, the hypothesis space is $ \{f_w : w = (w_1, w_2), w_1, w_2 \geq 0\} $ and $ f_w(x) $ calculates its $ i $-th logit by \[ f_w(x)_i = w_1 x_{E,i} + w_2 (x_{C,j_1} + x_{C,j_2}), \quad \text{where } \{j_1, j_2\} = \{1, 2, 3\} \setminus \{i\}. \] Now we consider adversarially training $ f_w $ with $ \ell_\infty $-norm perturbation bound $ \epsilon < \frac{\mu}{2} $. We also add a regularization term $ \frac{\lambda}{2} \\|w\\|_2^2 $ to the overall loss function, which can be modeled as \[ \mathbb{E}_{x \sim p_y} \left[ \max_{\\|\delta\\|_\infty \leq \epsilon} \ell(w; x + \delta) \right] + \frac{\lambda}{2} \\|w\\|_2^2, \] where \[ \ell(w; x + \delta) = \max_{\\|\delta\\|_\infty \leq \epsilon} (\max_{j \neq i} f_w(x + \delta)_j - f_w(x + \delta)_i). \] 4.2 MAIN RESULTS Cross-class features are more sensitive to robust loss We show that under the robust training loss (7), the model tends to abandon $ x_C $ by setting $ w_2 = 0 $ if $ \epsilon $ is larger than a certain threshold. However, any $ \epsilon \in (0, \frac{\mu}{2}) $ returns a positive $ w_1 $, as stated in Theorem 1. This result indicates that cross-class features are more sensitive to robust loss and are more likely to be eliminated in AT compared to class-specific features, even when they share the same mean value $ \mu $. Theorem 1 There exists a $ \epsilon_0 \in (0, \frac{1}{2} \mu) $, for AT by optimizing the robust loss (7) with $ \epsilon \in (0, \epsilon_0) $, the output function obtains $ w_2 > 0 $; for AT with $ \epsilon \in (\epsilon_0, \frac{1}{2} \mu) $, the output function returns $ w_2 = 0 $. By contrast, AT with $ \epsilon \in (0, \frac{1}{2} \mu) $ always obtains $ w_1 > 0 $. This claim is also consistent with our discussion on AT with different $ \epsilon $ in Section 3.3. Recall that AT with larger $ \epsilon $ tends to compress more cross-class features as shown in Figure 3. This observation can be verified by Theorem 1 that cross-class features are more likely to be eliminated during AT with larger $ \epsilon $, which causes more severe robust overfitting. Cross-class features are helpful for robust classification Although decreasing the value of $ w_2 $ may reduce the robust training error, we demonstrate in Theorem 2 that using a positive $ w_2 $ is always more beneficial for robust classification than simply setting $ w_2 $ to 0. Theorem 2 For any class $ y $, consider weights $ w_1 > 0, w_2 \in [0, w_1] $, and $ \epsilon \in (0, \frac{\mu}{2}) $. When sampling $ x $ from the distribution of class $ y $, increasing the value of $ w_2 $ enhances the possibility of the model assigning a higher logit to class $ y $ than to any other classes $ y' \neq y $ under adversarial attack. In other words, the probability $ \Pr_{x \sim D_y} [f_w(x + \delta))_y > f_w(x + \delta)_{y'}, \forall \delta : \\| \delta \\|_\infty \leq \epsilon ] $ monotonically increases with $ w_2 $ within the range $[0, w_1]$. Knowledge distillation preserves cross-class features Finally, we show that knowledge distillation helps preserve the cross-class features, which provide a justification on why this method can alleviate robust overfitting. Note that due to the symmetry of distributions and weights among classes, we apply label smoothing to simulate knowledge distillation (which we justify in Section E.4 in detail) and rewrite the robust loss as $ \mathbb{E}_{x \sim p_y} \left[ \max_{\\| \delta \\|_\infty \leq \epsilon} \ell_{LS}(w; x + \delta) \right] + \frac{\lambda}{2} \\| w \\|_2^2 $, where \[ \ell_{LS}(w; x + \delta) = (1 - \beta) \left[ \max_{\\| \delta \\|_\infty \leq \epsilon} (\max_{j \neq i} f_w(x + \delta)_j - f_w(x + \delta)_i) \right] - \frac{\beta}{2} \sum_{j \neq i} f_w(x + \delta)_j \] and $ \beta < \frac{1}{3} $ is the interpolation ratio of label smoothing. In Theorem 3 and Corollary 1, we show that not only the label smoothed loss (8) enables larger perturbation bound $ \epsilon $ for utilizing cross-class features, but also returns larger $ w_2 $. This explains that preserving the cross-class features is the reason why knowledge distillation helps mitigate robust overfitting. Theorem 3 Consider AT with knowledge distillation loss (8). There exists an $ \epsilon_1 \in (0, \frac{\mu}{2}) $ with $ \epsilon_1 > \epsilon_0 $ derived in Theorem 1, such that for $ \epsilon \in (0, \epsilon_1) $, the output function obtains $ w_2 > 0 $; for $ \epsilon \in (\epsilon_1, \frac{\mu}{2}) $, the output function returns $ w_2 = 0 $. Corollary 1 Let $ w_2^(\epsilon) $ be the value of $ w_2 $ returned by AT with (7), and $ w_2^{LS}(\epsilon) $ be the value of $ w_2 $ returned by label smoothed loss (8). Then, for $ \epsilon \in (0, \epsilon_1) $, we have $ w_2^{LS}(\epsilon) > w_2^(\epsilon) $. All proofs can be found in Appendix E. To summarize, our theoretical analysis demonstrates that cross-class features are more sensitive to robust loss, yet helpful for robust classification. We also show that knowledge distillation can mitigate robust overfitting by preserving the cross-class features. 5 Better Knowledge Distillation Further Improves Robustness In this section, we propose an improved knowledge distillation approach to further enhance adversarial robustness and mitigate robust overfitting in AT. 5.1 Weight Average Guided Knowledge Distillation Based on our understanding which explains that knowledge distillation can help alleviate robust overfitting by preserving cross-class features, we aim to introduce a better teacher model for knowledge distillation which can characterize more precise cross-class feature distribution. Motivated by the fact that weight-averaged models exhibit better robustness in AT (Wang & Wang, 2022), we propose leveraging the weight-averaged model as the teacher model for knowledge distillation, which also outperforms vanilla knowledge distillation in terms of computational cost since it does not require a pre-trained robust model. The loss function is similar to Equation (3), but with the robust-trained teacher replaced by an averaged model and the standard-trained teacher removed. The loss function can be formulated as \[ \max_{\\| \delta \\|_\infty \leq \epsilon} \tilde{\ell}(\theta ; \bar{\theta}, x+\delta, y), \quad \text{where } \tilde{\ell}(\theta ; \bar{\theta}, x+\delta, y) = (1-\lambda)\ell_{CE}(f(\theta, x+\delta), y) + \lambda K\mathcal{D}(f(\theta, x+\delta), f(\bar{\theta}, x+\delta)) \] where $ \bar{\theta} $ represents the parameter of the weight averaged model in AT, and $ \lambda $ is the interpolation ratio. However, some modifications are needed. First, the weight-averaged model requires warm-up before it is applied for knowledge distillation. Second, if we directly start to apply loss (9) at a specific checkpoint, we observe a catastrophic forgetting that the test accuracy drops significantly, which may be due to a drastic change in the loss function. Therefore, we introduce a (piecewise) linear Table 1: Comparison of our method with vanilla AT and AT+KDSWA. \| Dataset \| Method \| Robust Acc. (%) \| Clean Acc. (%) \| \|--------------\|------------\|-----------------\|----------------\| \| \| \| Best \| Last \| Best \| Last \| \| CIFAR-10 \| AT \| 47.8 ±0.2 \| 42.5 ±0.2 \| 82.7 ±0.5 \| 84.5 ±0.3 \| \| \| AT + KDSWA \| 49.8 ±0.4 \| 49.6 ±0.2 \| 83.8 ±0.6 \| 84.7 ±0.4 \| \| \| AT + WAKE \| 50.4 ±0.3 \| 50.1 ±0.2 \| 83.9 ±0.3 \| 84.9 ±0.3 \| \| CIFAR-100 \| AT \| 24.7 ±0.2 \| 19.6 ±0.3 \| 55.6 ±0.5 \| 57.4 ±0.2 \| \| \| AT + KDSWA \| 26.1 ±0.3 \| 25.7 ±0.2 \| 58.6 ±0.5 \| 59.1 ±0.2 \| \| \| AT + WAKE \| 26.8 ±0.3 \| 26.5 ±0.2 \| 59.5 ±0.4 \| 59.7 ±0.1 \| \| Tiny-Imagenet\| AT \| 18.0 ±0.3 \| 14.4 ±0.4 \| 45.5 ±0.6 \| 48.3 ±0.4 \| \| \| AT + KDSWA \| 19.9 ±0.3 \| 19.4 ±0.3 \| 49.7 ±0.4 \| 50.4 ±0.3 \| \| \| AT + WAKE \| 20.4 ±0.2 \| 19.9 ±0.2 \| 50.2 ±0.3 \| 50.8 ±0.2 \| scheduler to set the $\lambda$ in (9) to stabilize the training process. The $\lambda$ is set to 0 initially, and then gradually increases to the target after a certain checkpoint. Overall, we name our proposed method as Weight Average guided Knowledge distillation (WAKE), and the complete algorithm is elaborated in Algorithm 2 in Appendix G. 5.2 Experiment Settings We conduct experiment on CIFAR-{10, 100} (Krizhevsky et al., 2009) and Tiny-Imagenet (mmoustafa, 2017) datasets using PreActResNet-18 (PRN-18) (He et al., 2016) model. Following the best settings in (Rice et al., 2020), we train the model using SGD with a momentum of 0.9, weight decay of $5 \times 10^{-4}$, and an initial learning rate of 0.1. We compare our method with vanilla AT and KDSWA (Chen et al., 2021). Following the same settings as in AT+KDSWA, we train 200 epochs for CIFAR datasets and 100 epochs for Tiny-Imagenet. During AT, we apply a 10-step PGD attack with an $\ell_\infty$-norm perturbation bound $\epsilon = 8/255$ and a step size of $\alpha = 2/255$. For WAKE, we set the maximum interpolation ratio to $\lambda = 0.8$ and the knowledge distillation temperature to $T = 2$ for WAKE. The distillation warm-up starts and ends at epochs 90 and 110 for CIFAR datasets and at epochs 40 and 60 for Tiny-Imagenet. We use AutoAttack (AA.) (Croce & Hein, 2020) for reliable robustness evaluation, and conduct five independent experiments for each method and report the mean result and standard deviation. We also conduct experiments on AT with $\ell_2$-norm, CIFAR-100 and TinyImagenet dataset, and vision transformer architecture, and show the results in Appendix F. Improving robustness and alleviating overfitting Table 1 shows the overall comparison of our method and the baselines. From the results, we can see that in terms of adversarial robustness, our AT+WAKE outperforms vanilla AT and AT+KDSWA in all three datasets, both at the best and last checkpoints, since the better teacher models can characterize more precise cross-class features. In addition, regarding clean accuracy, AT+WAKE also outperforms vanilla AT and AT+KDSWA on these datasets, showing that our method achieves a better clean vs. robustness trade-off (Tsipras et al., 2018; Zhang et al., 2019). Overall, our proposed WAKE further improves adversarial robustness and mitigates robust overfitting in AT, and also has the advantage of lower computational cost. 6 Conclusion In this paper, we provide a novel interpretation of robust overfitting in AT through the lens of feature attribution. We point out that during AT, in order to achieve lower training robust loss, the model’s tendency to reduce its reliance on cross-class features is a key factor in robust overfitting. We empirically verify this claim by measuring the dependence on cross-class features of the model at different stages in AT under various settings, along with other empirical evidence including analysis of saliency maps and knowledge distillation-based methods. 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w5oP27fmYW	Test-Time Centering: The suggested method modifies both the training and testing processes, suggesting that the improvement is due to better utilization of network capacity. I wonder if this zero-centering could be applied solely as a test-time inductive bias instead?	CCD-3DR: Consistent Conditioning in Diffusion for Single-Image 3D Reconstruction Anonymous authors Paper under double-blind review Abstract In this paper, we present a novel shape reconstruction method leveraging a diffusion model to generate a 3D sparse point cloud for the object captured in a single RGB image. Recent methods typically guide a diffusion model with global shape information or local image features. However, such strategies fail to consistently align the denoised point cloud with the given image, leading to unstable conditioning and inferior performance. In this paper, we exploit a novel Centered Diffusion Probabilistic Model (CDPM) for consistent local feature conditioning. We constrain the noise and sampled point cloud from the diffusion model into a subspace where the point cloud center remains unchanged during both the forward and reverse diffusion process. Upon CDPM, we build CCD-3DR for single-image 3D reconstruction, where the stable point cloud center further serves as an anchor to align each point with its corresponding local projection-based features. Extensive experiments on synthetic benchmark ShapeNet-R2N2 demonstrate that CCD-3DR outperforms all competitors by a large margin, with over 40% improvement. We also provide results on the real-world dataset Pix3D to thoroughly demonstrate the potential of CCD-3DR in real-world applications. The code will be released soon. 1 Introduction Single-image object reconstruction is a well-known ill-posed problem. While deep learning methods have made remarkable strides in achieving high-quality reconstruction, further improvements are still necessary to meet the demands of real-world applications (Zhai et al., 2023; Yang and Scherer, 2019). Recently, a new wave of methods leveraging Denoising Diffusion Probabilistic Model (DDPM) (Ho et al., 2020) has emerged (Cheng et al., 2023; Melas-Kyriazi et al., 2023b; Luo and Hu, 2021; Melas-Kyriazi et al., 2023a; Poole et al., 2023), showcasing superior performance in various domains. For single-image 3D reconstruction with diffusion models, DMPGen (Luo and Hu, 2021) and PC$^2$ (Melas-Kyriazi et al., 2023b) are two representative baselines. In DMPGen, the condition is the global embedding of the target object, while in PC$^2$, in each step of the reverse process, the denoised point cloud is back-projected onto the feature map of the image to extract local feature for each point, which serves as the condition for the next reverse step. However, directly applying diffusion models in single-image 3D reconstruction suffers from an inevitable challenge: uncontrollable center deviation of the point cloud, as shown in Fig. 1. (a). Since each point inside the point cloud and predicted noise is independently modeled, under the single-image reconstruction setting, no geometric or contextual priors can be harnessed to control the point cloud center. After each step of the reverse process in DDPM, the centroid of the generated point cloud will be shifted slightly. Therefore, from a random sampled Gaussian noise towards the target object, in the reverse process, the center of the point cloud will continuously undergo disturbances until it reaches the center of the target object. Based on our experimental findings, we have identified two problems caused by this center deviation. First, the diffusion network needs to allocate capacity to handle the displacement of the point cloud center. It is crucial to ensure that the transition of the point cloud center from the initial Gaussian noise state to the final object reconstruction is appropriately managed. However, since the overall resource is limited, allocating network capacity to recover the center results in inferior performance in shape reconstruction. Second, the center deviation causes misalignment and inconsistency in the local feature conditioning, as used in PC$^2$ (Melas-Kyriazi et al., 2023b). The misaligned feature adversely affects the subsequent denoising process in DDPM and degrades the overall quality of the final reconstruction. We explain more details of these two points in Sec. 3.2. To address the aforementioned problems, in this paper, we present a simple but effective method, CCD-3DR, which takes a single RGB image with the corresponding camera pose as input and reconstructs the target object with a sparse point cloud. Instead of directly leveraging the off-the-shelf DDPM, we propose a novel Centered Diffusion Probabilistic Model (CDPM) that can enable consistent local feature conditioning in diffusion, which further significantly boosts the single-image reconstruction quality. Our core idea is to constrain the added noise in the diffusion process as well as the predicted noise and the sampled point cloud in the reverse process into a smaller subspace of the entire sampling space. With such constraints, CDPM sacrifices some of DDPM’s generation diversity, yet it stables the point cloud center in exchange. In this subspace, the center of the corresponding noise of the point cloud coincides with the origin throughout the diffusion and reverse processes, as shown in Fig. 1. (b). Thereby, the point cloud center serves as an anchor in local feature extraction to align the point cloud with its corresponding projections consistently. Based on CDPM, we design CCD-3D for single-image 3D object reconstruction. In CCD-3D, to ensure that the noise and point cloud lie in the subspace defined in CDPM, a straightforward strategy is to iteratively generate samples in the entire space until one sample lies in the subspace. However, this is time-consuming and infeasible in real implementations. Instead, we first sample the noise in the entire space and centralize it. Next, we denoise the point cloud after predicting the noise using the diffusion network. In the subsequent process, these are then transferred to the subspace. We follow PC$^2$ (Melas-Kyriazi et al., 2023b) to back-project the point cloud onto the feature map of the image to extract local features around each projection. In summary, our contributions are listed as follows, (i) We propose a novel centered denoising diffusion probabilistic model CDPM, which constrains the noise and point cloud in diffusion and reverse processes into a subspace where the point cloud center is forced to coincide with the origin. (ii) We present a new single-image 3D object reconstruction pipeline, CCD-3D, which leverages CDPM to consistently collect local features for the point cloud in diffusion, leading to superior performance in reconstruction quality. (iii) We evaluate CCD-3D on the synthetic dataset ShapeNet-R2N2 (Chang et al., 2015; Choy et al., 2016) to demonstrate its superiority over competitors. CCD-3D outperforms state-of-the-art methods by over 40% under F-Score. Additional experiments on the real-world dataset Pix3D (Sun et al., 2018) demonstrate the potential of CCD-3D in real applications. 2 RELATED WORKS 3D reconstruction of the object shape from a single image has been a research focus in the community (Kar et al., 2017; Wang et al., 2018; Wu et al., 2017; Kar et al., 2015; Li et al., 2019; 2018; Zhang et al., 2021; Mao et al., 2021). Although it is an ill-posed problem, the shape priors of large-scale training datasets can guide the reconstruction process with generalization ability. Non-Generative Reconstruction Models. Early methods use 2D encoders (Ronneberger et al., 2015; He et al., 2016; Simonyan and Zisserman, 2015) to encode features and use 3D decoders (Çiçek et al., 2016; Tran et al., 2015) to obtain shapes. The pioneering work such as 3D- R2N2 (Choy et al., 2016) uses the occupancy grids as object shape representations and a following LSTM (Hochreiter and Schmidhuber, 1997) to fuse inputs from multiple views for prediction. The 2D features are extracted by a 2D CNN and projected to the 3D occupancy grids with a 3D deconvolutional neural network. LSM (Kar et al., 2017) reprojects 2D features into voxel grids and decodes shapes from these grids using a 3D convolutional GRU (Cho et al., 2014). Pix2Vox series (Xie et al., 2019; 2020) enjoy a serial architecture composed of a pretrained 2D CNN backbone and 3D transposed convolutional layers with multi-scale fusion for enhanced voxelization. Since the voxel representations are limited by the resolution of voxel size, point cloud and mesh-based shape representations are favored to get rid of the limitation (Hu et al., 2021; Wang et al., 2020; Zhang et al., 2018; Henderson and Ferrari, 2019; Erler et al., 2020; Mandikal and Babu, 2019; Gkioxari et al., 2019; Wen et al., 2019; Pan et al., 2019; Huang et al., 2023). More recent works utilizes implicit representations such as signed distance functions (Park et al., 2019; Xu et al., 2019), occupancy networks (Mescheder et al., 2018; Chen and Zhang, 2019) or neural radiance fields for object shape generation (Yu et al., 2020; Wang et al., 2021; Jang and de Agapito, 2021). Despite the different shape representations, the above methods are restricted to auto-encoder architecture and suffer limited performances in comparison to generative models. Generative Reconstruction Models. Generative reconstruction models, in contrast to the routines mentioned above, estimate the shape distribution in a more explicit way to generate plausible shapes. For the first time to generate point clouds from single-view images, Fan et al. (Fan et al., 2017) build a point cloud generation network upon variational autoencoders (VAEs) (Kingma and Welling, 2014) to generate multiple plausible shapes. By incorporating both VAEs and generative adversarial networks (GANs) (Goodfellow et al., 2014), 3D-VAE-GAN (Wu et al., 2016) samples latent codes from a single-view image as the condition and outputs 3D shapes through 3D GAN generators. However, It heavily relies on class labels for reconstruction. 3D-aware GANs such as StyleSDF (Or-El et al., 2022) and Get3D (Gao et al., 2022) can simultaneously synthesize 2D images and 3D detailed meshes. However, these methods suffer from instabilities and mode collapse of GAN training. Recently, diffusion models (Song and Ermon, 2019; 2020; Ho et al., 2020) exhibit advanced generation ability in such as text-to-image (Rombach et al., 2021), text-to-shape (Nichol et al., 2022) areas, enjoying more stable training phase and elegant mathematical explainability. Thereby, various point cloud based tasks take advantage of diffusion models to get results of higher quality. DMPGen (Luo and Hu, 2021) firstly applies the diffusion process in the point cloud generation task. LION (Zeng et al., 2022) further generalizes the point cloud in the hierarchical latent space with diffusion. Similarly, Lyu et al. (Lyu et al., 2022) utilize the point diffusion for shape completion. Point-Voxel Diffusion (Zhou et al., 2021) combines multiple representations in the diffusion process to generate stable results. To get the texture information for the point cloud, (Nichol et al., 2022) generates colored point clouds as the diffusion output for better visualization. Theoretically, such methodology can be readily leveraged into the single-view reconstruction task by regarding the RGB information as the condition (Poole et al., 2023; Melas-Kyriazi et al., 2023b). The recent method PC$^2$ (Melas-Kyriazi et al., 2023b) projects point clouds in the reverse diffusion process onto the image plane to query 2D features as shape and color conditions. Our new diffusion paradigm CDPM can be compatible with recent work, such as DMPGen and PC$^2$, while providing more accurate results. ### 3 Method In the following sections, we outline our methodology. We start by providing a brief overview of point diffusion models, laying the groundwork for our approach. Subsequently, we explain the enhancements we have made to the traditional DDPM with the intention of augmenting its effectiveness in the realm of single-image reconstruction. These adaptations result in our innovative Centered Diffusion Probabilistic Model (CDPM). Lastly, we provide a comprehensive explanation of our single-image reconstruction pipeline CCD-3DR, which is constructed based on CDPM. #### 3.1 Preliminaries: Diffusion Models Diffusion denoising probabilistic models are a class of generative models inspired by non-equilibrium thermodynamics. It can iteratively move a set of Gaussian noise toward a uniform Figure 2: Pipeline of CCD-3D. Block (B) shows the local feature extraction process. Given a single RGB image (capturing the airplane) as the input, CCD-3D aims to reconstruct the object with CDPM. We first leverage a pre-trained MAE (He et al., 2022) model to extract feature maps from the image and interpolate them to the same size as the image (shown in the grey block). The feature maps provide local conditions for each point in the denoised centered point cloud $ x^t - \bar{x}^t $ during the reverse process of CDPM. We back-project the centered point cloud onto the image and collect features around the projections to serve as the local features. Block (A) demonstrates the reverse process of CDPM. At step $ t $, point cloud $ x^t $ is first centralized to $ x^t - \bar{x}^t $ and then concatenated with the local features out of Block (B). The U-Net denoiser $ \theta $ predicts noise $ \epsilon_\theta $ and centralizes it with $ \epsilon_\theta - \bar{\epsilon}_\theta $. The point cloud $ x^{t-1} $ can finally be recovered using Eq. 3. and clean point cloud, capturing the target object. DDPM contains two Markov chains called the diffusion process and the reverse process. The two processes share a length of $ T = 1K $ steps. Diffusion Process. Let $ p_0 $ be the potential distribution of the complete object point cloud $ x $ in the dataset and $ p_T $ be the standard Gaussian distribution $ p_T \sim \mathcal{N}(0_{3N}, I_{3N \times 3N}) $. The diffusion process iteratively adds Gaussian noise $ \epsilon $ into the clean data distribution $ p_0 $ according to the Markov Chain Rule until $ p_0 $ reaches $ p_T $. Formally, let $ x^0 \sim p_0 $, then \[ q(x^{1:T}\|x^0) = \prod_{t=1}^{T} q(x^t\|x^{t-1}), \] where $ q(x^t\|x^{t-1}) = \mathcal{N}(x^t; \sqrt{1-\beta_t}x^{t-1}, \beta_t I) $. The hyperparameter $ \beta_t $ is pre-defined small constants. We use the subscript to denote the diffusion step $ t $. Each $ q(x^t\|x^{t-1}) $ is a Gaussian distribution and $ q(x^t\|x^0) $ can be reparameterized as, \[ q(x^t\|x^0) = \sqrt{\alpha_t}x^0 + \epsilon \sqrt{1-\alpha_t}, \] where $ \alpha_t = 1 - \beta_t $, $ \bar{\alpha}_t = \prod_{s=0}^{t} \alpha_s $, and $ \epsilon \sim \mathcal{N}(0, I) $. From Eq. 2, for point diffusion, we can infer that if $ x^0 $ is sampled from a zero-mean distribution $ p_0 $, considering $ \epsilon $ is also zero-mean, $ q(x^t\|x^0) $ can be modeled as a zero-mean distribution, which implies that for any $ t \in [0, T] $, the diffusion process will generate a zero-mean distribution at this step. In this paper, we utilize this derivation to boost single-image 3D reconstruction. Reverse Chain. The reverse process is also a Markov process that removes the noise added in the diffusion process. In this paper, the reverse process is conditioned on an RGB image $ I $ capturing the object. We start with a sample $ x^T \sim p_T $, and then iteratively sample from $ q(x^{t-1}\|x^t, f(I)) $, where $ f(I) $ denotes features extracted from $ I $ to incorporate local or global supervision into the reverse process. When the number of sampling steps $ T $ is sufficiently large, $ q(x^{t-1}\|x^t, f(I)) $ can be well approximated with an isotropic Gaussian distribution with constant small covariance $ \sigma_t^2 $: \[ q(x^{t-1}\|x^t, f(I)) = \mathcal{N}(x^{t-1}; \mu_\theta(x^t, f(I)), \sigma_t^2 I), \] \[ \mu_\theta(x^t, f(I)) = \frac{1}{\sqrt{\alpha_t}}(x^t - \frac{\beta_t}{\sqrt{1-\alpha_t}}\epsilon_\theta(x^t, f(I))), \] where $ \mu_\theta $ is the estimated mean. Thus, we can use the network parameterized by $ \theta $ to directly learn $ \epsilon_\theta $ under the condition $ f(I) $. DDPM-Based Reconstruction Consider a 3D point cloud with $ N $ points. DDPM-based reconstruction methods (Luo and Hu, 2021; Melas-Kyriazi et al., 2023b) learn a diffusion model $ S_\theta : \mathbb{R}^{3N} \rightarrow \mathbb{R}^{3N} $ to denoise the randomly sampled point cloud from $ p_T $ into a recognizable object from target distribution $ p_0 $. Specifically, at each step $ t $, the noise is predicted as the offset of each point from Algorithm 1 CDPM: Training 1: repeat 2: $ x^0 \sim q(x^0), \quad x^0 = x^0 - \bar{x}^0 $ 3: $ t \sim \text{Uniform}(\{1, 2, ..., T\}) $ 4: $ \epsilon \sim \mathcal{N}(0, I), \quad \epsilon = \epsilon - \bar{\epsilon} $ 5: Take gradient descent step on: $ \nabla_\theta \\| \epsilon - \epsilon_\theta(x^t, f(I)) \\|^2 $ 6: until converged Algorithm 2 CDPM: Sampling 1: $ x^T \sim \mathcal{N}(0, I), \quad x^T = x^T - \bar{x}^T $ 2: for $ t = T, ..., 1 $ do 3: $ \epsilon_\theta = \epsilon_\theta - \bar{\epsilon}_\theta $ 4: $ x^{t-1} \sim q(x^{t-1}\|x^t), \quad x^{t-1} = x^{t-1} - \bar{x}^{t-1} $ 5: end for 6: return $ x^0 $ the current coordinate in $ x^t $ to $ x^{t-1} \sim q(x^{t-1}\|x^t, f(I)) $. Then we sample from $ q(x^{t-1}\|x^t, f(I)) $ to obtain $ x^{t-1} $. As for conditioning, DMPGen (Luo and Hu, 2021) encodes the given RGB image into a single global latent vector $ z $ and concatenates $ z $ with the obtained point cloud at each step during the reverse process. PC$^2$ (Melas-Kyriazi et al., 2023b) goes one step further by introducing local point-wise features for fine-grained geometry cues. It updates the local feature of each point at each step $ t $ by back-projecting the point cloud $ x^t $ onto the feature map using the known camera extrinsic $[R_c\|t_c]$ and perspective projection matrix $ \pi_c $, \[ \text{Proj}(x^t) = \pi_c(R_c x^t + t_c). \] Then local features $ f(I) $ around the projections $ \text{Proj}(x^t) $ are aggregated with rasterization. These two methods (Luo and Hu, 2021; Melas-Kyriazi et al., 2023b) are selected as our baselines. 3.2 Bottlenecks in DDPM-based Reconstruction We now analyze the limitations of directly applying DDPM in 3D reconstruction like in DMPGen and PC$^2$ (Luo and Hu, 2021; Melas-Kyriazi et al., 2023b). Two bottlenecks are deteriorating the performance of these methods. First, predicting the center bias is challenging for the network in the reverse process. Since we assume the variances are constant in all Gaussian distributions, we only need to analyze the center of each denoised point cloud. From $ x^t $ to $ x^{t-1} $, in Eq. 1 and 3, we have, \[ E(\bar{x}^{t-1}) = \frac{1}{\sqrt{\alpha_t}} E(\bar{x}^t), \quad E(\bar{\epsilon}_\theta(x^t, f(I))) = 0. \] Thus after sampling for $ x^{t-1} $, we can obtain, \[ \bar{x}^{t-1} = \frac{1}{\sqrt{\alpha_t}} \left( \bar{x}^t - \frac{\beta_t}{\sqrt{1 - \alpha_t}} \bar{\epsilon}_\theta(x^t, f(I)) \right) + \Delta_t, \] where $ \Delta_t $ is center bias generated by random sampling from Gaussian distribution for $ x^{t-1} $. When $ \bar{x}^T \neq \bar{x}^0 $, the network $ \theta $ needs to move the center of the denoised point cloud from $ \bar{x}^T $ towards $ \bar{x}^0 $ under the following handicaps. First, $ E(\bar{\epsilon}_\theta(x^t, f(I))) = 0 $, while the network needs to predict non-zero-mean noise $ \epsilon $ in several steps to move $ \bar{x}^T \rightarrow \bar{x}^0 $. Second, the network needs to overcome $ \Delta_t $. Last, each point in $ x^{T:0} $ is independently modeled in diffusion, and no constraints are incorporated to control the development of the point cloud center. Experiments in Sec. 4.1 demonstrate that accurately recovering $ \bar{x}^0 $ is a very hard job for the network. Wasting network capacity in the recovering center also results in poor performance in shape reconstruction. Second, the change of the point cloud center makes the local feature conditioning inconsistent. As in PC$^2$, the difference $ \Delta_{\text{Proj}} $ in projections of $ \text{Proj}(\bar{x}^{t-1}) $ and $ \text{Proj}(\bar{x}^t) $ can be derived as \[ \Delta_{\text{Proj}} = \pi_c(R_c (\bar{x}^{t-1} - \bar{x}^t) + t_c). \] If $ \Delta_{\text{Proj}} $ is sufficiently large, the features collected for the point center can be totally different from $ x^t $ to $ x^{t-1} $, which will mislead the following denoising steps. Moreover, since we only use a single RGB image as a conditioner, we have no contextual or geometric constraints to rectify this misalignment. 3.3 From DDPM to CDPM To address the aforementioned bottlenecks, we propose a novel CDPM model designed for single-view 3D reconstruction. The core idea of CDPM is simple and straightforward yet effective. To eliminate the influence of center bias in the reverse process, we add the following constraint, \[ \bar{x}^t = 0, \quad t = 0, 1, 2, \ldots, T. \] (8) This constraint enforces the denoised point cloud in each step to be zero-mean so that the center remains unchanged during the reverse process. As shown in Eq. 2 and Eq. 3, if Eq. 8 holds, we have $ \bar{\epsilon} = 0, \bar{\epsilon}_\theta(x^t, f(I)) = 0 $. Let $ S_{x^t} $ denote the space of all possible samplings from the distribution $ q(x^t\|x^{t+1}) $, then the space $ S_{x^t, \bar{x}^t=0} $ under the constraint Eq. 8 is a subspace, i.e., $ S_{x^t, \bar{x}^t=0} \subset S_{x^t} $. Similarly, we define $ S_\epsilon, S_{\epsilon, \bar{\epsilon}=0}, S_{\epsilon_\theta}, S_{\epsilon_\theta, \bar{\epsilon}_\theta=0} $. In summary, from DDPM to CDPM, we constrain $ x^t, \epsilon, \epsilon_\theta $ all in a smaller subspace, DDPM : $ x^t \in S_{x^t}, \epsilon \in S_\epsilon, \epsilon_\theta \in S_{\epsilon_\theta} \implies $ CDPM : $ x^t \in S_{x^t, \bar{x}^t=0}, \epsilon \in S_{\epsilon, \bar{\epsilon}=0}, \epsilon_\theta \in S_{\epsilon_\theta, \bar{\epsilon}_\theta=0}. $ (9) Therefore, we prioritize the stability of the point cloud center to a certain extent, sacrificing a portion of the diversity in diffusion models. For point cloud $ x^t $ in the reverse process, after obtaining $ q(x^t\|x^{t+1}) $, we can sample multiple times until the sampled point cloud lies in $ S_{x^t, \bar{x}^t=0} $. However, such a strategy is infeasible in real implementation. Thereby we simply first sample in $ S_{x^t} $ and then centralize the point cloud to project it into $ S_{x^t, \bar{x}^t=0} $. The same holds true for $ \epsilon $ and $ \epsilon_\theta $. Specifically, as explained in Alg. 1 and Alg. 2, we first build a dataset composed of $ M $ data pairs $ D = \{(x_i, I_i)\}_{1 \leq i \leq M} $, where $ x_i $ denotes the $ i $-th ground truth point cloud sampled from the object mesh, and $ I_i $ is the corresponding RGB image capturing the object. Compared to DDPM, CDPM mainly makes improvements in three points: First, the point clouds in $ D $ are centralized as $ x_i - \bar{x}_i $, where $ \bar{x}_i $ denotes the centroid of $ x_i $, establishing a new zero-mean dataset $ \tilde{D} = (\tilde{x}_i, I_i) $. Second, for noise $ \epsilon $ added in the diffusion process for training and the noise $ \epsilon_\theta $ predicted in the reverse process, we also centralize them with $ \epsilon - \bar{\epsilon} $ and $ \epsilon_\theta - \bar{\epsilon}_\theta $, where $ \bar{\epsilon} $ and $ \bar{\epsilon}_\theta $ denote the corresponding gravity centers. Third, during inference, for $ x^{t-1} $ sampled from $ q(x^{t-1}\|x^t, f(I)) $, we also centralize it with $ x^{t-1} - \bar{x}^{t-1} $. From Eq. 2, since we keep $ x^0 $ and $ \epsilon $ to be zero-mean, the diffused point cloud in each step $ t $ should be zero-mean. The advantages of CDPM over DDPM in single-image reconstruction can be summarized as follows: First, our reverse process starts with a zero-mean Gaussian noise and arrives at the zero-mean reconstruction $ x^0 $ after $ T $-step zero-mean denoising. This zero-mean nature of the reverse process provides a useful regularization for the network to focus more on the shape of the object rather than tracking the center of the point cloud. Therefore, our CDPM outperforms the previous DDPM-reconstruction methods even with only global embedding of the object, like in (Luo and Hu, 2021). Second, CDPM enables consistent local feature conditioning in the reverse diffusion process. As in PC$ ^2 $ (Melas-Kyriazi et al., 2023b), the point cloud is back-projected onto the image feature map to extract local point-wise features as conditioning. However, due to the uncontrollable center bias in the reverse process, the projection of each point may gradually deviate, making the local feature aggregation process fail and further deteriorating the final reconstruction quality. In contrast to DDPM-based PC$ ^2 $, our method CDPM keeps the centroid of the denoised point cloud in each step to coincide with the origin, which further serves as an anchor point in local feature collection. The projection of this anchor point remains the same in the reverse process and thus aligns the point cloud with the feature map to obtain consistent features. 3.4 CCD-3DR For a fair comparison with baseline methods, we follow PC$ ^2 $ (Melas-Kyriazi et al., 2023b) to use MAE (He et al., 2022) to extract 2D feature maps from the given RGB image. The feature maps are of equal length and width of the input image to facilitate point cloud projection. For the diffusion network $\theta$ used to predict the noise $\epsilon_\theta$, we adopt the Point-Voxel CNN (PVCNN) \citep{liu2019point}. We use the classic $L_2$ loss to supervise the training of $\theta$, as specified in Alg. 1. 4 EXPERIMENTS Datasets. We evaluate CCD-3DR on the synthetic dataset ShapeNet-R2N2 \citep{choy20163d,chang2015shapenet} and real-world dataset Pix3D \citep{sun2018pix3d}. ShapeNet contains a diverse collection of 3D models spanning various object categories, such as furniture, vehicles, and more. The dataset is meticulously annotated, providing not only the 3D geometry of the objects but also rich semantic information, making it an essential tool for the quantitative evaluation of single-view reconstruction methods. We follow baseline methods \citep{melas2023pc2,yagubbayli2021pc2,xie2020local} to use the R2N2 \citep{choy20163d} subset along with the official image renderings, train-test splits, camera intrinsic and extrinsic matrices. The R2N2 subset covers 13 categories in total. Pix3D \citep{sun2018pix3d} is a large-scale benchmark of diverse image-shape pairs with pixel-level 2D-3D alignment. Previous methods \citep{cheng2023local,xie2019local,xie2020local,sun2018pix3d} only harness the chair category and exclude the occluded samples. Since our method needs to use all data to demonstrate robustness towards occlusion, we leverage 3 categories: \{chair, table, sofa\} and randomly generate train-test split with about 90% samples as the training set and the remaining as the testing set. Details are provided in the Supplementary Material. Implementation Details. We implement CCD-3DR in PyTorch and evaluate the method on a single GeForce RTX 3090Ti GPU with 24GB memory. For ShapeNet-R2N2 \citep{choy20163d,chang2015shapenet}, we first resize the provided images of size $137 \times 137$ to $224 \times 224$ and adjust the focal length accordingly. We follow prior work to use 8192 points in training and inference for fairness in computing the F-Score. On Pix3D \citep{sun2018pix3d}, since the images are of different sizes, we first crop the image with the given bounding box to obtain an object-centric image and then resize it to $224 \times 224$. The camera intrinsic matrix is also adjusted correspondingly. During training, we train CCD-3DR with batch size 16 for 100K steps in total, following PC$^2$ \citep{melas2023pc2}. We use the AdamW optimizer with a dynamic learning rate with warmup which increases from $1 \times 10^{-9}$ to $1 \times 10^{-3}$ in the first 2K steps and then decays exponentially until 0 in the following 98K steps. Baselines. We select DDPM-based DMPGen \citep{luo2021ddpm} and PC$^2$ \citep{melas2023pc2} as our baseline methods. On ShapeNet-R2N2, we compare with the official results of PC$^2$. Since DMPGen doesn’t provide results of single-view reconstruction on ShapeNet-R2N2, we reimplement it by using pre-trained MAE \citep{he2022mae} to extract global shape code and then follow the diffusion process in the original paper to reconstruct the object, denoted as DMPGen. We provide three variants of CCD-3DR on ShapeNet-R2N2, in which Ours uses only local features like in PC$^2$, Ours-G leverages only global features as DMPGen and Ours-(G+L) incorporates both local and global features for reconstruction, as shown in Tab. 4. On Pix3D, we retrain PC$^2$ and DMPGen* under the same settings of CCD-3DR. Evaluation Metrics. We use Chamfer Distance (CD) and F-Score@0.01 following \citep{melas2023pc2,cheng2023local} as the evaluation metrics. CD quantifies the dissimilarity between two sets of points by measuring the minimum distance from each point in one set to its nearest point in the other set. To compensate for the problem that CD can be sensitive to outliers, we also report F-Score with the threshold 0.01, i.e., for each reconstructed point. If its nearest distance to the ground truth point cloud lies below the threshold, it is considered correctly predicted. Note that previous methods \citep{choy20163d,yagubbayli2021pc2,xie2020local} typically report the results using the voxelized $32^3$ volume as the shape representation, which quantizes the sampled points and fails to reflect the reconstruction quality of fine-grained structures. Therefore, we follow PC$^2$ \citep{melas2023pc2} to use sampled points from the object mesh as the ground truth. Results of other methods \citep{choy20163d,yagubbayli2021pc2,xie2020local} are re-evaluated using the same setting for fair comparisons. 4.1 Comparisons with State-of-the-Art Methods. Performance on Synthetic Dataset ShapeNet-R2N2. In Tab. 1, we compare CCD-3DR with state-of-the-art competitors on ShapeNet-R2N2 under the F-Score@0.01 metric. 3D-R2N2 \citep{choy20163d}, Figure 3: Qualitative comparisons on synthetic dataset ShapeNet-R2N2 (Choy et al., 2016; Chang et al., 2015) (left) and real-world dataset Pix3D (Sun et al., 2018) (right). Our method can recover fine-grained structures accurately, like the handle of the chair. \| Category \| 3D-R2N2 \| LegoFormer \| Pix2vox++ \| DMPGen* \| PC² \| Ours \| DMPGen(O) \| PC²(O) \| Ours(O) \| \|--------------\|---------\|------------\|-----------\|---------\|-----\|------\|------------\|--------\|---------\| \| airplane \| 0.225 \| 0.215 \| 0.266 \| 0.454 \| 0.473 \| 0.725 \| 0.565 \| 0.681 \| 0.785 \| \| bench \| 0.198 \| 0.241 \| 0.266 \| 0.175 \| 0.305 \| 0.480 \| 0.289 \| 0.444 \| 0.573 \| \| cabinet \| 0.256 \| 0.308 \| 0.317 \| 0.087 \| 0.203 \| 0.282 \| 0.111 \| 0.303 \| 0.371 \| \| car \| 0.211 \| 0.220 \| 0.268 \| 0.310 \| 0.359 \| 0.395 \| 0.402 \| 0.420 \| 0.466 \| \| chair \| 0.194 \| 0.217 \| 0.246 \| 0.171 \| 0.290 \| 0.335 \| 0.312 \| 0.377 \| 0.406 \| \| display \| 0.196 \| 0.261 \| 0.279 \| 0.211 \| 0.232 \| 0.381 \| 0.236 \| 0.357 \| 0.487 \| \| lamp \| 0.186 \| 0.220 \| 0.242 \| 0.207 \| 0.300 \| 0.438 \| 0.347 \| 0.399 \| 0.490 \| \| loudspeaker \| 0.229 \| 0.286 \| 0.297 \| 0.113 \| 0.204 \| 0.219 \| 0.126 \| 0.288 \| 0.291 \| \| rifle \| 0.356 \| 0.364 \| 0.410 \| 0.474 \| 0.522 \| 0.762 \| 0.663 \| 0.686 \| 0.828 \| \| sofa \| 0.208 \| 0.260 \| 0.277 \| 0.078 \| 0.205 \| 0.293 \| 0.106 \| 0.298 \| 0.349 \| \| table \| 0.263 \| 0.305 \| 0.327 \| 0.155 \| 0.270 \| 0.427 \| 0.310 \| 0.420 \| 0.488 \| \| telephone \| 0.407 \| 0.575 \| 0.582 \| 0.333 \| 0.331 \| 0.423 \| 0.464 \| 0.523 \| 0.598 \| \| watercraft \| 0.240 \| 0.283 \| 0.316 \| 0.201 \| 0.324 \| 0.475 \| 0.399 \| 0.424 \| 0.610 \| \| Average \| 0.244 \| 0.289 \| 0.315 \| 0.228 \| 0.309 \| 0.433 \| 0.333 \| 0.432 \| 0.519 \| Table 1: Performance on ShapeNet-R2N2. We compare our method with competitors under F-Score@0.01. The Oracle setting (marked as (O)) refers to predicting 5 samples of each image and selecting the best prediction as the final result. Legoformer (Yagubbayli et al., 2021), Pix2vox++ (Xie et al., 2020) are voxel-based methods, while DMPGen (Luo and Hu, 2021), PC² (Melas-Kyriazi et al., 2023b) are diffusion-based methods, serving as baselines of CCD-3DR. From Tab. 1, it can be clearly deduced that our method CCD-3DR achieves state-of-the-art performance in 10 out of 13 categories. Considering the Average performance, CCD-3DR outperforms previous best method Pix2vox++ with 0.433 vs. 0.315, about a 37.5% leap forward. Furthermore, compared with diffusion-based baseline method PC², CCD-3DR demonstrates superior performance under all the categories and improves PC² by 40.1%, with 0.433 vs. 0.309. We also report the Oracle results, following the setting in PC², where for each test image, we predict 5 possible reconstruction results and select the one with the highest F-Score@0.01 as the final result. Under the Oracle setting, our method surpasses all competitors by a large margin, with about a 20.1% improvement over PC² Oracle. Performance on Real-World Dataset Pix3D. In Tab. 2, we compare CCD-3DR with other DDPM-based reconstruction methods using Chamfer Distance and F-Score@0.01. Our method consistently outperforms competitors in all categories. On average, CCD-3DR surpasses the second-best method PC² by 20% on ShapeNet-R2N2 and 15% on Pix3D. Qualitative Comparisons. We provide visualization comparisons with previous methods in Fig. 3. It can be seen clearly that our method surpasses competitors with respect to the reconstruction quality. Particularly, due to our consistent feature conditioning scheme, our method showcases superiority in recovering fine-grained structures, like the hand of the chair. We provide more results in the Supplementary Material. 4.2 Ablation Studies We conduct several ablation studies on public datasets. Note that except for ablated terms, we leave all other terms and settings unchanged. \| Method \| Chair \| Table \| Sofa \| Average \| \|------------\|-------\|-------\|------\|---------\| \| DMPGen \| 0.188 \| 0.176 \| 0.243\| 0.202 \| \| PC$^2$ \| 0.336 \| 0.294 \| 0.377\| 0.336 \| \| Ours \| 0.439 \| 0.559 \| 0.489 \| 0.496 \| \| Chair \| Table \| Sofa \| Average \| \|-------\|-------\|------\|---------\| \| 53.30 \| 50.56 \| 21.04\| 41.63 \| \| 33.21 \| 13.13 \| 3.760\| 16.70 \| \| 14.98 \| 1.475 \| 0.712 \| 5.722 \| Table 2: Performance on Pix3d. F-Score@0.01 (left) and Chamfer Distance ($\times 10^{-3}$) (right) is reported in the table. Our method outperforms diffusion-based competitors. \| Occ. Ratio \| Method \| Chair \| Table \| Sofa \| \|------------\|---------------------------------\|-------\|-------\|------\| \| $\sim 20\%$\| PC$^2$ (Melas-Kyriazi et al., 2023b) \| 0.324 \| 0.280 \| 0.365\| \| \| Ours \| 0.424 \| 0.535 \| 0.421 \| \| $\sim 50\%$\| PC$^2$ (Melas-Kyriazi et al., 2023b) \| 0.310 \| 0.260 \| 0.337\| \| \| Ours \| 0.411 \| 0.520 \| 0.397 \| Table 3: Ablation studies of robustness towards occlusions. Occ. Ratio refers to occlusion ratio. We report the F-Score@0.01 after randomly masking about 20% and 50% visible pixels of the image. \| Category \| airplane \| bench \| cabinet \| car \| chair \| display \| lamp \| loudspeaker \| rifle \| sofa \| table \| tele-phone \| watercraft \| \|----------\|----------\|-------\|---------\|-----\|-------\|---------\|------\|-------------\|-------\|------\|-------\|------------\|------------\| \| Ours-G \| 0.599 \| 0.298 \| 0.204 \| 0.251\| 0.283 \| 0.223 \| 0.316\| 0.177 \| 0.653 \| 0.201\| 0.266 \| 0.355 \| 0.311 \| \| Ours-(G+L)\| 0.727 \| 0.463 \| 0.277 \| 0.398\| 0.341 \| 0.366 \| 0.429\| 0.214 \| 0.777 \| 0.287\| 0.433 \| 0.414 \| 0.469 \| \| Ours \| 0.725 \| 0.480 \| 0.282 \| 0.395\| 0.335 \| 0.381 \| 0.438\| 0.219 \| 0.762 \| 0.293\| 0.427 \| 0.423 \| 0.475 \| Table 4: Ablations on the effect of local and global features on ShapeNet-R2N2. We retrain and re-evaluate our method using different feature conditioning methods. Occlusions. In Tab. 3, we evaluate the performance of CCD-3DR with respect to different occlusion ratios on Pix3D. We randomly mask approximately 20% and 50% visible pixels of the object to test the robustness of CCD-3DR towards occlusions. From the table, it can be seen clearly that although the masked pixels increase from 20% to 50%, the performance of CCD-3DR only degrades very little, with 0.013 in chair, 0.015 in table and 0.024 in sofa. Moreover, in this experiment, PC$^2$ also demonstrates consistent and satisfactory results under different occlusion ratios, which verifies the capability of diffusion models in handling occlusions. Note that for fair comparisons, we retrain PC$^2$ and our method with the same augmented training data. We randomly mask 0% ~ 50% pixels of each image for training and then conduct the ablation study in Tab. 3. Local vs. Global Conditioning. In Tab. 4, we demonstrate the effect of local and global features in the diffusion-based reconstruction process. The global feature is obtained by averaging the pooling of the point-wise local features. And when the global feature is incorporated, we directly concatenate it to each point as the condition. Comparing Ours-(G+L) and Ours, it can be seen clearly that once a local feature is provided, an additional global feature is not necessary. Oracle Results. We report the oracle experiment results in Tab. 1. Following the setting in PC$^2$, we also predict 5 possible shapes for each image and select the one with the highest F-Score@0.01 as the final reconstruction result. It is obvious that under the oracle setting, all three diffusion-based methods, DMPGen*, PC$^2$, and Ours, showcase a significant leap forward. Thereby, although the centralization scheme in our method may influence the generalization capability of the diffusion model to a certain extent, in the single-view reconstruction case, our method still demonstrates the capability of generating multiple plausible results. We also provide the corresponding qualitative results in the Supplementary Material. 5 CONCLUSIONS In this paper, we propose CCD-3DR, a single-image 3D reconstruction pipeline that leverages a novel Centered Diffusion Probabilistic Model (CDPM) for consistent and stable local feature conditioning. We project the predicted noise and sampled point cloud from DDPM into a subspace where the point cloud center remains unchanged during the whole diffusion and reverse processes. Extensive experimental results and ablation studies on both synthetic and real-world datasets demonstrate that such a simple design significantly improves overall performance. We also analyze the influence of point cloud centralization with respect to diversity and point out the limitations of CCD-3DR. In the future, we plan to extend CCD-3DR with an advanced ordinary differentiable equation solver to enhance the inference speed. 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FLR7ElwD51	The authors argue that the OL agent is generalizable to large tasks based on the observation that the True Positive Rate (TPR) only decreases slightly (Table 4). However, in the same table, the SHD increases. If the SHD increases because of a large number of reversed edges, it is difficult to conclude that the OL agent generalizes to large tasks.	Learning Scalable Causal Discovery Policies with Adversarial Reinforcement Learning Anonymous authors Paper under double-blind review Abstract Learning the structure of causal graphs from observational data is a fundamental but challenging problem. Existing works focus on designing search-based methods for finding optimal causal graphs. However, search-based methods have proven low-efficient since they are naturally limited by the burdensome computation of decision criteria at every step. Consequently, they can hardly scale to larger tasks. This paper proposes a novel framework called AGCORL to learn reusable causal discovery policies, which can zero-shot generalize to related tasks with much larger sizes. Specifically, AGCORL employs an Ordering Learning (OL) agent to directly infer the order of variables taken from the observational data as input. To further improve the generalizability of the OL agent, an ADversarial (AD) agent is employed to actively mine tasks where the OL agent fails to find high-quality solutions. We theoretically prove that the AD agent significantly reduces the number of required tasks to achieve generalizability of the OL agent. Extensive empirical evaluations demonstrate the superiority of our method in both runtime and solution quality over the state-of-the-art baselines. 1 Introduction Discovering and understanding causal relations is a fundamental problem not only in machine learning but also in a variety of scientific disciplines such as computational biology [Friedman et al., 2000; Sachs et al., 2005], epidemiology [Robins et al., 2000; Vandenbroucke et al., 2016] and economics [Pearl, 2009; Peters et al., 2017], as well as industrial applications such as recommendations, marketing and stock [Liang et al., 2016; Varian, 2016; Zhang et al., 2017]. A common task of interest is causal structure learning also known as causal discovery [Pearl, 2009; Spirtes et al., 2000; Peters et al., 2017], which requires to identify the causal relationship of variables in observational data as a Directed Acyclic Graph (DAG). Score-based methods are a major class of causal discovery techniques, which aims to find a DAG that optimizes a certain criterion: $$\arg \min_{G} S(G), \text{ subject to } G \in \text{ DAGs},$$ where $S(\cdot)$ is a well-defined function scoring a DAG $G$ with observed data, such as Bayesian Information Criterion (BIC) score [Chickering, 2002]. However, Problem $\Pi$ is NP-hard as the space of DAGs increases super-exponentially with the number of graph nodes [Chickering, 1996; Chickering et al., 2004]. To search effectively, heuristic approaches like Greedy Equivalence Search (GES) add or delete edges greedily based on local heuristics which enforce the acyclicity [Chickering, 2002; Nandy et al., 2018]. Instead of directly searching over the DAG space, Causal Additive Models (CAM) divide the structure learning into two steps: firstly search the best variable ordering greedily, and then prune the extra edges from the fully-connected DAG derived from the ordering [Bühlmann et al., 2014]. These methods need to compute metrics like BIC at each searching step, which makes it challenging to scale up to large tasks. Recent works show that Reinforcement Learning (RL) [Sutton & Barto, 2018] has excellent potential in causal discovery tasks. RL-BIC [Zhu et al., 2020] is the first RL-based casual discovery algorithm that learns to explore the DAG search space via a regularized reward function. However, such a DAG regularizer often makes the algorithms prematurely converge to suboptimal solutions. To address the issue, CORL [Wang et al., 2021] avoids the acyclicity constraints by borrowing the two-stage scheme from CAM. Specifically, CORL trains an actor to output the ordering of variables in each search epoch and tries to find better ordering following the BIC reward. Unfortunately, CORL is still a search-based method, which can hardly scale up to realistic problems with more than hundreds of variables due to the computational cost of the BIC reward [Jensen & Kong (1999); Conati et al. (1997); Andreassen et al. (1991)]. Inspired by the recent successes of applying RL to combinatorial optimization problems [Bello et al. (2016); Khalil et al. (2017); Kool et al. (2019)], we aim to train causal discovery policies that can directly infer causal structure given the observational data as input. In such a way, a well-trained policy can be reused in a class of related tasks even with a much larger number of variables. The biggest challenge of training a reusable policy is the generalizability of the policy to new tasks. To address this challenge, we propose a novel adversarial reinforcement learning framework, where an Order learning agent (OL) and an Adversarial agent (AD) are mutually trained to mine adversarial tasks that the OL agent cannot solve and thus improve the generalizability of the OL agent. Specifically, our contributions fall into the following four parts. 1. We propose an Adversarially Generalizable Causal discovery with Ordering-based Reinforcement Learning framework (AGCORL), under which we can train causal discovery policies that directly infer the causal structures from the observational data. Different from existing works where training tasks are sampled from a pre-determined task distribution, we introduce an AD agent who actively mines the adversarial tasks for the OL agent. 2. We formulate the graph generation problem of the AD agent as a Markov Decision Process (MDP) and propose a novel Ground Truth Reward (GTR) as a fast surrogate of the computationally demanding BIC score. GTR measures the difference between the discovered structures and the ground truth structures in the generated tasks. 3. Theoretically, we show the sample complexity of training the OL agent can be improved by training on adversarial tasks mined by the AD agent. And extensive experimental results on linear and nonlinear synthetic data show that AGCORL generalizes better than pretrained baselines, and can scale to much larger tasks than baselines. Furthermore, the superiority of real-world data shows the potential of our method in practice. 2 RELATED WORK Most methods for structure learning from observational data belong to two classes: independence-based and score-based methods. Our method, AGCORL, is closely related to the second class. Score-based methods cast the causal discovery problem as a combinatorial optimization problem (Problem 1). To search effectively, heuristic approaches like Greedy Equivalence Search (GES) rely on local heuristics to enforce the acyclicity and add or delete edges greedily [Chickering (2002); Nandy et al. (2018)]. Instead of directly searching over the DAG space, Causal Additive Models (CAM) divide the structure learning into two steps: firstly search the best variable ordering greedily, and then prune the extra edges from the fully-connected DAG derived from the ordering [Bühlmann et al. (2014)]. RL-BIC [Zhu et al. (2020)] designed an RL agent to explore the DAG search space guided by a regularized reward function. CORL innovatively combines RL-BIC with CAM, formulating the ordering process as a Markov Decision Process (MDP) and employing an RL algorithm to search for the optimal BIC reward during testing. Inspired by CORL’s formulation, our approach in AGCORL advances this concept by training a generalizable ordering policy using reinforcement learning. This trained policy is capable of inferring the order directly in test time, without the need for further search. AGCORL’s key innovation, bypassing the search process during testing, significantly speeds up execution compared to CORL’s method of searching for optimal order in each test instance, thus greatly enhancing efficiency. The above heuristic and RL methods try to find the causal graph by searching. Another promising direction of research for scaling up causal discovery is continuous-optimization methods. The key that converts the discrete optimization problem into the continuous optimization problem is the differentiable DAG constraint proposed by [Zheng et al. (2018)] in NOTEARS. NOTEARS searches over the linear DAGs space using an augmented Lagrangian method. GOLEM [Ng et al. (2020)] studied the asymptotic role of the sparsity and DAG constraints in linear cases. DAGMA [Bello et al. (2022)] proposed a new DAG constraint via M-matrices and a log-determinant acyclicity characterization, which has better-behaved gradients and an-order-of-magnitude-faster runtime. In order to extend NOTEARS to nonlinear settings, DAG-GNN [Yu et al. (2019)], a graph neural network architecture... (GNN) was proposed, which can be used to learn DAGs via the maximization of evidence lower bound. By design, a DAG-GNN uses parameter sharing, which is not well suited for most DAG learning tasks. GraN-DAG \cite{lachapelle2020grann} also uses NN to model the nonlinear relationship between variables but applies the acyclicity constraint at the level of neural network paths, which achieves better performance than NOTEARS and DAG-GNN. 3 PRELIMINARY Causal Graphical Models (CGM). A CGM is defined by a joint distribution $P_X$ over $d$-dimensional random variable $X = (X_1, \ldots, X_d)$ and an underlying DAG $G = (d, V, E)$, where $V = \{X_1, \ldots, X_d\}$ is the set of nodes, and $E = \{(X_i, X_j) \| i, j = 1, \ldots, d\}$ is the set of directed edges from $X_i$ to $X_j$. The graph structure implies a canonical factorization of the joint distribution, which is referred to as causal factorization: $$P(X_1, \ldots, X_d) = \prod_{j=1}^{d} P(X_j \| Pa(X_j)),$$ where $Pa(X_j)$ represents the parents of node $X_j$ in the DAG $G$, i.e., $Pa(X_j) := \{X_k \| (k, j) \in E\}$. We assume that the observational data $x_j$ is generated by the Structural Causal Model (SCM) \cite{pearl2009causality} with additive noises: $$X_j := f_j(Pa(X_j)) + \epsilon_j, j = 1, \ldots, d$$ where $f_j$ represents the functional relationship between $X_j$ and its parents, and $\epsilon_1, \ldots, \epsilon_d$ denote mutually independent noises associated to each node. The SCM could be of various types, including the Linear Non-Gaussian Additive noise Model \cite{shimizu2006linear} and the Post Nonlinear Model \cite{zhang2009post}, based on reasonable assumptions regarding to different scenarios. Causal Discovery Task & BIC Score. A causal discovery task is a tuple with two elements: $M = (W, D) \in M$. $W \in \{0, 1\}^{d \times d}$ is the adjacency matrix of the underlying causal graph where $W_{ij} = 1$ denotes edge $(i, j) \in E$, and $D = [x_1, \ldots, x_d] \in \mathbb{R}^{m \times d}$ is the dataset of the nodes where $m$ is the number of samples. Given the dataset $D$, the goal of causal structure learning is to find the adjacency matrix $W$ by solving Problem[1]. In previous works, they usually consider BIC which is one of the most popular criterion defined as: $$S_{BIC}(G) = \sum_{j=1}^{d} \left[ \sum_{k=1}^{m} \log p(x_j^k \| Pa(x_j^k); \theta_j) - \frac{\|\theta_j\|}{2} \log m \right]$$ where $\theta_j$ represents the parameters of the likelihood function, which can be linear or a neural network according to $f_j$. The computational cost of the BIC score heavily depends on the size of $\theta_j$. Ordering-based Causal Discovery. The problem of finding a DAG can be cast as finding the order of variables \cite{wang2021ordering} and then prune the fully-connected DAG generated from the inferred order. Formally, let $\Omega$ be an ordered set of variables. We denote by $\Omega_{<X_j}$ the set of variables preceding $X_j$ in $\Omega$. CAM \cite{buhlmann2014causal} searches the order greedily and CORL \cite{wang2021ordering} formulate the ordering process as an MDP: at step $t$, CORL agent takes a action to pick a variable $X_j$ as the $t$-th element in $\Omega$. At the end of one episode, we have $\Omega_{<X_j}$ for all $j \in [d]$, so we can easily establish a unique fully-connected DAG $G^\Omega$ whose canonical factorization is $$P(X_1, \ldots, X_d) = \prod_{j=1}^{d} P(X_j \| \Omega_{<X_j}).$$ Then, the BIC reward can be calculated by Equation[4] to guild the searching of CORL. After all searching episodes, variable selection algorithms \cite{buhlmann2014causal, lachapelle2020grann, wang2021ordering} will be applied to prune the optimal $G^\Omega$ to get the final DAG. 4 ADVERSARIAL RL FRAMEWORK FOR CAUSAL DISCOVERY The existing search-based methods fail to scale up because they have to compute the BIC score at each iteration with a computational cost of $O(d^3)$, where $d$ is the number of variables. Search-based methods can hardly scale up to causal discovery tasks with a large number of variables. In this work, we aim to train a policy to directly infer the order of variables from the observational data without searching. This new approach to the causal discovery tasks has significant advantages in terms of both generalizability and scalability. Unfortunately, training such a policy is challenging because it could require massive number of training tasks. Thus, the quality of the training tasks also plays an important role. In fact, compared to easy counterparts, the tasks where the current policy fails to find high-quality solutions are more valuable. To this end, we propose Adversarially Generalizable Causal Discovery with Ordering-based Reinforcement Learning (AGCORL) framework. In the AGCORL framework, Order Learning (OL) agent and ADversarial (AD) agent are trained adversarially. OL agent is trained on a set of tasks $M_{train} = \{M_1, \ldots, M_n\}$ to directly infer the order of variables given the data $D$ of a causal discovery task $M$. Moreover, instead of training the policy with the tasks sampled from some pre-determined distributions, we train the other AD agent which actively mines the adversarial tasks $M_{adv}$ on which the current OL agent performs poorly and adds them to the training tasks pool $M_{train}$. 4.1 Inferring Order of Variables by OL Agent As is introduced in Section 3, the causal discovery task can be reduced to inferring the order of variables Bühlmann et al. (2014); Wang et al. (2021). Thus, we formulate the order search problem as an $d$-step MDP, where $d$ represents the number of variables. The basic elements of the MDP are illustrated as follows. Action. The action $a_t$ at each timestep is to select a variable from the candidate variable set $V = \{X_1, \ldots, X_d\}$. Once a variable is selected, it will be removed from the candidate variable set. Then, at the end of an episode, the actions will make up an ordered set $\Omega$ consisting of all variables. State and Transition. A state describes the current relationship between variables (nodes in the DAG). At the beginning of each episode, we will sample a batch of $N$ samples $[x_1, \ldots, x_d] \in \mathbb{R}^{N \times d}$ from dataset $D$. Each variable $X_i \in V$ can be represented by an embedding $s_i = \Phi(x_i)$, where $\Phi$ is a standard Transformer encoder. The overall state $S^t$ can be represented by a tuple $<S^+_t, S^-_t>$, where $S^+_t$ is the set of embeddings of variables that have not been selected, and $S^-_t$ is the set of embeddings of variables that have been selected. Obviously, at the initial state $S^0_+ = \{s_i \| i = [d]\}$ contains all node embeddings and $S^0_- = \emptyset$. At the end of episode, $S^T_+ = \emptyset$ and $S^T_- = \{s_i \| i = [d]\}$. Fig. 5 in Appendix illustrates how the policy network maps a state $S^t$ to an action $a_t$. Reward. As aforementioned, the computational cost of the BIC score prohibits the existing methods from scaling up to large problems. In fact, the computation requires performing linear regression, neural network training, or Gaussian process regression in every training step. On the other hand, since the training tasks are generated by the AD agent in our scenario, we have access to the corresponding ground truth DAGs during training. Therefore, we propose a fast Ground Truth Reward (GTR) to evaluate the discrepancy between the resulting DAG and the ground truth DAG. The goal of the OL agent is to minimize such a discrepancy by maximizing the GTR. Ground Truth Reward. The reward aims to evaluate the order $\Omega$ by comparing its corresponding DAG $G^\Omega$ with the ground truth DAG $G^$. However, many orders could correspond to the same DAG since exchanging two irrelevant variables in an order does not affect the resulting DAG. In other words, $G^\Omega$ only inherits a partial order from the full ordered set $\Omega$. Therefore, the principle of designing reward should capture the differences of partial orders between $G^\Omega$ and $G^$. Suppose two nodes in $\Omega$ satisfy partial order $X_i \prec X_j$. Then, at least one path from $X_j$ to $X_i$ must be in the corresponding DAG $G^\Omega$. If we reverse the partial order of the two nodes to $X_j \prec X_i$, at least one edge must be reversed in $G^\Omega$. Based on this observation, we design the ground truth reward function by punishing the reversed edges comparing $G^\Omega$ with $G^$. We denote by $e_{\text{rev}}^\Omega$ the number of reversed edges comparing $G^\Omega$ with $G^$. In addition, since we will train multiple tasks with different numbers of variables, it is necessary to balance the rewards of different tasks. Finally, we define the episodic Ground Truth Reward of task $M$ as $R_{OL}(M, \Omega) = -e_{\text{rev}}^\Omega / h$, where $\Omega$ is the final order outputted by the policy and $h$ is the total number of edges in $G^$. The negative sign indicates the punishment, as the goal of the OL agent is to maximize the reward. Fig. 2 shows an example of computing the GTR. Note that the computation of GTR requires only counting the edges and, therefore much more efficient than the computation of the BIC score. OL agent policy and training.* Since the policy of the OL agent sequentially selects variables at each time step, we choose the Pointer Net [Vinyals et al., 2015] as the backbone of our policy network $\pi_\phi$. Fig. 5 in Appendix shows the details of the network architecture. We adopt the actor-critic method [Konda & Tsitsiklis, 1999] to train the OL agent, where an additional critic network $V_\psi$ is introduced to estimate the baseline value of states. To improve the generalizability of the OL agent, we will iteratively train it over a set of tasks $M_{\text{train}}$. For each task $M \sim M_{\text{train}}$, the policy gradient for the actor is shown in Equation 5. Note that the reward $R_{OL}(M, \Omega)$ can only be computed at the end of an episode when all variables have been selected, therefore our critic $V_\psi$ is only used to estimate the value of the initial state. $$\nabla J(\phi) = \mathbb{E}_{S^0 \sim D_M} \left[ (R_{OL}(M, \Omega) - V_\psi(S^0)) \sum_{t=0}^{T} \nabla_\phi \log \pi_\phi (a_t \| S^t) \right]$$ The critic $V_\psi$ will be episodically updated by minimizing the following Mean Square Error (MSE). $$L(\psi) = \mathbb{E}_{S^0 \sim D_M} \left[ \text{MSE}(R_{OL}(M, \Omega), V_\psi(S^0)) \right].$$ Figure 3: An example of adversarial DAG generation: the blue nodes are the nodes generated in the previous timesteps and the green node is the newly generated node. At each timestep $ t $, the AD agent output $ a_{adv}^t $ to determine the parents of the green node. For example, $ a_{adv}^3 = \{0, 1, 0\} $ means only $ X_2 $ is the parent of $ X_4 $. Then we will generate data of $ X_4 $ following Equation [3]: $ X_j := f_4((X_2)) + \epsilon_4 $, where $ f_4 $ and $ \epsilon_4 $ are sampled from the SCM distribution and noise distribution. ### 4.2 Graph Generation with Adversarial Agent To improve the generalizability of the OL agent, we need to actively mine causal discovery tasks where the OL agent fails. We also formulate this causal discovery task generation process as an MDP. To generate a causal discovery task, one needs to determine its graph size, graph structure, type of Structural Causal Model (SCM) and the observational data. In real-world scenarios, the ground truth of SCM type is usually unknown, the SCM type used to model causal relationship is usually determined under some reasonable assumptions, such as Linear Non-Gaussian Additive noise Model (LiNGAM) and Post Nonlinear Model (PNL) [Shimizu et al., (2006); Zhang & Hyvärinen (2009)]. In addition, the policy trained on small tasks can generalize to large ones, we keep the sizes of the generated tasks the same with that in the original datasets. Hence, the graph generation problem of our AD agent is reduced to specifying the DAG structure and the observational data associated with the nodes in the DAG. In order to reduce the action space at each step, our AD agent is designed to generate nodes one by one. We formulate the sequential decision-making process as an MDP as follows. State. The state of the AD agent describes the set of variables generated so far. We denote by $ S_{adv}^t = \{s_0, \ldots, s_t\} $ the state of the AD agent at time $ t $, where $ s_t = \Phi_{adv}(x_t) $ is the embedding of the variable $ X_t $ and $ x_t \in \mathbb{R}^N $ is the batch of data associated with $ X_t $ generated at time $ t $. Action and Transition. An action $ a_{adv}^t $ is a length-$ t $ binary vector sampled from a $ t $-dimensional Bernoulli distribution, whose parameters are determined by the policy of the AD agent. The action specifies how a newly generated node $ X_t $ is added to the current adversarial DAG $ G_{adv}^t $. Fig. 3 shows an example of constructing the adversarial DAG. Once the SCM of Equation [3] is specified, we will have a function $ F $ that maps the data generated so far $ \{x_0, \ldots, x_t\} $ and the current $ G_{adv}^t $ to $ x_{t+1} $. At the end of the episode, we will have an adversarial task $ M_{adv} = \{G_{adv}^T, \{x_0, \ldots, x_T\}\} $. Reward. The AD agent aims to find tasks that the OL agent fails to solve, so the reward for the AD agent is based on the performance of the OL agent on the generated task $ M_{adv} $. Therefore, we define the reward for the AD agent as $ R_{AD}(M_{adv}, \Omega) = -R_{OL}(M_{adv}, \Omega) $, where $ \Omega $ is the ordered set of variables inferred by the OL agent. AD agent policy and training. The policy of the AD agent maps the current state to the parameters of the Bernoulli distribution used at the next time step. Fig. 6 in Appendix shows the architecture of the policy network of the AD agent. We also adopt the actor-critic framework [Konda & Tsitsiklis (1999)] to train the AD agent and reuse the critic $ V_\psi $ of OL agent. As the input of $ V_\psi $ should be the embedding of the full set of nodes, so here the baseline value $ V_\psi(S_{adv}^T) $ is estimated by the terminal state $ S_{adv}^T $. The policy gradient for the AD agent actor $ \pi_\theta $ is written as follows. \[ \nabla J(\theta) = \mathbb{E}_{S_{adv}} \left[ (V_\psi(S_{adv}^T) - R_{OL}(M_{adv}, \Omega)) \sum_{t=0}^{T} \nabla_\theta \log \pi_\theta(a_{adv}^t \| S_{adv}^t) \right] \] (7) 4.3 Adversarial Training and Deployment In this section, we introduce how to jointly train the OL agent and AD agent in our proposed adversarial training framework. All training tasks $M_{adv}$ mined by the AD agent will be stored in a set $M_{train}$. The adversarial training framework can be viewed as a zero-sum game. In each training epoch, the OL agent and the AD agent are trained in turn to maximize their own rewards and minimize opponents’ rewards. In each adversarial training epoch, the OL agent samples tasks from the tasks pool and learns to infer the correct order guided by GTR; then the AD agent is trained to find the tasks where the OL agent performs unsatisfactorily by minimize the performance of the OL agent on its generating tasks measured by GTR; finally the generated tasks will be added to task pool which will be learned by the OL agent in the following epochs. Please also refer to Algorithm 3 in Appendix C for details. Deployment. After adversarial training, the OL agent is supposed to zero-shot transfer to target tasks. However, the agent cannot take all data as input because of the data-sampling state space design. To get better performance from the probabilistic policy, we sample a batch of initial states in parallel and get a batch of ordered sets. Then we rank them by their BIC scores and select the best actions. Finally, we prune the fully-connected graph generated from the best order to obtain the final DAG. Alg. 4 in Appendix shows the details. 5 Experiment In this section, we conduct experiments to verify the generalizability of our methods to tasks with different sizes, noise types, and function types and compare our method with baselines in terms of performance and scalability on synthetic linear and nonlinear tasks as well as real data sets. Baselines. The baselines include random policy, the heuristic ordering-based searching approaches CAM [Bühlmann et al., (2014)] and CORL [Wang et al., (2021)], the gradient-based methods NOTEARS [Zheng et al., (2018)], DAG-GNN [Yu et al., (2019)] and GraN-DAG [Lachapelle et al., (2020)], and CORL-P which is CORL pretrained with presampled tasks. We use the code from the causal discovery toolbox [Zhang et al., (2021)]. Data generation. We generate testing synthetic data sets which vary along five dimensions: level of edge sparsity, graph type, number of nodes, causal functions, and sample size. We sample 10 data sets with 500 samples for each task: a ground truth DAG $G$ is firstly drawn randomly from either the Erdős–Rényi (ER) or scale-free (SF) graph model (5 from the ER graph model and the other 5 from the SF graph model) and the data are then generated according to different given Structural Equation Models (SEMs) model $X_j := f_j(\text{Pa}(X_j)) + \epsilon_j, j = 1, \ldots, d$. Metrics. We consider two common metrics to evaluate the performance: True Positive Rate (TPR) and Structural Hamming Distance (SHD). The former indicates the probability of finding the right edges, which is the higher, the better. The latter counts the total number of missing, false positive, or reversed edges, which is the smaller, the better. Pruning. We adopt the same variable selection methods for edge pruning as CORL. For linear tasks, we apply linear regression to the obtained fully-connected DAG and then use a threshold to prune edges with small weights, as similarly used by [Zheng et al., (2018)]. For the non-linear tasks, we adopt the CAM pruning [Bühlmann et al., (2014)] used by [Lachapelle et al., (2020)]. For each variable $X_j$, one can fit a generalized additive model against the current parents of $X_j$ and then apply significance testing of covariates, declaring significance if the reported p-values are no greater than 0.001. Other variable selection methods can also be considered, such as sparse candidate [Teyssier & Koller, (2005)] and group Lasso [Schmidt et al., (2007)]. 5.1 Linear Models with Gaussian Noise We further evaluate the proposed methods on linear-Gaussian (LG) tasks with equal variance Gaussian noise. We set $h \in \{2, 5\}$ and $d \in \{50, 100, 150, 200\}$ to obtain the ER and SF graphs with different levels of edge sparsity and different numbers of nodes. Then we generate 500 samples for each task following the linear SEM: $\mathbf{X} = \mathbf{W}^T \mathbf{X} + \epsilon$, where $\mathbf{W} \in \mathbb{R}^{d \times d}$ denotes the weighted adjacency matrix obtained by assigning edge weights independently sampled from a uniform distribution $Unif([-2, -0.5] \cup [0.5, 2])$. Here we present the evaluation result of the proposed method and baselines on LG tasks with 50- and 100-node tasks in Table 1. In this experiment, CORL is trained from scratch in each task for 2000 episodes. AGCORL is trained on 20-node tasks for 10 epochs. At the end of each epoch, the AD agent generates 10 adversarial tasks, which will be added to the training task pool. So the total number of training tasks is 100. CORL-P is trained for the same total of 40000 iterations as AGCORL on 200 uniformly sampled 20-node tasks. Across all settings, AGCORL is the best-performing method in terms of both TPR and SHD. For scalability, the running time of CORL is the longest due to its interactive search by training manner. AGCORL, which is trained on 100 actively mined tasks, is better than CORL-P, which is trained on 200 pre-sampled tasks, which shows the importance of adversarial training. We also present AGCORL’s performance on larger tasks in Fig. 7 in the Appendix, the SHD increases as the number of edges increases, but the TPR only decreases a little even on 200-node tasks, which shows that our method can generalize to very large tasks. To further illustrate the effect of adversarial training, we present the joint training curve of AGCORL on LG tasks in Fig. 4. The periodical downward spikes illustrate the adversarial training. As the amplitude of the spikes becomes smaller, the generalizability of the OL agent becomes better, and thus the testing performance becomes better too. ![Figure 4](image-url) The left figure is training curve of AGCORL on 20-node LG tasks for 10 epochs and the right figure is the evaluation on 30-node-5-edge tasks at each epoch. ### Table 1: Empirical results for DAGs of 50 and 100 nodes with LG data \| Method \| Random \| NOTEARS \| CORL \| CORL-P \| AGCORL \| \|--------\|--------\|---------\|------\|--------\|--------\| \| 50-node \| \| \| \| \| \| \| 2-edge \| TPR \| 0.37±0.03 \| 0.91±0.07 \| 0.92±0.04 \| 0.92±0.03 \| 0.94±0.03 \| \| \| SHD \| 161.1±21.6 \| 21.1±18.9 \| 21.4±7.4 \| 33.6±11.6 \| 16.1±6.5 \| \| 5-edge \| TPR \| 0.42±0.02 \| 0.70±0.17 \| 0.89±0.09 \| 0.87±0.12 \| 0.95±0.04 \| \| \| SHD \| 351.1±24.3 \| 130.8±42.5 \| 101.1±17.3 \| 172.3±33.5 \| 80.9±15.7 \| \| t \| - \| 12m \| 0.8h \| 4.7s \| 4.8s \| \| 100-node \| \| \| \| \| \| \| 2-edge \| TPR \| 0.39±0.04 \| 0.83±0.01 \| 0.91±0.01 \| 0.90±0.02 \| 0.93±0.01 \| \| \| SHD \| 394.6±27.8 \| 85.3±50.0 \| 87.9±14.6 \| 118.2±21.6 \| 79.3±11.3 \| \| 5-edge \| TPR \| 0.41±0.04 \| 0.64±0.20 \| 0.90±0.02 \| 0.88±0.02 \| 0.94±0.02 \| \| \| SHD \| 940.0±28.5 \| 303.5±128.6 \| 437.3±68.5 \| 504.3±89.2 \| 360±37.4 \| \| t \| - \| 1h \| 12h \| 19.8s \| 19.2s \| Table 2: Empirical results for DAGs of 10 and 30 nodes with GP data \| Method \| CAM \| GraN-DAG \| CORL \| CORL-P \| AGCORL \| \|--------\|-----\|----------\|------\|--------\|--------\| \| TPR \| 0.75±0.06 \| 0.59±0.12 \| 0.74±0.03 \| 0.64±0.08 \| 0.74±0.04 \| \| SHD \| 2.3±1.1 \| 5.2±3.3 \| 2.5±1.1 \| 3.3±1.4 \| 2.5±1.2 \| \| TPR \| 0.40±0.05 \| 0.64±0.11 \| 0.32±0.12 \| 0.32±0.14 \| 0.36±0.07 \| \| SHD \| 18.2±3.7 \| 25.3±4.8 \| 20.0±3.4 \| 21.4±3.8 \| 13.6±2.9 \| \| t \| 63s \| 17m \| 11m \| 49s \| 51s \| \| TPR \| 0.73±0.08 \| 0.35±0.04 \| 0.51±0.09 \| 0.57±0.15 \| 0.72±0.06 \| \| SHD \| 11.1±2.9 \| 20.1±5.7 \| 16.2±4.1 \| 13.8±3.7 \| 11.8±2.6 \| \| TPR \| 0.24±0.04 \| 0.31±0.03 \| 0.19±0.04 \| 0.20±0.05 \| 0.21±0.04 \| \| SHD \| 87.0±19.8 \| 97.4±11.5 \| 90.1±20.3 \| 85.2±16.5 \| 81.0±10.7 \| \| t \| 53m \| 30m \| 12h \| 11m \| 11m \| 5.2 Non-Linear Model with Gaussian Process In this set of experiments, we consider a causal relationship with $f_i$ being a function sampled from the Gaussian Process (GP) with radial basis function kernel of bandwidth one. The additive noise follows standard Gaussian distribution. The GP data sets with $h \in \{1, 4\}$ and $d \in \{10, 30, 80, 100\}$ are generated following $X_j = f_j(Pa(X_j)) + \epsilon_j$, where the function $f_j$ is a function sampled from a GP with radial basis function kernel of bandwidth one and $\epsilon_j$ follows standard Gaussian distribution. Presented in Table 2, AGCORL performs as well as CAM, but the deployment time of AGCORL is much less than CAM when the task is large. GraN-DAG gets the highest TPR in denser tasks, but the SHD is poor because it produces more edges than other methods. Besides, CORL is better than CORL-P on small tasks, but CORL-P performs better than CORL on 30-node tasks because CORL cannot converge in 2000 episodes on 30-node tasks. Like the result in LG tasks, the deployment time of CORL-P is close to AGCORL, but the performance is poor because of a lack of generalizability. Fig. 8 in Appendix shows the performance on large GP tasks, which is much more difficult than the linear case. 5.3 Real-world Data We test our agent trained in Section 5.2 on a real-world data sets: Sachs et al. (2005) with 11-node and 17-edge true graph, which is widely used for research on graphical models. The expression levels of protein and phospholipid in the data set can be used to discover the implicit protein signal network. The observational data set has $m = 853$ samples and is used to discover the causal structure. In this experiment, AGCORL and CORL achieve the best SHD 11, which shows our AGCORL can successfully generalize to real-world data. CAM, GraN-DAG, DAG-GNN and NOTEARS achieve SHDs 12, 13, 16, and 19 respectively. However, the running times of AGCORL and CORL are 56s and 12m, which shows the superiority of AGCORL in scalability. 6 Conclusion In this paper, we propose AGCORL, an adversarial training framework for training generalizable and scalable causal discovery policies. Compared to existing search-based methods, our causal discovery policies directly infer the causal graphs from the observational data, thus significantly reducing the computational cost. AGCORL employs an OL agent to infer the causal graph from data, and an AD agent to actively mine adversarial tasks where the OL fails. To further accelerate training, we design an efficient GTR function to evaluate the quality of inferred causal graphs, which provides reward signals for both agents. Our experiments show the advantages of the AGCORL framework, in terms of both solution quality and scalability. 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WZ6NY4JfFX	As demonstrated in Table 7 in the appendix, as the dataset size increases, the OTS scores progressively deteriorate, and the gap with ITM widens even when using the optimal alpha. How can this be explained? Might this indicate an inherent limitation of the method?	REVISITING THE ROLE OF LANGUAGE PRIORS IN VISION-LANGUAGE MODELS Anonymous authors Paper under double-blind review ABSTRACT Vision-language models (VLMs) are impactful in part because they can be applied to a variety of visual understanding tasks in a zero-shot fashion, without any fine-tuning. We study currently popular generative VLMs that are trained for next-word generation given the image. We explore their zero-shot performance on the illustrative task of image-text retrieval across 8 popular vision-language benchmarks. Our first observation is that they can be repurposed for discriminative tasks (such as image-text retrieval) by simply computing the match score of generating a particular text string given an image. We call this probabilistic score the Visual Generative Pre-Training Score (VisualGPTScore). While the VisualGPTScore produces near-perfect accuracy on some retrieval benchmarks, it produces poor accuracy on others. We analyze this behavior through a probabilistic lens, pointing out that some benchmarks inadvertently capture unnatural language distributions by creating adversarial but unlikely text captions. In fact, we demonstrate that even a “blind” language model that ignores any image evidence can sometimes outperform all prior art, reminiscent of similar challenges faced by the visual-question answering (VQA) community many years ago. We derive a probabilistic post-processing scheme that controls for the amount of linguistic bias in generative VLMs at test time without having to retrain or fine-tune the model. We show that the VisualGPTScore, when appropriately debiased, is a strong zero-shot baseline for vision-language understanding, oftentimes producing state-of-the-art accuracy. 1 INTRODUCTION Vision-language models (VLMs) trained on web-scale datasets will likely serve as the foundation for next-generation visual understanding systems. One reason for their widespread adoption is their ability to be used in an “off-the-shelf” (OTS) or zero-shot manner, without fine-tuning on any target application of interest. We study their OTS use on the task of image-text retrieval (e.g., given an image, predict which of $K$ possible captions is true) across a suite of 8 popular benchmarks. Challenges. While the performance of foundational VLMs is impressive, many open challenges remain. Recent analysis (Kamath et al., 2023; Yukselgonul et al., 2022) points out that leading VLMs such as CLIP (Radford et al., 2021) may often degrade to “bag-of-words” that confuse captions such as "the horse is eating the grass" and "the grass is eating the horse". This makes it difficult to use VLMs to capture compositions of objects, attributes, and their relations. But somewhat interestingly, large-scale language models (LLMs) trained for autoregressive next-token prediction (Brown et al., 2020) seem to be able to capture such distinctions, which we investigate below. A related but under-appreciated difficulty is that of benchmarking the performance of visio-linguistic reasoning. Perhaps the most well-known example in the community is that of the influential VQA benchmarks (Antol et al., 2015), which could be largely solved by exploiting linguistic biases in the dataset – concretely, questions about images could often be answered by “blind” language-only models that did not look at the image (Goyal et al., 2017). Notably, we find that such blind algorithms can still produce strong performance on many contemporary image-text retrieval benchmarks where VLMs may struggle. Generative models for discriminative tasks. We tackle the above challenges by revisiting the role of language priors through a probabilistic lens. To allow for a probabilistic treatment, we focus on generative VLMs that take an image as input and stochastically generate text via next-token Figure 1: Two train-test shifts encountered in image-to-text retrieval tasks. Scenario 1 constructs negative text captions by shuffling words in the true caption (as in ARO-Flickr), but this produces implausible text such as "white a duck spreads its wings while in the water." Here, exploiting the language bias of the training set will help since it will downweight the match score for negative captions. In fact, a blind language-only model can easily identify the correct caption. Scenario 2 constructs alternative text captions that are curated to be plausible (as in SugarCrepe). Here, the language bias of the training set may hurt, since it will prefer to match common captions (that score well under the language prior) as shown on the right. prediction (Li et al., 2022; 2023). We first demonstrate that such models can be easily repurposed for discriminative tasks (such as retrieval) by setting the match score for an image-text pair to be the probability that the VLM would generate that text from the given image. We call this probability score the Visual Generative Pre-Training Score, or VisualGPTScore. Computing the VisualGPTScore is even more efficient than next-token generation since given an image, all tokens from a candidate text string can be evaluated in parallel. Though conceptually straightforward, such an approach (to our knowledge) has not been proposed in the literature. In fact, the generative VLMs that we analyze train separate discriminative heads for matching/classifying image-text pairs (Li et al., 2022), but we find that their language generation head itself produces better scores for matching (since it appears to better capture compositions). Indeed, OTS VisualGPTScore by itself performs surprisingly well on many benchmarks, even producing near-perfect accuracy on ARO (Yuksekgonul et al., 2022). But it still struggles on other benchmarks such as Winoground (Thrush et al., 2022). We analyze this below. The role of language priors. We analyze the discrepancy in performance across benchmarks from a probabilistic perspective. Our key insight is that many benchmark biases can be formalized as mismatching distributions over text between train and test data - $P_{\text{train}}(\text{text})$ versus $P_{\text{test}}(\text{text})$. We use a first-principles analysis to account for distribution shift by simply reweighting the VisualGPTScore with the Bayes factor $P_{\text{test}}(\text{text})/P_{\text{train}}(\text{text})$, a process we call debiasing. To compute the Bayes reweighting factor, we need access to both the train and test language prior. We compute $P_{\text{train}}(\text{text})$ from an OTS VLM with Monte-Carlo samples of $P_{\text{train}}(\text{text}\|\text{image})$ computed on trainset or Gaussian noise images. Because $P_{\text{test}}(\text{text})$ may require access to the test set, we explore simplifying assumptions that assume it is (a) identical to $P_{\text{train}}(\text{text})$, (b) uninformative/uniform, or (c) tunable from a held-out val set. Our analysis helps explain the strong performance of the VisualGPTScore on certain benchmarks and its poor performance on others. Furthermore, this analysis provides simple strategies for improving performance with debiasing. We finally show a theoretical connection between debiasing and mutual information, which can be seen as a method for removing the effect of marginal priors when computing joint probability scores. Empirical Analysis. We present an exhaustive empirical analysis of the OTS VisualGPTScore (and its debiased variants) for open-sourced image-conditioned language models (Li et al., 2022; 2023) across 8 popular vision-language benchmarks. We first point out that VisualGPTScore by itself produces SOTA accuracy on certain benchmarks like ARO (Yuksekgonul et al., 2022) where its inherent language bias helps remove incorrect text caption candidates that are also unnatural (such as "a white duck the its wings while in water" as shown in Fig. 1). In fact, we show that blind baselines also do quite well on such benchmarks, since language-only models can easily identify such poor captions. However, such language biases do not work well on benchmarks where incorrect caption candidates are also realistic. Here, VisualGPTScore should be debiased so as not to naively prefer more common captions that score well under its language prior. When given access to a val set that reveals the amount of language bias in the benchmark, debiasing consistently improves performance on benchmarks such as Flickr30K (Young et al., 2014) and Winoground (Thrush et al., 2022). Interestingly, we find that debiasing can also improve accuracy on the train set used to learn the generative VLM, indicating that such models learn biased estimates of the true conditional distribution $P_{\text{train}}(\text{text}\|\text{image})$. We describe this further in our appendix. 2 RELATED WORKS Vision-language modelling. State-of-the-art VLMs like CLIP (Radford et al., 2021) are pre-trained on web-scale image-text datasets (Schuhmann et al., 2021; 2022) using discriminative objectives including image-text contrastive (ITC) (Radford et al., 2021; Jia et al., 2021) and image-text matching (ITM) (Li et al., 2021; 2022) loss, typically formulated as $P(\text{match}\|\text{image}, \text{text})$. These pre-trained models exhibit robust zero-shot and few-shot (Lin et al., 2023; Wortsman et al., 2022) performance on traditional discriminative tasks (Deng et al., 2009; Lin et al., 2014), often on par with fully-supervised models. More recently, image-conditioned language models like Flamingo (Alayrac et al., 2022) and BLIP (Li et al., 2022; 2023) incorporate generative objectives (Bengio et al., 2003) primarily for downstream tasks such as captioning (Agrawal et al., 2019) and VQA (Goyal et al., 2017). Visio-linguistic compositionality. Benchmarks like ARO (Yuksekgonul et al., 2022), Crepe (Ma et al., 2022), Winoground (Thrush et al., 2022), EqBen (Wang et al., 2023), VL-CheckList (Zhao et al., 2022), and SugarCrepe (Hsieh et al., 2023) show that discriminative scores of VLMs, such as ITCScore and ITMScore, fail on their image-text retrieval tasks that assess compositional reasoning. Concurrently, advances on these tasks often involve fine-tuning discriminative VLMs with more data. One of the most popular approaches, NegCLIP (Yuksekgonul et al., 2022), augments CLIP using programmatically generated negatives from original texts. Extending this, subsequent studies propose more expensive and heavily-engineered solutions. SyViC (Cascante-Bonilla et al., 2023) fine-tunes VLMs on million-scale synthetic images to augment spatial, attributive, and relation understanding. SGVL (Herzig et al., 2023) and Structure-CLIP (Huang et al., 2023) sample negatives using costly scene graph annotations. MosaiCLIP (Singh et al., 2023) and SVLC (Doveh et al., 2022) use linguistic tools such as scene graph parsers and LLMs to design better negative captions. The most recent DAC (Doveh et al., 2023) leverages a combination of foundation models including BLIP2, ChatGPT, and SAM to rewrite and augment image captions. Generative pre-training and scoring. Vision models trained with discriminative objectives often lack incentives to learn structure information (Brendel & Bethge, 2019; Tejankar et al., 2021). Similarly, early LLMs trained with discriminative approaches, such as BERT (Devlin et al., 2018) and RoBERTa (Liu et al., 2019), have also been criticized as bag-of-words models insensitive to word order (Bertolini et al., 2022; Hessel & Schofield, 2021; Papadimitriou et al., 2022; Sinha et al., 2021). Conversely, generative pre-trained LLMs (Radford et al., 2019) demonstrate exceptional compositional understanding while pre-trained solely with a next-token prediction (Bengio et al., 2003) loss. Furthermore, generative scores of LLMs (OpenAI, 2023; Chung et al., 2022; Zhang et al., 2022) have flexible usage in downstream tasks, such as text evaluation (Yuan et al., 2021; Fu et al., 2023) and reranking (Keskar et al., 2019). 3 THE ROLE OF LANGUAGE PRIORS In this section, we present a simple probabilistic treatment for analyzing the role of language priors in image-conditioned language models (or generative VLMs). Motivated by their strong but inconsistent performance across a variety of image-text retrieval benchmarks, we analyze their behavior when there exists a mismatch between training and test distributions, deriving simple schemes for addressing the mismatch with reweighting. We conclude by exposing a connection to related work on mutual information. Computing $P(t\|i)$. To begin our probabilistic treatment, we first show that image-conditioned language models (that probabilistically generate text based on an image) can be repurposed for computing a score between a given image $i$ and text caption $t$. The likelihood of a text sequence $t = \{t_1, t_2, \ldots, t_m\}$ conditioned on image $i$ is naturally factorized as an autoregressive product (Bengio et al., 2003): $$P(t\|i) = \prod_{k=1}^{m} P(t_k\|t_{<k}, i)$$ (1) Image-conditioned language models return back $m$ softmax distributions corresponding to the $m$ terms in the above expression. Text generation requires sequential token-by-token prediction, since token $t_k$ must be generated before it can be used as an input to generate the softmax distribution over token $t_{k+1}$. Interestingly, given an image $i$ and text sequence $t$, the above probability can be computed in parallel because the entire sequence of tokens $\{t_k\}$ are already available as input. We provide a visual illustration in Figure 2a. Train-test shifts. Given the image-conditioned model of $P(t\|i)$ above, we now analyze its behavior when applied to test data distributions that differs from the trainset, denoted as $P_{test}$ versus $P_{train}$. Recall that any joint distribution over images and text can be factored into a product over a language prior and an image likelihood $P(t,i) = P(t)P(i\|t)$. Our analysis makes the strong assumption that the image likelihood $P(i\|t)$ is identical across the train and test data, but the language prior $P(t)$ may differ. Intuitively, this assumes that the visual appearance of entities (such as a "white duck") remains consistent across the training and test data, but the frequency of those entities (as manifested in the set of captions $P(t)$) may vary. We can now derive $P_{test}(t\|i)$ via Bayes rule: $$P_{test}(t\|i) \propto P(i\|t)P_{train}(t)$$ $$= P(i\|t)\frac{P_{train}(t)}{P_{train}(t)}P_{test}(t)$$ $$\propto P_{train}(t\|i)\frac{P_{test}(t)}{P_{train}(t)}$$ The above shows that the generative pre-training score $P_{train}(t\|i)$ need simply be weighted by the ratio of the language priors in the testset versus trainset. Intuitively, if a particular text caption appears more often in the testset than the trainset, one should increase the score reported by the generative model. However, one often does not have access to the text distribution on the testset. For example, real-world deployments and benchmark protocols may not reveal this. In such cases, one can make two practical assumptions; either the language distribution on test is identical to train, or it is uninformative/uniform (see Figure 1): - Scenario 1: $P_{test}(t) = P_{train}(t)$ ⇒ Optimal score is $P_{train}(t\|i)$. - Scenario 2: $P_{test}(t)$ is uniform ⇒ Optimal score is $\frac{P_{train}(t\|i)}{P_{train}(t)}$. Tunable $\alpha$. In reality, a testset might be a mix of both scenarios. To model this, we consider a soft combination where the language prior on the testset is assumed to be a flattened version of the language prior on the trainset, for some temperature parameter $\alpha \in [0, 1]$: $$P_{test}(t) \propto P_{train}(t)^{1-\alpha} \Rightarrow \text{Optimal score is } \frac{P_{train}(t\|i)}{P_{train}(t)^\alpha}$$ By setting $\alpha$ to 0 or 1, one can obtain the two scenarios described above. Some deployments (or benchmarks) may benefit from tuning $\alpha$ on a val set. Implications for retrieval benchmarks. We speculate some benchmarks like ARO-Flickr (Yuksek et al., 2022) are close to scenario 1 because they include negative captions that are implausible, such as “a white duck the its wings while in water spreads”. Such captions will have a low score under the language prior $P_{train}(t)$ and so reporting the raw generative score $P_{train}(t\|i)$ (that keeps its language prior or bias) will improve accuracy. In fact, we show that applying a blind language model (that ignores all image evidence) can itself often identify the correct caption. On the other hand, for test datasets with more realistic negative captions (scenario 2), it may be useful to remove the language bias of the trainset, since that will prefer to match to common captions (even if they do not necessarily agree with the input image). This appears to be the case for SugarCrepe (Hsieh et al., 2023), which uses LLMs like ChatGPT to ensure that the negative captions are realistic. Relationship to prior approaches. Our approach to debiasing is reminiscent of mutual information, which can also be seen as a method for removing the effect of marginal priors when computing joint probability scores. In fact, our Appendix A derives that $\alpha$-debiasing is equivalent to a form of pointwise mutual information (PMI) known as $PMIk$ for $k = \frac{1}{\alpha}$. Figure 2: Estimating $P_{\text{train}}(t\|i)$ and $P_{\text{train}}(t)$ from generative VLMs. Figure (a) shows how image-conditioned language models such as \cite{li2022blip} that generate text based on an image can be repurposed for computing $P_{\text{train}}(t\|i)$, which is factorized as a product of $\prod_{k=1}^{m} P(t_k\|t_{<k}, i)$ for a sequence of $m$ tokens. These terms can be efficiently computed in parallel, unlike sequential token-by-token prediction for text generation. Figure (b) shows two approaches for Monte Carlo sampling of $P_{\text{train}}(t)$. While the straightforward approach is to sample trainset images, we find that using as few as three “null” (Gaussian noise) images can achieve more robust estimates. 4 EXPERIMENTAL RESULTS ON I-TO-T RETRIEVAL In this section, we verify our hypothesis on I-to-T retrieval benchmarks using state-of-the-art multimodal generative VLMs. In particular, we adopt image-conditioned language models such as BLIP \cite{li2022blip} as the learned estimator of $P_{\text{train}}(t\|i)$. Then, we discuss how we perform Monte Carlo estimation of $P_{\text{train}}(t)$, including a novel efficient sampling method based on “content-free” Gaussian noise images. Finally, we show the state-of-the-art results of our generative approach on existing I-to-T retrieval tasks. Preliminaries. We leverage OTS image-conditioned language models \cite{yu2022ots, alayrac2022alpaca, li2023vqa} to estimate $P_{\text{train}}(t)$. For ablation, we use the open-sourced BLIP models \cite{li2022blip}, trained on public image-text corpora using discriminative (ITC and ITM) and generative (captioning) objectives. Discriminative objectives typically model $P(\text{match}\|t,i)$. For example, ITCScore calculates cosine similarity scores between image and text features using a dual-encoder; ITMScore jointly embeds image-text pairs via a fusion-encoder and returns softmax scores from a binary classifier. Lastly, we term the generative score as Visual Generative Pre-Training Score (VisualGPTScore). While BLIP is pre-trained using all three objectives, this generative score has not been applied to discriminative tasks before our work. Implementing VisualGPTScore. Our method calculates an average of the log-likelihoods of $t_k$ at each token position $k$ and applies an exponent to cancel the log: $$\text{VisualGPTScore}(t,i) := e^{\frac{1}{m} \sum_{k=1}^{m} \log(P(t_k\|t_{<k},i))}$$ To condition on an input image, BLIP uses a multimodal casual self-attention mask \cite{li2022blip} in its image-grounded text decoder, i.e., each text token attends to all its preceding vision and text tokens. We emphasize that VisualGPTScore has the same computational cost as ITMScore, which uses the same underlying transformer but with a bi-directional self-attention mask to encode an image-text pair. We address potential biases of this estimator in Appendix C. Estimating $P_{\text{train}}(t)$ using Monte Carlo sampling (oracle approach). Given $P_{\text{train}}(t\|i)$, we can estimate $P_{\text{train}}(t)$ via classic Monte Carlo sampling \cite{shapiro2003monte}, by drawing $n$ images from the train distribution, such as LAION114M \cite{schuhmann2021laion} for BLIP: $$P_{\text{train}}(t) \approx \frac{1}{n} \sum_{k=1}^{n} P_{\text{train}}(t\|i_k)$$ Reducing sampling cost with content-free images (our approach). The above Equation 9 requires many trainset samples to achieve robust estimates. To address this, we draw inspiration from \cite{zhao2021calibrate}, which uses a content-free text prompt “N/A” to calibrate the probability of a text from LLMs, i.e., $P(t\|\text{“N/A”})$. To apply this to our generative VLMs, we choose to sample “null” inputs Table 1: OTS generative VLMs are SOTA on image-to-text retrieval benchmarks. We begin by evaluating blind language models (in red). Surprisingly, this already produces SOTA accuracy on certain benchmarks such as ARO-Flickr, compared to the best discriminative approaches (in gray). We also find that blind inference of generative VLMs, $ P_{\text{train}}(t) $ via sampling Gaussian noise images (in blue), often performs better and achieve above-chance performance even on the most recent SugarCrepe. Next, we show that simply repurposing a generative VLM’s language generation head for computing image-text scores (VisualGPTScore in yellow), which corresponds to $ \alpha = 0 $, consistently produces SOTA accuracy across all benchmarks. Finally, debiasing this score by tuning $ \alpha $ on val set (in green) further improves performance, establishing the new SOTA. as Gaussian noise images. As a result, our approach requires as few as three images to compute Eq. 9 by sampling from Gaussian noise images with a mean of 0.4 and a standard deviation of 0.25. We find this method to be less computationally demanding and just as effective as sampling thousands of images from trainset. We provide a visual illustration of this method in Figure 2b. We include sampling details in Appendix B. Benchmarks and evaluation protocols. We comprehensively report on four popular I-to-T retrieval benchmarks, including ARO (Yuksekgonul et al., 2022), Crepe (Ma et al., 2022), SugarCrepe (Hsieh et al., 2023), and VL-CheckList (Zhao et al., 2022). In these datasets, each image has a single positive caption and multiple negative captions. ARO (Yuksekgonul et al., 2022) has four datasets: VG-Relation, VG-Attribution, COCO-Order, and Flickr30k-Order. SugarCrepe (Hsieh et al., 2023) has three datasets: Replace, Swap, and Add. For Crepe (Ma et al., 2022), we use the entire productivity set and report on three datasets: Atom, Negate, and Swap. VL-CheckList (Zhao et al., 2022) has three datasets: Object, Attribute, and Relation. We visualize all datasets in Appendix Table 1. SOTA performance on all four benchmarks. In Table 1, we show that our OTS generative approaches, based on the BLIP model pre-trained on LAION-114M with ViT-L image encoder, achieves state-of-the-art results on all benchmarks. We outperform the best discriminative VLMs, including LAION5B-CLIP, and consistently surpass other heavily-engineered solutions, including NegCLIP, SyViC, MosaicCLIP, DAC, SVLC, SGVL, Structure-CLIP, all of which fine-tune CLIP on much more data. Details on how we report the baseline results can be found in Appendix E. For reference, we also include results of text-only Vera and Grammar from Hsieh et al. (2023). To show that even the most recent SugarCrepe is not exempt from language biases, we run two more text-only methods: 1. $ P_{LLM}(t) $: passing captions into a pure LLM, such as BART-base (Yuan et al., 2021), FLAN-T5-XL (Chung et al., 2022), and OPT-2.7B (Zhang et al., 2022), to compute a text-only GPTScore (Fu et al., 2023). 2. $ P_{train}(t) $: passing both captions and Gaussian noise images to BLIP as shown in Figure 2. Visualization of $ \alpha $-tuning. Finally, we observe that $ \alpha $-tuning can consistently improve the performance. For visualization, we attach the results of $ \alpha $-tuning in Table 2. We show side-by-side frequency charts of $ P_{train}(t) $ for positive and negative captions. ## 5 ADDITIONAL EXPERIMENTAL RESULTS In this section, we apply our OTS generative approaches to more benchmarks, including two compositionality benchmarks Winground (Thrush et al., 2022) and EqBen (Wang et al., 2023), and two classic large-scale retrieval benchmarks COCO (Lin et al., 2014) and Flickr30K (Young et al., 2014). While naively applying VisualGPTScore leads to bad performance on these benchmarks, our training-free debiasing solution can consistently improve its performance with a held-out validation set. Furthermore, we derive the optimal text-to-image (T-to-I) retrieval objective and show that OTS generative scores can achieve robust T-to-I performance without debiasing. Evaluation protocols of Thrush et al. (2022). While prior analysis (Diwan et al., 2022; Yuksekgonul et al., 2022) suggests that Winground is too out-of-distribution to evaluate compositionality, we argue that evaluation protocols of Winground and EqBen are more robust for future evaluations of VLMs. In these two benchmarks, each sample consists of two image-text pairs, ensuring uniform image and text priors. For simplicity, we consider a single Winground sample: $(i_0, t_0)$ and $(i_1, t_1)$. The joint probabilities are $ P_{test}(i_0, t_0) = P_{test}(i_1, t_1) = 0.5 $. Meanwhile, $ P_{test}(i_0, t_1) = P_{test}(i_1, t_0) = 0 $. Applying the law of total probability gives $ P_{test}(t_0) = P_{test}(t_1) = 0.5 $. A similar derivation can show that image priors are uniform too. In addition, Winground’s evaluation metrics (text score and image score) penalize unimodal shortcut solutions. For example, in I-to-T retrieval, the text score gets 1 point only if both images are matched to the correct caption. Therefore, “blind” solutions that choose the same text regardless of images will get 0 text score. Similarly, for T-to-I retrieval, the image score gets 1 point only if both captions are matched to the correct image. Tuning $ \alpha $ through cross validation. In Table 3a, we first show that OTS generative scores without debiasing ($ \alpha = 0 $) lead to inferior performance on these I-to-T benchmarks. This confirms the importance of $ \alpha $-tuning; even a simple $ \alpha = 1 $ can consistently and often significantly improve their I-to-T results. Furthermore, we try to use a held-out validation set to tune for optimal $ \alpha \in [0, 1] $. We sample half of the data as validation set to search for $ \alpha^_{val} $ (using a step size of 0.001) and report the performance on the other half. We repeat this process 10 times to and report the mean and std. We observe that the optimal alpha is usually stable under the same dataset, regardless of the sampled val set. For COCO and Flickr30K, we perform $ \alpha $-tuning using Recall@1 (R@1) on the official validation split. Because sampling additional Gaussian noise images can be too costly on these large-scale benchmarks, we directly approximate $ P_{train}(t) $ by averaging the scores of testset images, without incurring any computational cost. More ablation studies such as $ \alpha $-tuning using testset can be found in Appendix B. We also include the results of the ITMScore of BLIP for reference. While our debiasing solution can always boost performance, we observe that generative approaches still lag behind the ITMScore. This motivates us to study biases of generative scores towards more “common” texts in Appendix C. Table 2: $\alpha$-tuning on I-to-T benchmarks and $P_{train}(t)$ frequency charts of both positive and negative captions. Increasing $\alpha$ from 0 to 1 hurts performance on benchmarks with non-sensical negative captions such as ARO and Crepe. Such negative captions are easier to identify because of their low score under the language prior $P_{train}(t)$, implying such benchmarks may even be solved with blind algorithms that avoid looking at images. On the other hand, for benchmarks like SugarCrepe with more balanced $P_{train}(t)$ between positives and negatives, tuning $\alpha$ may lead to performance gain. Extending to T-to-I retrieval. Though not the focus of our work, we also show that image-conditioned language models can be applied to T-to-I retrieval. Given a text caption $t$, we can rewrite the Bayes optimal T-to-I retrieval objective as: $$P_{test}(i\|t) \propto P_{train}(t\|i) P_{train}(i)$$ Equation 10 is hard to implement because we do not have access to $P_{train}(i)$. However, when $P_{train}(i)$ is approximately uniform, one can directly apply $P_{train}(t\|i)$ for optimal performance. We report T-to-I performance on all four benchmarks in Table 3.b, where our generative approach obtain competitive results compared against ITMScore, presumably because T-to-I retrieval is less affected by language biases. Table 3: Additional results on Winoground/EqBen/COCO/Flickr30K retrieval benchmarks. Table (a) shows that tuning $\alpha$ can be essential for these compositionality and large-scale retrieval benchmarks. While OTS generative scores do not work well, debiasing with a larger $\alpha$ can consistently and often significantly improve I-to-T results on these tasks. To highlight the performance improvement, we mark results without debiasing ($\alpha = 0$) (in yellow), debiasing with a fixed $\alpha = 1$ (in pink), and cross-validation using held-out val sets ($\alpha = \alpha^_{val}$) (in green). Table (b) shows that OTS generative scores can obtain favorable results on classic T-to-I retrieval tasks, competitive with the ITMScore. 6 DISCUSSION AND LIMITATIONS Summary. Our study shows the efficacy of generative pre-training scores in solving discriminative tasks. With the rise of generative pre-training in recent models like GPT-4 [OpenAI, 2023], we see our work as a reliable starting point for future tasks. We present a first-principles analysis to account for mismatching distributions over text between train and test data. Based on this, we introduce a robust training-free (zero-shot) solution to debias linguistic priors in generative scores, achieving consistent and often significant improvement on all I-to-T retrieval tasks. Our thorough analysis also explains the performance discrepancy of generative scores on different benchmarks, and we hope it can encourage future work to revisit the issue of language biases in vision-language benchmarks. Limitations and future work. Our approach depends on generative VLMs pre-trained on noisy web datasets, which may result in inherited biases [Mehrab et al., 2021]. 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bcNwnuWMe0	How long does it take on average for the flow from one upstream gauge station to reach a neighboring downstream gauge station? I think these values should be taken into account for the history window size (24 hours currently) and the prediction horizon (6 hours currently).	Exploiting River Network Topology for Flood Forecasting with Graph Neural Networks Anonymous authors Paper under double-blind review Abstract Climate change exacerbates riverine floods, which occur with higher frequency and intensity than ever. The much-needed forecasting systems typically rely on accurate river discharge predictions. To this end, the SOTA data-driven approaches treat forecasting at spatially distributed gauge stations as isolated problems, even within the same river network. However, incorporating the known river network topology into the prediction model has the potential to leverage the adjacency relationship between gauges. Thus, we model river discharge for a network of gauging stations with a GNN, and compare the forecasting performance achieved by different adjacency definitions. Our results show that the model fails to benefit from the river network topology information, regardless of the number of layers and, thus, propagation distance. The learned edge weights correlate with neither of the static definitions and exhibit no regular pattern. Furthermore, a worst-case analysis reveals that the GNN struggles to predict sudden discharge spikes. This work may serve as a justification for the SOTA treating gauges independently and suggests that more improvement potential lies in anticipating spikes. 1 Introduction Floods are among the most destructive natural disasters that occur on Earth, causing extensive damage to infrastructure, property, and human life. They are also the most common type of disaster, accounting for almost half of all disaster events recorded (cp. Figure 1). In 2022 alone, floods affected 57.1 million people worldwide, killed almost 8000, and caused 44.9 billion USD in damages (CRED, 2022). With climate change ongoing, floods have become increasingly frequent over the last decades and are expected to be even more prevalent in the future (United Nations, 2022). Thus, early warning systems that can help authorities and individuals prepare for and respond to impending floods play a crucial role in mitigating fatalities and economic costs. Figure 1: Historical occurrence of natural disasters by disaster type. The number of events increased over time, with floods being the most common. (Ritchie et al., 2022). Operational forecasting systems such as Google’s Flood Forecasting Initiative (Nevo et al., 2022) typically focus on riverine floods, which are responsible for the vast majority of damages. A key component in these systems is the prediction of future river discharge at a gauging station based on environmental indicators such as past discharge and precipitation. The state-of-the-art data-driven approaches are based on Kratzert et al. (2019b) and consist in training an LSTM variant on multiple gauges jointly to exploit the shared underlying physics. However, even when some of those gauges are in the same river network, this topology information is not taken into account. One reason might be that the main benchmarking dataset family CAMELS-x (Addor et al., 2017; Alvarez-Garreton et al., 2018; Coxon et al., 2020; Chagas et al., 2020; Fowler et al., 2021) does not contain such information. Recently, Klingler et al. (2021) published a new benchmarking dataset LamaH-CE that follows the CAMELS-x framework but includes topology data. In this work, we investigate the effect of river network topology information on discharge predictions by employing a single end-to-end GNN to allow the network structure to be utilized during the prediction process. We train GNNs on LamaH-CE and, to assess the merit of incorporating the graph structure, compare the effect of different adjacency definitions: 1. no adjacency, which is equivalent to existing approaches with cross-gauge shared parameters but isolated gauges, 2. binary adjacency of neighboring gauges in the network, 3. weighted adjacency according to physical relationships like stream length, elevation difference, and average slope between neighboring gauges, and 4. learned adjacency by treating edge weights as a model parameter. Furthermore, we inspect how the learned edge weights from (4) correlate with the static weights in (3). We also explore the role of information propagation distance on predictive capabilities and analyze the model’s behavior on the worst-performing gauge. Our source code is publicly available at https://add-link-after-review. 2 RELATED WORK Classical approaches towards river discharge prediction stem from finite-element solutions to partial differential equations such as the Saint-Venant shallow-water equations (Vreugdenhil, 1994; Wu, 2007). However, these models suffer from scalability issues since they become computationally prohibitive on larger scales, as required in the real world (Nevo et al., 2020). Furthermore, they impose a strong inductive bias by making numerous assumptions about the underlying physics. On the other hand, data-driven methods and in particular deep learning provide excellent scaling properties and are less inductively biased. They are increasingly being explored for a plethora of hydrological applications, including discharge prediction (see surveys by Mosavi et al., 2018; Chang et al., 2019; Sit et al., 2020), where they tend to achieve higher accuracy than the classical models. The vast majority of studies employ Long Short-Term Memory models (LSTM; Hochreiter & Schmidhuber, 1997) due to their inherent suitability for sequential tasks and reliability in predicting extreme events (Frame et al., 2022). Whereas these studies usually consider forecasting for a single gauging station, Kratzert et al. (2019a,b) demonstrate the generalization benefit of training a single spatially distributed LSTM model on multiple gauging sites jointly. Their approach exploits the shared underlying physics across gauges but is still agnostic to the relationship between sites. Incorporating information from neighboring stations or even an entire river network into a spatially distributed model may improve prediction performance. Upstream gauges could “announce” the advent of significantly increased water masses to downstream gauges, which in turn could provide forewarning about flooding already ongoing further downstream. The input then becomes a graph whose vertices represent gauges and edges represent flow between gauges. The corresponding deep learning tool to capture these spatial dependencies is Graph Neural Networks (GNN). Kratzert et al. (2021) employ such a GNN as a post-processing step to route the per-gauge discharge predicted by a conventional LSTM along the river network, but it does not perform the actual prediction. 1 amount of water volume passing through a given river section per unit time 3 METHODOLOGY 3.1 DATA PREPROCESSING The LamaH-CE dataset (Klingler et al., 2021) contains historical discharge and meteorological measurements on an hourly resolution for 859 gauges in the broader Danube river network shown in Figure 2. Covering an area of 170,000 km$^2$, with diverse environmental conditions, Klingler et al. expect that results from investigations on this dataset carry over to other river networks. One caveat is that LamaH-CE does not provide any flood event annotations, so that we can only model continuous discharge but not floods as discrete events. The river network defined by LamaH-CE naturally forms a directed acyclic graph (DAG) $G = (\mathcal{V}, \mathcal{E})$. The nodes $\mathcal{V}$ represent gauges, and the edges $\mathcal{E}$ represent flow between a gauge and the next downstream gauges. Hence, $G$ is anti-transitive, i.e., no skip connections exist. We preprocess $G$ to distill a connected subgraph with complete data. Region Selection. Figure 2 shows that $G$ contains four different connected components, of which we restrict ourselves to the largest one, “Danube A”. Its most downstream gauge close to the Austrian-Hungarian border has complete discharge data for the years 2000 through 2017. Starting at this gauge, we determine all connected gauges of the Danube A region by performing an inverse depth-first search given by Algorithm A.1. Overall, 608 out of the original 859 gauges belong to this connected component. Gauge Filtering. While the meteorological data is complete, the discharge data contains gaps. Klingler et al. have filled any consecutive gaps of at most six hours by linear interpolation and left the remaining longer gaps unaltered. We only want to consider gauges that (a) do not have these longer periods of missing values and (b) provide discharge data for at least the same time frame (2000 to 2017) as the most downstream gauge. To this end, we remove all gauges that violate these requirements from the graph using Algorithm A.2. Predecessors and successors of a deleted node get newly connected so that network connectivity is maintained. Note that thanks to antitransitivity, a duplicate check is unnecessary when inserting the new edges. After this preprocessing step, we are left with 375 out of the previously 608 gauges. Overall, the reduced graph $G$ now consists of $n := \|\mathcal{V}\| = 375$ gauges with $T$ hours of discharge measurements for the years 2000 to 2017, which we can conceptually represent as a node signal $Q = [q^{(1)} \| q^{(2)} \| \ldots \| q^{(T)}] \in \mathbb{R}^{n \times T}$. This cleaned dataset needs to be prepared for training. Normalization. As is common practice in deep learning, we normalize the data to surrender all gauges to the same scale and accelerate the training process (LeCun et al., 2002). In particular, we normalize per gauge (i.e., element-wise) using the standard score: $$\mu = \frac{1}{T} \sum_{t=1}^{T} q^{(t)}, \quad \sigma^2 = \frac{1}{T-1} \sum_{i=1}^{T} (q^{(t)} - \mu)^2, \quad q^{(t)} \leftarrow \frac{q^{(t)} - \mu}{\sigma}$$ Train-test splits. To robustly assess the performance of a trained model on unseen data via cross-validation, we randomly partition the 18 available years of observations into six folds of three years. By choosing one fold as the test set and the remaining folds as the training set, we obtain six different train-test splits that we keep constant throughout experiments. --- 2LARGE-SAMPLE DATA FOR HYDROLOGY FOR CENTRAL EUROPE 3.2 The Forecasting Task We task the model with an instance of supervised node regression. Assume we are given a certain amount of $W$ ("window size") most recent hours of discharge and meteorological measurements, in particular precipitation, topsoil moisture, air temperature, and surface pressure, for all gauges. Our goal is to predict the discharge $L$ ("lead time") hours in the future. For simplicity, we restrict the following illustrations to the discharge data in the input since the meteorological data can be trivially added in an extra dimension. Features & Targets. To conduct supervised learning, we extract input-output pairs from the time series represented by $Q$ (cp. Section 3.1). For $t = W, W + 1 \ldots, T - L$, we define the feature matrix at time step $t$ and the corresponding target vector as $$X(t) := \begin{bmatrix} q^{(t-W+1)} \\ \vdots \\ q^{(t-1)} \\ q^{(t)} \end{bmatrix} \in \mathbb{R}^{n \times W}, \quad y(t) := q^{(t+L)} \in \mathbb{R}^n.$$ We collect all samples into the set $\mathcal{D} = \{(X(t), y(t))\}_{t=W}^{T-L}$ and partition it according to a given train-test split into $\mathcal{D} = \mathcal{D}_{\text{train}} \cup \mathcal{D}_{\text{test}}$. The extraction process can be illustrated as follows: Adjacency. Besides the input and target measurements, we feed the river network topology to the GNN in the form of an adjacency matrix $A \in \mathbb{R}^{n \times n}$. For the definition of matrix entries corresponding to an edge $(i, j) \in E$ (the rest being zero), we consider the following choices: 1. isolated: $A_{i,j} := 0$ equates to removing all edges and results in the augmented normalized adjacency matrix to be a multiple of the identity so that each GNN layer degenerates to a node-wise linear layer. 2. binary: $A_{i,j} := 1$ corresponds to the unaltered adjacency matrix as it comes with the LamaH-CE dataset. 3. weighted: $A_{i,j} := w(i,j)$ quantifies a physical relationship, for which LamaH-CE provides three alternatives: - the stream length along the river between $i$ and $j$, - the elevation difference along the river between $i$ and $j$, and - the average slope of the river between $i$ and $j$. 4. learned: $A_{i,j} := \omega(i,j)$ where $\omega \in \mathbb{R}^{\|E\|}$ is a learnable model parameter. The first two variants allow us to compare the effect of introducing the river network topology into the model at all. The last two variants enable insights into what kind of relative importance of edges is most helpful. As usual in GNNs, we define the normalized augmented adjacency matrix $$\tilde{A} := (\mathbf{D}_{\text{in}} + \text{diag}(\lambda))^{-\frac{1}{2}} (\mathbf{A} + \text{diag}(\lambda)) (\mathbf{D}_{\text{in}} + \text{diag}(\lambda))^{-\frac{1}{2}}$$ where self-loops for node $i$ with weight $\lambda_i$ are added and everything is symmetrically normalized based on the diagonal in-degree matrix $\mathbf{D}_{\text{in}}$. We generally set $\lambda_i$ as the mean of all incoming edge weights at node $i$ to make self-loops roughly equally important to the other edges. The only exception to this is option (1) above, where that mean would be zero and thus result in no information flow whatsoever, so that in this case, we set the self-loop weights to one instead. Model. Our desideratum is a GNN $ f_\theta : \mathbb{R}^{n \times W} \rightarrow \mathbb{R}^n $ parameterized by $ \theta $ which closely approximates the mapping of windows $ X $ to targets $ y $, i.e., $ \hat{y} := f_\theta(X) \approx y $. All our models have a sandwich architecture: a linear layer $ \text{Encoder}_{\Theta_0} : \mathbb{R}^{n \times W} \rightarrow \mathbb{R}^{n \times d} $ embeds the $ W $-dimensional input per gauge into a $ d $-dimensional latent space. On this space, a sequence of $ N $ layers $ \text{GNNLayer}_{\Theta_i} : \mathbb{R}^{n \times d} \times \mathbb{R}^{n \times n} \rightarrow \mathbb{R}^{n \times d} $ are applied. Finally, another linear layer $ \text{Decoder}_{\Theta_{N+1}} : \mathbb{R}^{n \times d} \rightarrow \mathbb{R}^n $ projects from the latent space to a scalar per gauge. In symbols: \[ H^{(0)} := \text{Encoder}_{\Theta_0}(X) \\ H^{(i)} := \text{GNNLayer}_{\Theta_i}(H^{(i-1)}, \bar{A}) \quad \text{for } i = 1, \ldots, N \\ \hat{y} := \text{Decoder}_{\Theta_{N+1}}(H^{(N)}). \] We consider three choices for $ \text{GNNLayer} $, with $ \sigma = \text{ReLU} $ as activation function: \[ \text{GCNLayer}_\Theta(H, \bar{A}) := \sigma(\bar{A}^\top H \Theta) \quad \text{(Kipf & Welling [2017])} \\ \text{ResGCNLayer}_\Theta(H, \bar{A}) := H + \text{GCNLayer}_\Theta(H, \bar{A}) \\ \text{GCNIIlayer}_\Theta(H, \bar{A}) := \sigma(((1 - \alpha)\bar{A}^\top H + \alpha H^{(0)})((1 - \beta)I + \beta \Theta)) \quad \text{(Chen et al. [2020])} \] where $ \alpha, \beta \in (0, 1) $ While the vanilla GCNLayer is the simplest definition, it famously suffers from a phenomenon known as oversmoothing (Oono & Suzuki [2020]) where the features of adjacent nodes converge with increasing depth. To alleviate this undesirable behavior, ResGCNLayer adds a residual connection, whereas GCNIIlayer introduces the notions of initial connection and identity mapping via weighted averages. Optimization Objective. To measure the error between a model prediction $ \hat{y} $ and the target $ y $, we use the multi-dimensional square loss $ L(\hat{y}, y) := \frac{1}{n} \\| \hat{y} - y \\|_2^2 $. Training is then defined as optimizing the expected loss over the empirical distribution of training samples in $ D_{\text{train}} $, i.e., the optimal model parameters are given by \[ \arg \min_\theta \mathbb{E}_{(x,y) \sim D_{\text{train}}} [L(f_\theta(X, \bar{A}), y)]. \] Metrics. Recall that we perform training on normalized samples. For evaluation, we must calculate metrics on the unnormalized version of the predictions and targets: \[ \hat{y}_{\text{orig}} := \sigma \circ \hat{y} + \mu, \quad y_{\text{orig}} := \sigma \circ y + \mu. \] The most intuitive regression metric is the Mean Squared Error (MSE). In our multi-dimensional regression problem, it is defined as the error vector \[ \text{MSE} := \frac{1}{\|D_{\text{test}}\|} \sum_{t=1}^{\|D_{\text{test}}\|} (\hat{y}_{\text{orig}}^{(t)} - y_{\text{orig}}^{(t)})^2 = \sigma^2 \circ \frac{1}{\|D_{\text{test}}\|} \sum_{t=1}^{\|D_{\text{test}}\|} (\hat{y}^{(t)} - y^{(t)})^2. \] Next to the MSE, a standard metric in hydrology is the Nash-Sutcliffe Efficiency (NSE; Nash & Sutcliffe [1970]). It compares the sum of squared errors of the model to the sum of squared errors of the constant mean-predictor and subtracts this value from one to obtain a percentage score in $[0, 1]$. An NSE of zero means that the model’s predictive capability is no better than that of the empirical mean, while an NSE of one means that all model predictions are perfect. \[ \text{NSE} := 1 - \frac{\sum_{t=1}^{\|D_{\text{test}}\|} (\hat{y}_{\text{orig}}^{(t)} - y_{\text{orig}}^{(t)})^2}{\sum_{t=1}^{\|D_{\text{test}}\|} (\mu - y_{\text{orig}}^{(t)})^2} = 1 - \frac{\text{MSE}}{\sigma^2} \] We straightforwardly obtain summary metrics for our experiments by averaging across gauges: \[ \text{MSE} := \frac{1}{n} \sum_{g=1}^n \text{MSE}_g, \quad \text{NSE} := \frac{1}{n} \sum_{g=1}^n \text{NSE}_g. \] 4 EXPERIMENTS 4.1 EXPERIMENTAL SETUP The code to reproduce our experiments is publicly available[^1]. Table 1 lists the relevant hyperparameters we use throughout all experiments unless stated otherwise, categorized into data, model, and training parameters. On the data side, we choose a window size of $W = 24$ hours as a compromise between sufficiently many past observations and computational efficiency. We set the lead time to $L = 6$ hours, which is a realistic choice. On the model side, we consider all three choices of layer definition detailed in Section [3.2] resulting in three model architectures GCN, ResGCN, and GCNII. We choose a depth of $N = 20$ layers to allow information propagation along the entire river graph, given that the longest path in the preprocessed graph consists of 19 edges. The latent space dimensionality of $d = 128$ was chosen large enough to allow an injective feature embedding but small enough to avoid memory issues. The edge direction and adjacency type hyperparameters will be explored in detail in Section 4.2. On the optimization side, all neural network parameters are randomly initialized using the standard Glorot initialization scheme (Glorot & Bengio, 2010). We then perform 20 epochs of stochastic mini-batch gradient descent, which is enough for the process to converge. The descent algorithm is Adaptive Moments (Adam) (Kingma & Ba, 2015), with a base learning rate of $5 \times 10^{-4}$, which results in stable training. To prevent overfitting, we randomly hold out 1/5 of the training set, which corresponds to three years of observations, and select the parameters from the epoch in which the loss calculated over this holdout set was the lowest. \| HYPERPARAMETER \| VALUE \| \|----------------\|-------\| \| WINDOW SIZE ($W$) \| 24 h \| \| LEAD TIME ($L$) \| 6 h \| \| NORMALIZATION? \| YES \| \| ARCHITECTURE \| [Res]GCN, GCNII \| \| NETWORK DEPTH ($N$) \| 20 \| \| LATENT SPACE DIM ($d$) \| 128 \| \| EDGE DIRECTION \| BIDIRECTIONAL \| \| ADJACENCY TYPE \| BINARY \| \| INITIALIZATION \| GLOROT \| \| OPTIMIZER \| ADAM \| \| # EPOCHS \| 20 \| \| BATCH SIZE \| 64 \| \| LEARNING RATE \| $5 \times 10^{-4}$ \| 4.2 RIVER TOPOLOGY COMPARISON Our main experiment compares the impact of the six different gauge adjacency definitions detailed in Section 3.2 on forecasting performance. In addition, we also consider three alternative edge orientations, which determine the direction of information flow in the GNN, as none of the options is a priori preferable. The downstream orientation is given by the dataset, the upstream orientation results from reversing all edges, and the bidirectional orientation from adding all reverse edges to the forward ones. We six-fold cross-validate all 18 topology combinations using the train-test splits established in 3.1 and the average MSE and NSE metrics defined in Section 3.2 and report the results for ResGCN and GCNII in Table 2. As the vanilla GCN suffers heavily from oversmoothing, we disregard it in the remaining discussions and only provide its results in Table A.2 for completeness. Surprisingly, model performance for ResGCN and GCNII shows almost no sensitivity to the choice of graph topology. Isolating the gauges does not harm performance beyond the standard deviation, and no combination outperforms a 20-layer MLP baseline by a margin. This indicates that the forecasting task for a gauge mainly benefits from the past discharge at that gauge but not from the discharge at neighboring gauges. The river graph topology makes no difference. Even when the model is allowed to learn an optimal edge weight assignment, it does not manage to outperform the baseline. However, a consistent pattern is that the GNNs achieve their best average NSE for a bidirectional edge orientation. [^1]: https://add-link-after-review Table 2: Forecasting performance on different river network topologies, given as mean and standard deviation of the respective metrics across folds. MSE is not scale-normalized per gauge, while NSE is (cp. Section [3.2]). A 20-layer MLP baseline achieves an NSE of 85.62% ± 4.90%. Bold indicates the best value per column. Note that results for the isolated adjacency type are not affected by the choice of edge orientation due to the absence of edges in this case. (a) ResGCN \| ADJACENCY TYPE \| DOWNSTREAM \| UPSTREAM \| BIDIRECTIONAL \| \|----------------\|------------\|----------\|---------------\| \| \| MSE ↓ \| NSE ↑ \| MSE ↓ \| NSE ↑ \| MSE ↓ \| NSE ↑ \| \| ISOLATED \| 899.80 ±1329.17 \| 80.85 % ±11.66 % \| 899.80 ±1329.17 \| 80.85 % ±11.66 % \| 899.80 ±1329.17 \| 80.85 % ±11.66 % \| \| BINARY \| 353.54 ±80.90 \| 83.53 % ±5.63 % \| 372.67 ±61.11 \| 84.99 % ±5.10 % \| 741.20 ±166.26 \| 85.34 % ±4.86 % \| \| STREAM LENGTH \| 524.03 ±100.46 \| 83.42 % ±5.59 % \| 435.66 ±60.49 \| 84.74 % ±5.02 % \| 785.38 ±171.49 \| 85.31 % ±4.92 % \| \| ELEVATION DIFFERENCE \| 407.67 ±95.16 \| 83.46 % ±5.60 % \| 456.32 ±63.80 \| 83.76 % ±4.80 % \| 773.95 ±182.22 \| 85.16 % ±4.93 % \| \| AVERAGE SLOPE \| 327.22 ±75.81 \| 83.45 % ±5.60 % \| 425.95 ±86.43 \| 84.10 % ±5.18 % \| 656.52 ±170.12 \| 85.23 % ±4.92 % \| \| LEARNED \| 345.57 ±199.76 \| 83.50 % ±5.40 % \| 366.94 ±80.72 \| 85.63 % ±4.65 % \| 567.39 ±160.84 \| 85.94 % ±4.52 % \| (b) GCNII \| ADJACENCY TYPE \| DOWNSTREAM \| UPSTREAM \| BIDIRECTIONAL \| \|----------------\|------------\|----------\|---------------\| \| \| MSE ↓ \| NSE ↑ \| MSE ↓ \| NSE ↑ \| MSE ↓ \| NSE ↑ \| \| ISOLATED \| 289.71 ±50.01 \| 85.95 % ±4.97 % \| 289.71 ±50.01 \| 85.95 % ±4.97 % \| 289.71 ±50.01 \| 85.95 % ±4.97 % \| \| BINARY \| 277.50 ±33.57 \| 86.17 % ±4.69 % \| 312.31 ±43.98 \| 85.75 % ±5.03 % \| 355.95 ±65.61 \| 86.44 % ±4.64 % \| \| STREAM LENGTH \| 343.86 ±29.33 \| 86.17 % ±4.66 % \| 311.32 ±43.91 \| 85.72 % ±5.01 % \| 393.39 ±81.15 \| 86.37 % ±4.67 % \| \| ELEVATION DIFFERENCE \| 302.76 ±48.07 \| 86.11 % ±4.69 % \| 314.72 ±42.75 \| 85.35 % ±5.28 % \| 411.96 ±80.55 \| 86.33 % ±4.71 % \| \| AVERAGE SLOPE \| 276.88 ±40.39 \| 86.08 % ±4.67 % \| 279.22 ±41.44 \| 85.44 % ±5.32 % \| 364.96 ±79.10 \| 86.26 % ±4.79 % \| \| LEARNED \| 169.93 ±33.40 \| 86.14 % ±4.87 % \| 280.07 ±46.97 \| 86.03 % ±4.80 % \| 323.54 ±83.12 \| 86.48 % ±4.69 % \| Table 3: Pearson correlation between learned and physical edge weights. \| PHYSICAL EDGE WEIGHTS \| LEARNED EDGE WEIGHTS \| \|-----------------------\|----------------------\| \| \| DOWNSTREAM \| UPSTREAM \| BIDIRECTIONAL \| \| \| ResGCN \| GCNII \| ResGCN \| GCNII \| ResGCN \| GCNII \| \| STREAM LENGTH \| 0.221 ±0.098 \| −0.23 ±0.012 \| 0.042 ±0.008 \| −0.14 ±0.006 \| −0.002 ±0.016 \| 0.054 ±0.031 \| \| ELEVATION DIFFERENCE \| 0.100 ±0.021 \| −0.17 ±0.003 \| −0.308 ±0.015 \| 0.027 ±0.007 \| −0.235 ±0.014 \| −0.103 ±0.035 \| \| AVERAGE SLOPE \| 0.168 ±0.038 \| −0.04 ±0.007 \| −0.293 ±0.009 \| 0.090 ±0.009 \| −0.24 ±0.012 \| −0.163 ±0.012 \| Figure 3: Model performance with varying depth, averaged over folds. Shaded areas correspond to 95% confidence intervals across folds. 4.3 Learning the Weights The case of learned edge weights is of particular interest. They were initialized by drawing from the uniform distribution in $[0.9, 1.1]$ to arrange them neutrally around one while still introducing sufficient noise to break symmetry. Whenever learned weights get negative during training, we clip them to zero. The distribution of the learned weights (cp. Table A.3) is still centered around one with minima close to zero and maxima below ten. To see if the learned weights exhibit any similarities with the physical weights, we calculate Pearson correlation coefficients for all topology combinations. Table 3 shows that none of the physical weight assignments correlate much with the learned weights. In multiple instances, the sign even flips when using a different model architecture. For instance, the largest positive correlation occurs with stream length for ResGCN, but in this same case GCNII achieves a negative correlation of the same magnitude. Hence, we conclude that none of the physical edge weights from the datasets are optimal context information for the predictor. 4.4 The Role of GNN Depth The rationale for setting the number of layers to $N = 20$ was to allow information to propagate across the entire river network. However, since removing all edges from the graph does not deteriorate the performance (cp. Table 2), we can also consider shallower neural networks. In particular, we want to exclude the possibility that the considerable depth is causing the GCN to not outperform the baseline MLP due to more general issues with training very deep networks. In this case, a GCN with fewer layers could profit more from the graph structure despite not achieving global information propagation. Hence, we train ResGCN and GCNII with the default hyperparameters from Table 1 where we only vary the number of layers in steps of one from 1 to 20. The resulting average MSE and NSE scores are shown in Figure 3. The experiment provides two insights. First, the inability of both GCN architectures to outperform the MLP baseline is consistent across network depths, so that we can rule out training issues. Second, the performance is independent of model depth, which means that the larger receptive field achieved by more layers does not help. Both corroborate the previous observations that GNNs fail to take advantage of the graph structure. 4.5 Worst Gauge Investigation The performance on gauge #80 of all trained models is considerably below the mean. For instance, the best overall performing model according to NSE (bidirectional-learned GCNII) achieves its worst NSE of only 24.78% on this outlier gauge. To better understand the scenarios that are challenging for the model, we determine the top disjoint time horizons of 48 hours (24 hours for past and future) in terms of deviation of model prediction from the ground truth. The resulting plots in Figure 4 reveal that the outlier gauge is characterized by sudden spikes, which are inherently hard to forecast for any predictor. The gauge might be located behind a floodgate. As a result, the forecasting performance is mediocre, with the forecast often missing spikes. Figure 4: Worst predictions of bidirectional-learned GCNII on its overall worst gauge #80. Negative time indicates past, and positive time indicates future discharge. 5 CONCLUSION In this work, we explored the applicability of GNNs to holistic flood forecasting in a river network graph. Based on the LamaH-CE dataset, we framed a supervised node regression task for predicting future discharge at all gauging stations in the graph given past observations. By modifying the adjacency matrix, we compared the impact of different adjacency definitions on the prediction performance. Our results reveal that the impact of river topology is negligible. The GNN performs equally well even when all edges are removed from the graph, which makes it act like an MLP. It does not benefit from weighted edges that resemble physical relationships between gauges. When the model is allowed to jointly learn the edge weights along with the other parameters, they correlate with neither constant weights nor any of the physical weightings given by the dataset. A depth study shows that the results are not caused by issues with training deep models but prove consistent throughout any number of layers. Investigations on a challenging outlier gauge show that the GNNs struggle to predict sudden discharge spikes. On a high level, future work is encouraged to investigate under which conditions including graph topology in neural predictors actually helps, which is not clear a priori. While the key could lie in employing more specialized model architectures such as DGCN (Tong et al., 2020), MagNet (Zhang et al., 2021), and DAGNN (Thost & Chen, 2021) for the dataset at hand, there might be more fundamental limitations to the use of GNNs for large-scale regression problems. Moreover, for the application of flood forecasting, our results suggest that focusing on accurate spike prediction is more promising than incorporating river network topology information. To this end, there is a broader issue: we used a river network dataset from central Europe as discharge measurements are readily available there for long time periods. However, the regions most affected by floods are typically in low-income countries where data is scarce. 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ynguffsGfa	If the LLM's output suggests a high correlation between certain marginal groups and a high prevalence of sexually transmitted diseases (STDs), how do you tell whether this is based on medical publications or social biases? Factual tracing for LLMs is currently known as a very hard problem. Active research is ongoing and there are not yet effective ways to relate the model's output to its training samples.	Curated LLM: Synergy of LLMs and Data Curation for Tabular Augmentation in Ultra Low-Data Regimes Anonymous authors Paper under double-blind review Abstract Machine Learning (ML) in low-data settings remains an underappreciated yet crucial problem. This challenge is pronounced in low-to-middle income countries where access to large datasets is often limited or even absent. Hence, data augmentation methods to increase the sample size of datasets needed for ML are key to unlocking the transformative potential of ML in data-deprived regions and domains. Unfortunately, the limited training set constrains traditional tabular synthetic data generators in their ability to generate a large and diverse augmented dataset needed for ML tasks. To address this technical challenge, we introduce CLLM, which leverages the prior knowledge of Large Language Models (LLMs) for data augmentation in the low-data regime. While diverse, not all the data generated by LLMs will help increase utility for a downstream task, as for any generative model. Consequently, we introduce a principled curation process, leveraging learning dynamics, coupled with confidence and uncertainty metrics, to obtain a high-quality dataset. Empirically, on multiple real-world datasets, we demonstrate the superior performance of LLMs in the low-data regime compared to conventional generators. We further show our curation mechanism improves the downstream performance for all generators, including LLMs. Additionally, we provide insights and understanding into the LLM generation and curation mechanism, shedding light on the features that enable them to output high-quality augmented datasets. CLLM paves the way for wider usage of ML in data scarce domains and regions, by allying the strengths of LLMs with a robust data-centric approach. 1 Introduction No data, No Machine Learning. Machine learning (ML) has transformed numerous industries, but its wider adoption is hindered by a pervasive roadblock: insufficient data. Specifically, the use of ML algorithms presumes the availability and access to large datasets for training, be it in the form of labeled or unlabeled data. Unfortunately, real-world domains are often data scarce: (i) in healthcare and finance, collecting annotations can be expensive or practically impossible; (ii) in developing and low-to-middle income countries (LMICs), digital infrastructure (such as electronic healthcare records (EHRs)) can be limited or nonexistent (Ade-Ibijola & Okonkwo, 2023; Asiedu et al., 2023; Owoyemi et al., 2020; Mollura et al., 2020; Alami et al., 2020; Ciecierski-Holmes et al., 2022) and (iii) within large datasets, there can be (ethnic) minorities that are underrepresented. This lack of data has serious consequences: to sideline these settings to the peripheries of ML advancements and prevent the development of accurate models. How can we build a reliable ML model in this low-data regime, where we have so few samples? Solving this problem is a major opportunity that would unlock the potential of ML across society, domains, and regions. Aim. To address this important yet undervalued low-data problem, we aim to augment the small labeled dataset ($n < 100$) with synthetic samples. We focus on tabular data, as defining augmentations is non-trivial and can easily result in nonsensical or invalid samples. Moreover, tabular domains like healthcare (of value in LMICs) are often where data scarcity is acute. Related work. Data augmentation is a widely used and different approach to address data scarcity in tabular data contexts. Methods are either based on generative models (Ghosheh et al., 2023; Biswas Figure 1: CLLM uses a small dataset $D_{\text{train}}$ and a frozen black-box LLM to generate a larger synthetic set $D_{\text{syn}}$. The curator computes the learning dynamics of samples in $D_{\text{syn}}$, assessing samples based on their aleatoric uncertainty and predictive confidence, then curates $D_{\text{syn}}$ with the goal that a downstream model trained on the curated $D_{\text{curated}}$ will have improved performance. et al., 2023; Wang & Pai, 2023; Machado et al., 2022; Tanaka & Aranha, 2019) such as GANs (Xu et al., 2019), VAEs (Xu et al., 2019), Normalizing Flows (Papamakarios et al., 2021), Score-based models (Kotelnikov et al., 2022; Kim et al., 2022), or alternatively traditional methods such as SMOTE (Chawla et al., 2002; Wang & Pai, 2023; Machado et al., 2022). However, in ultra low-data regimes ($n < 100$), the training data may not describe the full data distribution well, despite it being i.i.d. draws. Consequently, this harms conventional methods since the augmented data may not be sufficiently diverse and accurate, restricting the generalizability of predictive models trained on such data. Tangentially, prior works have tackled data scarcity in the tabular setting via the lens of transfer learning, where prior knowledge can be transferred from a pretrained model (Levin et al., 2022; Jin & Ucar, 2023) or a knowledge graph (Margeloiu et al., 2022; Ruiz et al., 2023), which might not be available in all settings. Recent work has shown the potential of fine-tuning Large Language Models (LLMs) for tabular data generation (Borisov et al., 2023). While LLMs offer some degree of prior knowledge, there are two challenges in our setting. First, it is computationally expensive to fine-tune LLMs, while needing specialized hardware — luxuries often not available in LMICs, thereby limiting applicability in such settings. Second, fine-tuning often assumes a large number of samples. In our low-data setting it could lead to overfitting and low-quality generated samples, and hence poor downstream models—as we show for Borisov et al. (2023) in Sec. 3. Curated LLMs. To address these challenges, we propose Curated LLM (CLLM). First, CLLM leverages the in-context capabilities of LLMs for generation, thereby reducing the computational burden. We also posit for the low-data regime; the diverse pretraining corpus of LLMs carries valuable prior knowledge, which may offer more diversity in their generation compared to other conventional tabular generators. Of course, LLMs are not perfect. Balancing the utility of LLMs against the risk of noisy, irrelevant data is important for downstream performance, hence requiring systemic assessment of the generated data. In fact, this issue is vital for any generative model. This motivates the second key aspect of CLLM, i.e. post-generation data curation. This addresses the overlooked aspect that not all of the synthetic samples are useful to downstream model performance, with some samples even harmful. We anchor our approach with ideas from learning theory that show the behavior of individual data samples during training, called learning dynamics, provides a salient signal about the value of samples to a learner (Arpit et al., 2017; Arora et al., 2019; Li et al., 2020). To provide intuition, samples with variable predictions might be considered ambiguous or other samples might never be learned correctly and could harm a model. In CLLM, we study the learning dynamics of the synthetic data samples, with respect to a model trained on the small real dataset. We then analyze these dynamics by computing two key metrics: confidence and aleatoric (data) uncertainty. These metrics form the basis for curating the synthetic samples. We aim to enable a highly performant downstream model when trained on the curated dataset. Contributions: CLLM is a novel data augmentation approach allying the strengths of LLMs with a robust data curation mechanism to improve data augmentation in the ultra low-data regime ($n < 100$), bringing several contributions: ① Improved performance: we empirically demonstrate on 7 real-world datasets that CLLM enables superior downstream performance compared to 6 widely used tabular data generative models and data augmentation techniques. ② Value of curation: we show the overlooked aspect of synthetic data curation improves downstream performance across the generative models. This highlights the flexibility and broad utility of our curation mechanism for data augmentation. ③ Insights: we dissect the two aspects of CLLM (LLM and data curation) along a variety of dimensions, providing insights and understanding into why the approach is beneficial. We show the largest gains are for underrepresented subgroups and in ultra low-data settings. These contributions pave the way towards wider usage of ML across society, domains and regions. Ethical considerations. LLMs may make errors and may reflect or exacerbate societal biases that are present in their data (Li et al., 2023). Though the curation in CLLM improves synthetic data quality, it does not directly aim to remove biases. The quality and fairness of generated data should always be evaluated. More research into LLM bias is required before methods like CLLM should be applied to real-world sensitive settings like healthcare and finance. 2 CLLM: Synergy of LLM Generation and Data Curation Set-up. Given feature space $ \mathcal{X} $, and label space $ \mathcal{Y} = \{1, \ldots, k\} $, we assume that we only have a small labeled dataset $ D_{\text{train}} = \{(x_i, y_i)\}_{i=1}^n $, with $ x_i \in \mathcal{X}, y_i \in \mathcal{Y} $ and $ n < 100 $ (ultra-low data setting). Assume $ D_{\text{train}} $ is drawn i.i.d. from the real distribution $ p_R(X,Y) $. We also assume access to a pretrained LLM to generate samples. We denote the output distribution of the LLM as $ p_\Phi(X,Y) $, with $ \Phi $ containing parameters that we control (e.g., input prompts). Our goal is to generate a dataset to augment the small $ D_{\text{train}} $, and subsequently use it to train a classifier $ f : \mathcal{X} \rightarrow \mathcal{Y} $. Successful augmentation will provide a better classifier $ f $, than if we had trained $ f $ on the small $ D_{\text{train}} $ itself. We measure downstream performance on a separate held-out dataset of real data, $ D_{\text{test}} $. Our Approach. To address this challenge, we introduce CLLM, an approach for data augmentation in low-data regimes. As shown in Figure 1, CLLM leverages LLMs to generate a synthetic dataset $ D_{\text{syn}} $ using a small dataset $ D_{\text{train}} $ (Sec. 2.1). It exploits the LLMs’ prior knowledge via in-context learning (ICL) and contextual information. CLLM then curates $ D_{\text{syn}} $ by analyzing the learning dynamics of samples in $ D_{\text{syn}} $ based on predictive confidence and aleatoric (data) uncertainty. These metrics are obtained by training a supervised model on $ D_{\text{train}} $. We leverage them to define a curated dataset $ D_{\text{curated}} $, which is used to train a downstream classifier (Sec. 2.2). In each sub-section we describe and motivate the design of the different aspects of CLLM (LLM and curation). Furthermore, we provide insights and understanding into their role in improving data utility, which we later quantify on multiple real-world datasets in Sec. 3. 2.1 Data generation with LLMs based on a small $ D_{\text{train}} $ As outlined in Sec. 1, in the ultra low-data regime, conventional tabular generative models (e.g. CTGAN, TVAE) are constrained by the limited $ D_{\text{train}} $ and may not generate sufficiently diverse and/or accurate synthetic data. To address this challenge, we propose to leverage LLMs, building on their large-scale pretraining. We first outline the desirable features of LLMs for tabular data generation when we have very few samples, then describe design choices to satisfy these. • Prior knowledge. LLMs have been pretrained with a vast corpus of information (Chowdhery et al., 2022; Singhal et al., 2023). When prompted to generate samples with limited real data, LLMs can leverage this encoded prior information about similar problems and feature-label relationships to enhance both accuracy and diversity of generation. • Contextual understanding. LLMs can process background and contextual information about the problem via natural language (Yang et al., 2023). For example, a high-level description of the task, features and their meanings can be conveniently described through natural language. Such information is unavailable to conventional generators that only utilize numerical examples. • Few-shot capabilities. LLMs have demonstrated proficiency in generalizing to tasks with just a few examples (Brown et al., 2020; Wei et al., 2023; Mirchandani et al., 2023). In the context of generation, we envision the idea of in-context generation using limited real examples. To benefit from these capabilities, we craft the LLM prompt with three different parts (see Fig. 1): (1) Background: text description of the dataset and task (e.g. predict Covid mortality). Additionally, we include a description of what each feature means, explicitly prompting the LLM to use prior knowledge about these features. (2) Examples: we serialize the samples in $ D_{\text{train}} $ as example demonstrations and provide both the features and the label in text format. (3) Instructions: To generate a synthetic dataset $ D_{\text{syn}} $, we instruct the LLM to leverage the contextual information and provided examples as an i.i.d. draw from the distribution. We instruct the LLM to identify structural and feature-label relationships in the data and generate diverse data following the structure and format of the provided examples. We provide more details on the prompts in Appendix B. Motivation for a frozen LLM. Using a frozen black-box LLM (e.g. GPT-4 or GPT-3.5) is computationally cheaper and requires less specialized hardware (i.e. GPUs) compared to fine-tuning. This relates to settings described in Sec. 1, such as LMICs, where we may not have the computational resources to fine-tune an LLM. Even in settings where fine-tuning is possible, we show empirically in Sec. 3 that LLM fine-tuning (e.g., GReaT baseline) is suboptimal in ultra-low data settings ($n < 100$) compared to providing in-context examples coupled with curation. Dissecting the LLM’s generative features. We now investigate various dimensions to understand and illustrate empirically the appealing features of LLMs as data generators in the low-data regime, and how our design choices unlock them. We take the Brazilian Covid-19 dataset (Baqui et al., 2020) as a running example and focus on GPT-4 as the LLM. ▶ GPT-4 extrapolates to unseen regions of the manifold. We compare the samples generated by GPT-4 to TVAE, a widely used tabular data generator. We consider $D_{\text{oracle}}$, a held-out dataset from the same distribution as $D_{\text{train}}$, such that $\|D_{\text{oracle}}\| \gg \|D_{\text{train}}\|$, thereby providing an approximation for the true manifold. The t-SNE plots in Fig. 2 shows, when $D_{\text{train}}$ is very small ($n = 20$ samples), that its samples do not cover all regions of $D_{\text{oracle}}$. For example, $D_{\text{train}}$ does not contain samples from specific demographic subgroups (e.g., people with age 40 or below). As expected, TVAE only generates samples constrained by the limited $D_{\text{train}}$. In contrast, GPT-4 is capable of extrapolating and generating samples even in unseen regions of $D_{\text{train}}$, thereby better covering $D_{\text{oracle}}$. This stems from its contextual understanding of the features, unlocking the use of its prior knowledge. It leads to better coverage in the low-data regime, consequently aiding in superior downstream performance, as we show in Table 3. As $n$ increases ($\geq 100$), $D_{\text{train}}$ provides better coverage, which naturally benefits both GPT-4 and TVAE. This result shows how prior knowledge encoded in LLMs addresses shortcomings of conventional generative approaches (e.g., TVAE) in the low-data regime. ![Figure 2](image) Figure 2: GPT-4 is able to extrapolate to regions of the oracle (true manifold) even where there is no training data covering them, as can be seen by the overlap with the turquoise dots, with the effect more pronounced when $D_{\text{train}}$ is small. ▶ GPT-4 benefits underrepresented groups the most. Having illustrated the extrapolation capabilities of GPT-4, we now ask: where does augmentation benefit downstream performance the most? We evaluate performance gains for different demographic subgroups, such as age groups and ethnic groups (Amarela, Prada). Fig. 3 shows the performance gain obtained by training a classifier on data generated by GPT-4 compared to training on the small $D_{\text{train}}$. The greatest gains, on average, are for subgroups for which we have no data in $D_{\text{train}}$, yet GPT-4 can extrapolate and generate samples for these subgroups. This further validates the rationale of extrapolation via prior knowledge being a key source of gain for GPT-4. Table 1 shows fine-grained results (across 10 different seeds) for the 5 subgroups that benefit the most from data augmentation, which are small-sized demographic subgroups. This finding has real-world implications for equity, as it shows we can improve performance for underrepresented subgroups even when we lack data or collecting data is difficult or costly. ![Figure 3](image) Table 1: Deep dive into the top 5 demographic subgroups in the Covid dataset with the largest gains, across 10 seeds, for $\|D_{\text{train}}\| = 20$. GPT-4 improves performance on the smallest groups. \| Subgroup \| $n_{\text{samples in } D_{\text{train}}}$ (min - max) \| Avg. Acc. Gain v. $D_{\text{train}}$ \| \|----------\|-----------------------------------------------\|-----------------------------------\| \| Age 40 \| 0-6 \| 6.38 ± 2.09 \| \| Liver \| 0-1 \| 3.85 ± 3.37 \| \| Renal \| 0-3 \| 4.52 ± 2.01 \| \| Amarela \| 0-1 \| 8.71 ± 1.40 \| \| Parda \| 3-11 \| 5.07 ± 1.50 \| Figure 3: Subgroups with fewest samples in $D_{\text{train}}$ benefit the most from data augmentation, on average. Importance of contextual information in the prompt. A natural question is: how important is the prompt to elicit the prior knowledge of the LLM? We explore two variants: (1) Prompt w/ context: provides contextual information including background about the dataset, feature names and descriptions (our approach) and (2) Prompt w/ no context: only provides the numerical in-context examples (ablation). Fig. 4 qualitatively shows that not including contextual knowledge in the prompt gives lower coverage of $D_{\text{extra}}$ with less extrapolation beyond $D_{\text{train}}$. We quantify this in Table 2 using Precision (Quality) and Recall (Diversity) metrics (Sajjadi et al., 2018), as well as Utility (Downstream performance). GPT-4 with contextual information has superior precision and recall in the ultra-low data setting. Furthermore, we show that the lack of contextual information in the prompt significantly harms the precision (quality) of the data even compared to TVAE. This highlights that LLMs need guidance, as we are only able to get the extrapolation and performance benefits by including contextual information, further motivating our design choices in the prompt. ![Figure 4: Contextual information in the prompt is important for extrapolation.](image) Table 2: Including contextual information in the prompt improves precision (P), recall (R), and utility (U) in low-sample settings (results shown for the Covid dataset). \| n_samples in $D_{\text{train}}$ \| GPT-4 w/ context \| GPT-4 no context \| TVAE \| \|-------------------------------\|------------------\|------------------\|------\| \| \| P \| R \| U \| P \| R \| U \| P \| R \| U \| \| 20 \| 0.41(±0.04) \| 0.78(±0.03) \| 0.74(±0.01) \| 0.13(±0.01) \| 0.82(±0.01) \| 0.59(±0.01) \| 0.26(±0.01) \| 0.52(±0.01) \| 0.32(±0.02) \| \| 40 \| 0.40(±0.01) \| 0.91(±0.01) \| 0.76(±0.01) \| 0.11(±0.01) \| 0.89(±0.01) \| 0.60(±0.01) \| 0.27(±0.01) \| 0.68(±0.01) \| 0.62(±0.03) \| \| 100 \| 0.42(±0.01) \| 0.88(±0.02) \| 0.75(±0.01) \| 0.11(±0.01) \| 0.90(±0.01) \| 0.70(±0.01) \| 0.30(±0.02) \| 0.67(±0.01) \| 0.64(±0.06) \| \| 200 \| 0.44(±0.02) \| 0.85(±0.02) \| 0.75(±0.01) \| 0.08(±0.01) \| 0.90(±0.01) \| 0.60(±0.01) \| 0.47(±0.01) \| 0.75(±0.01) \| 0.65(±0.02) \| 2.2 Data curation with learning dynamics When prompted with $\Phi$ (which contains the in-context samples of $D_{\text{train}}$), the LLM generates samples from a distribution $p_\Phi(X,Y)$ that approximates $p_R(X,Y)$, implicitly exploiting its large-scale pretraining and few-shot capabilities. LLMs are of course not perfect and could generate noisy samples, hence this distribution may be inaccurate. To make this distribution more relevant to the downstream task, we include a data curation mechanism. Specifically, we focus on the noisy feature-label relationship $p_\Phi(Y\|X)$, for which we expect $p_\Phi(Y\|X) \neq p_R(Y\|X)$ given the small size of $D_{\text{train}}$. This motivates us to curate $D_{\text{syn}}$ and discard likely mislabeled samples. We anchor our approach with ideas from learning theory that show the behavior of individual samples during model training (called learning dynamics) contains signal about the nature of the samples themselves (Arpit et al., 2017; Arora et al., 2019; Li et al., 2020). Some samples are easily and confidently predicted over different model checkpoints, whereas other samples might be challenging (e.g. due to mislabeling) and hence might be incorrectly predicted for the given label. Consequently, we operationalize learning dynamics as the basis of our curation mechanism. Specifically, we analyze samples in $D_{\text{syn}}$ by studying their learning dynamics computed with a classifier trained on $D_{\text{train}}$. We then categorize and filter samples in $D_{\text{syn}}$, and produce a curated dataset $D_{\text{curated}} \subset D_{\text{syn}}$. Learning dynamics. We now formalize how we compute learning dynamics for individual samples. Assume that a classifier $f$ is trained in an iterative scheme (e.g. neural networks or XGBoost trained over iterations) on $D_{\text{train}}$, which makes it possible to analyze the learning dynamics of samples in $D_{\text{syn}}$ over these iterations. The classifier $f$ should be at least as flexible as the model that the practitioner intends to use for the downstream task. $f$ is trained from scratch on $D_{\text{train}}$ and goes through $e \in [E]$ different checkpoints leading to the set $F = \{f_1, f_2, \ldots, f_E\}$, such that $f_e$ is the classifier at the $e$-th checkpoint. Let $[f_e(x)]_y$ denote the predicted probability for class $y$ and sample $x$. Our goal is to assess the learning dynamics of samples in $D_{\text{syn}}$ over these $E$ training checkpoints, while we train $f$ on $D_{\text{train}}$. For this, we define $H$, a random variable following a uniform distribution $U_F$ over the set of checkpoints $F$. Specifically, given $H = h$ and a sample $(x,y)$, we define the correctness in the prediction of $H$ as a binary random variable $\hat{Y}_F(x,y)$ with the following conditional: $$P(\hat{Y}_F(x,y) = 1 \| H = h) = [h(x)]_y \quad \text{and} \quad P(\hat{Y}_F(x,y) = 0 \| H = h) = 1 - P(\hat{Y}_F(x,y) = 1 \| H = h).$$ 1We could finetune the model on the scarce $D_{\text{train}}$ we have, but is likely to still lead to overfitting due to the extreme data scarcity and LLM parameter size. Curation metrics. Equipped with a probabilistic interpretation of the predictions of a model, we now define two characterization metrics that we use for curation: (i) average confidence and (ii) aleatoric (data) uncertainty, inspired by (Kwon et al., 2020; Seedat et al., 2022a). Definition 2.1 (Average confidence). For any set of checkpoints $ F = \{f_1, ..., f_E\} $, the average confidence for a sample $(x, y)$ is defined as the following marginal: \[ P_F(x, y) := P(\hat{Y}_F(x, y) = 1) = E_{H \sim U_F}[P(\hat{Y}_F(x, y) = 1 \| H)] = \frac{1}{E} \sum_{e=1}^{E} [f_e(x)]_y \] Definition 2.2 (Aleatoric uncertainty). For any set of checkpoints $ F = \{f_1, ..., f_E\} $, the aleatoric uncertainty for a sample $(x, y)$ is defined as: \[ v_{al,F}(x, y) := E_{H \sim U_F}[Var(\hat{Y}_F(x, y) \| H)] = \frac{1}{E} \sum_{e=1}^{E} [f_e(x)]_y (1 - [f_e(x)]_y) \] Intuitively, for binary classification $(k = 2)$, the aleatoric uncertainty for a sample $x$ is maximized when $[f_e(x)]_y = \frac{1}{2}$ for all checkpoints $f_e$, akin to random guessing. Recall aleatoric uncertainty captures the inherent data uncertainty, hence is a principled way to capture issues such as mislabeling. This contrasts epistemic uncertainty, which is model-dependent and can be reduced simply by increasing model parameterization (Hüllermeier & Waegeman, 2021). Having defined sample-wise confidence and aleatoric uncertainty, we characterize samples in $D_{syn}$ into two categories, namely Selected and Discarded. Given a sample $(x, y)$, a set of training checkpoints $F$, and two thresholds $\tau_{conf}$ and $\tau_{al}$, we define the category $c(x, y, F)$ as Discarded if $P_F(x, y) < \tau_{conf}$ and $v_{al,F}(x, y) < \tau_{al}$, and Selected otherwise. Hence, a Discarded sample is one for which we have a very low confidence in predicting its associated label whereas we also have low inherent data uncertainty. Finally, given a function $f$ associated with the set of checkpoints $F$, we define the curated set $D_{curated} = \{(x, y) \| (x, y) \in D_{syn}, c(x, y, F) = Selected\}$. We also define $D_{discarded} = D_{syn} \setminus D_{curated}$. To summarize, the objective of the curation step is that training on the curated synthetic data leads to a better classifier $f_{D_{curated}}$ for the downstream task, compared to training on the uncurated synthetic data, i.e. $M(f_{D_{curated}}) > M(f_{D_{syn}})$, where $M$ is a performance measure (for example accuracy). In Sec. 3, we empirically show how performance on this curated dataset is superior both for LLM generated data as well as other classes of generative models. Dissecting the role of curation. We now empirically demonstrate the role of curation in correcting the noisy feature-label relationship present in $D_{syn}$, highlighting two insights: (i) curation discards samples which are atypical in their label with respect to their neighbors in $D_{syn}$ (ii) discarded samples can be considered “mislabeled”, and we quantify their atypicality using a large held-out dataset $D_{oracle}$. - Discarded samples conflict on the label with their neighbors in $D_{syn}$. We audit every synthetic sample $(x, y)$ generated by GPT-4 (across 7 datasets) and compute the proportion of its $k$ nearest neighbors in $D_{syn}$ which share the same label $y$. The agreement with the neighbors assesses the typicality of a sample’s $y$ given $x$, where naturally lower agreement is linked to mislabeling, which we aim to detect via curation. Taking $k = 10$, we obtain an average agreement of $a_{curated} = 0.74$ for $D_{curated}$, compared to $a_{discarded} = 0.58$ for $D_{discarded}$. This shows that the samples removed are those which, despite having similar features $x$, do not agree with their surrounding neighbors’ labels. This corroborates ideas in (Ashmore et al., 2021) of how proximity violations are useful to guide remedial action to improve models. Not removing these mislabeled samples injects noise into the downstream classifier, thus reducing performance. - Assessing discarded samples with $D_{oracle}$. Ideally, the samples we select should better align with the true feature-label distribution. Since we don’t have access to this distribution explicitly, we compute a proxy for $\eta(x) = \arg \max_y p(Y = y \| X = x)$, which we call $\hat{\eta}$. It is obtained by training a classifier on a held-out dataset $D_{oracle}$—the same size as $D_{test}$ and an order of magnitude larger than $D_{train}$. For each synthetic method, we then report the accuracy of $\hat{\eta}$ on both the curated $D_{curated}$ and discarded $D_{discarded}$ datasets —see Fig. 5. We highlight two key observations. First, the curated datasets, for all the generative models, exhibit a higher agreement with the proxy $\hat{\eta}$ than the discarded datasets. This aligns with the desideratum of only keeping samples that exhibit the correct feature-label relationships. This provides a rationale for why curation helps improve discriminative performance, as samples in $D_{curated}$ are much more likely to have the correct feature-label relationship. Second, GPT-4 has a higher agreement with $\hat{\eta}$ on $D_{discarded}$ compared to the other generators. This illustrates that GPT-4’s prior knowledge enables it to better capture the distribution $p(Y\|X = x)$. Note that generative baselines (e.g. TVAE) model the joint $p(X, Y)$, without any context of which is the set of features and which is the label. In contrast, we can define in the LLM prompt which column is the target $Y$, allowing the LLM to better capture the feature-label relationships. This complements the findings from Fig. 2, which showed that GPT-4 extrapolates to unseen regions of the feature manifold, captured by the support of $p(X)$. 3 CURATED LLMs FOR BETTER DATA AUGMENTATION We now perform an end-to-end quantitative evaluation of CLLM across multiple real-world datasets, for downstream utility, demonstrating the value of allying the generative capabilities of LLMs with our curation mechanism. Sec. 3.1 compares the performance of GPT-4 and our curation approach with respect to a variety of state-of-the-art tabular augmentation baselines. Having evaluated CLLM on a range of datasets, we also demonstrate how we can leverage information extracted during curation to characterize datasets via a hardness proxy. Sec. 3.2 illustrates how our characterization of samples during the curation step can help to flag synthesized datasets (e.g via the LLM) which, if used for training, will result in poor downstream performance. Experimental setup. We compare CLLM (with GPT-4 (OpenAI, 2023) and GPT-3.5 (Brown et al., 2020)) against a variety of baselines for tabular data generation and augmentation: CTGAN (Xu et al., 2019), TVAE (Xu et al., 2019), Normalizing Flows (Papamakarios et al., 2021), TabDDPM (Kotelnikov et al., 2022), SMOTE (Chawla et al., 2002) and GReaT (Borisov et al., 2023), which fine-tunes an LLM. We evaluate performance on 7 real-world datasets with different feature counts and vary the number of samples available in $D_{train}$, repeating each experiment across 10 seeds. While we do not know the exact makeup of the pretraining data of LLMs like GPT-4, there is the possibility that open-source data might be included. This poses a risk of memorization as the primary source of performance gain. To disentangle the role of memorization, we select 4 real-world medical datasets (Maggic (Pocock et al., 2013), Covid (Baqui et al., 2020), SEER (Duggan et al., 2016), CUTRACT (PCUK, 2019)) that require an authorization process to access, hence are unlikely to form part of the LLMs training corpus. We use common open-source datasets (Adult and Drug from the UCI repository (Asuncion & Newman, 2007) and Compas (Angwin et al., 2016)) that are highly reflective of data scarce domains. Further experimental details can be found in Appendix B. 3.1 OVERALL PERFORMANCE: DOWNSTREAM UTILITY We assess overall performance based on Utility of the augmented data, which we evaluate in terms of AUC on the real $D_{test}$, when using four different types of downstream models (see Appendix B). This setup mirrors the widely adopted Train-on-synthetic-Test-on-real (TSTR) (Esteban et al., 2017). Additionally, we compare the performance to training on the small $D_{train}$, as well as training on the large held-out $D_{oracle}$, the latter serving as an upper bound. GPT-4 + Curation has best overall performance. Table 3 shows the performance of the proposed CLLM (GPT-4 and GPT-3.5) against baselines — both with and without our curation mechanism. We find that the GPT-4 + Curation variant of CLLM outperforms baselines in almost all settings (20/28). Interestingly, its performance is close to or even exceeds the performance of $D_{oracle}$. Table 4 further shows that GPT-4 + Curation ranks first on average among all the generative methods. Table 3: AUC averaged over 4 downstream models on $D_{\text{test}}$, where curation improves performance for all methods across all sample sizes $n$, as indicated by $\uparrow$. CLLM w/ GPT-4 (Curated) dataset provides the strongest performance for both private/proprietary datasets and public datasets. \| Dataset \| $D_{\text{train}}$ \| $D_{\text{test}}$ \| Real data \| GPT-4 \| GPT-3.5 \| CTGAN \| Tab-DDPM \| GReaT \| NFLOW \| SMOTE \| TVAE \| \|--------------\|---------------------\|-------------------\|-----------\|-------\|---------\|-------\|----------\|-------\|-------\|-------\|------\| \| covid (n=20) \| 74.41 \| 68.50 \| 73.78 \| 73.87 \| 69.85 \| 71.41 \| 59.93 \| 63.67 \| 66.84 \| 66.85 \| 57.38 \| 66.64 \| 62.87 \| 68.36 \| 66.95 \| 66.82 \| 61.69 \| 66.11 \| \| cuttract (n=20) \| 72.23 \| 70.12 \| 71.15 \| 72.50 \| 69.97 \| 71.54 \| 64.14 \| 67.98 \| 66.05 \| 66.59 \| 52.38 \| 67.02 \| 64.44 \| 70.42 \| 68.41 \| 69.24 \| 68.94 \| 70.22 \| \| maggic (n=20) \| 67.41 \| 57.13 \| 60.70 \| 61.48 \| 57.54 \| 58.69 \| 52.75 \| 54.51 \| 54.59 \| 55.39 \| 50.29 \| 55.64 \| 54.72 \| 57.38 \| 55.84 \| 56.15 \| 54.08 \| 56.19 \| \| seer (n=20) \| 87.92 \| 80.67 \| 84.53 \| 84.82 \| 83.34 \| 83.71 \| 74.34 \| 78.73 \| 80.59 \| 80.60 \| 47.57 \| 74.43 \| 76.06 \| 79.98 \| 79.23 \| 80.02 \| 74.53 \| 78.73 \| \| compas (n=20) \| 67.51 \| 63.11 \| 68.01 \| 67.91 \| 62.07 \| 64.43 \| 55.67 \| 62.56 \| 57.67 \| 60.87 \| 53.33 \| 63.39 \| 59.49 \| 64.62 \| 61.06 \| 61.59 \| 58.30 \| 62.58 \| \| adult (n=20) \| 84.17 \| 77.45 \| 50.39 \| 71.48 \| 49.23 \| 73.37 \| 72.23 \| 76.86 \| 74.35 \| 75.04 \| 67.00 \| 77.25 \| 67.46 \| 76.48 \| 73.75 \| 73.67 \| 73.20 \| 76.30 \| \| drug (n=20) \| 77.81 \| 70.84 \| 75.08 \| 75.29 \| 71.68 \| 72.14 \| 68.31 \| 72.65 \| 68.12 \| 69.68 \| 58.78 \| 68.89 \| 62.13 \| 67.75 \| 70.16 \| 70.16 \| 66.09 \| 69.18 \| Sample size sensitivity. We now investigate the performance gains of CLLM as we vary the number of samples $n$ in $D_{\text{train}}$, in Table 3 and Table 4. Performance improvements and high ranking across datasets for CLLM (GPT-4+Curation) are especially noticeable in the ultra low-data regime (i.e. $n < 100$). In this regime, the limited size of $D_{\text{train}}$ severely constrains the other baseline methods. In contrast, as illustrated in Sec. 2.1, CLLM can leverage GPT-4’s prior knowledge to extrapolate beyond the small $D_{\text{train}}$, thereby improving downstream performance. As expected, the performance gap between CLLM and other methods decreases as the size of $D_{\text{train}}$ grows (e.g. $n = 200$), where sufficient training data helps other generators achieve good performance. Curation generally helps all generative models. Our curation mechanism consistently benefits all generative models for the different $n$. It ensures only high quality samples are retained, which is crucial for good data augmentation and downstream performance and has been overlooked in previous works. This explains why the combination of the best generative model and curation, which is CLLM, gives the best results and highest rankings in the low-data regime (e.g. $n = 20$). Performance benefits maintained for private and public datasets. One may hypothesize that the strong LLM (e.g. GPT-4) performance is explained by datasets being part of the LLMs’ training corpus, hence possibly being memorized. We show in Table 3 that it is unlikely, as we retain strong performance for both open-source datasets, as well as private medical datasets which require authorization processes for access and are unlikely to be part of the LLM pretraining dataset. Table 4: Average rank of approaches across the different datasets and seeds. CLLM w/ GPT-4 ranks first across all $n$ and curation improves all the generative models. \| Method \| n=20 \| n=40 \| n=100 \| n=200 \| \|-----------------\|------\|------\|-------\|-------\| \| CLLM w/GPT-4 \| 2.71 ± 1.44 \| 2.14 ± 1.06 \| 2.29 ± 1.19 \| 3.29 ± 1.38 \| \| GPT-4 \| 3.86 ± 1.73 \| 4.29 ± 1.83 \| 6.00 ± 1.77 \| 7.57 ± 1.65 \| \| CLLM w/GPT-3.5 \| 4.14 ± 0.94 \| 4.14 ± 0.71 \| 6.86 ± 1.24 \| 7.57 ± 0.70 \| \| NFLOW (curated) \| 6.00 ± 1.21 \| 4.71 ± 0.80 \| 4.00 ± 0.57 \| 4.71 ± 0.63 \| \| GPT-3.5 \| 6.71 ± 1.52 \| 7.29 ± 1.26 \| 11.57 ± 0.94 \| 12.57 ± 0.57 \| \| TVAE (curated) \| 7.14 ± 1.17 \| 7.86 ± 1.30 \| 6.43 ± 0.40 \| 6.71 ± 0.52 \| \| SMOTE (curated) \| 7.71 ± 0.33 \| 8.14 ± 0.91 \| 7.71 ± 1.19 \| 7.43 ± 1.07 \| \| SMOTE \| 7.86 ± 0.55 \| 9.57 ± 0.80 \| 9.57 ± 1.09 \| 9.00 ± 1.03 \| \| Tab-DDPM \| 8.29 ± 0.98 \| 8.00 ± 0.93 \| 6.00 ± 0.95 \| 3.14 ± 1.68 \| \| CTGAN (curated) \| 8.29 ± 1.42 \| 7.14 ± 0.91 \| 4.14 ± 0.62 \| 3.71 ± 0.39 \| \| GReaT (curated) \| 8.29 ± 1.42 \| 7.14 ± 0.91 \| 4.14 ± 0.62 \| 3.71 ± 0.39 \| \| NFLOW \| 10.14 ± 1.19 \| 9.86 ± 1.15 \| 10.00 ± 1.03 \| 10.29 ± 1.02 \| \| TVAE \| 12.14 ± 0.89 \| 14.00 ± 0.70 \| 13.71 ± 0.39 \| 14.43 ± 0.40 \| \| NFLOW \| 12.86 ± 0.47 \| 14.14 ± 0.37 \| 14.00 ± 0.45 \| 15.29 ± 0.33 \| \| CTGAN \| 13.86 ± 0.68 \| 13.14 ± 0.47 \| 12.86 ± 0.37 \| 12.00 ± 0.53 \| \| GReaT \| 15.71 ± 0.26 \| 15.00 ± 0.53 \| 14.57 ± 1.03 \| 12.71 ± 0.96 \| Remark on ICL versus fine-tuning. Our results in Table 3 and Table 4 indicate that ICL is better than fine-tuning (GReaT baseline) in the low-data regime. This highlights the difficulty of fine-tuning in this regime, where it is easy to overfit to $D_{\text{train}}$. As we increase the number of samples, this baseline coupled with curation improves toward the level of CLLM (GPT-4). 3.2 Hardness: A Proxy Signal to Flag Poor Quality Synthetic Datasets Having a systematic way to assess datasets generated by LLMs like GPT-4 is important because their black-box nature provides little control on their generation quality. This contrasts conventional generators for which training loss is an exploitable signal. Hence, we ask: could we have a signal to identify a potential problematic dataset generated by GPT-4 without an exhaustive manual review? For example, GPT-4 produced low-quality synthetic data for the Adult dataset (across the different sample sizes) resulting in poor downstream performance. While curation improves it, downstream performance is still suboptimal. Addressing this question is important, since datasets are rarely created by the ML model builder in real-world ML workflows, but rather by specialist data teams or data owners (Gebru et al., 2021; Sambasivan et al., 2021; Goncalves et al., 2020). Hence, having a signal to preemptively flag a potentially suboptimal generated dataset spares investment in both storing the subpar data and/or training a model likely to underperform on real data. $D_{\text{syn}}$ should intuitively be considered imperfect if curation discards many of its samples, since the number of discarded samples measures the quality of samples with respect to the small but gold-standard $D_{\text{train}}$. Hence, we investigate the relationship between test performance (AUC) and the proportion of samples discarded by the curation. Fig. 6, where each point is a synthetic dataset generated by GPT-4 (e.g. Adult, Compas), shows a strong negative linear relationship between these two quantities. This holds across the different $n$ with slopes fairly stable around $-1.4$. This relationship corroborates the poor quality of the dataset generated by GPT-4 on the Adult dataset, providing a useful proxy that $D_{\text{syn}}$ is unlikely to lead to good downstream performance. Figure 6: The proportion of discarded samples $D_{\text{syn}}$ is a proxy for test performance. This negative linear relationship where each point is a synthetic dataset generated by GPT-4 (e.g. Adult, Covid, Compas) allows us to flag datasets that will lead to unreliable downstream performance. 4 Discussion We introduce CLLM, an approach for data augmentation in the ultra low-data setting. CLLM exploits the prior knowledge of LLMs along with curation for improved downstream performance. As empirically shown, CLLM outperforms traditional generative models—most noticeably on under-represented subgroups, for which data augmentation is of utmost importance. CLLM is grounded in the ICL capability of LLMs, and benefits from its simplicity. We studied GPT-3.5 and GPT-4 as backbones for CLLM. The cost of the API access pose limitations, e.g. on wide accessibility, on knowing which data was used for training the models, and on understanding the LLM’s output better. Using smaller and open LLMs could overcome these limitations, though this could come with a reduction in performance. We leave this as a promising direction for future work. Further improvements may be achieved through different tuning and prompting of the LLM, as shown in different domains (Meng et al., 2023; Liu et al., 2023). Improving LLM tuning and prompting is beyond the scope of our work, but we regard this as a promising avenue for future work. Data scarcity and computational limitations are deterrents for developing ML. These challenges should inspire cutting-edge ML research (De-Arteaga et al., 2018). We believe CLLM takes a step in this direction toward improving the use of ML in low-data settings, across society (e.g. underrepresented subgroups (Suresh & Guttag, 2021)), domains (e.g. healthcare (Alami et al., 2020; Owoyemi et al., 2020)) and regions (e.g. LMICs). ETHICS AND REPRODUCIBILITY STATEMENTS Ethics. In this work, we evaluate CLLM using multiple real-world datasets. The private datasets are de-identified and used in accordance with the guidance of the respective data providers. We follow recommendations to use the Azure OpenAI service when using GPT-4 and GPT-3.5 models, where via the agreement we ensure the medical data is not sent for human review or stored, hence respecting the guidelines given by the dataset providers. LLMs may make errors and may reflect or exacerbate societal biases that are present in their data (Li et al., 2023). Though the curation in CLLM improves synthetic data quality, it does not directly aim to remove biases. The quality and fairness of generated data should always be evaluated. More research into LLM bias is required before methods like CLLM should be applied to real-world sensitive settings like healthcare and finance. Finally, increasing access to ML across regions, domains and societies is also about more than just technology. We believe broader engagement and discussion with various stakeholders is crucial to responsibly expand ML access, thereby realizing the benefits of ML in an equitable way. Reproducibility. Experiments are described in Section 4 with further details of the method, experimental setup and datasets included in Appendix B. Code will be released upon acceptance. REFERENCES Abejide Ade-Ibijola and Chinedu Okonkwo. 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85gNpcUhmx	The conclusion of ‘with more advanced lane detection methods, e.g., anchor-based methods’ is also lack of context. Firstly, in the related work, author stated that ‘In this paper, we consider segmentation-based domain-adaptive lane detection.’, which contradict with the conclusion.	Unsupervised Domain Adaptive Lane Detection via Contextual Contrast and Aggregation Anonymous authors Paper under double-blind review Abstract This paper focuses on two crucial issues in domain-adaptive lane detection, i.e., how to effectively learn discriminative features and transfer knowledge across domains. Existing lane detection methods usually exploit a pixel-wise cross-entropy loss to train detection models. However, the loss ignores the difference in feature representation among lanes, which leads to inefficient feature learning. On the other hand, cross-domain context dependency crucial for transferring knowledge across domains remains unexplored in existing lane detection methods. This paper proposes a Domain-Adaptive lane detection via Contextual Contrast and Aggregation (DACCA), consisting of two key components, i.e., cross-domain contrastive loss and domain-level feature aggregation, to realize domain-adaptive lane detection. The former can effectively differentiate feature representations among categories by taking domain-level features as positive samples. The latter fuses the domain-level and pixel-level features to strengthen cross-domain context dependency. Extensive experiments show that DACCA significantly improves the detection model’s performance and outperforms existing unsupervised domain adaptive lane detection methods on six datasets, especially achieving the best accuracy of 92.24% when using RTFormer on TuLane. 1 Introduction Lane detection is crucial in autonomous driving and advanced driver assistance systems. Benefitting from developing convolutional neural networks, deep learning-based lane detection methods (Pan et al., 2018; Xu et al., 2020) demonstrate greater robustness and higher accuracy than traditional methods (Liu et al., 2010). To train a robust lane detection model, a high-quality dataset is necessary. However, acquiring high-quality labeled data is laborious and costly. Simulation is a low-cost way to obtain training pictures. Nevertheless, the detection performance may be degraded after transitioning from the virtual (source domain) to the real (target domain). Unsupervised domain adaptation (UDA) has been proposed to solve this problem (Saito et al., 2018; Vu et al., 2019). Recently, UDA has been successfully applied in the image segmentation task (Vu et al., 2019; Tarvainen & Valpola, 2017), significantly improving the segmentation performance. However, applying existing unsupervised domain-adaptive segmentation methods to lane detection does not yield satisfactory results, even inferior to those of supervised training, as revealed in (Li et al., 2022). We consider the cross-entropy loss adopted in these methods only focuses on pulling similar features closer but ignores different features across categories, making these methods inefficient in learning discriminative features of different categories (Vayyat et al., 2022). Contrastive learning (He et al., 2020; Chen et al., 2020) is expected to solve this problem by appropriately selecting positive and negative samples. However, segmentation models may generate false pseudo-labels on the input image for the unlabeled target domain, causing false assignments of positive samples. On the other hand, cross-domain context dependency is essential for adaptive learning of cross-domain context information (Yang et al., 2021), which is overlooked by many existing domain adaptive lane detection methods, e.g. (Garnett et al., 2020) and (Gebele et al., 2022). In MLDA (Li et al., 2022), an Adaptive Inter-domain Embedding Module (AIEM) is proposed to aggregate contextual information, but it is limited to performing on a single image and disregards useful contextual information. from other images. How to effectively leverage the potential of cross-domain context dependency in domain-adaptive lane detection remains a challenging topic. This paper presents a novel Domain-Adaptive lane detection via Contextual Contrast and Aggregation (DACCA) to address the aforementioned issues. As shown in Figure 1, two positive sample memory modules (PSMMs) are adopted to save domain-level features for each lane in both source and target domains. We select two corresponding domain-level features as positive samples from both source and target PSMMs for each lane pixel in an input image. Subsequently, the selected domain-level features are aggregated with the original pixel feature to enrich the cross-domain contextual information. In addition, we pair the aggregated features with the source and target positive samples to avoid the false assignment of positive samples in the cross-domain contrastive loss. The main contributions of this paper are as follows. (1) We propose a novel cross-domain contrastive loss to learn discriminative features and a novel sampling strategy to fully utilize the potential of contrastive loss without modifying an existing contrastive loss. (2) A novel domain-level feature aggregation module combining pixel-level and domain-level features is presented to enhance cross-domain context dependency. Aggregating domain-level features, instead of feature aggregation of mini-batches or individual images, is a fresh perspective. (3) Extensive experiments show that our method can significantly improve the baseline performance on six public datasets. Remarkably, we achieve the best results on TuLane using RTFormer (Wang et al., 2022). 2 RELATED WORK Lane detection. Traditional lane detection mainly depends on image processing operators, e.g., Hough transforms (Liu et al., 2010). Although they can quickly achieve high detection accuracy in specific scenarios, their generalization ability is too poor to apply to complex scenarios. Deep learning-based lane detection has received increasing attention, including segmentation-based methods (Pan et al., 2018; Zheng et al., 2021) and anchor-based methods (Torres et al., 2020; Liu et al., 2021). SCNN (Pan et al., 2018) is one of the typical segmentation-based methods using a message-passing module to enhance visual evidence. Unlike pixel-wise prediction in segmentation-based methods, anchor-based methods regress accurate lanes by refining predefined lane anchors. For example, using a lightweight backbone, UFLD (Qin et al., 2020) pioneers row anchors in real-time lane detection. In this paper, we consider segmentation-based domain-adaptive lane detection. Unsupervised domain adaptation. Domain adaptation has been widely studied to address the domain discrepancy in feature distribution, usually, implemented through adversarial training and self-training. Adversarial training (Gong et al., 2019) eliminates the differences in feature distribution between the source and target domains by adversarial approaches. Different from adversarial training, self-training (Sajjadi et al., 2016; Tarvainen & Valpola, 2017) trains a model in the target domain using generated pseudo labels. On the other hand, the contrastive loss is introduced as an auxiliary loss to improve the model’s robustness. CDCL (Wang et al., 2023) takes labels and pseudo-labels as positive samples in the source and target domain, respectively. However, the model may generate false pseudo labels in the unlabeled target domain, leading to false positive sample assignments. There exists some works (Li et al., 2023; Wang et al., 2021; Jiang et al., 2022; Zhang et al., 2022; Melas-Kyriazi & Manrai, 2021) taking positive samples from the prototypes to achieve accu- Figure 2: An overview of DACCA’s framework. (a) Training pipeline of DACCA. (b) Student/Teacher model structure. The source domain-level feature assignment shares the same structure with the target domain-level feature assignment, except that a PSMM saves features from the source domain. The representation head $U$ is used to obtain the pixel-wise feature representation. rate positive sample assignments. CONFETI (Li et al., 2023) adopts the pixel-to-prototype contrast to enhance the feature-level alignment. CONFETI only uses a prototype to save source and target domain features, but we think this way is inappropriate because the feature distribution between the two domains is different. In our work, we use two PSMMs to save features of two domains separately and take the domain-level features as positive samples. In addition, we also optimize the sample selection policy in the contrastive loss but most works ignore it. Unsupervised domain adaptive lane detection. Due to the lack of a domain adaptive lane detection dataset, early studies (Garnett et al., 2020; Hu et al., 2022) focus on synthetic-to-real or simulation-to-real domain adaptation. Their generalizability in real-world scenarios is not satisfactory with low-quality synthetic and simulation images. Gebele et al. (2022) establishes a specific dataset for domain adaptive lane detection and directly apply a general domain adaption segmentation method to this dataset. However, it does not yield good results, since conventional domain adaptive segmentation methods generally assume the presence of salient foreground objects in the image, occupying a significant proportion of the pixels. On the other hand, lane lines, which occupy a relatively small proportion of the image, do not exhibit such characteristics. To solve this problem, MLDA (Li et al., 2022) introduces an AIEM to enhance the feature representation of lane pixel by aggregating contextual information in a single image. Unfortunately, in this way, useful contextual information from other images may be ignored. Instead, we propose to aggregate the domain-level features with pixel-level features. Context aggregation. Performing contextual information aggregation for pixel-level features can effectively improve segmentation performance in semantic segmentation. In supervised methods, common context information aggregation modules, e.g., ASPP (Chen et al., 2017), PSPNet (Zhao et al., 2017), OCRNet (Yuan et al., 2020), and MCIB1 (Jin et al., 2021), only aggregate features within a single domain instead of both target and source domains. In UDA, some methods try to design modules to aggregate contextual features by attention mechanisms, such as cross-domain self-attention (Chung et al., 2023), and context-aware mixup (Zhou et al., 2022). However, all existing cross-domain feature aggregation methods only fuse a mini-batch of contextual features. In contrast to previous works, our method tries to simultaneously fuse features from the whole target and source domains to enhance the cross-domain context dependency. 3 Method As illustrated in Figure 2, the network is self-trained in our DACCA, where the student model is trained in both the labeled source domain and the unlabeled target domain with pseudo-labels generated by the teacher model. DACCA has two key components, i.e., cross-domain contrastive loss and domain-level feature aggregation. 3.1 Self-Training In UDA, a segmentation-based lane detection model $s_\theta$ is trained using source images $X^s = \{x^s_k\}_{k=1}^{N_s}$ with labels $Y^s = \{y^s_k\}_{k=1}^{N_s}$, to achieve a good performance on the unlabeled target images $X^t = \{x^t_k\}_{k=1}^{N_t}$, where $N_s$ and $N_t$ are the number of source and target images, respectively. $y^s_k$ is a one-hot label. Pixel-wise cross-entropy loss $L^s_k$ is adopted to train $s_\theta$ in the source domain. $$L^s_k = - \sum_{i=1}^{H} \sum_{j=1}^{W} \sum_{c=1}^{C+1} (y^s_k)_{(i,j,c)} \times \log(s_\theta(x^s_k)_{(i,j,c)}),$$ where $C$ is the number of lanes and class $C + 1$ denotes the background category. $H$ and $W$ are the height and width of $x^s_k$. However, when transferred to the target domain, $s_\theta$ trained in the source domain suffers from performance degradation due to the domain shift. In this paper, we adopt a self-training method (Tarvainen & Valpola [2017]) to address this issue. As shown in Figure 2(a), in the self-training process, we train two models, i.e., student model $s_\theta$ and teacher model $t_\theta$ to better transfer the knowledge from the source domain to the target domain. Specifically, $t_\theta$ generates the one-hot pseudo-label $y^t_k$ on the unlabeled target image $x^t_k$. $$(y^t_k)_{(i,j,c)} = \begin{cases} c = \argmax_{c' \in c^} (t_\theta(x^t_k)_{(i,j,c')}) & , i \in [0, H], j \in [0, W], \end{cases}$$ where $[.]$ denotes the Iverson bracket and $c^$ represents the set of all categories. To ensure the quality of pseudo-labels, we filter low-quality pseudo-labels by setting the confidence threshold $\alpha_c$, i.e., $$(y^t_k)_{(i,j,c)} = \begin{cases} (y^t_k)_{(i,j,c)}, & \text{if } (t_\theta(x^t_k)_{(i,j,c)}) \geq \alpha_c \\ 0, & \text{otherwise} \end{cases}$$ $s_\theta$ is trained on both labeled source images and unlabeled target images with pseudo-labels. The same pixel-wise cross-entropy loss $L^t_k$ is used as the loss function in the target domain. $$L^t_k = - \sum_{i=1}^{H} \sum_{j=1}^{W} \sum_{c=1}^{C+1} (y^t_k)_{(i,j,c)} \times \log(s_\theta(x^t_k)_{(i,j,c)}).$$ During training, no gradients are backpropagated into $t_\theta$ and the weight of $t_\theta$ is updated by $s_\theta$ through Exponentially Moving Average (EMA) at every iteration $m$, denoted by, $$t_\theta^{m+1} = \beta \times t_\theta^m + (1 - \beta) \times s_\theta^m,$$ where the scale factor $\beta$ is set to 0.9 empirically. After the training, we use the student model $s_\theta$ for inference and produce the final lane detection results. 3.2 Cross-domain Contrastive Loss Since the cross-entropy loss is ineffective in learning discriminative features of different lanes, we introduce the category-wise contrastive loss (Wang et al. [2021]) to solve this problem. The formulation of category-wise contrastive loss $L_{CL}$ is written as, $$L_{CL} = - \frac{1}{C \times M} \sum_{c=1}^{C} \sum_{p=1}^{M} \log \left[ \frac{e^{-<V_{cp}, V^+_c>/\tau}}{e^{-<V_{cp}, V^+_c>/\tau} + \sum_{q=1}^{N} e^{-<V_{cp}, V^-_{cq}>/\tau}} \right],$$ where $M$ and $N$ represent the numbers of anchors and negative samples, respectively. $V_{cp}$ is the feature representation of the $p$-th anchors of class $c$, used as a candidate for comparison. $V^+_c$ is the feature representation of the positive sample of class $c$. $V^-_{cq}$ denotes the feature representation of the $q$-th negative samples of the $p$-th anchors of class $c$. $\tau$ is the temperature hyper-parameter and $<\cdot, \cdot>$ is the cosine similarity between features from two different samples. In the target domain, existing methods either focus on improving the form of contrastive loss (Wang et al. [2023]), introducing extra hyper-parameters, or only select $V^+_c$ from the current input images (Wang et al. [2021]). However, the false pseudo-labels generated by $t_\theta$ cause the incorrect positive samples assignment, making the contrastive loss ineffective in learning discriminate features of different categories. We develop a sample selection policy without modifying the existing contrastive loss to overcome the difficulty. Anchor Selection. We choose anchors for each lane from a mini-batch of samples. The anchors of the $ c $-th lane, $ A_c $ can be selected according to, \[ A_c = \{(i, j) \| GT_{(i,j)} = c, s_\theta(x^{in}_{(i,j,c)}) \geq \mu_c, i \in [0, H], j \in [0, W]\}, \] \[ V_c = \{V_{(i,j)} \| (i, j) \in A_c\}, \] where $ GT $ denotes the labels in the source domain or pseudo-labels in the target domain, $ x^{in} $ represents an input image, and $ \mu_c $ is the threshold. We set pixels whose GT are category $ c $ and whose predicted confidence are greater than $ \mu_c $ as anchors to reduce the effect of hard anchors. $ V \in R^{H \times W \times D} $ is the pixel-wise representation and $ D $ is the feature dimension. As illustrated in Figure 2(b), we achieve $ V $ by exploiting an extra representation head $ U $. $ U $ shares the input with the prediction head and is only used in the training process. $ V_c $ is the set of feature representation of anchors and $ V_{cp} \in R^D $ is randomly selected from $ V_c $. Positive sample selection. To ensure the appropriate assignment of positive samples, we establish a positive sample memory module (PSMM) for each lane in both the source and target domains to save its domain-level feature, denoted as $ B_{so} \in R^{C \times D} $ and $ B_{ta} \in R^{C \times D} $. We initialize and update the domain-level features saved in PSMM, following MCIBI (Lin et al., 2021). This process can be found in Appendix A.2. For the $ c $-th lane, we take its domain-level feature as the feature representation of the positive sample. \[ V_c^+ = B_o(c), \] where $ o $ is the source domain ($ so $) or the target domain ($ ta $). Negative sample selection. We directly use pixels of a lane not labeled $ c $ as the negative samples in the source domain. On the other hand, in the target domain, pixels with the lowest predicted conference for category $ c $ are selected as negative samples. \[ neg_{loc_c} = \left\{(i, j) \| \argmin_{c' \in C^} s_\theta(x^k_T(i,j,c')) = c, i \in [0, W], j \in [0, H]\right\}, \] \[ neg_c = \{V_{(i,j)} \| (i, j) \in neg_{loc_c}\}, \] where $ neg_{loc_c} $ and $ neg_c $ denote the location and the set of feature representation of negative samples of class $ c $, respectively. $ V_{cpq} \in R^D $ is also randomly selected from $ neg_c $. To compare intra-domain and inter-domain features at the same time, we propose a Cross-domain Contrastive Loss (CCL), consisting of an intra-domain contrastive learning loss $ L_{inter} $ and an inter-domain contrastive learning loss $ L_{intra} $. \[ CCL = L_{inter} + L_{intra}, \] where $ L_{inter} $ and $ L_{intra} $ are the same as Eq. 6. CCL is applied in both source and target domains. For the source cross-domain contrastive loss (SCCL), the positive samples in $ L_{inter} $ are the domain-level features saved in $ B_{ta} $, and the positive samples in $ L_{intra} $ are the domain-level features saved in $ B_{so} $. The positive samples in the target cross-domain contrastive loss (TCCL) are opposite to SCCL. The overall loss of DACCA is, \[ Loss = \frac{1}{N_s} \sum_{k=1}^{N_s} (\lambda_c \times SCCL^k + L_S^k) + \frac{1}{N_t} \sum_{k=1}^{N_t} (\lambda_c \times TCCL^k + L_T^k), \] where $ \lambda_c $ is the scale factor, which is set to 0.1 empirically. 3.3 Domain-level Feature Aggregation Cross-domain context dependency is essential to transfer knowledge across domains. Cross-domain Contextual Feature Aggregation (CCFA) is an effective way to achieve cross-domain context dependency. Existing CCFA methods (Yang et al., 2021; Zhou et al., 2022; Chung et al., 2023) only aggregate a mini-batch of features. We argue that aggregating features from a whole domain is more beneficial. As shown in Figure 2(b), Domain-level Feature Aggregation (DFA) aims to fuse the domain-level features into the pixel-level representation. DFA contains two key components, i.e., source and target domain-level feature assignment. The process is the same for both. We take the target domain-level feature assignment as an example to depict the process. Figure 3: Location of unreliable background pixels in green. Pixel feature selection.* To select the corresponding domain-level feature for each lane pixel, we propose the pixel feature selection. We first obtain the predicted category at location $(i,j)$ by, \[ P = \argmax_{c' \in C^} (\text{Softmax}(\text{Conv}(E))(i,j,c')), i \in [0,W], j \in [0,H], \] (14) where $E \in R^{H \times W \times D}$ represents the feature map, containing the pixel-level feature representation. 1×1 convolution (termed as Conv) is adopted to change the channels of $E$ to $C + 1$. $P \in R^{H \times W}$ saves the predicted category at each location of $E$. Then, we build a feature map $Z$ whose pixel values are zero and whose size and dimension are the same as $E$. We assign the pixel-wise feature to $Z$ using the domain-level feature. \[ Z(i,j) = B_{ta}(P(i,j)), P(i,j) \neq C + 1, i \in [0,W], j \in [0,H]. \] (15) After the assignment, $Z$ is a domain-level feature map. Here, the lane pixels on $E$ predicted as the background in training are called unreliable background pixels (UBP). For example, as illustrated in Figure 3, UBP is mainly located at the edge of the lane. However, the features of UBP cannot be augmented since domain-level features are only aggregated for the foreground pixels. To refine the features of UBP, we also perform further feature aggregation on UBP. Specifically, the predicted confidence of the UBP is usually low, hence we distinguish UBP from reliable background pixels by setting confidence threshold $\varepsilon$. The UBP is defined as, \[ UBP = \{(i,j)\|\text{pred}_{(i,j)} < \varepsilon, P(i,j) = C + 1, i \in [0,W], j \in [0,H]\}, \] (16) where $\text{pred}_{(i,j)}$ is the confidence of the predicted category at location $(i,j)$. $\text{pred}_{(i,j)}$ is obtained by: \[ \text{pred}_{(i,j)} = \max_{c' \in C^} (\text{Softmax}(\text{Conv}(E))(i,j,c')) . \] We choose the category with the lowest Euclidean distance as the pseudo category of UBP and use domain-level feature of pseudo category to instantiate UBP in $Z$. \[ P(i,j) = \argmin_{c' \in C^} (\text{dis}(E_{UBP}^{(i,j)}, B_{ta}(c'))), (i,j) \in UBP, \] (17) \[ Z(i,j) = B_{ta}(P(i,j)), (i,j) \in UBP, \] (18) where $E_{UBP}^{(i,j)}$ is the feature representation of UBP at location $(i,j)$ in $E$, and $\text{dis}$ is used to calculate the Euclidean distance between the feature representation of UBP and the domain-level feature. Thereafter, we adopt a linear layer to extract features along the channel dimension in $Z$ to obtain the output of target domain-level feature assignment $F_T$. In the same process, we replace the target PSMM with the source PSMM to obtain the feature $F_S$. $F_S$, $F_T$, and $E$ are concatenated along the channel dimension and fused by a 1×1 convolution to enrich the cross-domain context information of $E$. \[ F_{aug} = \text{Conv}(\varphi(E,F_S,F_T)), \] (19) where $F_{aug} \in R^{H \times W \times D}$ is the aggregated features and $\varphi$ is the concatenate operation. ### 4 EXPERIMENTS #### 4.1 Experimental Setting We provide the experimental setting including datasets and implementation details in Appendix A.1. Table 1: Results of critical components. \| Source-only \| SCCL \| Self-Training \| TCCL \| DFA \| UBP \| Accuracy(%) \| FP(%) \| FN(%) \| \|-------------\|------\|---------------\|------\|-----\|-----\|-------------\|-------\|-------\| \| ✓ \| \| \| \| \| \| 77.42 \| 58.29 \| 54.19 \| \| ✓ \| ✓ \| \| \| \| \| 79.63 \| 53.41 \| 50.00 \| \| ✓ \| ✓ \| ✓ \| \| \| \| 80.76 \| 49.39 \| 47.50 \| \| ✓ \| ✓ \| ✓ \| ✓ \| \| \| 81.77 \| 48.36 \| 45.06 \| \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| \| 82.43 \| 44.53 \| 42.89 \| \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| 83.99 \| 42.27 \| 40.10 \| 4.2 Ablation Study We ablate the key components of DACCA and use SCNN with ResNet50 (He et al., 2016) as the detection model. If not specified, all ablation studies are conducted on TuLane. Additional ablation study can be found in Appendix A.3. Effectiveness of cross-domain contrastive learning (CCL). In Table 1, when only source domain data are used in supervised learning, SCCL prompts the accuracy from 77.42% to 79.63%. It also indicates that our SCCL works for supervised training. On the other hand, the accuracy increases by 1.01%, i.e., from 80.76% to 81.77%, if TCCL is adopted. T-SNE visualization in Figure A4(c) of Appendix A.4 shows that the model with CCL can learn more discriminative features. Effectiveness of domain-level feature aggregation (DFA). In Table 1, DFA can improve the detection accuracy from 81.77% to 82.43%. As for feature aggregation of UBP, the accuracy is further increased by 1.56% (83.99% vs. 82.43%). Also, we can observe a significant adaptation of the source and target domain features in Figure A2(c) of Appendix A.4 which validates the effectiveness of domain-level feature aggregation. Table 2: Generalizability of different methods. The symbol indicates source domain only. \| Model \| Backbone \| Accuracy/% \| FP/% \| FN/% \| \|----------------\|--------------\|------------\|------\|------\| \| SCNN* \| ResNet50 \| 77.42 \| 58.29\| 54.19\| \| SCNN+DACCA \| ResNet50 \| 83.99 \| 42.27\| 40.10\| \| ERFNet (Romera et al., 2017)* \| ERFNet \| 83.30 \| 37.46 \| 37.55 \| \| ERFNet+DACCA \| ERFNet \| 90.47 \| 30.66\| 18.16\| \| RTFormer (Wang et al., 2022)* \| RTFormer-Base \| 87.24 \| 26.78 \| 25.17 \| \| RTFormer+DACCA \| RTFormer-Base \| 92.24 \| 15.10\| 12.58\| Generalizability of different methods. As shown in Table 2, our method can be integrated into various segmentation-based lane detection methods. In SCNN, using our method can increase the accuracy by 6.57% and decrease FP and FN by 16.02% and 14.09%, respectively. Also, in the lightweight model ERFNet, the accuracy rises by 7.17%, and FP and FN drop by 6.8% and 19.39%. Finally, in the Transformer-based method RTFormer, our method significantly improves the detection performance, in terms of accuracy, FP, and FN. Comparison with existing contrastive loss variants. In Figure 4(a), CCL is evaluated against other contrastive loss variants in UDA. In turn, we replace CCL in DACCA with CDCL, ProCA (Jiang et al., 2022), CONFETI (Li et al., 2023), and SePiCo (Xie et al., 2023). Compared with ProCA and CONFETI, CCL increases the accuracy by 2.58% (81.77% vs. 79.19%) and 1.9% (81.77% vs. 79.87%), respectively. The reason may be that both ProCA and CONFETI ignore the differences in feature distribution between the source domain and target domain and only use a prototype to represent the features of the two domains. Moreover, CCL overwhelms SePiCo regarding accuracy. It attributes to SePiCo only taking domain-level features from the source domain as the positive samples but ignoring the samples from the target domain. Comparison with existing cross-domain context aggregation. We substitute the DFA with Cross-domain (Yang et al., 2021) and Self-attention module (SAM) (Chung et al., 2023)—the latter aggregate features in a mini-batch. The superiority of the DFA is shown in Figure 4(b). DFA performs better than Cross-domain and SAM, e.g., prompts the accuracy by 0.46% (83.51% vs. 83.05%) and 0.72% (83.51% vs. 82.79%), respectively. From the T-SNE visualization in Figure A3 of Appendix A.4, we can see that DFA aligns the features of two domains better. The results demonstrate that aggregating features from the whole domain is more effective than from a mini-batch. Figure 4: Accuracy comparison with counterparts of key peer components. (a) Comparison among existing contrastive loss variants. (b) Comparison among existing cross-domain context aggregation. Figure 5: Visualization result comparison among cross-domain, SGPCS, and our method. Results on (a) MuLane, (b) MoLane, and (c) TuLane. Table 3: Performance comparison on TuLane. \| Method \| Detection model \| Backbone \| Accuracy/% \| FP/% \| FN/% \| \|-----------------\|-----------------\|------------\|------------\|------\|------\| \| DANN \| ERFNet \| ERFNet \| 86.69 \| 33.78\| 23.64\| \| ADDA \| ERFNet \| ERFNet \| 87.90 \| 32.68\| 22.33\| \| SGADA \| ERFNet \| ERFNet \| 89.09 \| 31.49\| 21.36\| \| SGPCS \| ERFNet \| ERFNet \| 89.28 \| 31.47\| 21.48\| \| LD-BN-ADAP \| RTFormer \| RTFormer-Base \| 90.78 \| 28.44\| 15.66\| \| MLDA \| UFLD \| ResNet18 \| 91.55 \| 28.52\| 16.16\| \| PyCDA \| ERFNet \| ERFNet \| 88.43 \| 31.69\| 21.33\| \| Cross-domain \| ERFNet \| ERFNet \| 89.00 \| 30.53\| 20.42\| \| Maximum Squares \| ERFNet \| ERFNet \| 86.73 \| 31.26\| 24.13\| \| DACCA \| ERFNet \| ERFNet \| 90.47 \| 30.66\| 18.16\| \| DACCA \| RTFormer \| RTFormer-Base \| 92.24 \| 15.10\| 12.58\| 4.3 Comparison with State-of-the-Art Methods Performance on TuLane. The results on TuLane are shown in Table 3. When ERFNet is used as the detection model, our method performs better than other methods. For instance, our method Table 4: Performance comparison on "OpenLane" to "CULane". \| Method \| Normal \| Crowded \| Night \| No line \| Shadow \| Arrow \| Dazzle \| Curve \| Cross \| Total \| \|-----------------\|--------\|---------\|-------\|---------\|--------\|-------\|--------\|-------\|-------\|-------\| \| Advent (Li et al., 2022) \| 51.2 \| 24.5 \| 21.5 \| 19.9 \| 16.9 \| 34.7 \| 27.2 \| 35.3 \| 5789 \| 31.7 \| \| PyCDA (Lian et al., 2019) \| 42.4 \| 20.6 \| 14.7 \| 15.9 \| 14.4 \| 28.6 \| 19.5 \| 30.8 \| 4452 \| 26.3 \| \| Maximum Squares (Chen et al., 2019) \| 51.4 \| 28.4 \| 22.1 \| 19.7 \| 20.9 \| 40.8 \| 28.1 \| 39.3 \| 9813 \| 31.8 \| \| MLDA (Li et al., 2022) \| 62.0 \| 38.0 \| 28.5 \| 21.9 \| 24.1 \| 50.3 \| 31.7 \| 44.5 \| 11399 \| 38.8 \| \| DACCA \| 64.9 \| 39.6 \| 29.3 \| 25.1 \| 26.3 \| 52.8 \| 34.1 \| 43.5 \| 7158 \| 43.0 \| Table 5: Performance comparison on "CULane" to "Tusimple". \| Method \| Detection model \| Backbone \| Accuracy/% \| FP/% \| FN/% \| \|-----------------\|-----------------\|----------\|------------\|------\|------\| \| Advent (Li et al., 2022) \| ERFNet \| ERFNet \| 77.1 \| 39.7 \| 43.9 \| \| PyCDA (Lian et al., 2019) \| ERFNet \| ERFNet \| 80.9 \| 51.9 \| 45.1 \| \| Maximum Squares (Chen et al., 2019) \| ERFNet \| ERFNet \| 76.0 \| 38.2 \| 42.8 \| \| MLDA (Li et al., 2022) \| ERFNet \| ERFNet \| 89.7 \| 29.5 \| 18.4 \| \| DACCA \| ERFNet \| ERFNet \| 92.1 \| 26.7 \| 14.6 \| outperforms MLDA in terms of accuracy by 2.04% (90.47% vs. 88.43%). Besides, using our CCL and DFA, the performance of MLDA gains consistent improvement. It indicates our sample selection policy is more effective than designing complicated loss functions, and DFA has a stronger domain adaptive ability than AIEM in MLDA. Regarding FN metrics, our method is 5.97% and 4.11% lower than PyCDA and Cross-domain, respectively. Significantly, when using the Transformer model RTFormer, DACCA outperforms the state-of-the-art SGPCS (92.24% vs. 91.55%) and achieves the best experimental results on TuLane in similar settings. Performance on OpenLane to CULane. To further validate our method’s generalization ability, we carry out experiments transferring from OpenLane to CULane to demonstrate a domain adaptation between difficult real scenarios. As shown in Table 4, our method delivers 4.2% enhancement (43.0% vs. 38.8%) compared to the state-of-the-art MLDA. Our DACCA surpasses the existing methods in most indicators and also all these results reflect its outperformance. Performance on CULane to Tusimple. As presented in Table 5, our DACCA achieves the best performance on "CULane to Tusimple". For instance, DACCA increases the accuracy from 89.7% to 92.1% compared with the state-of-the-art method MLDA. It indicates our DACCA can perform well on the domain adaptation from difficult scene to simple scene. Qualitative evaluation. We display the visualization comparison results between Cross-domain, SGPCS, and our method in Figure 5. In Figure 5(c), our method predicts more smooth lanes than the other methods in the urban scenario. Our method can detect the complete lanes in the real-world scene in Figure 5(a) and 5(b). Qualitative results demonstrate that our method can effectively transfer knowledge across different domains. 5 CONCLUSION This paper presents a novel unsupervised domain-adaptive lane detection via contextual contrast and aggregation (DACCA), in which learning discriminative features and transferring knowledge across domains are exploited. Firstly, we create the positive sample memory module to preserve the domain-level features of the lane. Then, we propose a cross-domain contrastive loss to improve feature discrimination of different lanes by a novel sample selection strategy without modifying the form of contrastive loss. Finally, we propose the domain-level feature aggregation to fuse the domain-level features with the pixel-level features to enhance cross-domain context dependency. Experimental results show that our approach achieves the best performance on the TuLane dataset. On the MuLane and MoLane datasets, our method outperforms existing unsupervised domain-adaptive segmentation-based lane detection methods. Although DACCA is implemented upon the segmentation-based lane detection, it holds potential for application in other lane detection methods, e.g., keypoint-based and transformer-based approaches. Our future work is to explore this aspect. REFERENCES Tusimple dataset. https://github.com/TuSimple/tusimple-benchmark Accessed on 11th August 2023. Kshitij Bhardwaj, Zishen Wan, Arijit Raychowdhury, and Ryan Goldhahn. Real-time fully unsupervised domain adaptation for lane detection in autonomous driving. In 2023 Design, Automation & Test in Europe Conference & Exhibition (DATE), pp. 1–2, 2023. Li Chen, Chonghao Sima, Yang Li, Zehan Zheng, Jiajie Xu, Xiangwei Geng, Hongyang Li, Conghui He, Jianping Shi, Yu Qiao, et al. Persformer: 3d lane detection via perspective transformer and the openlane benchmark. 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zIJFG7wW2d	- I also didn’t fully understand how the agent uses the question-answering API (Section A.3.1) to add context about the task? I think it’s one of the most crucial steps of the pipeline, and it’d be great to explain how it’s implemented in more detail (and perhaps in the main manuscript). Concretely, how are the retrieved documents to generate the instruction? Are the retrieved documents added as prompt to the agent to generate task-specific instruction? Is that in addition to the name of the dataset, task information and few input examples? Why are they added to a vector database?	AGENT INSTRUCTS LARGE LANGUAGE MODELS TO BE GENERAL ZERO-SHOT REASONERS Anonymous authors Paper under double-blind review ABSTRACT We introduce a method to improve the zero-shot reasoning abilities of large language models on general language understanding tasks. Specifically, we build an autonomous agent to instruct the reasoning process of large language models. We show this approach further unleashes the zero-shot reasoning abilities of large language models to more tasks. We study the performance of our method on a wide set of datasets spanning generation, classification, and reasoning. We show that our method generalizes to most tasks and obtains state-of-the-art zero-shot performance on 20 of the 29 datasets that we evaluate. For instance, our method boosts the performance of state-of-the-art large language models by a large margin, including Vicuna-13b (13.3%), Llama-2-70b-chat (23.2%), and GPT-3.5 Turbo (17.0%). Compared to zero-shot chain of thought, our improvement in reasoning is striking, with an average increase of 10.5%. With our method, Llama-2-70b-chat outperforms zero-shot GPT-3.5 Turbo by 10.2%. 1 INTRODUCTION Large language models (LLMs) (Brown et al., 2020; Wang & Komatsuaki, 2021; Zhang et al., 2022a; Smith et al., 2022; Chowdhery et al., 2022; Hoffmann et al., 2022; Scao et al., 2023; OpenAI, 2023; Anil et al., 2023; Touvron et al., 2023a,b; Penedo et al., 2023) have significantly advanced the state-of-the-art on a wide range of language understanding tasks. These models have led to widespread deployment and adoption in applications (Araci, 2019; Huang et al., 2020; Bolton et al., 2022; Wu et al., 2023; Driess et al., 2023; Huang et al., 2023). In particular, the emerging capabilities of LLMs such as complex reasoning (Wei et al., 2022b; Wang et al., 2023c; Kiciman et al., 2023) have made them the subject of research in recent years. Among these, zero-shot reasoning (Kojima et al., 2022; Wan et al., 2023) has drawn substantial public interest and achieved promising results in specific task domains. However, the reasoning ability of LLMs on general tasks remains unclear. In this paper, we improve the zero-shot reasoning abilities of LLMs on general language understanding tasks. To solve a task, we build an agent to instruct the reasoning process of LLMs for the task (Figure 1). More specifically, our autonomous agent generates task-specific instructions to better align the chain of thought reasoning process of LLMs with each task. We refer to this approach as zero-shot agent instructed reasoning (AgentInstruct). The basic idea of our approach is motivated by two lines of work. First, language agents (Yao et al., 2023b; Shin et al., 2023; Park et al., 2023; Wang et al., 2023a; Xi et al., 2023) have been developed to automatically complete a task. Instead of completing the task, our agent produces instructions on how to complete the task. We enable this by adapting an existing agent to access a wide range of task-relevant knowledge on the web, given basic task information such as the name of the dataset and several input examples. As a result, the agent synthesizes high-quality step-by-step instructions for tasks verified by the web resources. We follow the recent design of agents where an LLM is used to produce the plans for this process. Second, zero-shot chain of thought (CoT) reasoning of LLMs has obtained promising results on tasks such as arithmetic reasoning (Kojima et al., 2022; Wang et al., 2023b). Standard in-context zero-shot learning prompts an LLM to directly output predictions without task examples. In contrast, CoT decomposes a task into intermediate steps and solving each will lead to the final output. We further align the CoT reasoning steps with a particular task by prompting with the task-specific agent instructions. The design of zero-shot AgentInstruct is important: We generalize the zero-shot reasoning abilities of LLMs to more tasks with the combination of task-specific instructions from a language agent and task-specific reasoning of LLMs. We empirically evaluate the zero-shot reasoning abilities of LLMs on a wide set of language understanding tasks across 29 datasets (including 53 subsets), spanning generation, classification, and reasoning. Zero-shot AgentInstruct obtains state-of-the-art performance on 20 datasets. We conduct our evaluation on three state-of-the-art LLMs, namely, Vicuna (Chiang et al., 2023), Llama-2-chat (Touvron et al., 2023b), and GPT-3.5 Turbo (OpenAI, 2022). We show that zero-shot AgentInstruct boosts the performance of these models by 17.8% on average. When compared to zero-shot CoT, the overall performance improvement is significant (6.5%), and in particular, the improvement in reasoning is substantial with an average increase of 10.5%, leading to the best performance on 10 out of 12 reasoning tasks. Notably, Llama-2-70b-chat with zero-shot AgentInstruct outperforms standard zero-shot GPT-3.5 Turbo by an average of 10.2%. We hope the results help foster future research on further unlocking the zero-shot reasoning abilities of large foundation models and exploring the broader usage of agents. 2 APPROACH We present zero-shot AgentInstruct in this section. To solve a task, zero-shot AgentInstruct employs an agent to instruct an LLM to reason toward the final prediction. Intuitively, humans often rely on specific instructions to more effectively guide their thought process as they work towards a solution to a problem. For instance, to understand the sentiment in the movie reviews, instructions such as “1. Understand the Dataset: ... Movie Reviews dataset ... 2. Analyze the Passage: Pay attention to ... the tone of the review ...” help humans decompose the problem into task-specific reasoning steps and solve each to deliver the final answer (Figure 1). Zero-shot AgentInstruct follows this intuition. Agent Instructions Instead of handwriting task-specific instructions, we build an agent to automate the process. The intuition is that an agent is able to synthesize high-quality instructions with access to a wide range of existing task knowledge on the web. We design our agent based on ReAct (Yao et al., 2023b). This is motivated by the recent developments of language agents for task solving. Our agent highlights two features (Figure 2): (i) Instruction generation. Our agent follows ReAct which uses an LLM to propose a series of thoughts. The agent then receives observations and takes actions following the thoughts. Different from ReAct which aims to directly solve the task, our agent outputs step-by-step instructions on how to solve the task. The major advantage of this is that we only need to generate instructions once per dataset instead of running the agent on all dataset instances. We use GPT-4 (OpenAI, 2023) as the default agent. Once the action is finish, the corresponding output is our task-specific instructions. (ii) Action space. We constrain our action space to contain two types of actions that support the instruction generation: (a) ask_about_dataset[string], which returns the top relevant web pages containing information about the dataset. To do this, we construct and utilize a question answering API as a tool. This API is meant to answer questions about the document by interfacing with a vector database storing the web pages to provide an answer. (b) finish[instructions], which finishes the instruction generation with the task-specific instructions. As shown in Figure 2, to produce the instructions, our agent takes basic dataset information such as the name of the dataset (e.g., IMDB), a few input-only examples (examples without ground truth labels), and the set of output labels for the dataset (if applicable, else the type of dataset like generation) as its input. Using the task knowledge from the web, our agent forms observations (e.g., “Observation 1: … labeled as either positive or negative …”) and thoughts (e.g., “Thought 2: … creating instructions …”) which trigger the agent to perform actions, such as the finish action to output the task-specific instructions. Chain of Thought Reasoning Chain of thought (CoT) (Wei et al., 2022b; Kojima et al., 2022) prompts LLMs to break down the task into intermediate reasoning steps that lead to the final answer. Unlike zero-shot CoT which uses a fixed prompt “Let’s think step by step”, we prepend our task-specific agent instructions to the input to prompt the LLMs to optimize their reasoning processes for the task. LLMs will then follow our task-specific instructions to decompose the task into a chain of more specific intermediate steps to solve the task. As shown in Figure 1, the agent instructions “… Pay attention to … explicit or implicit expressions of sentiment towards the movie …” are the key to producing the critical reasoning path “… the movie is worth a view only for the performances of the three actors …”, which leads to the correct prediction where standard zero-shot and zero-shot CoT fail. We follow zero-shot CoT, which consists of a reasoning extraction prompt to produce the intermediate reasoning steps, and an answer extraction prompt to collect the answers. For simplicity of implementation, we replace zero-shot CoT’s fixed prompt with zero-shot AgentInstruct’s task-specific instructions in the reasoning extraction prompt. AgentInstruct is zero-shot as there are no task-specific examples involved in the pipeline. It enjoys several unique properties: (i) Zero-shot AgentInstruct is a new way to improve the zero-shot reasoning of LLMs. Zero-shot AgentInstruct decouples the language agent and reasoning process of LLMs, which helps zero-shot AgentInstruct generalize to more tasks. Additionally, the agent instructions provide more task-specific controls to the reasoning paths of LLMs, which benefits human alignment and improves the safety of LLMs. (ii) Our agent instructions are customized for different tasks and verified by existing task knowledge. For each task, different LLMs use the same set of instructions, and we find the instructions transfer well among these models. This is important in practice as the agent LLMs are often more powerful and costly than the reasoning LLMs, so our approach is a cost-effective alternative to using agents directly. (iii) By providing task-specific instructions, chain of thought reasoning abilities are further generalized to more tasks beyond reasoning tasks. We show general language understanding tasks such as generation and classification also benefit from chain of thought reasoning. To better embrace the emerging reasoning capabilities of LLMs, our task-specific instructions align the reasoning process with a particular task better than general or fixed instructions. 3 EXPERIMENT We show that zero-shot AgentInstruct successfully improves the zero-shot reasoning abilities of LLMs, namely, Vicuna (Chiang et al., 2023), Llama-2-chat (Touvron et al., 2023b), and GPT-3.5 Turbo (OpenAI, 2022), by a large margin on average. We evaluate zero-shot AgentInstruct on an exhaustive selection of 29 benchmarking datasets containing 53 subsets. As shown in Figure 3, each dataset is either a generation or classification task and a portion of the datasets in each category are also reasoning tasks. The datasets consist of all HELM core scenarios from Liang et al. (2023), as well as the reasoning datasets from Kojima et al. (2022). The details of the experimental setup, including datasets and models, are described in Appendix A. 3.1 MAIN RESULTS Results are shown in Figure 5. We compare zero-shot AgentInstruct to standard zero-shot and zero-shot CoT. We focus our analysis on three models: Vicuna-13b, Llama-2-70b-chat, and GPT-3.5 Turbo. ![Figure 3: Datasets for generation (blue), classification (green), and reasoning (orange). Reasoning contains generation and classification tasks.](image) We first compare zero-shot AgentInstruct to standard zero-shot prompting (Figure 1). The zero-shot prompt design follows Liang et al. (2023). On each model, zero-shot AgentInstruct wins on the majority of datasets, with no less than a 13.0% increase on average. Average performance versus the zero-shot setting is best on Llama-2-70b-chat with a 23.2% improvement. Figure 5a and Figure 5b show the results for generation and classification tasks respectively. On average, with zero-shot AgentInstruct, the three models beat the zero-shot setup by 23.1% for generation and 13.5% for classification. We hypothesize that generation datasets generally require more specific instructions than classification datasets, as the model does not know the best format of the generation output unless it has sufficient task information. This shows that our agent is able to instruct the reasoning process to improve the final outputs for different tasks, and zero-shot AgentInstruct is able to generalize LLMs’ reasoning abilities across tasks. Note that we only run the agent 53 times, resulting in 53 agent generated instructions, as we evaluate on 53 subsets. With zero-shot AgentInstruct, we also observe a large margin of improvement for zero-shot performance across different models. Significantly, Llama-2-70b-chat beats the performance of zero-shot GPT-3.5 Turbo by 10.2% on average across all datasets. This indicates our agent instructions are the key to improving the reasoning performance of LLMs. The most immediate comparison to zero-shot AgentInstruct is zero-shot CoT, as zero-shot AgentInstruct uses task-specific instructions instead of a fixed manual instruction. On average, across all three models, zero-shot AgentInstruct beats zero-shot CoT by 6.5%, with the largest growth being Vicuna-13b at 9.5%. On both generation and classification datasets, across three models, zero-shot AgentInstruct wins by 5.0% and 7.8% on each category respectively. This suggests that zero-shot AgentInstruct is able to generalize the zero-shot reasoning abilities of LLMs to both generation and classification tasks, and optimize the performance of specific tasks. ![Figure 4: Winning rate (%) between zero-shot, zero-shot CoT, and zero-shot AgentInstruct based on the average results over three models.](image) In particular, we look into the performance of reasoning tasks (Figure 5c). Of our three models, the average difference between zero-shot AgentInstruct and the zero-shot setting on reasoning tasks is 31.3%, whereas the difference between zero-shot AgentInstruct and zero-shot CoT is 10.5%. This shows that our task-specific instructions are more helpful for LLMs to break down tasks into more specific intermediate reasoning steps compared to the task-agnostic instructions in zero-shot CoT, which leads to improved final predictions. Overall, zero-shot AgentInstruct wins on 9 of the 13 generation datasets, 11 of the 16 classification datasets, and 10 of the 12 reasoning datasets (Figure 4). See Appendix B for additional results and analysis, including results on individual datasets and subsets. ![Figure 5](image-url) Results on Vicuna-13b, Llama-2-70b-chat, and GPT-3.5 Turbo across tasks. Top: generation. Middle: classification. Bottom: reasoning. ### 3.2 Ablation Study \| \| AddSub \| IMDB \| NarrativeQA \| \|------------------\|--------\|------\|-------------\| \| Zero-Shot AgentInstruct \| 79.5 \| 94.0 \| 65.0 \| \| w/o Agent Instructions \| 73.2 \| 89.0 \| 62.3 \| \| w/o Input Examples \| 72.4 \| 88.0 \| 60.1 \| \| w/o Labels \| 74.9 \| 93.8 \| 63.9 \| \| w/o GPT-4 \| 75.2 \| 92.6 \| 63.5 \| Table 1: Ablation over different facets of zero-shot AgentInstruct with Llama-2-70b-chat. We examine how different components of zero-shot AgentInstruct impact its zero-shot reasoning performance. Results are shown in Table 1 on AddSub (reasoning), IMDB (classification), and NarrativeQA (generation). We use Llama-2-70b-chat for reasoning. The four settings examine the importance of agent instructions in zero-shot AgentInstruct. Descriptions of each setting are as follows: (i) w/o Agent Instructions: We compare the zero-shot AgentInstruct methodology to zero-shot CoT. (ii) w/o Input Examples: We remove the input-only examples from the input to the agent. (iii) w/o Labels: We remove the description of the labels from the input to the agent. (iv) w/o GPT-4: We use GPT-3.5 Turbo, instead of GPT-4, as the agent to generate instructions. ![Figure 6](image-url) Comparison on GPT-4 using zero-shot, zero-shot CoT, ReAct, and zero-shot AgentInstruct on AddSub. results suggest that all components of zero-shot AgentInstruct are effective in providing high-quality instructions and eliciting high-quality reasoning steps before making predictions. See Appendix C.3 for further descriptions of each setting. Next, we focus our analysis on GPT-4. We test the following methods: zero-shot, zero-shot CoT, ReAct (Yao et al., 2023), and zero-shot AgentInstruct. Figure 6 shows the performance of each method on AddSub. Zero-shot AgentInstruct outperforms zero-shot GPT-4 by 8.6% and ties the performance of zero-shot CoT GPT-4 for approximately one-tenth of the cost. Zero-shot AgentInstruct is using GPT-3.5 Turbo for the CoT reasoning. This indicates that both task-specific instructions and CoT reasoning help improve the zero-shot performance of LLMs. Though ReAct narrowly outperforms zero-shot AgentInstruct, it costs nearly 100 times more since the zero-shot AgentInstruct agent is run only once per dataset rather than per instance. Each run of our agent to generate instructions cost less than $1. We also add results on IMDB, namely zero-shot GPT-4, zero-shot CoT GPT-4, and zero-shot AgentInstruct GPT-4 (the reasoning step also uses GPT-4), scoring 87.4, 96.1, and 96.6 respectively. This result implies that decoupling the agent instruction generation and reasoning steps further unleashes the zero-shot reasoning abilities of LLMs, and that zero-shot AgentInstruct is a cost-effective alternative to using agents directly, resulting in only a minimal performance loss. 3.3 Truncated Context Length In Figure 7, we test how zero-shot AgentInstruct performs on various context lengths. We artificially reduce the context length of Llama-2-70b-chat from 4,000 (maximum context length) to 2,048, 1,024, 768, and 640. The results suggest that the performance of zero-shot AgentInstruct is worse for models with smaller context lengths. While the impact is minimal on AddSub and IMDB, performance on NarrativeQA steeply declines below a context length of 2,048, which is due to truncation. This is because the instances of NarrativeQA are much longer, with an average length of 855 tokens compared to the instances of AddSub and IMDB, with average lengths of 330 tokens and 48 tokens respectively. ![Figure 7: Truncating context lengths Llama-2-70b-chat with zero-shot AgentInstruct on AddSub, IMDB, and NarrativeQA.](image) 3.4 Model Scaling As it is often the case that larger models substantially improve the reasoning capabilities of LLMs (Wei et al., 2022), we test zero-shot AgentInstruct’s performance on models of various sizes. Specifically, we test on three Llama-2-chat models with 7 billion, 13 billion, and 70 billion parameters. Figure 8 shows the average score across all 29 datasets for zero-shot, zero-shot CoT, and zero-shot AgentInstruct. These results confirm that the average performance of all three methods increases with model size. Each time model size increases, zero-shot CoT and zero-shot AgentInstruct show consistent gains of around 6%, while zero-shot has smaller gains near 2%. This is because reasoning steps are best produced with more powerful models. In fact, at just 13b parameters, Llama-2-13b-chat with zero-shot AgentInstruct surpasses the performance of zero-shot GPT-3.5 Turbo by over 2%. Additionally, zero-shot AgentInstruct’s superiority over zero-shot and zero-shot CoT appears independent of model size. ![Figure 8: Model scaling results of zero-shot, zero-shot CoT, and zero-shot AgentInstruct with Llama-2-chat on all datasets.](image) \| CoT Reasoning \| Prompt \| Quasi-Exact Match (%) \| \|---------------\|------------------------------------------------------------------------\|-----------------------\| \| Reasoning extraction \| Follow the instructions to generate an explanation that reasons towards the correct answer to the task above. End the explanation with the correct answer.\n\nExplanation: \| 79.5 \| \| \| Let’s think step-by-step according to the instructions. End with the correct answer. Explanation: \| 77.5 \| \| \| Let’s think step-by-step according to the instructions. First, \| 75.2 \| \| \| Use the instructions to guide you towards your answer.\nExplanation: \| 79.0 \| \| \| Explanation: \| 78.5 \| \| Answer extraction \| Therefore, the answer to the task is below. Give the answer in the shortest form possible that will still be correct. \nAnswer: \| 79.5 \| \| \| Therefore, the answer to the task is below.\nAnswer: \| 79.7 \| \| \| Therefore, the answer is \| 79.2 \| \| \| Answer: \| 77.7 \| Table 2: Prompt sensitivity analysis of chain of thought reasoning of zero-shot AgentInstruct with Llama-2-70b-chat on AddSub. The default prompts are highlighted. Higher scores are better. ### 3.5 Manual Prompt Sensitivity Zero-shot AgentInstruct has two manual prompts in the CoT reasoning step: (1) the reasoning extraction prompt, which asks for intermediate reasoning steps, and (2) the answer extraction prompt, which collects the final answer. To test the sensitivity of each prompt, we vary a single prompt while keeping the default zero-shot AgentInstruct prompt for the other. Results are shown in Table 2 based on Llama-2-70b-chat on AddSub. Overall, zero-shot AgentInstruct’s performance does not appear particularly sensitive to changes in the manual prompts, suggesting that the methodology behind zero-shot AgentInstruct is robust. Additional prompt sensitivity experiments are conducted in Appendix C.2. ### 3.6 Error Analysis To investigate the reasons for errors made by zero-shot AgentInstruct, we select 25 samples from AddSub, IMDB, and NewsQA respectively where zero-shot AgentInstruct results in incorrect predictions on Llama-2-70b-chat. We define incorrect predictions as those with a quasi-exact match or F1 score less than 1. Table 3 shows our error analysis. The most common error across datasets is incorrect reasoning, i.e., not correctly reasoning through the problem when applying the accurate agent instructions. For example, on AddSub, zero-shot AgentInstruct chooses the wrong operation due to a misleading verb. On IMDB, zero-shot AgentInstruct misreads the sentiment due to emphasizing words describing the movie, not the review. See Figure 9 for an example of incorrect reasoning on IMDB. The answer ambiguity is another main source of errors. For example, in a review where the reviewer clearly enjoyed the movie even though the reviewer acknowledged it was a typical bad movie, our prediction is “Positive” while the ground truth is “Negative” on IMDB. For many errors, either the instructions are taken too literally or partially ignored. As larger models become better at reasoning, such errors should be minimized. More thorough error analysis and full examples for each error category are in Appendix C.1. ### 3.7 Case Study Next, we analyze the quality of the CoT reasoning steps when predictions are correct. On three datasets (AddSub, IMDB, and NewsQA), we randomly select 25 examples from each dataset with a perfect quasi-exact match or F1 score. We find that the reasoning capabilities are further enhanced by the combination of effective agent instructions and the task-specific reasoning process of LLMs. An example is in Figure 10. \| Error Type \| Percentage \| \|------------------\|------------\| \| Reasoning \| \| \| Incorrect reasoning \| 32.0 \| \| Not factual \| 12.0 \| \| Answer \| \| \| Ambiguity \| 22.7 \| \| Invalid label \| 14.7 \| \| Short answer \| 10.6 \| \| Incorrect format\| 8.0 \| Table 3: Error analysis for zero-shot AgentInstruct Llama-2-70b-chat on AddSub, IMDB, and NewsQA. Passage: As an avid Disney fan, I was not totally impressed by this movie, certainly not motivated to catch it in the theaters. I am, however, so glad that I caught it on DVD and watched the special features. You MUST check out the “Moose commentary” The enjoyment I got from this commentary completely made up for the tepid reaction I had to the film itself. CoT Reasoning: Based on the language used in the passage, it is clear that the reviewer has a positive sentiment towards the movie. The reviewer uses positive adjectives such as “enjoyment” to describe their experience with the movie… Answer: Positive Figure 9: An incorrect reasoning example for Llama-2-70b-chat with zero-shot AgentInstruct on IMDB for error analysis. Here, the model mistook the love of the commentary as the love of the movie (highlighted). 3.8 Comparison to Related Methods Few-Shot We compare Llama-2-70b-chat zero-shot AgentInstruct results with few-shot results on AddSub, IMDB, and NarrativeQA in Figure 11. Surprisingly, zero-shot AgentInstruct reaches competitiveness with few-shot prompting. Zero-shot AgentInstruct, without any few-shot examples, outperforms few-shot performance on AddSub and NarrativeQA, by over 4.3% and 23.7% respectively, and loses by 0.7% on IMDB. Ideally, all the information encoded in the few-shot examples can be presented in a clear manner within the agent instructions to better utilize the reasoning capabilities of models. As shown, zero-shot AgentInstruct has the potential to reach or even beat few-shot performance. Self-Consistency Finally, we compare zero-shot AgentInstruct results with self-consistency (Wang et al., 2023c) results of Llama-2-70b-chat on AddSub, IMDB, and NarrativeQA in Figure 12. We adapt self-consistency to the zero-shot setting as follows: We sample three responses using a temperature of 0.7, top-$k$ sampling with $k = 40$, and a randomly generated seed for each request. After cleaning the output, we use a majority vote to determine the consensus answer, choosing at random to break ties. On AddSub, IMDB, and NarrativeQA, zero-shot AgentInstruct outperforms self-consistency by 5.8%, 7.5%, and 1.9% respectively. Besides, zero-shot AgentInstruct is more computationally efficient than self-consistency as there is no need for sampling the reasoning paths multiple times. 4 Related Work Large language models, such as GPT-4 (OpenAI, 2023), GPT-3 (Brown et al., 2020), PaLM (Chowdhery et al., 2022), PaLM-2 (Anil et al., 2023), BLOOM (Scao et al., 2023), OPT (Zhang et al., 2022a), LLaMA (Touvron et al., 2023a), Llama-2 (Touvron et al., 2023b), and many others (Radford et al., 2019; Wang & Komatsuizaki, 2021; Black et al., 2021; Smith et al., 2022; Hoffmann et al., 2022; Penedo et al., 2023) have shown remarkable performance on natural language processing... (NLP) tasks. Following the pretraining phase, additional finetuning enables models (e.g., FLAN (Wei et al., 2022a), FLAN-T5 (Chung et al., 2022), InstructGPT (Ouyang et al., 2022)) to better align with human instructions to complete tasks. Moreover, instruction-tuning has enabled LLMs (e.g., GPT-3.5 Turbo (OpenAI, 2022), Llama-2-chat (Touvron et al., 2023b), Self-Instruct (Wang et al., 2023d), Alpaca (Taori et al., 2023), Vicuna (Chiang et al., 2023), Koala (Geng et al., 2023)) to better engage with users through single-turn or multi-turn dialogue. Zero-shot AgentInstruct builds on these instruction-following language models, enabling better zero-shot reasoning abilities through the use of agent instructions. Language agents (Yao et al., 2023b; Shinn et al., 2023; Xu et al., 2023; Park et al., 2023; Zhou et al., 2023b; Andreas, 2022; Wang et al., 2023a; Xi et al., 2023; Sumers et al., 2023; Chan et al., 2023) have recently emerged due to the task planning capabilities of LLMs. Given a task that is often demonstrated in natural language, these agents aim to complete the task directly. Unlike existing agents, the goal of our agent is to generate task-specific instructions on how to complete the given task, decoupling the agent planning and reasoning steps. Besides reaching competitive performance with using agents directly, our design turns out to be more cost-effective. Finetuning is an effective method for generating higher-quality responses from LLMs on downstream tasks (Liu et al., 2019; Howard & Ruder, 2018). As model scale increases, finetuning becomes less practical, which lightweight tuning methods (such as prefix tuning (Li & Liang, 2021), prompt learning (Lester et al., 2021; Liu et al., 2023), LoRA (Hu et al., 2022)) have tried to solve. Even with such methods, in-context prompting techniques have gained attention as an alternative. Few-shot learning, which involves providing a few examples demonstrating the task before prompting the models during inference, is often effective on a range of tasks (Brown et al., 2020; Dong et al., 2023). Chain of thought (CoT) prompting (Wei et al., 2022b) involves generating a series of intermediate reasoning steps, which can dramatically increase the performance of LLMs on complex reasoning tasks. While this reasoning behavior is traditionally learned from few-shot demonstrations, Kojima et al. (2022) extends CoT prompting to the zero-shot setting. More recently, new approaches such as self-consistency (Wang et al., 2023c), plan-and-solve prompting (Wang et al., 2023b), tree of thought (Yao et al., 2023a), and graph of thought (Besta et al., 2023) have further improved the reasoning quality. On the other hand, zero-shot AgentInstruct focuses on the zero-shot setup. It generalizes the reasoning abilities of LLMs to more tasks by utilizing task-specific instructions generated from our agent to better align the reasoning process with a particular task. NLP benchmarking datasets provide a standardized interface to evaluate LLMs on specific downstream tasks. Common benchmarks (e.g., HELM (Liang et al., 2023), MMLU (Hendrycks et al., 2021), and many others (Wang et al., 2019a,b; Raipurkar et al., 2016; Srivastava et al., 2023; Zhong et al., 2023; Suzgun et al., 2023; Chen et al., 2021)) have become part of the standard evaluation of LLMs. We benchmark our method on 29 datasets, including the core scenarios from HELM (Liang et al., 2023) and the reasoning datasets from Kojima et al. (2022). We also include results from these benchmarks as task-specific model results for comparison purposes. Besides reasoning tasks, our method generalizes to general language understanding benchmark tasks including generation and classification. For reasoning tasks, our method outperforms existing zero-shot approaches (Brown et al., 2020; Kojima et al., 2022) by a large margin. 5 CONCLUSION Our work proposes a new way of improving the zero-shot reasoning abilities of large language models on general language understanding tasks. We build an agent to instruct the reasoning process of LLMs. Our agent automatically generates task-specific instructions for a wide set of tasks. The instructions are used to guide LLMs to reason better across these tasks to make high-quality predictions. Our method is zero-shot so no input-output examples are required to solve the task. Our results confirm the overall efficacy of our approach, leading to substantial improvements across various NLP tasks spanning generation, classification, and reasoning. Average score enhancements of 13.3%, 23.2%, and 17.0% are achieved over the standard zero-shot setting across 29 datasets for Vicuna-13b, Llama-2-70b-chat, and GPT-3.5 Turbo respectively. Our method wins on 20 of the 29 datasets used for evaluation. We believe zero-shot AgentInstruct’s style of human-understandable reasoning, along with its utilization of an autonomous agent with access to a wide range of dataset knowledge, can replace more traditional styles of zero or few-shot prompting as models become equipped with stronger reasoning capabilities. ETHICS STATEMENT We hereby acknowledge that all of the co-authors of this work are aware of the provided ICLR Code of Ethics and honor the code of conduct. Our method is built on LLMs, for which the risks and potential harms are discussed in Brown et al. (2020); OpenAI (2023); Touvron et al. (2023). A concern is LLMs’ tendency to generate non-factual responses with a high degree of confidence. Our approach is advantageous in this regard, as guiding the model to output step-by-step reasoning leading to its answer offers researchers the opportunity to analyze cases of erroneous outputs. Moreover, our method shows notable improvement on safety benchmarks such as CivilComments and TruthfulQA, suggesting that the use of instructions to ground the reasoning along a task-specific path can reduce the risk of harmful outputs. However, better performance on benchmarks does not guarantee safe outputs, which requires future research from the community. REPRODUCIBILITY STATEMENT The source code is available at https://anonymous.4open.science/r/AgentInstruct_ICLR2024. Our datasets are based on existing benchmarks (Liang et al., 2023; Kojima et al., 2022). We provide specific model details in Appendix A.2 and additional dataset information in Appendix A.4. REFERENCES Neel Alex, Eli Lifland, Lewis Tunstall, Abhishek Thakur, Pegah Maham, C. Riedel, Emmie Hine, Carolyn Ashurst, Paul Sedille, Alexis Carlier, Michael Noetel, and Andreas Stuhlmüller. RAFT: A Real-World Few-Shot Text Classification Benchmark. In NeurIPS, 2021. Jacob Andreas. Language Models as Agent Models. 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PJVUWpPnZC	Symbolic Regression is the task of distilling equations from data. The sentence gives the impression that SR is something else and that the authors are using SR to solve that something else. Please clarify.	REINFORCEMENT SYMBOLIC REGRESSION MACHINE Yilong Xu\textsuperscript{1}, Yang Liu\textsuperscript{2}, Hao Sun\textsuperscript{1,} \textsuperscript{1}Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China; \textsuperscript{2}School of Engineering Science, University of Chinese Academy of Sciences, Beijing, China; Emails: xuyilong88@ruc.edu.cn liuyang22@ucas.ac.cn haosun@ruc.edu.cn ABSTRACT In nature, the behavior of many complex systems can be described by parsimonious math equations. Symbolic Regression (SR) is defined as the task of automatically distilling equations from limited data. Keen efforts have been placed on tackling this issue and demonstrated success in SR. However, there still exist bottlenecks that current methods struggle to break, when the expressions we need to explore tend toward infinity and especially when the underlying math formula is intricate. To this end, we propose a novel Reinforcement Symbolic Regression Machine (RSRM) that masters the capability of uncovering complex math equations from only scarce data. The RSRM model is composed of three key modules: (1) a Monte Carlo tree search (MCTS) agent, designed for exploration, that explores optimal math expression trees consisting of pre-defined math operators and variables, (2) a Double Q-learning block, designed for exploitation, that helps reduce the feasible search space of MCTS via properly understanding the distribution of reward, and (3) a modulated sub-tree discovery block that heuristically learns and defines new math operators to improve representation ability of math expression trees. Binding of these modules yields the SOTA performance of RSRM in SR as demonstrated by multiple benchmark datasets. The RSRM shows clear superiority over several representative baseline models. 1 INTRODUCTION The pursuit of mathematical expressions through data represents a crucial undertaking in contemporary scientific research. The availability of quantitative mathematical expressions to depict natural relationships enhances human comprehension and yields more precise insights. Parsing solutions offer superior interpretability and generalization compared to numerical solutions generated by neural networks. Additionally, simple expressions exhibit computational efficiency advantages over the latter. As a result, these techniques have found applications across diverse fields, e.g., discovering fundamental physical laws \cite{Udrescu & Tegmark 2020, Liu & Tegmark 2021} or governing equations \cite{Schmidt & Lipson 2009, Chen et al. 2021, Sun et al. 2023}, modeling material constitutive relations \cite{Wang et al. 2019}, and TCP congestion control \cite{Sharan et al. 2022}, among many others. The process of fitting expressions in early years involves polynomial interpolation to derive an equation, followed by the appearance of the SINDy method \cite{Kaiser et al. 2018}, which utilizes sparse regression to identify appropriate mathematical expressions based on a predefined library of candidate terms. These methods e.g., \cite{Sun et al. 2021, Chen et al. 2021, Champion 2019}, effectively reduce the search space from an infinitely large set of possibilities to a limited fixed set of expressions, thereby narrowing down the search process. However, the applicability of this approach is limited, since the compositional structure of many equations cannot be predefined in advance. Therefore, there is a need for more comprehensive methods to search for expressions. The Equation Learner \cite{Martius & Lampert 2016, Sahoo et al. 2018} model was then introduced as a novel method in symbolic learning, which incorporates symbolic operators as activation functions. This modification enabled the neural network to generate more precise and interpretable functional relationships, allowing for the discovery of intricate math expressions. However, given the compact structure of EQL, optimizing the sparse network to distill parsimonious equations is a key challenge. Another approach involves generating optimal expression trees \cite{Hopcroft et al. 2006}, where internal nodes correspond to operators and each leaf node represents a constant or variable. By recursively Corresponding author computing the expressions of the sub-trees, these expression trees can be transformed into math expressions. Initially, genetic programming (GP) (Schmidt & Lipson [2009], Augusto & Barbosa [2000], Gustafson et al. [2005]) was employed to address these problems. Although GP showed promise, its sensitivity to parameter settings leads to instability. Deep learning methods emerged then to tackle the problem. SymbolicGPT (Valipour et al. [2021]) utilizes a generative model like GPT to create expression trees, while AIFeynman (Udrescu & Tegmark [2020]) uses neural networks to analyze the relationships and dependencies between variables and search for relevant expressions. Despite its ad-hoc characteristic, the AIFeynman (Udrescu et al. [2020]) method was further improved, offering faster and more precise expression search capabilities. Additionally, reinforcement learning (RL) (Sun et al. [2023]) has been employed, which utilizes the Monte Carlo tree search method to explore and discover expressions, along with a module-transplant module that generates new expressions based on existing ones. Deep (RL) methods, e.g., DSR (Petersen et al. [2019]), utilize recurrent neural networks to learn expression features and generate probabilities. A policy gradient search algorithm samples the probabilities to generate a batch of expressions, which are subsequently evaluated for performance. Combining DSR and GP leads to a new model called NGGP (Mundhenk et al. [2021a]), which achieves better performance. Then uDSR (Landauela et al. [2022]), a comprehensive framework that combines DSR, AIFeynman, LSPT (Large-scale pre-training), GP, and LM (Linear models) emerged to enhance the efficiency and accuracy of symbolic regression. Pre-trained generative models (Holt et al. [2022]) and end-to-end transformer modules (Kamienny et al. [2022], Li et al. [2022]) also achieved satisfactory expression search results. Nevertheless, the existing methods still struggle with generating lengthy and complex equations, and are faced with issues related to overfitting, e.g., poor generalizability. To overcome these challenges, we propose a model named Reinforcement Symbolic Regression Machine (RSRM) that masters the capability of uncovering complex math equations from only scarce data, composed of an RL-search agent, a GP-based expression tuning element, and a modulated sub-tree discovery (MSDB) block. The RL-search agent is designed based on the synergy between Monte Carlo tree search (MCTS) (Coulom [2006]) and double Q-learning (Hasselt [2010]) for enhanced exploration and exploitation. The GP learner is employed to fine-tune the generated expression trees (e.g., see the demonstration in Mundhenk et al. [2021a]), while the (MSDB) block heuristically learns and defines new math operators to improve the representation ability of math expression trees. We would like to emphasize that MSDB addresses a crucial observation that models often struggle to generate complete expressions but excel in capturing certain components. For instance, NGGP (Mundhenk et al. [2021a]) may discover an expression like $x^4 - x^3 + \cos(y) + x - 1$, while the ground truth is $x^4 - x^3 - 0.5y^2 + x$. Notably, it successfully recovers the simplified expression $x - 0.5y^2$ with the same distribution. To this end, MSDB offers a new alternative to simplify expressions by subtracting specific components in the context of a sub-tree, as exemplified by the subtraction of $x^4 - x^3$ in the aforementioned case. Such an MSDB module takes the divide-and-conquer concept and could significantly improve the overall search performance of the RSRM model. The aforementioned aspects form the main contributions of this paper: Our proposed RSRM model offers a novel solution to the search for mathematical expressions. By incorporating double Q-learning into MCTS, we effectively balance exploration and exploitation of SR tasks. The proposed (MSDB) block can handle equations with symmetry (reducing the complexity), and assist in dealing with long equations by identifying common patterns and defining new math operators on the fly. As a result, the RSRM model demonstrates clear superiority over several baseline models, which surpasses that of the baseline models in terms of accuracy and generalization ability. 2 BACKGROUND Genetic Programming: Genetic programming (Stephens [2016], Koza [1994], Schmidt & Lipson [2009]) is employed to iteratively improve expression trees in order to approximate the optimal expression tree. The mutation step in GP enables random mutations in the expression tree, while genetic recombination allows for the exchange of sub-trees between expression trees, leading to the creation of new expression trees based on the knowledge acquired from previous generations. This “genetic evolution” process progressively yields highly favorable outcomes after a few generations. Double Q-Learning: Double Q-learning (Hasselt [2010]) is a reinforcement learning algorithm designed to overcome the overestimation bias issue in traditional Q-learning. The key idea behind Double Q-learning is to use two sets of Q-values to independently estimate the value of each action in a given state. By using two separate Q-functions, Double Q-learning can mitigate the overestimation bias of traditional Q-learning and provide more accurate value estimations, leading to better policy learning and performance in various reinforcement learning tasks. Monte Carlo Tree Search: MCTS (Coulom, 2006) is a decision-making search algorithm that constructs a search tree representing possible game states and associated values. It employs stochastic simulations to explore the tree and determine the value of each node. This algorithm gained prominence via its adoption by the AlphaZero team (Silver et al., 2017). MCTS consists of four steps in each iteration: (1) selection, (2) expansion, (3) simulation, and (4) backpropagation. During selection, the best child node is chosen based on certain criteria. If an expandable node lacks children, it is extended by adding available children. The simulation step involves simulating the current state before selecting the next node, often using the Upper Confidence Bound for Trees (UCT) algorithm to calculate the selection probabilities, defined as $ UCT(v') = Q(v') + c \sqrt{\ln(N(v))/N(v')} $. Here, $ Q(v') $ means the average reward of the child node, $ N(v) $ and $ N(v') $ represents the number of visits to the current node and its child node, respectively, and $ c $ typically represents the exploration-exploitation tradeoff parameter. The first part of the equation makes the nodes with high reward visit more often, and the second part ensures that the nodes with fewer visits have a higher probability of being selected. Finally, in the backpropagation step, the reward function evaluates child nodes, and their values are used to update the values of parent nodes in the tree. The theoretical analysis (e.g., convergence, guarantees) of the UCT-based MCTS algorithm can be found in Shah et al. (2022). 3 Method The RSRM model consists of a three-step symbolic learning process: RL-based expression search, GP tuning, and MSDB. With these steps, our model effectively learns and represents the relationship present in the data, facilitating accurate and interpretable modeling. The schematic representation of RSRM is depicted in Figure 1. The full settings of our model are in Appendix A. The RL search consists of a double Q-learning empowered MCTS agent. Here, MCTS is employed for exploration (global search) that aids in generating unexplored expressions, while double Q-learning enables exploitation that captures the local distribution of equations. Additionally, we adopt a method that involves visiting each child node a specific number of times before activating double Q-learning. This approach aims to avoid excessive reliance on historical information, mitigating the risk of overfitting and promoting a more robust learning process. To address the challenge of lengthy and hard equations, we introduce an interpolation method (e.g., data pre-processing) to identify whether the equation exhibits symmetry prior to each search, followed by a modulated sub-tree discovery block (MSDB). If symmetry is present, we pre-process the equation accordingly to simplify the subsequent search process. This approach effectively reduces the difficulty associated with specific equations. The MSDB examines whether the few expressions that perform well adhere to a specific form. This divide-and-conquer algorithm enables a step-by-step search for equations, facilitating the generation of long expressions. 3.1 Expression Tree The objective of SR can be transformed into the generation of an optimal expression tree (Hopcroft et al., 2006), which represents a mathematical expression. The expression tree consists of internal Figure 2: Schematic of the proposed RL search. MCTS selects functions based on the maximum reward, expands them using the results of double Q-learning, simulates node selection through the UCT function, randomly fills the current tree, and provides rewards to double Q-learning to train. Once the generation is complete, the rewards are back-propagated to the parent node. nodes that correspond to operators (e.g., +, −, ×, ÷, log, exp, sin, cos) and leaf nodes that correspond to constants (e.g., 1, 2) or variables (e.g., x). By recursively computing the expressions of the subtrees, the expression tree can be transformed into mathematical expressions. The process of generating an expression tree follows a recursive method where operators are added until no more can be added. This approach simplifies the task of creating expressions as it focuses on constructing the expression tree, which can be easily generated using recursive techniques. In contrast to previous methods, we employ a hierarchical traversal strategy for generating expression trees. This is motivated by the Monte Carlo tree search algorithm, where conducting more searches on vertices that are filled earlier is deemed more beneficial. In the context of constructing expression trees, this implies that higher-level nodes in the tree carry greater significance. Consequently, we use a hierarchical construction method to build the expression tree layer by layer, similar to the hierarchical traversal of trees. 3.2 Reinforcement Learning Guided Search The search step relies on the double Q-learning and MCTS algorithms, which are shown in Figure 2. The specific algorithm is shown in Algorithm 1. Reward function: The reward function used in our approach is based on the root mean square error (RMSE) and is designed to evaluate the fit of the generated equations to the measured data. It promotes concise and accurate expressions by assigning higher rewards to shorter and more precise functions. Inspired by the SPL approach [Sun et al., 2023], the reward function is computed by: \[ R = \frac{\eta^l}{1 + \sqrt{\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}}, \] where $ \eta $ is a discount factor promoting concise trees, and $ l $ is the number of nodes in the expression tree. $ y_i $ and $ \hat{y}_i $ are the true and the predicted values generated by the MSDB with the output of Reinforcement Learning Search of the $ i $th data point, respectively. Using this reward function, our approach encourages the discovery of equations that minimize the RMSE and favors shorter and more concise expressions, leading to higher reward values for functions that provide better fits to the data. Algorithm 1 Expression generation by RSRM Input: dataset $S_{data}$, expression form $\mathcal{F}$ Parameters: discount rate $\eta$, UCT const $c$, minimum selected times $n_0$ Outputs: best expression Initiate $S$ as top of MCTS Selection: $a \leftarrow$ children of $S$ with maxium $R$ ▷ Greedy selection $S$ take action $a$ Simulation: $S' \leftarrow S$ repeat if children of $S$ is empty then Expand $S'$ end if if $\exists x \in$ children of $S' \rightarrow (N(x) < n_0)$ then $a' \leftarrow x$ ▷ Select child with visit times $< n_0$ else $a' \leftarrow$ randomly choose child of $S'$ by UCT ▷ Select through UCT function end if $S'$ take action $a'$, $S'' \leftarrow S'$, Fill up randomly $S''$ double Q-learning $\leftarrow S'$, $a'$, $R$ of $S''$ ▷ train double Q-learning by simulated reward until $S'$ is full Expansion children of $S \rightarrow$ double Q-learning $\rightarrow p_{children}$ ▷ estimate initial possibility of each child Back-propagate Back-propagate $R$ of $S'$ based on $\mathcal{F}$ Greedy selection: Our method employs greedy selection, similar to Sun et al. (2023). Instead of selecting the token with the highest UCT score, we choose the token that currently yields the best reward (Eq. 1). This ensures the selection of tokens leading to expressions resembling the current best one, potentially resulting in improved expressions, yet, increasing the possibility of overfitting. Note that UCT is employed during the MCTS simulation while the greedy selection of the maximum reward is applied to choose the optimal expression tree. Simulated reward: At each token generation, the entire expression tree is randomly completed based on the current tree. The reward is then computed using the reward function and fed back to double Q-learning for training. This approach avoids excessive rounds of learning at the top node and filters out irrelevant nodes initially. Parameter optimization: After an expression tree is built, we need to fill the parameter (i.e., equation coefficients) placeholders in it. We treat each placeholder as an unknown variable, which is optimized to maximize the reward. The BFGS (Roger Fletcher & Sons, 2013) algorithm, available in the scipy (Virtanen et al., 2020) module in Python, is used for optimization. In contrast to the approach in DSR (Petersen et al., 2019), we find that Gaussian random numbers with a unit mean and variance provide more effective initial values for optimization (see further information in Appendix Section C.G where we test the performance of the model with different initial values). 3.3 Modulated Sub-tree Discovery We incorporate three specific sub-tree expression forms to enhance the exploration and analysis of equations, where $A$ represents a fixed form and $f(x)$ a learnable part, explained as follows: - $A + f(x)$: This search form focuses on identifying expressions of the form like $e^x - x$ and $e^x + x$. By recognizing this pattern, we can effectively explore and analyze equations that follow the structure of $e^x + f(x)$. - $A \times f(x)$: In this search form, we obtain good expressions such as $1.57e^x$ and $1.56e^x + x$, aiming to detect equations of the form $e^x \times f(x)$. - $A^{f(x)}$: The search form $A^{f(x)}$ is designed to recognize equations like $(e^x)^{2.5}$ and $(e^x)^e$, indicating the presence of expressions in the form $(e^x)^{f(x)}$. Our approach involves the establishment of these forms based on the initial token of the expression tree, because the root of an expression tree serves as a focal point, indicating the primary operation or function in the expression. Thus, we separate the sub-tree forms based on it. Specifically, if the first token corresponds to addition (+) or subtraction (−), the method proceeds to learn the generation of the left and right sides of the respective operators. Similarly, for tokens such as multiplication (×), division (÷), or exponentiation (^), a similar procedure is followed. In the case of unary expressions, such as trigonometric functions (sin and cos), the MCTS and GP models effortlessly derive the complete expression. Therefore, while our method involves a degree of empirical design in identifying the sub-tree expression forms, it possesses a universal nature. The complete form-discovery algorithm, which outlines the procedure for selecting and generating the search form among the three options, is provided in Algorithm 2 and Appendix Figure S1. Algorithm 2 Search for the form of the expression through the generated expressions Input: best expression set $S_{\text{best}}$ Parameters: selection ratio $k_s$, expression percentage ratio $k_p$, maximum select number $N$ Output: the form of the expression $F$ $l \leftarrow$ length of $(S_{\text{best}})$ Sort $S_{\text{best}}$ by $\mathcal{R}$ decent for $i$ in $1, 2...l$ do if $i \leq N$ and $\mathcal{R}(S_{\text{best}}[i]) > k \times \mathcal{R}_{\text{max}}$ then $\mathcal{D} = \mathcal{D} + \text{Split}(S_{\text{best}}[i])$ ▶ If number of $G$ exceeds or $\mathcal{R}$ is low, break out end if end for $G_0 \leftarrow \mathcal{D}$ with maximum number of occurrences. if $\exists C \notin Z \rightarrow G_0 = A^C$ then $F = A^{f(x)}$ ▶ The form is $A^{f(x)}$, $Z$ means integer set. else if $\exists C \notin Z \rightarrow G_0 = A \times C$ then $F = A \times f(x)$ ▶ The form is $A \times f(x)$ else $F = f(x)$ for $G$ in $\mathcal{D}$ do if Occurrences of $G \geq l \times k_p$ then $F = F + G$ ▶ Add $G$ to $A$ end if end for end if Splitting by Addition: In this step, we convert the formula, which is represented as a token set, into a string using a library like sympy [Meurer et al., 2017]. Then we expand the expression into a sum of simpler expressions. Next, we split the expanded expression into multiple simple expressions using sum or difference notation. In this way, we convert $[+, \times, -, x, y, z, \log, x]$ to $x \times y + z - \log(x)$, and then transforms it to $xy, z, -\log(x)$. Once the expression is refined into its desired form, the subsequent search becomes more manageable. For instance, in the case of aiming to derive $\exp(x^2) + x^4 + x^3 + 0.5 \log(x)$, we can break down the search. Initially, we generate $\exp(x^2) + ...$ to identify the form $\exp(x^2) + f(x)$, and then extend this to $\exp(x^2) + x^4 + x^3 + f(x)$, simplifying the process of obtaining $\exp(x^2) + x^4 + x^3 + 0.5 \log(x)$. Inspired by the approach proposed by Udrescu & Tegmark [2020], we introduce a data pre-processing module to determine the potential parity of the underlying equation. The cubic splines [Catmull & Rom] [1974] are applied for equation fitting, generating a function. Subsequently, this function is used to compute the relationship between $y(-x)$ and $y(x)$, enabling the determination of whether $y(x)$ is an odd, even, or neither function, where $y(x)$ is the relation of $x$ and $y$ in given data. When the error (RMSE) between $y(-x)$ and $y(x)$ remains below constant $E_{\text{sym}}$, the function is considered even with respect to $x$. Negative values of the independent variables are transformed to their absolute values, while retaining the dependent variable values. Further exploration is conducted using the form of $\hat{y} = (g(x) + g(-x))/2$ to make it discover specific forms. Similarly, if the error between $y(-x)$ and $y(x)$ is within the limit constant $E_{\text{sym}}$, the function is classified as odd relative to $x$. Negative values of the independent variables are converted to absolute values, and the dependent variable values are inverted. The search continues employing the form of $\hat{y} = (g(x) - g(-x))/2$ to make it odd in discovering specific forms. Once the expression is refined to its parity form, the difficulty of searching for the expression is reduced. If we want to get $\cosh(x)$, we only need to generate $\exp(x)$ after parity determination. Such a partitioning strategy has been used in the past, e.g., Petersen et al. [2019] using sub-trees as new tokens, Sun et al. [2023] using transplanted sub-trees, Udrescu & Tegmark [2020] using problem Table 1: Recover rate (%) of several difficult equations in symbolic regression: trigonometric functions and sum of multiple power functions with parameter 1/2 in Nguyen; power functions and trigonometric functions in Nguyenc; trigonometric functions and hyperbolic function and functions with weird power in Livermore; rational functions in R and R0 (where x = 0, y = 0 add to dataset). \| BenchMark \| Equation \| Ours \| SPL \| uDSR \| NGGP \| DSR \| GP \| \|-----------------\|--------------------------------------------------------------------------\|------\|-----\|------\|------\|-----\|----\| \| Nguyen-5 \| sin(x_1^2)cos(x_1) - 1 \| 100 \| 95 \| 55 \| 80 \| 72 \| 12 \| \| Nguyen-12 \| x_1^4 - x_1^3 - 0.5x_2^2 + x_2 \| 100 \| 28 \| 30 \| 21 \| 0 \| 0 \| \| Nguyen-2c \| 0.48x_1^4 + 3.39x_1^3 + 2.12x_1^2 + 1.78x_1 \| 100 \| 94 \| 100 \| 98 \| 90 \| 0 \| \| Nguyen-9c \| sin(1.5x_1) + sin(0.5x_2) \| 100 \| 96 \| 0 \| 90 \| 65 \| 0 \| \| Livermore-3 \| sin(x_1^3)cos(x_1^2) - 1 \| 55 \| 15 \| 0 \| 2 \| 0 \| 0 \| \| Livermore-7 \| sinh(x_1) \| 100 \| 18 \| 0 \| 24 \| 3 \| 0 \| \| Livermore-16 \| x_1^2/5 \| 100 \| 40 \| 60 \| 26 \| 10 \| 5 \| \| Livermore-18 \| sin(x_1^2)cos(x_1) - 5 \| 100 \| 80 \| 59 \| 33 \| 0 \| 0 \| \| AlFeynman-9 \| x_1 + x_2 + 2\sqrt{x_1}x_2\cos(x_3) \| 67 \| 0 \| 8 \| 7 \| 0 \| 0 \| \| AlFeynman-10 \| \frac{1}{2}x_1(x_2 + x_3 + x_4) \| 15 \| 0 \| 0 \| 0 \| 0 \| 0 \| \| R-10 \| (x_1 + 1)^3/(x_1^2 - x_1 + 1) \| 49 \| 0 \| 17 \| 2 \| 0 \| 0 \| \| R-20 \| (x_1^3 - 3x_1^2 + 1)/(x_1^2 + 1) \| 89 \| 0 \| 0 \| 0 \| 0 \| 0 \| \| R-30 \| (x_1^3 + x_1^2)/(x_1^4 + x_1^3 + x_1^2 + x_1 + 1) \| 91 \| 0 \| 0 \| 4 \| 0 \| 0 \| Our main innovation is the use of partial sub-trees separated according to the plus sign as part of the new expression. Also, we conduct a parity determination performance test to compare the efficiency and effectiveness of form discovery by AlFeynman and our method. The experiment setting and results are given in Appendix C. It shows that our method (based on cubic splines) outperforms AlFeynman (based on MLP) in terms of higher accuracy and smaller data requirements. 4 RESULTS We test the performance of our method on multiple different datasets and compare it with the following baseline models in symbolic learning: SPL (Sun et al., 2023), DSR (Petersen et al., 2019), NGGP (Mundhenk et al., 2021a), uDSR (Landauela et al., 2022), DGSR (Holt et al., 2022), gplearn (Stephens, 2016), and AFP-FE (Schmidt & Lipson, 2010). The description of each baseline along with parameter setting is found in Appendix B. 4.1 BASIC BENCHMARKS To evaluate the efficiency of our model, we first utilize four basic benchmark datasets (see Appendix C for details): Nguyen (Uy et al., 2011), Nguyenc (McDermott et al., 2012), R (Mundhenk et al., 2021b), Livermore (Mundhenk et al., 2021b), and AlFeynman (Udrescu & Tegmark, 2020). Note that parameter optimization (e.g., calibration of the equation coefficients) is prohibited in this experiment except the Nyugenc dataset. Figure 3: Recover rates of benchmark datasets. a, Basic benchmarks (detailed results shown in Appendix C). b, SRbench dataset (detailed results found in Appendix D and Appendix Figure S3), where the symbol # denotes the presence of noise with a mean of $10^{-3}$ added to the target values and * represents missing data in the literature. We employ the recovery rate as the evaluation metric, which measures the number of times the correct expression is recovered across multiple independent repetitions of a test. Note that this metric ensures that the model’s output exactly matches the target expression. We summarize the comparison of recovery rates for several difficult expressions on the five benchmark datasets listed in Table 1. The results demonstrate that our model performs well on these complex expressions. Furthermore, we compared the mean recovery rates of all equations on each benchmark (see Figure 3a and Appendix C). Our method outperforms other approaches, achieving the highest recovery rates for all benchmark expressions. We also conducted an experiment on the trade-off between accuracy and the number of evaluations in the Nyugen Benchmark. Details of this experiment are given in Appendix Section C.7. 4.2 SRBench Dataset We further tested our model’s ability to learn more complex equations with noisy training data using the SRbench dataset [La Cava et al., 2021], where parameter optimization is allowed. This dataset comprises 252 datasets sourced from [Romano et al., 2021]. We specifically concentrated on 131 of these datasets equipped with ground truth equations, which were drawn from two primary sources, namely, AlFeynman [Uldrescu & Tegmark, 2020] and Strogatz [Strogatz, 2014] datasets. Our evaluation encompassed a spectrum of baseline models from SRBench, as well as more contemporary approaches like uDSR [Landajuela et al., 2022] and DGSR [Holt et al., 2022]. The recovery rate results of each method for different testing datasets (e.g., all datasets, only Feynman, only Strogatz) and noise effect can be found in Figure 3b. Here, the results for the AlFeynman dataset in Figure 3b are inconsistent with Figure 3a, since parameter optimization is prohibited in a but engraved in b. Notably, our MSDB module demonstrated proficiency in handling intricate equations, such as the challenging example $-(32x_1^4x_2^2x_3^2(x_3 + x_4))/(5x_2^5x_5^5)$, which can be effectively discovered through the form $(Cx_4^4)/(x_2^2x_5^5) \times f(x)$. Additionally, numerous equations, like $-10/3x_1^3 - 10/3x_1 + 10x_2$, were successfully identified by aggregating multiple smaller equations, a feat achievable through the form $Cx_3^3 + f(x)$. Consequently, our model exhibited proficiency in uncovering a wide range of equations. The comparative analysis in Figure 3b clearly illustrates the superior performance of our approach in comparison with other baseline models. We further tested our model by discovering a surrogate formula to approximate the cumulative density function (CDF) of a normal distribution, e.g., $F(x; \mu, \sigma) = \int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}\sigma} \exp\left[-(t - \mu)^2/(2\sigma^2)\right] dt$, where $\mu$ and $\sigma$ denote the mean and standard deviation. Since this equation lacks an explicit elementary expression, finding a parsimonious equation for approximation based on a small training set is intractable. Nevertheless, our model shows a great capability of approximating the CFD with generalizability (see Appendix Section F for more details). 4.3 Free-falling Balls Dataset We conducted an experimental evaluation on the free-falling balls dataset to assess the parametric learning capability of our model. The dataset consisted of experimental data of balls dropped from a bridge, as described in [de Silva et al., 2020]. The dataset comprised 20-30 observations of a ball throw height within the first 2 seconds, aiming to learn the equation governing the ball’s drop and predict the height between 2 and 3 seconds. Since an exact solution for this dataset is not available, we employed the mean squared error (MSE) as our evaluation metric. \| BenchMark \| Ours \| Ours* \| SPL \| M-A \| M-B \| M-C \| \|-----------------\|--------\|--------\|--------\|--------\|--------\|--------\| \| baseball \| 0.053 \| 0.068 \| 0.300 \| 2.798 \| 94.589 \| 3.507 \| \| blue basketball \| 0.008 \| 0.027 \| 0.457 \| 0.513 \| 69.209 \| 2.227 \| \| bowling ball \| 0.014 \| 0.034 \| 0.003 \| 0.33 \| 87.02 \| 3.167 \| \| golf ball \| 0.006 \| 0.041 \| 0.009 \| 0.214 \| 86.093 \| 1.684 \| \| green basketball\| 0.094 \| 0.045 \| 0.088 \| 0.1 \| 85.435 \| 1.604 \| \| tennis ball \| 0.284 \| 0.068 \| 0.091 \| 0.246 \| 72.278 \| 0.161 \| \| volleyball \| 0.033 \| 0.025 \| 0.111 \| 0.574 \| 80.965 \| 0.76 \| \| whiffle ball 1 \| 0.038 \| 0.660 \| 1.58 \| 1.619 \| 65.426 \| 0.21 \| \| whiffle ball 2 \| 0.041 \| 0.068 \| 0.099 \| 0.628 \| 58.533 \| 0.966 \| \| yellow whiffle ball \| 1.277 \| 1.080 \| 0.428 \| 17.341 \| 44.984 \| 2.57 \| \| orange whiffle ball \| 0.031 \| 0.368 \| 0.745 \| 0.379 \| 36.765 \| 3.257 \| Average MSE of free-falling balls dataset. Details of the equations generated by different models are shown in Appendix E. We consider two sets of RSRM models, the standard one and the one named RSRM* (denoted by Ours* in Table 2) that fixes the expression form $c_4x^3 + c_3x^2 + x_2x + c_1 + f(x)$. We compared these models with the baseline method SPL (Sun et al., 2023), since other models tend to have large generalization errors due to the limited data points (20-30 per training set) in the falling balls benchmark given the fact that the exact solution is unknown. Three physics models derived from mathematical principles were selected as baseline models for this experiment, and the unknown constant coefficient values were estimated using Powell (1964). The equations of the baseline models are presented as follows. M-A: $h(t) = c_1t^3 + c_2t^2 + c_3t + c_4$, M-B: $h(t) = c_1 \exp(c_2t) + c_3t + c_4$, and M-C: $h(t) = c_1 \log(\cosh(c_2t)) + c_3$. The results (see Table 2) show that in most cases, the RSRM model performs better than SPL. The RSRM model can successfully find the equation of motion for uniformly accelerated linear motion ($c_1x^2 + c_2x + c_3 + f(x)$) and search for additional terms to minimize the training error. This leads to improved results compared to SPL. However, there are cases where RSRM makes mistakes, such as obtaining expressions in the form of $c_1 \cos(x)^2 + c_2 + f(x)$ when searching for the yellow whiffle ball. This increases the generalization error and reduces the overall effectiveness compared to SPL. Overall, RSRM outperforms SPL in physics equation discovery, demonstrating its effectiveness in solving parametric learning tasks on the free-falling balls dataset. ## 5 Ablation Study We conducted ablation studies on the Livermore dataset, namely, ablations of the double Q-learning (M-A), the MCTS algorithm (M-B), the MSDB (M-C), the pre-processing step (M-D), and GP (M-E), respectively. We list some of expressions affected by the performance of the model in Table 3. \| Equation \| Ours \| M-A \| M-B \| M-C \| M-D \| M-E \| \|---------------------------\|------\|-----\|-----\|-----\|-----\|-----\| \| $\sin(x_1^2)\cos(x_1) - 2$\| 100 \| 100 \| 100 \| 6 \| 100 \| 100 \| \| $\sin(x_1^3)\cos(x_2^2) - 1$\| 55 \| 20 \| 0 \| 0 \| 55 \| 0 \| \| $\sinh(x_1)$ \| 100 \| 100 \| 100 \| 100 \| 10 \| 100 \| \| $\sum_{k=1}^{9} x_k^k$ \| 100 \| 83 \| 100 \| 88 \| 100 \| 67 \| \| $x_1^{1/3}$ \| 100 \| 100 \| 100 \| 67 \| 100 \| 100 \| \| $x_1^{2/5}$ \| 100 \| 100 \| 100 \| 12 \| 100 \| 33 \| Average: 97.95 94.36 93.64 80.45 89.45 84.95 When the double Q-learning module is removed, Model A with only MCTS experiences a decrease in knowledge from previous iterations. This results in reduced search efficiency but increased diversity. As a result, we observe a decrease in performance for equations like $\sum_{k=1}^{9} x_k^k$, while equations like $\sin(x_1^3)\cos(x_2^2) - 1$ show improved performance. On the other hand, when the MCTS module is removed, Model B with pure double Q-learning tends to overfit more quickly. Consequently, it struggles to produce the most challenging equations, such as $\sin(x_1^3)\cos(x_2^2) - 1$. Similarly, the absence of the expression form search module in Model C limits its ability to discover complex expressions with simple forms, such as $x_1^{1/3}$ and $\sin(x_1^3)\cos(x_2^2) - 1$. Lastly, Model D, without the preprocessing module, suffers a significant reduction in its ability to search for odd and even functions like $\sinh(x_1)$. The removal of the genetic algorithm (M-E) resulted in decreased efficiency across all expression searches. While simpler expressions such as $x_1^{1/3}$ still performed adequately, the performance for complex expressions notably deteriorated. These observations highlight the importance of all the modules in RSRM. Each module contributes to the overall performance and enables the model to tackle different types of equations effectively. ## 6 Conclusion We have proposed a novel model RSRM that integrates RL techniques, GP, and a modulated sub-tree discovery block to improve the search process for mathematical expressions. Our model outperforms the state-of-the-art baselines in the context of accurately recovering the exact equations for various datasets, and demonstrates superior generalization capabilities. However, one limitation of our current model is the lack of flexibility in setting the expression form as it currently encompasses only three fixed types, which restricts its adaptability to different problem domains. We anticipate future advancements in more flexible methods, e.g., potentially incorporating neural networks to generate slots for SR, utilizing other Q-learning techniques such as prioritized experience replay (Schaul et al., 2015) to enhance the exploitation, etc. Furthermore, we believe that our approach has the potential to be extended to other domains, such as reinforcement learning control tasks. By applying our method to diverse areas, we aim to enhance the performance and applicability of SR techniques. ACKNOWLEDGMENTS The work is supported by the National Natural Science Foundation of China (No. 92270118), which is greatly acknowledged. Code and models of Reinforcement Symbolic Regression Machine (RSRM) are available at https://github.com/intell-sci-comput/RSRM. REFERENCES Douglas Adriano Augusto and Helio JC Barbosa. Symbolic regression via genetic programming. In Proceedings. Vol. 1. Sixth Brazilian Symposium on Neural Networks, pp. 173–178. IEEE, 2000. 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wHLDHRkmEu	Does the side network (Global Shortcut Tuning Network) not operate in parallel with the vision/language encoder? And is it required to cache each $F_v^i$ for the side network? If so, what is the extra time and memory cost for this approximately?	BarLeRia: An Efficient Tuning Framework for Referring Image Segmentation Yaoming Wang\textsuperscript{1,\dagger} Jin Li\textsuperscript{1,\dagger} Xiaopeng Zhang\textsuperscript{2}\textsuperscript{\S} Bowen Shi\textsuperscript{1} Chenglin Li\textsuperscript{1} Wenrui Dai\textsuperscript{1}\textsuperscript{\S} Hongkai Xiong\textsuperscript{1} Qi Tian\textsuperscript{2} \textsuperscript{1}Shanghai Jiao Tong University \textsuperscript{2}Huawei Cloud \{wang.yaoming, deserve1j, lcl1985, daiwenrui, xionghongkai\}@sjtu.edu.cn; zxphistory@gmail.com, tian.qi1@huawei.com Abstract Pre-training followed by full fine-tuning has gradually been substituted by Parameter-Efficient Tuning (PET) in the field of computer vision. PET has gained popularity, especially in the context of large-scale models, due to its ability to reduce transfer learning costs and conserve hardware resources. However, existing PET approaches primarily focus on recognition tasks and typically support uni-modal optimization, while neglecting dense prediction tasks and vision language interactions. To address this limitation, we propose a novel PET framework called Bi-directional Intertwined Vision Language Efficient Tuning for Referring Image Segmentation (BarLeRia), which leverages bi-directional intertwined vision language adapters to fully exploit the frozen pre-trained models’ potential in cross-modal dense prediction tasks. In BarLeRia, two different tuning modules are employed for efficient attention, one for global, and the other for local, along with an intertwined vision language tuning module for efficient modal fusion. Extensive experiments conducted on RIS benchmarks demonstrate the superiority of BarLeRia over prior PET methods with a significant margin, i.e., achieving an average improvement of 5.6%. Remarkably, without requiring additional training datasets, BarLeRia even surpasses SOTA full fine-tuning approaches. The code is available at https://github.com/NastrondAd/BarLeRia. 1 Introduction In recent years, large-scale models have made significant contributions to advancements in NLP and CV. However, the cost associated with full fine-tuning of large models has become prohibitively expensive. To address this challenge, Parameter-Efficient Tuning (PET) approaches have emerged as a prevalent paradigm (Houlsby et al., 2019; Jie & Deng, 2022; Jia et al., 2022; Wang et al., 2023). By freezing a majority of the pre-trained model and fine-tuning only a small subset of parameters, PET approaches offer high efficiency while maintaining performance comparable to full fine-tuning, and are increasingly favored for language dialogue (Karimi Mahabadi et al., 2021; Sung et al., 2021) as well as visual recognition tasks (Chen et al., 2022b; Jia et al., 2022). Despite these advancements, limited research has explored the effectiveness of PET pipelines for adapting to dense prediction tasks (Ding et al., 2022; Qian et al., 2023) or facilitating cross-modal fusion. This paper investigates the generalization ability of Parameter-Efficient Tuning (PET) and examines its affordability for a challenging cross-model dense prediction task Referring Image Segmentation (RIS). RIS is a fundamental segmentation task designed to segment target objects from input images based on given text descriptions (Hu et al., 2016). Different from vanilla segmentation tasks, RIS needs to extract not only spatial and semantic information from images, but also key semantics from textual descriptions, and merge them in order to get the correct segmentation results. Previous studies have approached this task by either concatenating textual embeddings with visual features and incorporating vision-language attention mechanisms to facilitate interactions (Yu... Figure 1: Comparison between BarLeRia and state-of-the-art PET RIS method ETRIS. We perform experiments using two different referring expressions: detailed or abstracted. In the first row, the expression is detailed and two methods can locate the object given sufficient knowledge, though BarLeRia outperforms ETRIS a lot. In the second row, only brief expressions are provided, ETRIS locates wrong contours while BarLeRia still segments the target objects well. Best viewed in color. et al., 2018] [Li et al., 2018] [Chen et al., 2019], or by pursuing vision-language alignment using unimodal pre-trained models supplemented with additional training [Liu et al., 2023] [Yan et al., 2023]. More recently, leveraging the advancements in vision-language pre-training [Radford et al., 2021], Wang et al. (2022) propose to transfer multi-modal knowledge from CLIP through text-to-pixel contrastive learning, leading to remarkable performance gains. However, these approaches rely on computationally extensive full fine-tuning, which raises concerns about scalability and affordability. Few works explore integrating PET into RIS task, a pioneering work [Xu et al., 2023] introduces a vision-language bridge that combines vision inductive biases and language information, and achieves comparable performance to full fine-tuning. However, this approach primarily focuses on alignment between the vision and language modalities, while overlooking the core aspect of PET, namely, adapting the biased feature from pre-trained models [Jia et al., 2022] [Wang et al., 2023]. Besides, local modal fusion is adopted in the proposed bridge network as well as the pre-trained vision language models and the segmentation head [Wang et al., 2022]. Consequently, all components of the model repetitively fuse local visual features with textual embeddings without incorporating a global prior from the text input to regularize the visual features, which leads to off-target visual information interference and sub-optimal performance. Considering the above two issues when incorporating PET into RIS task. Firstly, we propose a novel technique to address the feature adaption problem. The highlight is an intertwined vision language efficient tuning framework for better modal fusion along with feature adaption as a basic design. For both visual and textual branches, we fuse the visual and textual input in front of each frozen layer and adapt each layer’s shortcut feature distribution via normalizing flow [Wang et al., 2023]. In this way, we keep the backbone frozen, employ modal fusion via the original self-attention mechanism, and are able to adapt the biased features for segmentation tasks. Second, in order to address the global regularization issue, we extract a global prior from the text input to regulate the vision features. This regularization is achieved with a limited number of parameters in an end-to-end manner. Our proposed method consists of a bi-directional efficient tuning framework, which comprises a global prior module and a global tuning network. The global prior module leverages the cosine similarity between visual features and textual embeddings to enforce regularization. Moreover, to ensure that the global prior regularization does not conflict with the local intertwined vision language tuning, we introduce global shortcut tuning modules that are detached from the pre-trained backbone. By doing so, we establish a parallel shortcut tuning network alongside the backbone. Similarly, we extend the intertwined vision language tuning to the shortcut tuning network to facilitate better fusion of the models. Incorporating the proposed intertwined vision language efficient tuning and the bi-directional efficient tuning modules, we produce a novel PET framework, namely Bi-directional Intertwined Vi- Vision Language Efficient Tuning for Referring Image Segmentation (BarLeRIa), to fully exploit the potential of frozen pre-trained vision-language models. BarLeRIa exhibits remarkable performance improvement with only 0.4% to 2.5% tuning parameters compared with the backbone when utilizing CLIP ViT-B as the pre-trained model. Compared to the state-of-the-art PET approach ETRIS [Xu et al., 2023], BarLeRIa shows a significant improvement, e.g., +2.01 IoU on RefCOCO, +5.19 IoU on RefCOCO+, and +3.74 IoU on G-Ref, respectively. Compared with the fully fine-tuned large visual language model LISA-7B [Lai et al., 2023], BarLeRIa achieves comparable performance with only about 2M learnable backbone parameters and significantly outperforms the untuned 7B model. Besides, BarLeRIa also outperforms full fine-tuning state-of-the-art approaches, e.g., PolyFormer [Liu et al., 2023] and UNINEXT [Yan et al., 2023], which need pre-training on extra region-level datasets. As comparison, without extra pre-training, BarLeRIa achieves a new SOTA performance when adopting EVA-CLIP [Radford et al., 2021] as the pre-trained vision language model. In a nutshell, our contributions can be summarized as follows: • We find that previous PET methods for RIS task focus on modal fusion and ignore feature distribution adaptation and propose a novel intertwined vision language efficient tuning algorithm for both feature adaptation and modal fusion with only 0.4M (ViT-B) learnable parameters. • We reveal that repeating fusing the local visual features with the textual embeddings is another problem for previous approaches and propose a bi-directional efficient tuning framework that enables both local feature fusion and global prior regularization. • We design a novel global shortcut tuning module that tunes only 1.8M (ViT-B) parameters and learns the global prior regularization in parallel with the backbone to avoid conflicts with our proposed local intertwined vision language efficient tuning. 2 METHODOLOGY 2.1 PRELIMINARIES Adapting Shortcut with Normalizing Flow (SNF) [Wang et al., 2023] adjusts the shortcut to adapt pre-trained models into downstream tasks. For a given skip connection inside the transformer, it can be depicted as $ y = x + f(x) $ where $ x $ is the input feature, $ f $ is a certain architecture of the transformer and $ y $ is the output. During fine-tuning, SNF only operates on the shortcut $ x $ while keeping other parts frozen, i.e., $ y = s(x) + f(x) $. For a given feature $ x \in \mathbb{R}^{N \times d} $, the transformation imposed by SNF is given by: \[ s(x) = x + \lambda \cdot h(\gamma^T \cdot x + \beta) \] where $ \lambda, \gamma, \beta \in \mathbb{R}^d $, $ \cdot $ is the Hadamard product and $ h(\cdot) $ is a smooth non-parameteric non-linearity. Note that SNF allows for multiple concatenated transformations, i.e., $ y = s(s(\cdots s(x))) + f(x) $. The number of transformations is denoted as the depth of SNF. 2.2 FRAMEWORK OVERVIEW The framework of BarLeRIa is depicted in Fig. 2. The fundamental design of BarLeRIa is the proposed intertwined vision language efficient tuning algorithm, which is used to enhance modal fusion. Along with it, we employ a bi-directional efficient tuning framework that simultaneously adjusts local features and extracts global priors from the text input, thereby regularizing the visual features. This framework consists of two distinct efficient tuning modules. The first module, known as the local intertwined module, utilizes the intertwined vision language efficient tuning approach to enable efficient modal fusion and multi-modal feature adaptation. The second module, referred to as the global shortcut tuning module, incorporates a parallel shortcut module and leverages the global prior generated from the global prior module to complement the local vision features. Finally, the complete vision features, alongside the textual embeddings, are inputted into the learnable referring image segmentation head and generate the corresponding segmentation masks. 2.3 INTERTWINED VISION LANGUAGE EFFICIENT TUNING For an input tokenized referring expression $ T \in \mathbb{R}^{L \times D} $ and an input tokenized image $ I \in \mathbb{R}^{H \times W \times C} $, along with a visual encoder $ \phi : \{\phi_1, \cdots, \phi_N\} $ composed of $ N $ transformer blocks, we Figure 2: The framework of BarLeRia. GST is the abbreviation of global shortcut tuning. For the visual branch, we fuse the textual embeddings with the visual input in the frozen visual block and further adapt the feature distribution with the normalizing flow. For the language branch, we concatenate the visual class token to the textual input and achieve modal fusion and feature adaption similarly. Besides, a global shortcut tuning module along with a global prior module is proposed in parallel with the backbone for global visual regularization. begin by projecting the tokenized expression $T$ as: $T' \leftarrow TW_{proj}$, where $W_{proj} \in \mathbb{R}^{D \times C}$. Next, we concatenate the projected expression $T'$, the tokenized image $I$ (we reshape it into $\mathbb{R}^{(H \cdot W) \times C}$ before), and the class token $cls$ as $[cls, T', I]$. These intertwined embeddings are then fed into the frozen visual encoder $\phi$, and the output is given as: $$[cls, embed] = F([cls, T'_{i-1}, embed]) + \phi_i([cls, T'_{i-1}, embed])$$ where $F = f_J \circ \cdots \circ f_1$ represents a chain of $J$ invertible feature mappings: $f_i(z) = z + \lambda_i \cdot h(\gamma_i^T \cdot z + \beta_i)$, and $T'_{i-1}$ represents the output from the $(i - 1)$-th textual block. Note that we take the projected expression $T'$ as input for the first layer. With the Multi-Head Self-Attention (MHSA) module employed in each visual block for information interaction between tokens, we successfully achieve modal fusion through these frozen visual blocks. Additionally, the shortcut normalizing flow $F$ is applied in each visual block for feature adaptation. As for the textual block $\psi_i$, we project the visual class token $cls$ into $cls'$ to align its feature dimension with the textual embeddings $T$. Then, we concatenate the textual embedding $T$ with the projected class token $cls'_{i-1}$ from the previous visual block to form the input. Furthermore, we leverage the shortcut normalizing flow to adapt the shortcut textual embeddings and employ the frozen transformer block to fuse the textual and visual features. Consequently, we obtain the output of the textual block $\psi_i$ as follows: $$[T] = F([cls'_{i-1}, T]) + \psi_i([cls'_{i-1}, T])$$ 2.4 Bi-DIRECTIONAL EFFICIENT TUNING As discussed in Sec. 1, bi-direction refers to the combination of local efficient tuning and global prior regularization, and the bi-directional efficient tuning framework consists of two modules: the Global Prior Module and the Global Shortcut Tuning Network. Global Prior Module. As language is more semantic-rich, the produced language embeddings tend to be more robust compared to the visual ones. Therefore, we propose regularizing visual features through the global prior generated by the language embeddings. Specifically, given the visual encoder $\phi$ and its output vision feature $F_v$, as well as the language encoder $\psi$ and its output language embeddings $F_l$, we concatenate the vision features with the language embeddings to obtain intertwined features: $[F_l, F_v]$. Next, we calculate the cosine similarity between the intertwined feature $[F_l, F_v]$ and the language embeddings $F_l$. This cosine similarity serves as an attention mask, which is then multiplied with the intertwined feature to produce the global prior $p$: $$p = \cos([F_l, F_v], F_l) \cdot [F_l, F_v]$$ Global Shortcut Tuning Network. To ensure that the global prior regularization does not conflict with our local intertwined vision language efficient tuning and to achieve an end-to-end PET pipeline, we introduce a global shortcut tuning network $ G $ that operates in parallel with the vision encoder. This network consists of $ M $ modules $ \{G_1, \cdots, G_M\} $, each following the design of the transformer block with MHSA and MLP, but with smaller feature dimensions (default setting: 144). Firstly, we transform the global prior $ p $ using $ M $ linear transformations to obtain adapted priors $ p_1, \cdots, p_M $ for each global shortcut tuning module. Then, given the tokenized input image $ I $ and the language embedding $ F_l $, we concatenate them with learnable query tokens $ q $ as $ [q, F_l, I] $, which serves as the input for the global shortcut tuning network. For the first global shortcut tuning module $ G_1 $, we feed these concatenated tokens along with the global prior $ p_1 $ as follows: \[ I_1 = G_1([q, F_l, I], p_1) \] where $ I_1 $ represents the output of the first module, and $ G_1(I, p) $ utilizes the global prior $ p $ to regularize the MHSA feed-forward in the global shortcut tuning module. In elaboration, we project the input $ I $ into query, key, and value tokens using projection matrices $ W_q, W_k, $ and $ W_v $, respectively. Additionally, we project the global prior $ p $ into complementary value tokens using learnable projection matrices $ W_p $. The two sets of value tokens are then added together, resulting in the new value tokens $ IW_v + IW_p $. Consequently, we redefine the self-attention mechanism as: \[ \text{MHSA}(I) = \text{Softmax}\left(\frac{IW_qIW_k}{\sqrt{C}}\right)(IW_v + IW_p) \] Here, $ C $ denotes the feature dimension, and for simplicity, we omit the multi-head division. By modifying the value tokens in the MHSA block, we successfully incorporate global prior regularization into the global shortcut tuning module through the introduced global attention. For the remaining $ M - 1 $ modules, we repeat the global prior regularization process as follows: \[ I_i = G_i(I_{i-1}, p_i), i = 2, \cdots, M \] The global shortcut tuning network comprises very few parameters and is primarily employed for parameter-efficient tuning. Subsequently, we proceed to fuse the output shortcut features $ I_1, \cdots, I_M $ from each global shortcut tuning module with the corresponding vision features $ F_v $. For a given $ i $-th output shortcut feature $ I_i $ and its corresponding vision features $ F_v^i $, we first interpolate $ I_i $ to match the height and width of the original vision features $ F_v^i $. Subsequently, we add these two sets of vision features to obtain the output feature $ F_{out}^i $ as follows: \[ F_{out}^i = \text{Interpolate}(I_i) + F_v^i \] To prevent conflicts with the local intertwined module, we detach $ F_v^i $ during fine-tuning. 2.5 Final Objective Following Wang et al. (2022) and Xu et al. (2023), we incorporate a learnable referring image segmentation head composed of a cross-modal neck, vision-language decoder, and an up-sample projector to extract the cross-modal intertwined feature $ F_{ci} $ and the transformed textual feature $ F_t $: \[ F_{ci}, F_t = \text{Head}(F_{out}^M, F_v, F_t) \] where $ F_{out}^M $ represents the output from the last global shortcut tuning module, while $ F_t $ and $ F_v $ denote the textual embeddings and the vision encoder features adapted by the local intertwined modules. To train our model, we employ a text-to-pixel contrastive loss (Wang et al., 2022) as our training objective, which encourages the alignment of textual embeddings with their corresponding visual pixels, while pushing textual embeddings away from other irrelevant visual pixels. The text-to-pixel contrastive loss is formulated as follows: \[ L_{tp}(F_{ci}, F_t) = \begin{cases} -\log \left( \sigma(F_{ci} \cdot F_t) \right), & i \in P \\ -\log \left( 1 - \sigma(F_{ci} \cdot F_t) \right), & i \in N \end{cases} \] \[ L_{tp}(F_{ci}, F_t) = \frac{1}{\|P \cup N\|} \sum_{i \in P \cup N} L_{tp}(F_{ci}, F_t) \] where $ \sigma $ denotes the sigmoid function, $ P $ and $ N $ represent the classes of 1 and 0, respectively. Table 1: Comparison with SOTA RIS methods and the PET RIS SOTA method without additional datasets evaluated using the IoU metric on RefCOCO-related datasets. \| Method \| RefCOCO \| RefCOCO+ \| G-Ref \| \|-----------------\|---------\|----------\|-------\| \| \| val \| testA \| testB \| val \| testA \| testB \| val(u) \| test(u) \| val(g) \| Avg \| \| Traditional \| \| \| \| \| \| \| \| \| \| \| \| MAttNet (Yu et al., 2018) \| 56.5 \| 62.4 \| 51.7 \| 46.7 \| 52.4 \| 40.1 \| 47.6 \| 48.6 \| - \| 50.5 \| \| RRN (Li et al., 2018) \| 55.3 \| 57.3 \| 54.0 \| 39.8 \| 42.2 \| 36.1 \| - \| - \| 36.5 \| 43.8 \| \| CMSA (Ye et al., 2019) \| 58.3 \| 60.6 \| 55.1 \| 43.8 \| 47.6 \| 37.9 \| - \| - \| 40.0 \| 47.0 \| \| CAC (Chen et al., 2019) \| 58.9 \| 61.8 \| 53.8 \| - \| - \| - \| 46.4 \| 47.0 \| 44.3 \| - \| \| BRINet (Hu et al., 2020) \| 61.4 \| 63.4 \| 59.6 \| 48.6 \| 52.9 \| 42.1 \| - \| - \| 48.0 \| 52.5 \| \| CMPC+ (Liu et al., 2021a) \| 61.4 \| 64.5 \| 59.6 \| 49.6 \| 53.4 \| 43.2 \| - \| - \| - \| - \| \| CGAN (Luo et al., 2020) \| 64.9 \| 68.0 \| 62.1 \| 51.0 \| 55.5 \| 44.1 \| 51.0 \| 51.7 \| - \| 55.5 \| \| LTS (Jing et al., 2021) \| 65.4 \| 67.8 \| 63.1 \| 54.2 \| 58.3 \| 48.0 \| - \| - \| - \| - \| \| VLT (Ding et al., 2021) \| 65.7 \| 68.3 \| 62.7 \| 55.5 \| 59.2 \| 49.4 \| - \| - \| 49.8 \| 56.7 \| \| PCAN (Chen et al., 2022a) \| 69.5 \| 71.6 \| 64.2 \| 58.3 \| 63.7 \| 48.9 \| 60.0 \| 60.8 \| 57.5 \| 61.6 \| \| ReSTR (Kim et al., 2022) \| 67.2 \| 69.3 \| 64.5 \| 55.8 \| 60.4 \| 48.3 \| 54.5 \| - \| 54.5 \| 58.8 \| \| CRIS (Wang et al., 2022) \| 70.5 \| 73.2 \| 66.1 \| 62.3 \| 68.1 \| 53.7 \| 59.9 \| 60.4 \| - \| 63.8 \| \| LAVT (Yang et al., 2021) \| 72.7 \| 75.8 \| 68.8 \| 62.1 \| 68.4 \| 55.1 \| - \| - \| 60.5 \| 64.9 \| \| WiCo (Cheng et al., 2023) \| 73.5 \| 76.9 \| 68.1 \| 63.4 \| 69.2 \| 55.8 \| - \| - \| 60.2 \| 65.3 \| \| Parameter Efficient-Tuning \| \| \| \| \| \| \| \| \| \| \| \| ETRIS (Xu et al., 2023) \| 70.5 \| 73.5 \| 66.6 \| 60.1 \| 66.9 \| 50.2 \| 59.8 \| 59.9 \| 57.9 \| 62.8 \| \| Ours \| 72.4 \| 75.9 \| 68.3 \| 65.0 \| 70.8 \| 56.9 \| 63.4 \| 63.8 \| 61.6 \| 66.5 \| 3 EXPERIMENTS 3.1 EXPERIMENTAL SETUP Datasets. We employ three challenging referring image segmentation benchmarks in our experiments: RefCOCO (Kazemzadeh et al., 2014), RefCOCO+ (Kazemzadeh et al., 2014), and G-Ref (Yu et al., 2016). Please refer to the appendix A.1 for details. Implementation Details. We train the whole network in an end-to-end manner for 50 epochs using the Adam optimizer with a learning rate of 0.0001. A learning rate decay is employed at the 35th epoch with a decay factor of 0.1. We train the model using 2 Tesla V100 GPUs with a batch size of 32. For ViT-L/14, we train the model using 8 Tesla V100 GPUs with a batch size of 64 and an initial learning rate of 0.0002. Following previous works (Ding et al., 2021; Liu et al., 2017; Wang et al., 2022; Xu et al., 2023), we adopt IoU as the metric to evaluate the performance. More details can be referred to in the appendix A.2. 3.2 MAIN RESULTS We conducted a comprehensive comparison between our BarLeRia and a series of previous RIS approaches. The results, presented in Tab.1, demonstrate that our approach significantly outperforms state-of-the-art RIS methods on three commonly used datasets, achieving 6 SOTA and 3 sub-sub-SOTA performance across 9 evaluation tasks. In particular, we surpass the performance of WiCo (Cheng et al., 2023), which utilizes an additional ResNet-50 to extract top-down segmentation proposals as a pre-stage. In contrast, our BarLeRia model achieves superior results using only 2.2M parameters and is trained in an end-to-end manner. Furthermore, we compare our proposed approach with the state-of-the-art parameter-efficient tuning RIS method, ETRIS (Xu et al., 2023). To ensure a fair comparison, we employ the same CLIP pre-trained vision language model of ViT-B/16 as used in ETRIS and freeze the visual and textual encoders. The tuning backbone parameters in BarLeRia are only 2.2M, which is comparable to ETRIS. It is worth noting that we can achieve superior performance to ETRIS only using the local intertwined module with much fewer tuning parameters, and more details are shown in Sec.3.5. Overall, BarLeRia achieves a significant improvement of +3.7 IoU on average across the three RefCOCO-related datasets, demonstrating its superiority over existing RIS methods. Table 2: Comparison with full fine-tuning SOTA RIS methods and these methods either utilize large language models or are pre-trained with additional datasets. IoU is utilized as the metric. † denotes that the model is tuned using the mixed RefCOCO datasets. \| Method \| RefCOCO \| RefCOCO+ \| G-Ref \| \|-----------------\|---------\|----------\|-------\| \| \| val \| testA \| testB \| val \| testA \| testB \| val(u) \| test(u) \| Avg \| \| LISA-7B (Lai et al., 2023) \| 74.1 \| 76.5 \| 71.1 \| 62.4 \| 67.4 \| 56.5 \| 66.4 \| 68.5 \| 67.9 \| \| LISA-7B (†) (Lai et al., 2023) \| 74.9 \| 79.1 \| 72.3 \| 65.1 \| 70.8 \| 58.1 \| 67.9 \| 70.6 \| 69.9 \| \| PolyFormer-B (Liu et al., 2023) \| 74.8 \| 76.6 \| 71.1 \| 67.6 \| 59.3 \| 72.9 \| 67.8 \| 69.1 \| 69.9 \| \| UNINEXT-R50 (Yan et al., 2023) \| 77.9 \| 79.7 \| 75.8 \| 66.2 \| 71.2 \| 59.0 \| 70.0 \| 70.5 \| 71.3 \| \| ETRIS (Xu et al., 2023) \| 72.4 \| 74.6 \| 69.3 \| 64.5 \| 70.4 \| 56.9 \| 62.6 \| 63.1 \| 66.1 \| \| BarLeRIa \| 75.0 \| 77.1 \| 71.2 \| 68.6 \| 73.2 \| 61.2 \| 65.9 \| 66.4 \| 69.8 \| \| BarLeRIa-Mixed† \| 77.6 \| 79.4 \| 75.3 \| 71.7 \| 75.7 \| 66.0 \| 70.9 \| 71.4 \| 73.5 \| ### 3.3 Comparison to Full Fine-tuning Methods We further conducted a comparison between our proposed approach and existing SOTA full fine-tuning methods. These methods either utilize large language models or are pre-trained with additional datasets that contain region-level information. Without using additional datasets, we select a superior CLIP version, EVA-CLIP, which is still pre-trained using general-purpose datasets. For fair comparisons, we also use the EVA-CLIP pre-trained vision language model as the backbone for ETRIS. As shown in Tab. [2], BarLeRIa outperforms ETRIS by a significant margin, achieving an average improvement of +3.7 IoU using the same EVA-CLIP pre-trained backbone. Compared to LISA-7B (Lai et al., 2023), a large vision language model with 7 billion parameters, our approach demonstrates a significant improvement when LISA-7B is not fine-tuned and achieves comparable performance when LISA-7B is fully fine-tuned. Compared with PolyFormer-B (Liu et al., 2023) that utilizes Swin-B (Liu et al., 2021b) as the visual encoder and the BERT transformer as the textual encoder, our proposed BarLeRIa achieves comparable performance without additional region-level pre-training and mixed fine-tuning. It is worth noting that PolyFormer introduces a second pre-training phase to incorporate region-level information using additional datasets, including Visual Genome, three RefCOCO-related datasets, and Flickr30k-entities. Furthermore, BarLeRIa achieves a +3.6 IoU improvement over PolyFormer-B when we additionally employ mixed fine-tuning. UNINEXT (Yan et al., 2023) leverages pre-training on Objects365 to learn region-level information and also employs mixed fine-tuning. BarLeRIa achieves a +2.2 IoU improvement over UNINEXT-R50 when we also employ mixed fine-tuning with much fewer tuning parameters. We also conduct experiments using the ViT-Large visual encoder to verify the generalization ability of our method across different architectures. As shown in Tab. [3], BarLeRIa-L outperforms PolyFormer-L without additional region-level pre-training and mixed fine-tuning. Moreover, compared to the best-performing RIS method, UNINEXT, BarLeRIa-L-Mixed achieves a clear margin of +1.0 IoU averaged improvement across RefCOCO-related datasets, demonstrating its effectiveness. ### 3.4 Visualization As illustrated in Fig. [3], we present visualization results with different settings under easy scenarios and hard scenarios, respectively. In the figure, (d) SNF means we just use normalizing flow to adapt the visual features without the bridge used in ETRIS, and (e) SNF+ETRIS means we combine SNF with ETRIS. We use these two settings to determine whether SNF is the key to PET RIS approaches. We find that both (d) and (e) lag much compared with our BarLeRIa and prove that our proposed two PET modules provide great improvement (more details of the ablation are shown in Sec. [3.5]). The first two rows of Fig. [3] represent the easy scenario and all methods can segment objects correctly. The difference is only in the detail and the finesse of the contours. BarLeRIa and BarLeRIa-L-mixed achieve the best segmentation IoU while ETRIS performs worst. For the hard scenario, i.e., the last two rows of Fig. [3], ETRIS fails to locate the object correctly, SNF and SNF+ETRIS introduce overly large outlines, indicating that they do not fully understand the text description, while our BarLeRIa fully understands the meaning of the text and accurately segments the target objects. Table 3: Comparison with full fine-tuning SOTA RIS methods using ViT-Large as the visual backbone. These methods either utilize large language models or are pre-trained with additional datasets. IoU is utilized as the metric. † denotes that the model is tuned using the mixed RefCOCO datasets. \| Method \| RefCOCO \| RefCOCO+ \| G-Ref \| Avg \| \|-------------------------\|---------\|----------\|-------\|-----\| \| \| val \| testA \| testB \| val \| testA \| testB \| val(u) \| test(u) \| \| PolyFormer-L† (Liu et al., 2023) \| 76.0 \| 78.3 \| 73.3 \| 69.3 \| 61.9 \| 74.6 \| 69.2 \| 70.2 \| 71.6 \| \| UNINEXT-L† (Yan et al., 2023) \| 80.3 \| 82.6 \| 77.8 \| 70.0 \| 74.9 \| 62.6 \| 73.4 \| 73.7 \| 74.4 \| \| BarLeRIa-L \| 76.8 \| 79.0 \| 74.0 \| 71.5 \| 76.2 \| 65.4 \| 68.7 \| 69.7 \| 72.7 \| \| BarLeRIa-L-Mixed† \| 79.0 \| 80.8 \| 77.0 \| 74.2 \| 77.8 \| 68.3 \| 72.7 \| 73.3 \| 75.4 \| Language: “kid running” Language: “horse closest to us” Language: “cut banana in bowl” Language: “bear with face most showing” (a) Image (b) GT (c) ETRIS (d) SNF (e) SNF+ETRIS (f) BarLeRIa (g) BarLeRIa-L* Figure 3: Qualitative results with different settings. (a) the input image. (b) the ground truth. (c) ETRIS. (d) SNF without local intertwined module. (e) SNF+ETRIS. (f) our proposed BarLeRIa. (g) BarLeRIa-L using mixed datasets. Best viewed in color. 3.5 Ablation Study To establish the efficacy of our proposed approach, we perform ablation studies on the components of our proposed BarLeRIa. We only briefly document the averaged performance of different test splits for RefCOCO, RefCOCO+, and G-Ref respectively (please refer to the appendix B for detailed results). Illustrated in Tab. 4, SNF means we use the normalizing flow to adjust the feature, LIM is the abbreviation of Local Intertwined Module, GST denotes Global Shortcut Tuning, and No Global means we just use the Local Intertwined Module without the Global Shortcut Tuning. As we can see, just employing existing SNF or combining SNF with ETRIS does not improve segmentation performance. Besides, if we only use the local intertwined module (No Global in the table), we can outperform ETRIS with +2.6 averaged IoU improvement with nearly one-tenth the number of tuning parameters. This result demonstrates that BarLeRIa can greatly surpass existing PET state-of-the-art with fewer learnable parameters and showcases its superiority. Finally, with the proposed Global Shortcut Tuning, BarLeRIa achieves further enhancements to +1.1 averaged IoU. 4 Related Work Parameter Efficient Tuning (PET) adjust only a fraction of the parameters and alleviate the computational challenges associated with fine-tuning the entire model. One prominent research direction focuses on incorporating lightweight architectures into the frozen backbone and updating only these newly added architectures during fine-tuning (Houlsby et al., 2019; Mahabadi et al., 2021; Lester... Table 4: Ablation study on the components of BarLeRlA. LIM is the abbreviation of Local Intertwined Module, GST denotes Global Shortcut Tuning, and No Global means we just use the Local Intertwined Module without the Global Shortcut Tuning. \| Method \| SNF \| LIM \| GST \| Params(M) \| RefCOCO \| RefCOCO+ \| G-Ref \| Avg \| \|-------------\|-----\|-----\|-----\|-----------\|---------\|----------\|-------\|-----\| \| ReSTR \| - \| - \| - \| 86.19 \| 67.0 \| 54.8 \| 54.5 \| 58.8\| \| ETRIS \| × \| × \| × \| 1.39 \| 70.2 \| 59.1 \| 59.2 \| 62.8\| \| SNF \| ✓ \| × \| × \| 0.18 \| 70.6 \| 59.6 \| 59.3 \| 63.2\| \| SNF+ETRIS \| ✓ \| × \| × \| 1.57 \| 70.2 \| 59.9 \| 60.1 \| 63.4\| \| No Global \| ✓ \| ✓ \| × \| 0.39 \| 71.4 \| 63.1 \| 61.6 \| 65.4\| \| BarLeRlA \| ✓ \| ✓ \| ✓ \| 2.21 \| 72.2 \| 64.2 \| 62.9 \| 66.5\| et al., 2021; Li & Liang, 2021; Karimi Mahabadi et al., 2021; Chen et al., 2022b; Jie & Deng, 2022; Jia et al., 2022). For instance, AdaptFormer (Chen et al., 2022b) and ConvPass (Jie & Deng, 2022) introduce bottleneck or convolution modules along the skip connections within transformer layers and adapt the residuals for downstream tasks. Recently, Wang et al. (2023) proposed leveraging normalizing flows to adjust the shortcuts rather than the residuals within transformer layers, offering an easily implementable and accessible approach for various architectures. Another line of the PET method involves updating only a subset of the parameters in the original model (Sung et al., 2021; Zaken et al., 2021). Zaken et al. (Zaken et al., 2021), for example, demonstrate that updating only the bias terms can achieve competitive or even superior performance compared to full fine-tuning. Additionally, some researchers have explored matrix decomposition techniques to reduce the number of learnable parameters by factorizing the weights of pre-trained models (Hu et al., 2021; Jie & Deng, 2023), which also yield satisfactory performance. Unfortunately, PET approaches for referring image segmentation are less investigated. Recently, Xu et al. (2023) introduced PET to referring image segmentation by leveraging the bridge module for information fusion between visual and textual modalities. However, their proposed ETRIS lacks feature adaption and global visual regularization, resulting in unsatisfactory performance. Besides these works, some researchers factorize weights of the pre-trained model based on the low-rank assumption, such that parameters that need to be tuned can be largely reduced (Hu et al., 2021). Referring Image Segmentation (RIS) aims to segment a target instance or region referred by the given text query and is initially introduced by Hu et al. (2016). Early methods were predominantly based on the CNN+LSTM approach (Liu et al., 2017; Li et al., 2018), where the image and text inputs were encoded separately using their respective backbones. However, in recent years, transformer architectures have gained popularity due to their flexibility and scalability (Vaswani et al., 2017; Dosovitskiy et al., 2020), allowing RIS methods to employ a unified architecture across different modalities (Kim et al., 2022; Yang et al., 2021; Liu et al., 2023; Yan et al., 2023). Additionally, the advent of multi-modal pre-training (Radford et al., 2021) has provided RIS models with the advantage of leveraging large-scale pre-training data (Wang et al., 2022). Besides, recent work (Cheng et al., 2023) has proven that the global prior can help the referring segmentation. However, these methods require full fine-tuning of an additional over-parameterized model and divide the segmentation process into two stages without end-to-end training. 5 CONCLUSION In this paper, we pay attention to parameter efficient tuning for referring image segmentation. We reveal that previous approaches focus on vision and language modal alignment, but ignores adapting the biased feature from pre-trained models. Besides, previous approaches fuse the local visual features with the textual embeddings without introducing global prior from text input to regularize the visual feature. To address these issues, we propose a novel PET framework BarLeRlA: Bi-directional Intertwined Vision Language Efficient Tuning for Referring Image Segmentation, which leverages intertwined vision language adapters and bi-directional tuning framework to fully exploit the frozen pre-trained models’ potential. We conduct extensive experiments on three RefCOCO-related benchmarks. BarLeRlA consistently outperforms prior parameter efficient tuning methods with a clear margin. Moreover, BarLeRlA also surpasses full fine-tuning state-of-the-art approaches without pre-training using additional training datasets. Acknowledgment This work was supported in part by the National Natural Science Foundation of China under Grant 62125109, Grant 62250055, Grant 61931023, Grant 61932022, Grant 62371288, Grant 62320106003, Grant 62301299, Grant T2122024, Grant 62120106007. REFERENCES Bo Chen, Zhiwei Hu, Zhilong Ji, Jinfeng Bai, and Wangmeng Zuo. Position-aware contrastive alignment for referring image segmentation. arXiv preprint arXiv:2212.13419, 2022a. Shoufa Chen, Chongjian Ge, Zhan Tong, Jiangliu Wang, Yibing Song, Jue Wang, and Ping Luo. Adaptformer: Adapting vision transformers for scalable visual recognition. arXiv preprint arXiv:2205.13535, 2022b. Yi-Wen Chen, Yi-Hsuan Tsai, Tiantian Wang, Yen-Yu Lin, and Ming-Hsuan Yang. Referring expression object segmentation with caption-aware consistency. arXiv preprint arXiv:1910.04748, 2019. Zesen Cheng, Peng Jin, Hao Li, Kehan Li, Siheng Li, Xiangyang Ji, Chang Liu, and Jie Chen. Wico: Win-win cooperation of bottom-up and top-down referring image segmentation. arXiv preprint arXiv:2306.10750, 2023. Henghui Ding, Chang Liu, Suchen Wang, and Xudong Jiang. Vision-language transformer and query generation for referring segmentation. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 16321–16330, 2021. Shuangrui Ding, Weidi Xie, Yabo Chen, Rui Qian, Xiaopeng Zhang, Hongkai Xiong, and Qi Tian. Motion-inductive self-supervised object discovery in videos. arXiv preprint arXiv:2210.00221, 2022. Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020. Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski, Bruna Morrone, Quentin De Laroussilhe, Andrea Gesmundo, Mona Attariyan, and Sylvain Gelly. Parameter-efficient transfer learning for nlp. In International Conference on Machine Learning, pp. 2790–2799. PMLR, 2019. Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. Lora: Low-rank adaptation of large language models. arXiv preprint arXiv:2106.09685, 2021. Ronghang Hu, Marcus Rohrbach, and Trevor Darrell. Segmentation from natural language expressions. In Computer Vision–ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11–14, 2016, Proceedings, Part I 14, pp. 108–124. Springer, 2016. Zhiwei Hu, Guang Feng, Jiayu Sun, Lihe Zhang, and Huchuan Lu. Bi-directional relationship inferring network for referring image segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 4424–4433, 2020. Menglin Jia, Luming Tang, Bor-Chun Chen, Claire Cardie, Serge Belongie, Bharath Hariharan, and Ser-Nam Lim. Visual prompt tuning. arXiv preprint arXiv:2203.12119, 2022. Shibo Jie and Zhi-Hong Deng. Convolutional bypasses are better vision transformer adapters. arXiv preprint arXiv:2207.07039, 2022. Shibo Jie and Zhi-Hong Deng. Fact: Factor-tuning for lightweight adaptation on vision transformer. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pp. 1060–1068, 2023. Ya Jing, Tao Kong, Wei Wang, Liang Wang, Lei Li, and Tieniu Tan. Locate then segment: A strong pipeline for referring image segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9858–9867, 2021.
cB9bAFGFAA	Why the client IDs are private information? Why FedSRC can protect the client IDs? The models are still sent for weighted aggregation like FedAvg. The server can infer the client ID by the data size. Moreover, the server usually knows who is the sender in a communication protocol.	FedSRC: Federated Learning with Self-Regulating Clients Anonymous authors Paper under double-blind review Abstract Federated Learning (FL) has emerged as a prominent privacy-preserving decentralized paradigm for collaborative machine learning across many devices. However, FL suffers from performance degradation in the global model due to heterogeneity in clients’ locally generated data. Some prior studies address this issue by limiting or even discarding certain clients’ contributions to the global model, resulting in unnecessary computation and communication for the discarded clients. Alternatively, selectively choosing clients to participate in FL may avoid such resource waste. But, such active client selection requires client-level profiling that violates privacy. In this paper, we present a novel FL approach, called FedSRC: Federated Learning with Self-Regulating Clients, that can save clients’ resources while preserving their anonymity. In FedSRC, clients can determine themselves if their local training is favorable to the global model and whether they should participate in an FL round using a lightweight checkpoint based on a local inference loss on the global model. Through comprehensive evaluations using four datasets, we show that FedSRC can improve global model performance, all the while reducing communication costs by up to 30% and computation costs by 55%. 1 Introduction Motivation. Federated Learning (FL) is a popular privacy-preserving distributed Machine Learning (ML) approach that has been implemented in widely used applications like Google’s Gboard [Hard et al., 2019] and Apple’s Siri [Granqvist et al., 2020]. In FL, many clients collaboratively train a shared ML model through iterations where clients locally train the shared model using their private data and anonymously send back their updated model. FL enjoys the advantages of training with a larger dataset from many clients, yet clients’ data never leaves their devices, offering enhanced privacy. A central FL server facilitates the iterations by aggregating the clients’ model updates into the shared global model. A well-documented drawback of FL’s siloed decentralized training is the slow convergence and poor performance of the global model due to statistical differences (i.e., not independent or identically distributed (non-IID)) among the clients’ data [Li et al., 2020b]. On top of the naturally occurring data variation among FL clients, another source of such statistical differences is the data quality. The quality and reliability of the locally collected client data by different hardware and sensors may vary significantly, especially in mobile and wearable devices [Cho et al., 2021a,b]. Device manufacturers utilize hardware/sensors of varying qualities to meet their own goals of device functionality and price points. Moreover, continued innovation in mobile devices is yielding increasingly high-quality data with newer generations of sensors and hardware [Haghi et al., 2017; Cheng et al., 2021]. A recent Facebook study identifies thousands of different types of hardware among the devices using their application [Wu et al., 2019]. In addition, malfunctioning devices can also be responsible for feeding FL with bad-quality data [Liu et al., 2020]. Worse yet, the FL clients’ devices and sensors are also subject to malicious attacks that may intentionally corrupt client data to poison the global model, exacerbating FL’s data quality issue [Tolpegin et al., 2020]. Limitations of existing approaches. A prominent line of prior work aims at handling the aforementioned data heterogeneity/quality issue by controlling the contributions from different clients [Li et al., 2022; Karimireddy et al., 2020; Yin et al., 2018; Talukder & Islam, 2022]. This approach is built upon the idea that the updates from certain clients (e.g., clients with bad data quality) are un- favorable for the global model and should be given a lower weight in FL’s centralized model update aggregation. While this can improve FL performance, a major limitation of the aggregation-weight-based approach is that clients whose model updates receive a lower weight and, hence, contribute little to the central model still go through the computationally hungry model training and communicate the updated model to the central FL server. This unnecessary use of clients’ computation and communication resources makes FL training inefficient. An alternative to the wasteful weight-based approach to handle FL’s data heterogeneity is “active client selection” where the central server profiles the quality of clients’ model updates and selects only “favorable” clients to participate in FL training [Cho et al., 2022; Goetz et al., 2019]. While active client selection does not waste resources, this approach needs to tag client updates with client IDs for the active selection. Therefore, it cannot maintain clients’ anonymity and diminishes FL privacy. Our contribution. We recognize that both the above-mentioned resource waste and breach of anonymity can be avoided (while still handling data quality issues) if the clients themselves can anticipate the resource waste and refrain from participating in model updates. To enable this client-side active client selection, we propose a novel FL approach where the clients actively regulate their own participation. We call this Federated Learning with Self-Regulating Clients or FedSRC in short. In FedSRC, clients implement a “checkpoint” in its local training path to determine whether it should continue and finish training and send the model update back to the FL central server. A client saves the computation cost of local training and the communication cost of sending the model update if it decides to exit the FL round. On the other hand, client anonymity is not violated, as the central server does not need any client profiling for client selection. Furthermore, FedSRC’s client-side implementation can still be paired with heavier centralized FL techniques that tackle model and data poisoning attacks at the central server. To the best of our knowledge, FedSRC offers the first variation of FL that allows clients to make strategic decisions to aid the FL global model. However, implementing FedSRC’s active client selection (i.e., participation checkpoint) that handles FL’s data quality issue is challenging. In FL, clients only have access to their own data and, therefore, cannot statistically determine their data quality and employ strategic FL participation. Moreover, our selection strategy must be lightweight since it needs to be deployed on the client device. With these constraints in mind, we develop an inference loss-based participation policy where the clients utilize the global shared model as a “litmus test” for their data quality and exit the FL training if they have high local inference loss on the global model. Our design is motivated by our empirical observation that, in general, clients’ low-quality data results in higher local inference loss on the global model. Fig. 1 illustrates the main working principle of FedSRC while the details are presented in Section 3.1. We also offer the first-ever theoretical analysis of FL convergence under clients with data quality issues. Our analysis reveals that FedSRC’s strategic selection can boost both the performance and convergence rate of FL. We evaluate FedSRC using four different datasets and show that with the same number of communication rounds, FedSRC can save as much as 30% on communication and 55% on computational cost. 2 PRELIMINARIES 2.1 FEDERATED LEARNING Problem formulation. Suppose in a federated setup, there are $K$ clients, each with their own dataset $\mathcal{D}_k$. The objective of FL is to minimize the global loss $F(w)$, which can be expressed as $$F(w) = \frac{1}{\sum_{k=1}^{K} \|\mathcal{D}_k\|} \sum_{k=1}^{K} \sum_{\xi \in \mathcal{D}_k} f(w, \xi) = \sum_{k=1}^{K} p_k F_k(w)$$ (1) where $f(w, \xi)$ is the composite loss function for sample $\xi$ and model parameter $w$, $p_k = \frac{\|\mathcal{D}_k\|}{\sum_{k=1}^{K} \|\mathcal{D}_k\|}$ is the fraction of data at the $k$-th client, and $F_k(w) = \frac{1}{\|\mathcal{D}_k\|} \sum_{\xi \in \mathcal{D}_k} f(w, \xi)$ is the local loss function of client $k$. Solution. The FedAVG McMahan et al. (2017) algorithm minimizes Eq. equation 1 efficiently by dividing training into multiple rounds. In each round $t$, a fraction $C$ of clients ($m = CK$) is randomly selected from $K$, and the selected clients are denoted by $S(t)$. Selected clients perform $\tau$ local SGD iterations and update their models, which are then aggregated into a new global model. Accordingly, the model update for a client can be written as follows: $$w_k^{(t+1)} = \begin{cases} w_k^{(t)} - \eta_t g_k(w_k^{(t)}, \xi_k^{(t)}) & \text{if } (t + 1) \mod \tau \neq 0 \\ \frac{1}{m} \sum_{l \in S(t)} \left( w_l^{(t)} - \eta_t g_l(w_l^{(t)}, \xi_l^{(t)}) \right) = \bar{w}^{(t+1)} & \text{if } (t + 1) \mod \tau = 0 \end{cases}$$ where $w_k^{(t+1)}$ denotes the local model parameters of client $k$ at iteration $t$, $\bar{w}^{(t+1)}$ is the global model, $\eta_t$ is the learning rate, and $g_k(w_k^{(t)}, \xi_k^{(t)}) = \frac{1}{b} \sum_{\xi \in \xi_k^{(t)}} \nabla f(w_k^{(t)}, \xi)$ is the stochastic gradient over mini-batch $\xi_k^{(t)}$ of size $b$ which is randomly sampled from local dataset $\mathcal{D}_k$ of client $k$. The global model, $\bar{w}^{(t)}$, is only updated after every $\tau$ iteration. But, for the purpose of our analysis, we consider a virtual sequence of $\bar{w}^{(t)}$ that is updated at each iteration as follows: $$\bar{w}^{(t+1)} = \bar{w}^{(t)} - \eta_t \bar{g}^{(t)} =: \bar{w}^{(t)} - \frac{\eta_t}{m} \sum_{k \in S(t)} g_k(w_k^{(t)}, \xi_k^{(t)}).$$ 2.2 IMPROVING FL WITH DATA QUALITY ISSUES Biased aggregation. In FL, treating every client equally (e.g., model aggregation of FedAVG) when they have data quality issues may lead to severe performance degradation of the global model Talukder & Islam (2022). Unlike centralized training, FL suffers more from data quality issues. FL clients train their ML models locally, and their model updates can deteriorate significantly due to bad data, eventually affecting the aggregated (with equal weights) global model. To mitigate this, several biased FL aggregation policies have been developed Li et al. (2022); Karimireddy et al. (2020); Talukder & Islam (2022). However, while a biased aggregation improves global performance, it can also be seen as “unfair” to clients who get low or zero weights and, consequently, do not benefit from federation. After all, FL is intended for collaborative training across many participating clients. Nevertheless, we argue in favor of biased aggregation since clients with bad data quality suffer from worse model performance on their local data anyway; including them in model aggregation only harms the global model for everyone else. Self-regulating clients. Implementing the biased aggregation in the central server is inefficient since it requires every client, even those with data quality issues, to complete the local training and model update. To avoid this wasteful (for clients with bad data) model training and updating the central server, we adopt biased aggregation through client selection, i.e., we select the clients with good data to participate in FL training. However, client selection in such a manner requires profiling of clients’ data quality and, hence, if implemented at the central server, breaks clients’ anonymity. Consequently, to maintain client anonymity even through the client selection process, we need the clients to be able to profile and apply the selection to themselves; in other words, clients need to self-regulate their FL participation. Figure 2: Histograms of clients’ test losses on the global model of one training round of FedAVG for different dataset in presence of bad clients. Figure 3: Accuracy of inference loss-based detection (cutoff) of good and bad clients across the training rounds. Challenges. Implementing self-regulating clients with strategic client selection is non-trivial. First, clients only have access to their own data, which may not be enough to apply statistical methods to determine the quality of data effectively. Second, unlike the central server, the clients also do not have access to other clients’ model updates. Third, the selection strategy must be lightweight with low overhead since FL clients are resource-constrained. In what follows, we develop a lightweight client selection policy to address these challenges. ### 3 Our Solution #### 3.1 FedSRC: Federate Learning with Self-Regulating Clients Client classification. To implement a client selection strategy, we first need to define who should be considered as a “bad client” and discarded from model aggregation. Since FL clients’ data is private, we cannot directly assess clients’ data quality. Hence, we classify bad clients based on the impact of their inclusion in the global model as follows: Definition 1 (ε-Bad Client). An FL client is ε-bad if its inclusion in the unbiased global aggregation increases the converged global objective loss by more than ε. The parameter ε in our definition serves two purposes. First, it allows us to set the degree of negative impact that constitutes a bad client. Second, it can absorb the variation of global objective loss (for good client participation) due to non-IID data distribution and the sequence of client participation in the training rounds. Our definition, however, can only be an approximate definition of bad clients since we cannot distinguish the impact of bad data (from bad clients) and non-IID data (from good clients) on the global loss. Nevertheless, our definition serves to develop a client selection strategy for improving the global model performance, albeit there is a possibility (tunable through ε) of treating some good clients with non-IID data as bad clients. Client selection strategy. The client classification in Definition 1 requires N + 1 complete FL training for N clients, rendering it impractical due to its huge computation and communication overheads. Moreover, this classification also breaks client anonymity at the central server. Hence, we need to develop a lightweight approach for identifying good and bad clients at the client level that can serve as a proxy for Defintion 1. We devise FedSRC’s client selection strategy based on our empirical observation that clients with bad quality data suffer from worse performance when vetted against the global model. More specifically, we find that the bad clients tend to have a higher inference loss on the shared global model across training rounds, even when they are included in the model aggregation. To demonstrate this, we run experiments on several data sets with 30% of clients suffering from noisy data (more details... Algorithm 1 FedSRC Input: Initial global model $ w_0 $ 1: for each round $ i = 0 $ to $ t $ do 2: Global model sharing (Server): The central server randomly selects a subset of FL clients and sends them the latest global model. 3: Local inference (Client): The clients run inference on the global model with a random subset of their training data set and send back the inference loss to the central server. 4: Setting participation threshold (Server): The central server collects the test losses from the clients, determines the participation threshold, and then broadcasts the threshold to the participating clients. 5: Self-Regulating participation (Client): The clients check their test loss against the server’s threshold. A client stops training and drops from the FL round if its test loss exceeds the participation threshold. 6: Local training (Client): Participating clients complete the training and send the updated model to the central server. 7: Model aggregation (Server): Central server aggregates the model updates and prepares the global model for the next FL round. 8: end for of data sets and how we add the noise can be found in Section 4.1 and Appendix B. Fig. 2 shows the histograms of clients’ test losses on the global model of one training round of FedAvg for different data sets. We can clearly see that the test losses of the bad clients are distinguishably higher than those of good clients, and we can set a cutoff/threshold to separate the good and the bad clients. To investigate the efficacy of an inference loss threshold-based approach, we then vary the clients’ data quality by changing the amount of noise in the bad clients’ data. We track the accuracy of an inference loss-based detection of good and bad clients and show them in Fig. 3. We see that, even when there is only 10% Gaussian noise added to the bad clients, an interference loss threshold can identify the good and bad clients with ~80% accuracy. The inference on the client side is not computation-heavy and can be done on a randomly chosen subset of a client’s training data. An inference loss-based approach satisfies our requirements for client-side regulation since it can be a reasonably accurate proxy for Definition 1. Consequently, we set our client selection strategy as follows: during FL training, we select the clients with inference loss lower than a given threshold. Threshold-based participation. While we would like the clients to implement our selection strategy and set the participation threshold themselves, they do not have access to the inference losses of other clients. Hence, in FedSRC, we engage the central server to anonymously collect the inference losses of participating clients’ of a particular FL round and determine the participation threshold. This is an additional step we introduce in FedSRC. We defer the discussion of the overhead associated with FedSRC to the end of this section. Setting the participation threshold. Since we have the inference losses of both good and bad clients, the central server can set the threshold autonomously by running unsupervised clustering to break the inference losses into two groups and setting the cluster boundary as the threshold. A computationally lighter alternative to the autonomous approach above is utilizing insight into user data sets, such as the expected percentage of clients with bad data or user-defined participation policy, such as discarding a certain percentage of clients every round. The central server can then set the threshold accordingly to satisfy the externally determined participation percentage. Ideally, with perfect separation between good and bad clients, the user-supplied percentage should match the percentage of bad clients participating in the FL round. While determining the percentage of bad clients in a real-world scenario is non-trivial, we find in our evaluation that overestimating the percentage of bad clients is more favorable than underestimating (Fig. 1(a) in the Appendix C.3). The intuition behind this observation is that including a few bad clients is more harmful to the global model than missing the contribution from a few good clients. After setting the threshold, the central server broadcasts the threshold loss to all clients selected for that specific training round. At no point in FedSRC does the central server need to track the source of the data (i.e., client ID) it collects from clients. We summarize the implementation of FedSRC in Algorithm 1. Overhead of FedSRC’s implementation. FedSRC’s checkpoint adds minor computational overhead to the client as we add one additional inference on a subset of the client’s training data. But the added inference cost is negligible compared to the training cost savings. We can also tap the initial minibatch error before the global weight is modified to estimate the inference loss of the clients when the minibatch is randomly sampled from the training data. In FedSRC, clients also have additional communication with the central server to send their test losses. However, the clients send only one value to the server, hence, the extra communication cost is negligible. Nevertheless, to collect the test losses from all clients reliably, the server may need to offer longer response deadlines, thereby leaving FL clients waiting for the participation threshold. ### 3.2 Theoretical Analysis Here, we prove the convergence of FedSRC and discuss how our client selection policy affects the convergence. To facilitate our analysis, We make the following assumptions: Assumption 1. $ F_1, \ldots, F_k $ are all $ L $-smooth, i.e., for all $ v $ and $ w $, \[ F_k(v) \leq F_k(w) + (v - w)^T \nabla F_k(w) + \frac{L}{2} \\|v - w\\|_2^2. \] Assumption 2. $ F_1, \ldots, F_k $ are all $ \mu $-strongly convex, i.e., for all $ v $ and $ w $, \[ F_k(v) \geq F_k(w) + (v - w)^T \nabla F_k(w) + \frac{\mu}{2} \\|v - w\\|_2^2. \] Assumption 3. For the mini-batch $ \xi_k $ uniformly sampled at random from $ D_k $ of user $ k $, the resulting stochastic gradient is unbiased; that is, $ \mathbb{E}[g_k(w_k, \xi_k)] = \nabla F_k(w_k) $. Also, the variance of stochastic gradients is bounded: $ \mathbb{E}[\\|g_k(w_k, \xi_k) - \nabla F_k(w_k)\\|^2] \leq \sigma^2 $ for all $ k = 1, \ldots, K $. Assumption 4. The stochastic gradients’ expected squared norms are uniformly bounded, i.e., $ \mathbb{E}[\\|g_k(w_k, \xi_k)\\|^2] \leq G^2 $ for $ k = 1, \ldots, K $. Denote by $ B $ the set of $ \epsilon $-bad clients for a fixed $ \epsilon > 0 $, and let $ G $ be the set of good clients (i.e., those that are not $ \epsilon $-bad. By Definition 1 these sets are fixed. Since our assumption is that there are bad clients whose updates adversely affect the global model, our convergence analysis takes this into account by separating the good and bad clients in all terms defined below. We utilize similar ideas to Cho et al. (2022) by defining a local-global objective gap and a skewness of biased selection of clients who send their model update to the central server. In contrast to prior work, our definitions are in terms of the good (or potentially bad) clients, which allows us to understand the effect of our client selection strategy in the context of our problem setup. We define the global loss for two client sets: $ F_g(w) = \sum_{k \in G} p_k F_k(w) $ for the good clients in $ G $, and similarly define $ F_b $ for the bad clients in $ B $. The optimal global losses for good and bad clients are $ F^_g = \min_w F_g(w) $ and $ F^_b = \min_w F_b(w) $. Additionally, we define the global model optimum $ w^* = \arg \min_w F(w) $, and the client-level optima $ w^_k = \arg \min_w F_k(w) $ for each client $ k $. Definition 2 (Local-Global Objective).* We define the local-global objective gap for the set of good clients as follows: \[ \Gamma_g = F^_g - \sum_{k \in G} p_k F^_k = \sum_{k \in G} p_k (F_k(w^) - F_k(w^_k)) \geq 0. \] For highly non-iid data, $ \Gamma_g $ is non-zero, and larger $ \Gamma_g $ implies higher data heterogeneity. $ \Gamma_g = 0 $ implies consistent optimum models among the clients and the central server. Definition 3 (Selection Skewness). Let $ w $ be the current weights of the global model, and $ \pi $ be any client selection strategy. We let $ S(\pi, w) $ denote the selected clients using selection strategy $ \pi $ and define the skewness of the client selection strategy $ \pi $ for good and bad clients via \[ \rho_g(S(\pi, w), w') = \frac{\mathbb{E}_{S(\pi, w)} \left[ \frac{1}{p} \sum_{k \in S(\pi, w) \cap G} (F_k(w') - F^_k) \right]}{F_g(w') - \sum_{k \in G} p_k F^_k}, \] \[ \rho_b(S(\pi, w), w') = \frac{\mathbb{E}_{S(\pi, w)} \left[ \frac{1}{q} \sum_{k \in S(\pi, w) \cap B} (F_k(w') - F^_k) \right]}{F_g(w') - \sum_{k \in G} p_k F^_k}, \] where $ p $ is the number of selected good clients, $ q $ is the number of selected bad clients, and $ m = p + q $. Above, the current global model weights $ w $ influences the selection strategy $ \pi $, while $ w' $ is the global model weight at which the selection skewness is evaluated. $ \mathbb{E}_{S(\pi, w)}[\cdot] $ represents the expectation over the randomness from the selection strategy $ \pi $ in determining $ S(\pi, w) $. Note that the denominator of both $ \rho_g $ and $ \rho_b $ are the same, and represent the current gap between the local and global models for good clients only. This is because we do not wish to select the bad clients, and their local-global objective gap should not influence our convergence analysis. The following terms are useful for providing a concrete error bound in the main theorem below. \[ \tilde{\rho}_g = \min_{w,w'} \rho_g(S(\pi, w), w'), \] \[ \tilde{\rho}_g = \max_w \rho_g(S(\pi, w), w^). \] We define $ \tilde{\rho}_b $ and $ \tilde{\rho}_h $ similarly. Theorem 1.* Under the Assumptions stated above, for a learning rate $ \eta_t = \frac{1}{\mu(t+\gamma)} $ with $ \gamma = \frac{4L}{\mu} $, and for client selection strategy $ \pi $ that selects the same number of good and bad clients ($ p $ and $ q $, respectively) after time $ T $, the error of federated learning with self-regulating clients satisfies, for every $ t \geq T $, \[ \mathbb{E}[F(\bar{w}(t))] - F^* \leq \frac{1}{(t + \gamma)} \left[ \frac{4L(32\tau^2G^2 + \sigma^2/m)}{3\mu^2\tilde{\rho}_g} + \frac{8L^2\Gamma_g \tilde{\rho}_b}{\mu^2 \tilde{\rho}_g} + \frac{L(\gamma + 1)(\\|\bar{w}(1) - w^\\|^2)}{2} \right] \] Vanishing Term \[ + \frac{8L\Gamma_g}{3\mu} \left( \frac{p\tilde{\rho}_g + q\tilde{\rho}_b}{m\tilde{\rho}_g} - 1 \right) \] Bias Term To the best of our knowledge, Theorem 1 provides the first theoretical bound of convergence for federated averaging in the presence of bad clients. The complete proof of the theorem can be found in Appendix A. Effect of the client selection strategy.* First, note that for an unbiased client selection strategy (clients participate in the model update uniformly at random), both good and bad clients will provide a model update. As the training of the model progresses, the loss of the good clients decreases, whereas the loss of the bad clients does not improve. This results in a decreasing $ \rho_g $, but increasing $ \rho_b $, both of which negatively affect both the rate of convergence of the vanishing term and the magnitude of the bias term in equation 7. A biased client selection strategy that is able to discard clients with higher loss will ensure an increase in the number of good clients selected and decrease in number of bad clients selected, which reduces the value of $ \rho_b $ and increases the value of $ \rho_g $, resulting in both faster convergence and smaller bias. Reducing $ \rho_b $ and increasing $ \rho_g $ for faster convergence. Under our model for good and bad clients, if our selection strategy prioritizes client updates for those with small testing loss value $ F_k $, the number of bad clients selected in $ S(\pi, w) $ will be smaller, which results in larger $ \rho_g $ but smaller $ \rho_b $. Consequently, the first two terms in the vanishing term of Theorem 1 will be smaller leading to faster convergence compared to an unbiased selection strategy. Bias Term. Similarly, a client selection strategy that prioritizes lower-loss clients will reduce the bias term as $ p $ increases. Indeed, $ \tilde{\rho}_b $ should be larger than $ \tilde{\rho}_g $ for a given selection strategy, so decreasing $ q $, the number of bad clients selected, decreases the numerator significantly, even as $ p $ increases. Likewise, as $ p $ increases based on the selection strategy, the denominator increases as well, thereby decreasing the bias term. 4 EVALUATION 4.1 SETTINGS Dataset and model description. We utilize four prominent datasets: MNIST [LeCun et al., 2010], CIFAR10 [Krizhevsky, 2009], FEMNIST [Caldas et al., 2018], and SHAKESPEARE [Caldas et al., 2018] which are widely utilized in the literature [McMahan et al., 2017; Li et al., 2020c]. For the MNIST and CIFAR10 datasets, we create non-IID settings by assigning each client a dominant class of 50% data and the remaining classes with the rest of the data. FEMNIST and SHAKESPEARE datasets are naturally non-IID. For the handwriting classification of MNIST and FEMNIST, we implement multilayer perceptron (MLP). Convolutional Neural Network (CNN) is used for CIFAR10 image classification, and Recurrent Neural Network (RNN) is used for the next character prediction in SHAKESPEARE. More details of our model description and dataset can be found in Appendix B. Evaluation scenarios. We consider three scenarios reflecting potential data corruption due to sensor quality, malfunction, and aging. Label shuffling: It can be referred to as random sensor malfunction, leading to assigning random labels to data. Label flipping: It refers to mislabeling data, leading to the same mislabel across all the client data. Noisy data: It results from hardware quality in the feature space. To simulate this, we added Gaussian noise to the feature and then clipped the value within the desired feature space level. As the default configuration for our evaluation, we use a mix of 70% good clients with 30% bad clients. The bad clients are equally divided among the three cases. More details of our evaluation scenarios can be found in Appendix B. Benchmark algorithms. To assess the performance of FedSRC, we compare it with the following benchmark algorithms. FedAVG [McMahan et al., 2017]: the standard federated averaging technique which assigns client weights based on dataset size. Median [Yin et al., 2018]: a Byzantine robust aggregation rule that independently aggregates each model parameter. For each $i^{th}$ parameter, the server sorts the $i^{th}$ parameters of the selected clients and takes the median as the global parameter. Trimmed Mean [Yin et al., 2018]: another Byzantine robust aggregation rule that independently aggregates each model parameter. It sorts the parameters and removes a percentage of the largest and smallest values, then averages the remaining for each parameter. FedASL [Talukder & Islam, 2022]: automatically assigns weights to clients based on the median of their training losses. Clients within a predefined “good zone” around the median have higher contributions to the global model, while those outside this zone have inversely proportional contributions of the distance from the median. Krum [Blanchard et al., 2017]: operates by calculating the Euclidean distance norms between each client’s model weights and those of other clients. It removes the highest value for each client, averages the rest, and selects the client with the lowest scores as the next global model. 4.2 RESULTS Comparison with the benchmark algorithms. We compare FedSRC with the benchmark algorithms under our default setting. Here, we block 30% clients in FedSRC, Trimmed Mean, and Krum, while FedASL discards clients falling outside one standard deviation (i.e., discards $\sim$32% clients). FedAVG does not discard any clients. We show the test loss of the global model against an uncorrected test data set in Fig. 4. The comparison of their Figure 5: Client side savings of FedSRC. accuracy can be found in Fig. 6 in Appendix B. Our experiments reveal that FedSRC consistently outperforms other benchmark algorithms resulting in better global performance. For instance, for the FEMNIST data set, after 300 training rounds, FedSRC has 33%, 33%, 33%, 28% and 34% lower loss compared to FedASL, Trimmed Mean, Median, Krum, and FedAVG. Extended evaluation of FedSRC can be found in Appendix B for IID and extreme Non-IID cases for MNIST and CIFAR10 and in all the cases FedSRC outperforms the benchmark algorithms. Computation and communication savings. To evaluate the client-side computation and communication savings achieved by FedSRC, we conduct experiments across various proportions of corrupted clients for different datasets. Clients need to do a forward pass (inference loss) of the first batch only to make the decision of participation. Fig. 5(a) demonstrates that FedSRC yields substantial savings of up to 55% in local computational expenses for higher proportions of malicious clients. Turning to communication savings, when a client abstains from participating, the model upload cost for the trained model is spared. Meanwhile, there is negligible extra communication for sending the test loss to the server. Fig. 5(b) shows that FedSRC can save up to 30% of the communication cost with 60% bad clients. Note here that, unlike computation savings, the communication savings do not depend on the dataset. Extended results. We also evaluate FedSRC’s integration with centralized FL approaches and varying degrees of data quality issues. We defer these results to the Appendix C due to space limitations. ## 5 RELATED WORK The performance of FL deteriorates in the presence of corrupted clients [Tolpegin et al., 2020]. Notably, FedAVG lacks mechanisms to mitigate the impact of bad clients [Fang et al., 2020; Tolpegin et al., 2020]. Consequently, various FL algorithms have emerged to defend against data corruption [Pillutla et al., 2022; So et al., 2020; Sattler et al., 2020]. Statistics-based algorithms. Krum [Blanchard et al., 2017] selects a global model based on similarity to local models. Bulyan [Guerrraoui et al., 2018] augments Krum with a Trimmed Mean [Yin et al., 2018] variant. However, Bulyan’s computational burden arises from dual computations in each training round. Byzantine Robust Algorithms. Median [Yin et al., 2018] sorts outliers from individual models before global averaging. Geometric Median (GM) [Chen et al., 2017; Pillutla et al., 2022] is another technique, but computational intensity hampers its edge feasibility. Client Selection Algorithms. Loss-based client selection methods, like AFL [Goetz et al., 2019] and Power-of-Choice [Cho et al., 2022], assess and prioritize high-loss clients. However, these methods compromise privacy due to ID tagging. Re-weighting Algorithms. Zhao et al. [2020] adjusts aggregation weights based on cross-validation. Zhao et al. [2019] and Talukder & Islam [2022] reweight models using auxiliary data, but online detection raises privacy concerns. Other Data Poisoning Approaches. Some methods rely on trusted client subsets [Cao et al., 2021; Han & Zhang, 2020; Li et al., 2020a; Sattler et al., 2020; Ghosh et al., 2020] or cluster-based approaches [Sattler et al., 2020] for defense. Yet, trustworthiness and communication constraints pose challenges. In contrast, FedSRC can manage client-side data corruption without reliance on validation datasets or identity disclosure. Moreover, client-side blocking reduces local computation and communication costs. To the best of our knowledge, FedSRC is the pioneering algorithm that addresses data corruption directly from the client side. ## 6 CONCLUDING REMARKS In this paper, we presented FedSRC, a novel solution for handling data corruption from the client side to enhance the efficiency of federated learning. Our approach saves communication and computation costs while enhancing global model accuracy and preserving client anonymity. To the best of our knowledge, this is the first attempt to regulate client participation from the client side. Limitations. FedSRC relies on client-level statistics to implement its checkpoint, and therefore, it cannot operate if a significant portion of clients is corrupted. 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ghyeMoj1gK	My main concern with the work is its practical relevance. The authors claim to put clients at the forefront. But, if the server estimates the distribution of the clients, this is a violation of the client data privacy? How can this be reconciled since privacy is one of the privacy motivations for doing FL in the first place?	CLIENT-CENTRIC FEDERATED LEARNING UNDER DYNAMIC MIXTURES OF DISTRIBUTIONS Anonymous authors Paper under double-blind review ABSTRACT Conventional federated learning (FL) frameworks follow a server-centric model where the server determines session initiation and client participation. We introduce Client-Centric Federated Learning (CCFL), a novel client-centric FL framework that puts clients as the driving role of FL sessions. In CCFL, each client independently and asynchronously updates its model by uploading a locally trained model to the server and receiving a customized model tailored to its local task. The server maintains a repository of cluster models, iteratively refining them using received client models. Our framework accommodates complex dynamics in clients’ data distributions, characterized by time-varying mixtures of cluster distributions, enabling rapid adaptation to new tasks with high performance. We propose novel strategies for accurate server estimation of clients’ data distributions. CCFL offers clients complete autonomy for model updates, enhances model accuracy, and significantly reduces client computation, communication, and waiting time. We provide a theoretical analysis of CCFL’s convergence. Extensive experiments across various datasets and system settings highlight CCFL’s substantial advantages in model performance and computation efficiency over baselines. 1 INTRODUCTION Federated Learning (FL) [McMahan et al., 2017] is a distributed learning framework that allows for collaborative training of a global model across multiple clients while keeping their raw data local. To tackle the problem of clients’ non-iid data distributions, personalized FL [Tan et al., 2022] frameworks have emerged to offer each client a tailored model. However, in nearly all works within personalized FL, and even in the broader FL context, the central locus of control invariably resides with the server. That is, the server typically initiates training sessions and determines which clients should participate and when. Astonishingly, the following question has been conspicuously absent from the discourse: Why should a client always comply with the server’s directives regarding model uploads? Are there not situations where network connectivity issues arise, or, indeed, a client simply does not want to share the model when server calls? In this paper, we raise a brand-new FL framework: Client-Centric Federated Learning (CCFL), which empowers each individual client to assume a dominant role in the FL process. In CCFL, each client device collects data from a mixture of distributions, whose mixing ratios may vary over time. Once a distribution shift is observed, the client may seek help from the server, who acts as a service provider, in updating its local model to match the new distribution. In real-life scenarios, this setting is commonplace. Consider a skincare maintenance application, where users’ skin types exhibit complexity — perhaps featuring a combination of oiliness and dryness in different areas of skin, reflecting a mixture of distributions. Additionally, users’ skin conditions may vary with seasons, leading to shifts in distributions. Another example is a retail chain with various branches, each of which sell commodities of different store categories. The commodities offered by these branches may evolve based on changing customer preferences, creating a dynamic mixture of various distributions. Note that in CCFL, each client possesses complete autonomy in deciding when to update its model, and the servers plays an assistive role for the clients to accommodate their new distributions. To tackle clients’ data variations across multiple distributions, CCFL adopts the clustered FL setting where $K$ base cluster models are maintained at the server [Sattler et al., 2020a], which are used to update clients’ models. In existing clustered FL works, a crucial consideration is to measure the data distributions of clients. Many works distribute all cluster models to clients, leaving it to clients to determine the distribution based on local empirical loss (Ghosh et al., 2020; Mansour et al., 2020; Ruan and Joe-Wong, 2022). However, such an approach poses several challenges. Firstly, it places a significant communication burden to send all the cluster models. Secondly, it imposes substantial computational demands on clients, requiring them to calculate losses for each cluster and make comparisons. Some other approaches leverage distances between uploaded models to form client groups (Duan et al., 2021a), imposing impractical synchronization requirements on clients for data uploads. In sharp contrast, as illustrated in Figure 1, CCFL assigns the task of evaluating client data distribution to the server. Based on the model uploaded by a client, the server analyzes its data distribution, and updates the cluster models. Subsequently, the server generates a personalized model and sends it to the client. This significantly simplifying clients’ communication and computation compared with previous clustered FL solutions. In the context of above mentioned clustered FL, and building upon the client-centric foundation, we develop an asynchronous CCFL framework that focuses on maximizing clients’ performance and minimizing clients’ complexity. Notably, we introduce an effective newcomer cold start mechanism, a feature conspicuously absent in the majority of related works (Duan et al., 2021a; Zeng et al., 2023). Furthermore, our framework exhibits adaptability in addressing client distribution drift, a challenge specifically addressed in only one previous study (Duan et al., 2021b) within the context of clustered FL. CCFL is the first clustered FL framework that focuses on client’s autonomy, efficiency, and performance. Compared to existing clustered FL works, client involvement remains minimal, as they only need to conduct local model training—a computationally modest task; users’ communication overhead is equally reduced, with solely uploading and downloading one single model, and when to upload is left at their discretion. We provide convergence analysis that theoretically validates our framework. Extensive experiments over different datasets and network settings attest to the outstanding performance of CCFL. Notably, it significantly alleviates both communication and computational costs compared to existing works. 2 RELATED WORK Clustered Federated Learning (clustered FL). Hard clustering algorithms assume clients in the same group have identical data distribution (Briggs et al., 2020; Ghosh et al., 2020; Mansour et al., 2020); while soft clustering methods assume the data of each client follows a mixture of multiple distributions (Ruan and Joe-Wong, 2022; Li et al., 2021). In most cases, expectation-maximization (EM) methods are used to compute clients’ distribution (Long et al., 2023; Ma et al., 2022; Ghosh et al., 2022), and global updates leverage methods based on FedAvg (Briggs et al., 2020). Some works add proximal terms on clients’ objectives for personalization (Tang et al., 2021). Asynchronous Federated Learning (asynchronous FL). Asynchronous FL operates on resource-constrained devices (Xu et al., 2021). In typical asynchronous setups, the central server conducts global aggregation immediately upon receiving a local model (Xie et al., 2019; Wang et al., 2022; Chen et al., 2020), or a set of local models (Nguyen et al., 2022; Wu et al., 2020). These asynchronous clients may be grouped into tiers for updating based on factors like staleness or model similarities (Park et al., 2021; Wang and Wang, 2022), referred to as semi-asynchronous. However, this clustering typically contributes to a single global model, and sometimes, the server still selects the clients (Zhang et al., 2021). Existing clustered FL frameworks primarily operate within a synchronous setting. In the context of asynchronous FL, clients are sometimes grouped only to control staleness. Our framework is the first, to the best of our knowledge, to integrate clustered FL within an asynchronous setting. User-centric FL frameworks. Few works have studied FL from a comprehensive user’s perspective. Mestoukirdi et al. (2021, 2023) claim to be user-centric, but are indeed personalized FL frameworks dealing with communication burdens. In Khan et al. (2023), the authors point out that existing FL works take away clients’ autonomy to make decisions themselves, and propose a token-based incentive mechanism that rewards personalized training. However, this work fails to consider the asynchrony among clients, making it insufficient to provide full autonomy to clients. Note that the shift in clients’ distribution is distinct from Federated Continual Learning (FCL) (Yoon et al., 2021). which primarily aims to minimize catastrophic forgetting. Our focus lies solely in enabling clients to seamlessly adapt their models to new data during distribution shifts. 3 Problem Definition Consider an FL system with one central server and many distributed clients. The server maintains $K$ cluster models, each with a validation dataset $D_k$ corresponding to different distributions $P_1, \ldots, P_K$. The value of $K$ is determined a priori, according to the type of service (e.g., genders or ethnicities in the skincare service), or is deducted from a small amount of validation data collected in advance at the server. Given a loss function $l(w; x, y)$, each cluster $k \in [K]$ aims to find an optimal model $w_k$ that minimizes the objective $$F_k(w_k) = \mathbb{E}_{(x,y) \sim P_k}[l(w_k; x, y)].$$ The training takes $T$ global epochs. For each epoch $t \in [T]$, some client $m$ collects local data following a mixture of distribution $P_m^t = \sum_{k=1}^{K} \mu_{mk}^t P_k$, with $\mu_{mk}^t \in [0, 1]$ and $\sum_{k=1}^{K} \mu_{mk}^t = 1$. Here $\mu_{mk}^t$ is the importance weight of cluster $k$ to client $m$ at epoch $t$. The importance weights may vary over time, and are unknown to the client. Each time when client $m$’s data distribution shifts, it may choose to fit the local model $w_m^t$ to the new distribution, by optimizing the local objective $$h_m^t(w_m^t; w_m^\tau) \triangleq \frac{1}{m_t} \mathbb{E}_{(x^i,y^i) \sim P_m^t} \sum_{i=1}^{m_t} l(w_m^t; x^i, y^i) + \frac{\rho}{2} \\| w_m^t - w_m^\tau \\|^2.$$ Here $m_t$ is the number of data samples; $\rho$ is some scaling parameter; $\tau < t$ is the last epoch when client $m$ uploads its model $w_m^\tau$ to the server, and the server returns a model $w_m^\tau$. 4 Client-Centric Federated Learning Figure 2: CCFL workflow. Client $m$ uploads model and timestamp tuple $(w_m, \tau)$ to the server. Server labels it at epoch $t$. In this figure, server estimates little distribution of $P_1$, and would not update cluster 1. An aggregated model based on client’s estimated distribution is sent back after update. 4.1 Client Update The user-centric architecture of CCFL empowers users to initiate the uploading process autonomously. To begin, client $m$ receives an initialization tuple from the server, comprising the global model and a timestamp, denoted as $(w, t)$. Subsequently, the user adapts the global model $w$ to its own dataset to obtain a personalized model $w_m$. After initialization, client $m$ retains the discretion to select when to upload the tuple of their local model and timestamp $(w_m, t)$, and then awaits the server’s response, which serves to enhance their local performance. Client Data Shifts. We assume the distribution shifts of clients between different epochs, i.e. for client $m$, it is possible that $\mu_{mk}^t \neq \mu_{mk}^{t'}$ for all $t \neq t', t, t' \in [T]$. Training and Uploading. In order to establish a mutually beneficial system, clients are required to perform local training prior to model uploading (refer to Algorithm 2). The decision of when to upload rests entirely with the clients themselves. Furthermore, clients are advised to do training and uploading when there are shifts in data distribution to better align with the new data stream; or when a substantial amount of time has elapsed since the last upload to ensure synchronization with the server’s state. Through this preliminary training session before uploading, the server gains valuable insights from the clients, facilitating the performance of cluster models. Scalable Client Integration. We do not presuppose a fixed total number of clients. Our system is designed to be fully open and dynamic. A new user simply fetches an initialization tuple from the server, and starts the training and uploading process, seamlessly integrating into the system. 4.2 SERVER UPDATE Algorithm 1: DistributionEstimation & UpdateRaTioCompute Function DistributionEstimation $(w_m, w_0, ..., w_K, D_0, ..., D_K)$: \[ \text{foreach } k \in [K] \text{ do} \\ l_k \leftarrow F(w_{tk}; D_k); d_{1k} \leftarrow \\|F(w_{tk}; D_k) - F(w_{tm}; D_k)\\|_1; d_{2k} \leftarrow \\|w_m - w_k\\|_2 \\ /* l_{bar}, d_{1bar}, d_{2bar} are hyperparameters to control the scale / \\ l_k \leftarrow l_k - l_{bar}, d_{1k} \leftarrow d_{1k} - d_{1bar}, d_{2k} \leftarrow d_{2k} - d_{2bar} \\ / hyperparameters c_1, c_2, c_1 + c_2 \in [0, 1], u_{tk}^t \in [0, 1], \sum_k u_{tk}^t = 1 / \\ u_{tk}^t \leftarrow \frac{1}{K-1} \cdot \left( c_1 \cdot \frac{\sum_{i \neq k} l_i}{\sum_i l_i} + c_2 \cdot \frac{\sum_{i \neq k} d_{1i}}{\sum_i d_{1i}} + (1 - c_1 - c_2) \cdot \frac{\sum_{i \neq k} d_{2i}}{\sum_i d_{2i}} \right) \\ / A > 0 is the amplifier, helping magnify the difference of distribution estimation among clusters. / \\ u_{t0}^t, ..., u_{tK}^t \leftarrow \text{softmax}(u_{t0}^t \cdot A, ..., u_{tK}^t \cdot A) \\ \text{return } u_{t0}^t, ..., u_{tK}^t \] Function UpdateRaTioCompute $(u_{t0}^t, ..., u_{tK}^t, \alpha_0, \tau)$: \[ \text{foreach } k \in [K] \text{ do} \\ \alpha_{10}, ..., \alpha_{1K} \leftarrow u_{t0}^t, ..., u_{tK}^t \\ / If distribution content is less than preset bar $\alpha_{1bar}$, do not update the cluster. / \\ \alpha_{1max} \leftarrow \max(\alpha_{1k}). \text{if } \alpha_{1k} < \alpha_{1bar} \text{ then } \alpha_{1k} \leftarrow 0; \text{else then } \alpha_{1k} \leftarrow \alpha_{1k}/\alpha_{1max} \\ / a, b are hyper-parameters to control staleness. / \\ \text{if } t_k - \tau < b \text{ then } \alpha_{2k} \leftarrow 1; \text{else then } \alpha_{2k} \leftarrow 1/(a(t_k - \tau) + 1) \\ / Hyper-parameter $\alpha_0$ governs the maximum extent of local model modification to the global cluster model. / \\ \alpha_{tk}^t \leftarrow \alpha_0 \cdot \alpha_{1k}\alpha_{2k} \quad / \alpha_{tk}^t \in [0, \alpha_0] / \\ \text{return } \alpha_{t0}^t, ..., \alpha_{tK}^t \] Throughout the entire process of CCFL process, the server passively waits for the clients’ uploads. Upon receipt of an upload, the server first updates and labels the client with global epoch $t$, then the server initiates a two-step evaluation process. Firstly, it checks if the client is too stale, i.e., when client $m$ uploads $(w_m; \tau)$ at epoch $t$. If $t - \tau > \tau_0$ ($\tau_0$ is a preset staleness threshold), the server refrains from updating and instead transmits a personalized model aggregated by cluster models. Otherwise, the server proceeds to estimate client $m$'s data distribution. Subsequently, it updates each cluster using a cluster-specific updating parameter and dispatches the personalized model back to the client. Distribution Estimation. For each cluster $k$, a small public dataset $D_k$ derived from $P_k$ is stored at the server to do the clients’ distribution estimation. Upon client $m$ uploading $w_m$ at epoch $t$ (referred to as $w_{tm}$ for clarity), the estimation of client $m$'s data distribution hinges on several components, including $w_{tk}$, the latest models of clusters denoted as $w_{tk}^t (k \in [K])$, where $t_k$ is the last epoch when cluster $k$ is updated, and the validation dataset $D_k$. For distribution $k$, this estimation involves two distinct considerations. First, it takes into account the loss incurred by $w_{tm}$ on distribution $P_k$, which is quantified by the empirical loss on validation dataset $D_k$, i.e., $F(w_{tm}; D_k) = E_{(x,y) \sim D_k} l(w_{tm}; x, y)$. If $F(w_{tm}; D_k) < F(w_{tm}; D_{k'})$, it signifies that client $m$'s distribution $P_m^t$ may have a higher composition of distribution $P_k$ compared to $P_{k'}$. Second, if client $m$ is not too stale ($t - \tau < \tau_0$), it is likely to resemble the latest global cluster model. This similarity is discernible either through the loss difference between the latest cluster model and the client’s model on validation data, denoted as $\\|F(w_{tk}^t; D_k) - F(w_m^t; D_k)\\|_1$, or through the model distance, such as the $l_2$-norm distance, $\\|w_{tk}^t - w_m^t\\|_2$. Smaller values of these metrics signify a higher degree of similarity. Drawing from these observations, we employ Algorithm 1 to calculate the distribution estimation $u_{mk}^t, k \in [K]$. Based on the analysis presented in Section 5.2, we can reasonably posit that $u_{m0}^t, \ldots, u_{mK}^t$ serve as accurate estimations of the true importance weights $\mu_{m0}, \ldots, \mu_{mK}$. It’s important to note that due to the potential distribution shifts on the client side, the server must recompute these weights every time a client initiates an upload. Clusters Updating.* The server updates the model of each cluster $k$ as follows $$w_k^t = (1 - \alpha_{mk}^t)w_k^{tk} + \alpha_{mk}^tw_m^t,$$ where $\alpha_{mk}^t$ is the updating ratio contributed by client $m$ to cluster $k$ at epoch $t$. The calculation of $\alpha_{mk}^t$ considers whether the client model predominantly originates from distribution $P_k$ (by the estimated proportion $u_{mk}^t$), and whether the client model is too stale (by $t_k$ and the timestamp $\tau$ to assess the degree of staleness). Detailed procedures for computing the updating ratio are elucidated in Algorithm 1. Note that only clusters with a non-zero updating rate ($\alpha_{mk}^t > 0$) undergo updates facilitated by client $m$’s model $w_m^t$. Aggregation and Feedback. If client $m$ is not so stale ($t - \tau < \tau_0$), when all corresponding models finish updating, the server sends back the aggregated model $w_m^t = \sum_{k=1}^K u_{mk}^tw_k^t$ to client $m$. Otherwise, the new distribution would not be measured, and the server only sends back model $w_m^t = \sum_{k=1}^K u_{mk}^\tau w_k^t$ based on the measures at the last upload epoch $\tau$. Algorithm 2: CCFL Input: Server pre-trained model $w_0^k$, server validation dataset $D_k \sim P_k$ ($k \in [K]$), staleness threshold $\tau_0 < T$, server update threshold $\alpha_0 \in (0, 1)$ Output: Local model parameter $w_m$, global model parameter $w_k$ Initialization: Server sends $(w_0^0, 0)$ to each client, $w_0 = \frac{1}{K}\sum_{k=1}^K w_0^k$. Global epoch $t \leftarrow 0$. Run Client() thread and Server() thread asynchronously in parallel. Thread Server(): ```plaintext foreach k ∈ [K] do t_k ← 0. while t ≤ T do while no client uploads do /* Server passively waits for upload from clients. / Wait for client update. if client m uploads (w_m, τ) then t ← t + 1; w_m^t ← ServerUpdate (w_m, τ, t); send (w_m^t, t) to client m. ``` Thread Client():* ```plaintext foreach client m in parallel do Receive pair (w_m, 0) from server. set local model w_m ← w_m, local timestamp t_m ← 0. while active do if choose to upload then Define h_m(w_m; w) = f_m(w_m; D_m) + $\frac{\rho}{2}\\|w_m - w_m\\|^2$ foreach local iteration h do w_{m,h} ← w_{m,h-1} − γ∇h_m(w_{m,h-1}; w_m) /* learning rate γ / Upload (w_m, t_m) and wait for server response (w_m, t); t_m ← t ``` Function ServerUpdate (w_m, τ, t):* ```plaintext /* If client deprecated, do not update global model. / if t − τ > τ_0 return w_m^t = \sum_{k=1}^K u_{mk}^\tau w_k^t. u_{m0}^t, ..., u_{mK}^t ← DistributionEstimate(w_m, w_0, ..., w_K, D_0, ..., D_K) α_{m0}^t, ..., α_{mK}^t ← UpdateRatioCompute(u_{m0}^t, ..., u_{mK}^t; α_0, τ) foreach k ∈ [K] do if α_{mk}^t > 0 then w_k^t ← (1 − α_{mk}^t)w_k^{tk} + α_{mk}^tw_m, t_k ← t return w_m^t = \sum_{k=1}^K u_{mk}^t w_k^t. ``` The entire workflow of CCFL is depicted in Figure 2 and described in Algorithm 2. 4.3 Convergence Analysis We make some universal assumptions to assist the convergence analysis of CCFL. Assumption 1.* $F_k$ is $L_k$-smooth and $\mu_k$-strongly convex and for some $L_k, \mu_k > 0$ for all $k \in [K]$. Assumption 2. Each client executes at least $H_{\text{min}}$ and at most $H_{\text{max}}$ local updates before updating. Assumption 3. Denote $h^t_m(w; w) = f(w) + \frac{\rho}{2} \\|w - w\\|^2$, where $w, w \in \mathbb{R}^d$ are respectively local and global models, we assume $\forall m, \forall t \in T$, we have $\\|\nabla f^t_m(w)\\|^2 \leq V_1$ and $\\|\nabla h^t_m(w; w)\\|^2 \leq V_2$. Assumption 4. The distance of different clusters are bounded by $a_0 \Delta \leq \\|w_k^* - w_{k'}^\\| \leq \Delta$ for all $k \neq k', k, k' \in [K]$, where $\Delta \geq 0, 0 \leq a_0 \leq 1$ and $w_k^ := \arg\min_{w_k} F_k(w_k)$. Assumption 5. We assume there is always an upper bound on the $l_2$-norm of cluster $k$’s model $w_k$, i.e., $\forall k \in [K], \\|w_k\\| \leq a_k \Delta, a_k > 0$. Theorem 1. With above assumptions, for a small constant $\epsilon > 0$, assume we choose $\rho \geq \frac{2V_1 + \frac{\rho}{2}\\|w-w\\|^2 + \frac{\rho}{2}\\|w-w\\|^2(1+V_1)\epsilon}{2\\|w-w\\|^2}$ for all possible $w, w$ in global and local iterations, then if cluster $k$ undergoes $S_k$ updates, Algorithm 2 would converge to: $$E[\\|\nabla F_k(w)\\|^2] \leq \frac{E[F_k(w_0) - F_k(w_{S_k})]}{\alpha_0 \gamma S_k H_{\text{min}}} + \left(\frac{L_k + \rho H_{\text{max}} + \frac{\rho}{2} H_{\text{max}}^2}{\epsilon H_{\text{min}}}\right) \gamma H_{\text{max}} V_2 + \sqrt{V_1 \left(2 \sum_{i=1}^{K} a_i + (2K+1)a_k + K\right) \Delta} + \frac{(L_k + \rho)(2 \sum_{i=1}^{K} a_i + (2K+1)a_k + K)^2 \Delta^2}{\gamma \epsilon H_{\text{min}}}$$ Discussions. The theorem indicates that if a client’s model $w$ undergoes continuous training on data from distribution $k$, meaning that a portion of the client’s data consistently originates from distribution $k$, then the $l_2$-norm of the gradient of the model loss on cluster $k$ will converge to a specific point (always less than $\infty$). For any data distribution $k$ continuously sampled by a client, the proposed algorithm guarantees the client’s model to have good performance on this particular distribution $k$. 5 Experiments 5.1 Setup We create FL clustered datasets via three commonly used public datasets: FashionMNIST (Xiao et al., 2017), CIFAR-100 (Krizhevsky et al., 2009), MiniImageNet-100 (Winyals et al., 2016). In order to simulate different distributions, we augment the datasets using rotation, and create the Rotated FashionMNIST, Rotated CIFAR-100 and Rotated MiniImagenet-100 datasets. Each dataset is applied by $i \times \frac{360}{K} (i = 0, ..., K - 1)$ degrees of rotation to the images, resulting in $K$ clusters. In our experiment, we try $K = 2, 3, 4, 6$ to simulate an FL setup with clear cluster structure. Rotated FashionMNIST: Each rotated cluster has 60,000 training images and 10,000 testing images containing 10 classes. Rotated CIFAR-100: Each rotated cluster has 50,000 training images and 10,000 testing images containing 100 classes. Rotated MiniImagenet-100: Each rotated cluster has 48,000 training images and 12,000 testing images containing 100 classes. 2,000 images of testing images from each cluster of each dataset are used to pre-train the cluster models. Training models structures are listed in Appendix A.1. All the experiments are conducted using PyTorch version 1.9 on a single machine equipped with two Intel Xeon 6226R CPUs, 384GB of memory, and four NVIDIA 3090 GPUs. We compare our CCFL method with below baseline methods: - FedSoft-Async. An asynchronous adaptation of the soft-clustering baseline Ruan and Joe Wong (2022) is employed. Clients receive all global models from the server, and distribution is assessed by identifying the model with the smallest loss for each data point. Distribution values $\mu_{m0}, \ldots, \mu_{mK}$ are transmitted to the server alongside the local model for global updates. The clusters’ update ratio, denoted as $\alpha_{mk}^t$, integrates the locally computed distribution $\mu_{mk}$ and staleness, given by $\alpha_{mk}^t := \alpha_0 \cdot \mu_{mk} \alpha_{2k}$, with $\alpha_{2k}$ computed in a similar manner as in CCFL. As there are no existing works addressing both asynchrony and soft-clustering concurrently in FL, FedSoft-Async serves as the most suitable baseline method. - Local. The clients only do local optimizations and never upload the local models. In the initialization phase, clients perform computations using the averaged cluster model. Each client possesses a dataset ranging from 500 to 2000 data points, with 40% to 90% originating from a primary distribution and the remainder from other cluster distributions. Upon completing the initialization, clients autonomously decide when to upload their models. After uploading, an accuracy evaluation is conducted initially on a test set matching the client’s data distribution. Subsequently, upon receiving the updated model from the server, a second accuracy evaluation is conducted to compare the local and global model improvements. Each upload-download cycle prompts clients to receive new data, necessitating recalculations and interactions with the server for updates. In the experiments presented in Table 1, the number of clients is 20 times the number of models. The value of global staleness control $\tau_0$ equals to the number of clients. In the FashionMNIST experiments, on average, each client undergoes 25 upload-download cycles, while in the CIFAR-100/MiniImagenet-100 experiments, each client averages 20 cycles. We set cluster aggregation parameter $\alpha_0 = 0.025$, $a = 10$, $b = 5$, client personalization parameter $\rho = 0.1$. Other parameters and explanations are left at Appendix A.1. ### 5.2 Behavior of Clusters and Clients Table 1: Client and Cluster Accuracy of FashionMNIST, CIFAR100, and MiniImagenet-100. Client accuracy is represented as the average accuracy, along with the standard deviation, of all clients at the final upload-download cycle. Cluster accuracy is depicted as the average accuracy, along with the standard deviation, of all clusters across all cycles averaged across repeated experiments. "Cli Bfr" denotes the accuracy of the local model before uploading; "Cli Aft" represents the accuracy of the model received by the client. Average accuracy over 3 trials are reported. \| Dataset (cluster No.) \| CCFL ACC. \| FedSoft–Async ACC. \| Local Client ACC. \| \|-----------------------\|-----------\|--------------------\|-------------------\| \| \| Cli Bfr \| Cli Aft \| Cluster \| Cli Bfr \| Cli Aft \| Cluster \| .784±.014 \| \| FashionMNIST (2) \| .799±.011 \| .836±.003 \| .840±.008 \| .798±.012 \| .836±.003 \| .833±.008 \| \| FashionMNIST (3) \| .783±.015 \| .822±.003 \| .833±.003 \| .780±.015 \| .819±.004 \| .822±.005 \| .741±.057 \| \| FashionMNIST (4) \| .768±.020 \| .801±.006 \| .830±.005 \| .763±.020 \| .785±.006 \| .795±.020 \| .693±.076 \| \| FashionMNIST (6) \| .760±.021 \| .779±.019 \| .811±.009 \| .753±.025 \| .750±.024 \| .740±.071 \| .694±.072 \| \| CIFAR-100 (2) \| .373±.022 \| .398±.006 \| .423±.015 \| .374±.026 \| .404±.004 \| .420±.001 \| .279±.030 \| \| CIFAR-100 (3) \| .292±.037 \| .313±.029 \| .370±.031 \| .281±.033 \| .301±.008 \| .354±.023 \| .21±.033 \| \| CIFAR-100 (4) \| .354±.029 \| .371±.012 \| .427±.017 \| .330±.037 \| .355±.017 \| .425±.022 \| .259±.035 \| \| CIFAR-100 (6) \| .302±.032 \| .319±.009 \| .373±.024 \| .278±.041 \| .303±.016 \| .382±.031 \| .212±.035 \| \| MiniImagenet (2) \| .345±.022 \| .372±.004 \| .388±.009 \| .346±.026 \| .378±.003 \| .393±.003 \| .226±.032 \| \| MiniImagenet (3) \| .290±.030 \| .311±.017 \| .358±.010 \| .275±.034 \| .306±.011 \| .352±.005 \| .184±.029 \| \| MiniImagenet (4) \| .346±.025 \| .371±.013 \| .406±.007 \| .323±.034 \| .366±.008 \| .403±.007 \| .215±.028 \| \| MiniImagenet (6) \| .312±.028 \| .336±.008 \| .387±.012 \| .283±.037 \| .325±.011 \| .383±.016 \| .192±.027 \| Figure 3: Accuracy of clients and clusters on FashionMNIST(3 clusters) and MiniImagenet (4 clusters). Average accuracy of clients is shown for equal times of upload-download cycles. Shaded areas represent variances across 3 trials. Accuracy Behavior. Table 1 presents a comprehensive overview of client and cluster accuracy. Notably, both CCFL and FedSoft–Async exhibit significant enhancements in client performance compared to only local training, underscoring the importance of clients staying synchronized with the server. Across most experiments, CCFL outperforms FedSoft–Async for both clients and clusters, particularly when dealing with larger $K$. Figure 3 provides a performance analysis for a subset of experiments. Additional details can be found in Appendix A.3. In FashionMNIST experiments, both CCFL and FedSoft–Async require a few training epochs for the downloaded global model to surpass the performance of their locally uploaded counterparts. During this period, clusters may experience a temporary dip in performance, and we refer to it as the "preparation period". This preparatory phase can be executed effectively through limited-scale $\alpha$-testing before software release. It’s worth noting that this phenomenon is not observed in CIFAR-100 and MiniImagenet datasets due to their more complex prediction tasks, where the upload-download cycles with the server significantly aid clients in mitigating overfitting issues arising from limited data availability. Distribution Estimation. To assess the accuracy of the distribution estimation outlined in Algorithm 1 in representing the true distribution, we conduct empirical comparisons between the estimation outcomes of CCFL and those obtained using FedSoft-Async. To quantify this assessment, we employ the KL-divergence metric, which measures the information loss when one distribution approximates another, denoted as $KL(P\|\|Q) = \sum P(x) \log \left( \frac{P(x)}{Q(x)} \right)$, where $P$ represents the true distribution, and $Q$ represents the estimated distribution. Lower KL divergence values signify more accurate estimation. The KL-divergence results for all the aforementioned experiments are depicted in Figure 4(b). We normalize the divergence rate of FedSoft-Async to 1 and record the proportional ratio of CCFL. Across all experiments, CCFL exhibits superior distribution estimation performance compared to FedSoft-Async, whose estimation method is commonly utilized in clustered FL works for distribution analysis. ![Figure 4](image) (a) M-I(6clusters) Contrast of Distribution (b) KL-Divergence of Distribution on CCFL and FedSoft-Async (c) Communication and Computation Overhead contrast of FedSoft-Async with CCFL. FM($k$) denotes FashionMNIST ($k$ clusters), Ci as CIFAR-100, M-I as MiniImagenet-100. Communication and Computation Overhead. We conduct a comparative analysis of the communication and computation overhead between FedSoft-Async and CCFL, as illustrated in Figure 4(c). Specifically, we focus on download sessions for communication overhead evaluation, as both methods upload one local model during upload sessions. We normalize both the communication and computation overhead of CCFL to 1, and record the proportional ratio of FedSoft-Async. Due to the fact that clients in CCFL solely download an aggregated global model and do not engage in additional computations beyond local model training, the communication and computation overhead is significantly reduced compared to FedSoft-Async. This highlights the lightweight and client-centric nature of our approach. ### 5.3 Ablation Study In order to comprehensively evaluate the robustness and performance of our framework, we conduct an ablation study on the FashionMNIST (4 clusters) and CIFAR100 (4 clusters) datasets. The results of this study are depicted in Figure 5. Multiple Clients: We conduct experiments with varying numbers of clients of 100, 250, 500, 1000. Remarkably, the average accuracy of both clients and clusters exhibited minimal variation across different client counts. This observation underscores the robustness of our system. Different $\rho$ Values: We experiment with $\rho$ values set to 0.01, 0.1, 0.5, and 1. The results on both FashionMNIST and CIFAR100 datasets reveal that smaller $\rho$ values consistently lead to improved cluster accuracy. However, smaller $\rho$ values, as observed in CIFAR-100, result in suboptimal client local training performance before uploading, presenting a challenge. This can be attributed to similarities among cluster models, arising from generating clusters via various degrees of image rotation. These inherent similarities improve the aggregated data performance. across diverse distributions, consistent with Ruan and Joe-Wong (2022). Additionally, smaller $\rho$ values increase the risk of client overfitting to local data, further degrading local performance. Global Adjustments: To better regulate clients’ contributions to global models, we introduce an adjustment technique in our experiments. During each client’s update session, we record the values of $l_k$, $d_{1k}$, and $d_{2k}$ for each cluster $k$. Over time, this data accumulation creates a reference database resembling normal distributions. Subsequently, after a certain number of epochs, the uploaded models undergo adjustments based on thresholds derived from the aforementioned database: if any of the uploaded model’s $l_k$, $d_{1k}$, and $d_{2k}$ for given cluster $k$ exceeds 70% of the database, this client model is refused by the server to update global model $k$. This adjustment begins either after half of the training session, after 7/10 of the session, or not at all. Though accuracy does not change, we believe this adjustment mechanism acts as a filter, potentially preventing certain clients’ models from negatively impacting the server’s model due to the non-iid nature of clients’ data distribution. Ablation study with different size of public dataset on the server and data distribution without changes can be found in [A.4]. This section sheds light on the versatility and robustness of our CCFL framework, showcasing its adaptive ability to various scenarios and configurations while maintaining stable performance. Figure 5: Ablation study on FashionMNIST (4 clusters) and CIFAR-100 (4 clusters). The clients undergo average 20 (FashionMNIST) / 10 (CIFAR-100) upload-download cycles in every experiment. Average accuracy of clients and clusters are recorded. ### 6 CONCLUSION In summary, our paper introduces the Client-Centric Federated Learning (CCFL) framework, an approach that redefines the traditional server-centric FL paradigm. In this setting, clients independently decide when to upload their local models, resulting in rapid and personalized model updates from the server, who maintains multiple cluster models. Compared to existing clustered FL works, it significantly reduces computation and communication costs for clients. Moreover, CCFL accommodates dynamic clients’ data distributions. Our experiments on FashionMNIST, CIFAR100 and MiniImagenet-100 datasets underscore CCFL’s robustness and performance across different configurations. In conclusion, CCFL bridges the gap between user-centricity and model refinement, making it a pioneering framework in the FL landscape. REFERENCES Briggs, C., Fan, Z., Andras, P., 2020. Federated learning with hierarchical clustering of local updates to improve training on non-iid data, in: 2020 International Joint Conference on Neural Networks (IJCNN), IEEE. pp. 1–9. Chen, Y., Ning, Y., Slawski, M., Rangwala, H., 2020. 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EE75tyB5Ay	It is common for training-based methods to have more limited generalizability since they somewhat overfit a specific data distribution. From another perspective, would any unsupervised OOD detection methods [1,2] apply to detecting LLM-generated content?	ON THE GENERALIZATION OF TRAINING-BASED CHATGPT DETECTION METHODS Anonymous authors Paper under double-blind review ABSTRACT ChatGPT is one of the most popular language models which achieve amazing performance on various natural language tasks. Consequently, there is also an urgent need to detect the texts generated ChatGPT from human written. One of the extensively studied methods trains classification models to distinguish both. However, existing studies also demonstrate that the trained models may suffer from distribution shifts (during test), i.e., they are ineffective to predict the generated texts from unseen language tasks or topics. In this work, we aim to have a comprehensive investigation on these methods’ generalization behaviors under distribution shift caused by a wide range of factors, including prompts, text lengths, topics, and language tasks. To achieve this goal, we first collect a new dataset with human and ChatGPT texts, and then we conduct extensive studies on the collected dataset. Our studies unveil insightful findings which provide guidance for developing future methodologies or data collection strategies for ChatGPT detection. 1 INTRODUCTION ChatGPT (OpenAI) is one of the most popular language models, which demonstrates a great versatility to handle diverse language tasks, including question answering (Tan et al., 2023), creative writing (Bishop, 2023) and personal assistance (Shahriar & Hayawi, 2023). Meanwhile, it also gives rise to an urgent need for detecting ChatGPT generated texts from human written texts to regulate the proper use of ChatGPT. For example, ChatGPT can be misused to accomplish the tasks such as producing fake news or generating fake reviews (Li et al., 2023), leading to public deception. Similarly, ChatGPT can be also used for plagiarism, offending people’s intellectual property (Falati, 2023). These misuses of ChatGPT can cause severe negative consequences for our society. Since the model parameters of ChatGPT are not publicly available, many detection techniques for open-source language models (e.g., DetectGPT (Mitchell et al., 2023), Watermarks (Kirchenbauer et al., 2023)) cannot be utilized for ChatGPT detection. Therefore, a major stream of works (Guo et al., 2023; Chen et al., 2023; Tian et al., 2023b) propose to train classification models on the collected human texts and ChatGPT texts to distinguish each other, which we called “training-based methods” in this work. The empirical studies also demonstrate that the trained classifiers can achieve high detection performance under their studied datasets. However, it is evident from recent works (Yu et al., 2023; Guo et al., 2023) that these training-based methods tend to be overfitted to their training data distribution. For instance, Guo et al. (2023) show that a RoBERTa classification model (Liu et al., 2019) trained on HC-3 dataset (Guo et al., 2023) for detecting ChatGPT answered questions will exhibit a notable accuracy decrease, when it is tested on some specific topics (i.e., finance and medicine). Yu et al. (2023) also find that the detection models trained on HC-3 struggle to detect ChatGPT written news or scientific paper abstracts. In addition, in our work, we also notice that other types of distribution shifts (between training and test distribution) can occur and cause detection performance decrease, which are not identified or adequately discussed in previous works. These distribution shifts include: • Prompts to Inquire ChatGPT outputs: The ChatGPT user can have various prompts to obtain the ChatGPT outputs. For example, when asking ChatGPT to write a movie review, a user can ask “Write a review for the movie <MovieTitle>”. Alternatively, they can also let ChatGPT give comments to the movie, via asking ChatGPT to complete a dialogue which reflects the preference of the talkers towards this movie (see Section 3 for more details). The detection models that trained on texts obtained from certain prompts may face texts from other unknown prompts. • Length of ChatGPT outputs: The ChatGPT user can designate and control the length of the output to inquire longer or shorter generated outputs. It is also possible that the (distribution of) lengths of test samples differ from training ones. In reality, because only a limited number of training data can be collected, the training data cannot fully cover the distribution of test data. Thus, it is critical to deeply understand the detection models’ generalization behaviors when the distribution shifts occur. To achieve this goal, we first collect a new text dataset, named HC-Var (Human ChatGPT Texts with Variety), which contains human texts and ChatGPT outputs by considering multiple types of variety, including prompts, lengths, topics and language tasks (see Section 3). Facilitated with HC-Var, we can conduct comprehensive analysis on the models’ generalization, when facing the aforementioned distribution shifts. Through extensive experiments, we draw key findings and understandings, which provide guidance for developing better methodologies and data collection strategies to assist the success of ChatGPT detection: • From the pessimistic side, we identify one possible reason that can hurt the detection models’ generalization. For the training-based methods, the trained models tend to overfit to some “irrelevant features” which are not principal for ChatGPT detection. This overfitting issue can be originated from the “incautious and insufficient” data collection process, which collects ChatGPT texts that are distinct from human texts in these “irrelevant features”. In Section 4.3, we conduct a theoretical analysis to deeply understand this phenomenon. • From an optimistic side, we find the trained models are also capable to extract “transferable features”, which are shared features that can help detect the ChatGPT generated texts from various topics and language tasks. For example, in Section 5, we show that the models trained on existing topics or language tasks can be leveraged as a source model to accommodate transfer learning (Pan & Yang, 2009; Hendrycks et al., 2019), when it is adapted to unforeseen topics and language tasks. ## 2 RELATED WORKS In this section, we introduce background knowledge about existing methods for ChatGPT generated text detection, as well as other detection methods for open-source language models. We also discuss existing research findings about the generalization of ChatGPT detection methods. ### 2.1 OPEN-SOURCE LANGUAGE MODEL DETECTION AND CHATGPT DETECTION For open-source language models such as GPT-2 (Solaiman et al., 2019), and LLaMa (Touvron et al., 2023), since their model parameters are publicly available, information such as model probability scores can be leveraged for detection. For example, DetectGPT (Mitchell et al., 2023) assumes that LLMs always generate the texts with high probability scores. Thus, it manipulates the candidate texts (by editing or paraphrasing) to check whether the model gives a lower probability score. Besides, there are watermarking strategies (Kirchenbauer et al., 2023) which intervene the text generation process to inject watermarks into the generated texts to make them identifiable. Detecting ChatGPT generated texts is also an important task because of the extraordinary prevalence of social-wide usage of ChatGPT. However, many previously mentioned methods are not applicable due to the lack of access to ChatGPT’s model & probability scores. Therefore, plenty of works leverage the Training-based Methods (Guo et al., 2023; OpenAI, 2019; Chen et al., 2023), to train classification models to predict whether a text $x$ is human-written or ChatGPT generated: $$\min_f \mathbb{E}\left[1(f(x) \neq y)\right], \quad y \sim \{0, 1\}, \quad x \sim \begin{cases} D_H & \text{if } y = 0 \\ D_C & \text{if } y = 1 \end{cases}$$ where $D_H$ and $D_C$ represent the collected human and ChatGPT texts, respectively. Besides, there are “similarity-based” methods, such as GPT-Pat (Yu et al., 2023) and DNA-GPT (Yang et al., 2023) to compare the similarity of a text $x$ with its ChatGPT re-generated texts. Besides, “score-based methods” such as GPT-Zero (GPTZero.com) and GLTR (Gehrmann et al., 2019) detection ChatGPT texts based on their specific traits. More details of these methods are in Appendix D. ### 2.2 TRAINING-BASED CHATGPT DETECTION UNDER DISTRIBUTION SHIFT Notably, our work is not the first work studying or identifying the generalization issues of the training-based ChatGPT detection models. For example, the works (Wang et al., 2023; Yang et al., 2023; Yu et al., 2023) have discovered that it is challenging for the detection models to generalize to unseen language tasks and topics. Different from these existing works, we collect a new dataset to include different varieties to support a comprehensive analysis on their generalization. In Section 5, we discuss potential strategies to overcome the distribution shift. Besides, there are also previous works claiming that the models can struggle to predict texts with shorter lengths (Tian et al., 2023b; Guo et al., 2023). While, our paper finds that it could be related to a poor HC-Alignment (see Section 4.2) and we provide theoretical understandings (Section 4.3) about this issue. 3 PRELIMINARY In this section, we first introduce the details of our proposed dataset, HC-Var: Human and ChatGPT texts with Variety. Then we discuss the general experimental setups and evaluation metrics used in the paper. Next, we conduct a preliminary comparison on existing methods under the “in-distribution” setting, before we discuss their generalization behaviors. 3.1 HC-Var: Human and ChatGPT texts with Variety As discussed, we are motivated to study the generalization of ChatGPT detection when faced with various distribution shifts, including prompts, lengths, topics and language tasks. Refer to Table 1, existing datasets do not sufficiently support this analysis, because they don’t cover all types of considered varieties. Therefore, in HC-Var, we create a new dataset, collecting human and ChatGPT generated texts to include these varieties. Overall, as shown in Table 2, the dataset contains 4 different types of language tasks, including news composing (news), review composing (review), essay writing (writing) and question answering (QA). Each task covers 1 to 4 different topics. In HC-Var, human texts are from different public datasets such as XSum, IMDb. Variety in Prompts & Lengths. In each task, we design 3 prompts to obtain ChatGPT outputs to ensure the variety of generated outputs and their lengths. For example, to ask ChatGPT to compose a review for a movie with title <MovieTitle>, we have the prompts: • P1: Write a review for <MovieTitle> in [50, 100, 200] words. • P2: Develop an engaging and creative review for <MovieTitle> in [50, 100, 200] words. Follow the writing style of the movie comments as in popular movie review websites such as imdb.com. • P3: Complete the following: I just watched <MovieTitle>. It is [enjoyable, just OK, mediocre, unpleasant, great]. [It is because that, The reason is that, I just feel that, ...]. The design of P3 will make ChatGPT texts look much more casual and conversational than P1 and P2 (see Appendix A for some examples). Notably, previous studies (Guo et al., 2023; Kabir et al., 2023) observe that ChatGPT texts are much more formal and official compared with human texts. However, our dataset includes the instances to employ ChatGPT to produce texts, which are casual and close to spoken language. This can greatly enriches the collection of ChatGPT generated outputs. Similarly, under “QA”, given a question <Q>, we have the following prompts: • P1: Answer the following question in [50, 100, 150] words. <Q> • P2: Act as you are a user in Reddit or Quora, answer the question in [50,100,150] words. <Q> • P3: Answer the following question in [50, 100, 150] words. <Q> Explain like I am five. The P3 (which is also used in (Guo et al., 2023)) also encourages the generated answers to be closer to spoken language. Besides, for tasks such as essay writing and news writing where human texts are originally formal, we design various prompts by assigning different writing styles. For example, in essay writing, one of the prompt is “Writing an article with following title like a high school student”. More details about the prompt design are in Appendix A. --- 1 We follow existing datasets to take public available datasets as human texts. 2 Each word / phrase in the gray list has the same chance to be randomly selected. In P3, the each generated text is randomly truncated to 50-200 tokens. 3.2 IN-DISTRIBUTION EVALUATION In this subsection, under our proposed dataset HC-Var, we verify that the training-based detection methods can indeed achieve advantageous detection performance under the “in-distribution” setting, when compared with other detection methods. This part of experiments is also consistent with previous experimental studies (Guo et al., 2023; Chen et al., 2023) which are conducted in other datasets. The extraordinary in-distribution performance motivates us to study its generalization behavior. Experimental Setup. Generally, each experiment is focused on a specified language task, so the detection models are trained and tested on the texts from the same task. For example, under QA, we train the detection models on human and ChatGPT answered questions, and test whether they can distinguish these answers. Under each task, we randomly sample from the datasets to obtain class-balanced training, validation and test subsets (each has an equal number of human and ChatGPT samples). Thus, all training, validation and test datasets contain various topics, prompts and lengths, so distribution shift between training and test set is negligible, namely “in-distribution” evaluation. Evaluation Metrics. We evaluate the detection performance using different metrics: True Positive Rate (tpr) shows the detector’s power to identify ChatGPT generated texts, 1 - False Positive Rate (1-fpr) shows the detector’s accuracy on human texts, F1 score considers the tpr and 1-fpr trade-off. All F1 score, tpr and fpr are calculated under a fixed decision threshold 0.5. We also include AUROC which considers all possible thresholds for decision making. Performance Comparison. In Table 3, we report the performance of trained classification models, which are based on model architectures RoBERTa-base, RoBERTa-large and T-5. We also include representative “similarity-based” methods DNA-GPT (Yang et al., 2023) and GPT-PAT (Chen et al., 2023), and “score-based” methods including GLTR (Gehrmann et al., 2019) and GPTZero (GPTZero.com). From the table, we can see that training-based methods outperform non-training based methods under the in-distribution evaluation. \| Model \| News \| Auc \| f1 \| tpr \| 1-fpr \| Review \| Auc \| f1 \| tpr \| 1-fpr \| Writing \| Auc \| f1 \| tpr \| 1-fpr \| QA \| Auc \| f1 \| tpr \| 1-fpr \| \|----------------\|------\|-----\|-----\|-----\|-------\|--------\|-----\|-----\|-----\|-------\|---------\|-----\|-----\|-----\|-------\|--------\|-----\|-----\|-----\|-------\| \| GPTZero \| 0.99 \| 0.94\| 1.00\| 0.94\| 0.99 \| 0.90 \| 0.82\| 1.00\| 0.98 \| 0.89 \| 0.97 \| 0.90 \| 0.95 \| 0.90 \| 0.98 \| 0.91 \| \| GLTR \| 0.94 \| 0.87\| 0.88\| 0.86\| 0.90 \| 0.82 \| 0.85\| 0.80\| 0.99 \| 0.95 \| 0.94 \| 0.98 \| 0.88 \| 0.81 \| 0.78 \| 0.82 \| \| DNA-GPT \| 0.92 \| 0.90\| 0.89\| 0.89\| 0.93 \| 0.90 \| 0.88\| 0.89\| 0.97 \| 0.92 \| 0.88 \| 0.95 \| 0.87 \| 0.82 \| 0.86 \| 0.80 \| \| GPT-PAT \| 1.00 \| 0.99\| 1.00\| 0.99\| 1.00 \| 1.00 \| 1.00\| 1.00\| 1.00 \| 0.99 \| 0.99 \| 0.99 \| 0.99 \| 0.95 \| 0.97 \| 0.94 \| \| RoBERTa-b \| 1.00 \| 1.00\| 1.00\| 1.00\| 1.00 \| 0.99 \| 0.99\| 1.00\| 1.00 \| 1.00 \| 1.00 \| 1.00 \| 1.00 \| 0.98 \| 0.99 \| 0.98 \| \| RoBERTa-l \| 1.00 \| 1.00\| 1.00\| 1.00\| 1.00 \| 0.99 \| 1.00\| 0.99\| 1.00 \| 1.00 \| 1.00 \| 1.00 \| 1.00 \| 0.99 \| 0.99 \| 0.99 \| \| T-5 \| 1.00 \| 1.00\| 0.99\| 0.99\| 1.00 \| 0.99 \| 0.99\| 0.99\| 1.00 \| 1.00 \| 0.99 \| 1.00 \| 1.00 \| 0.98 \| 0.98 \| 0.96 \| The training-based methods present extraordinary “in-distribution” detection performance. This motivates us to have a further exploration on their generalization performance under out-of-distribution scenarios. In the following, we design experiments to analyze them when the training data cannot fully cover the distribution of test data. Our analysis contains two major scenarios. In Section 4, we consider the scenario that the model trainer aims to detect the texts from their interested language tasks and topics. In this case, the possible distribution shifts can be due to the variation of prompts and lengths. In Section 5, we discuss the cases that the models encounter unforeseen tasks or topics. 4 HOW PROMPT & LENGTH AFFECT DETECTION GENERALIZATION 4.1 GENERALIZATION TO UNSEEN PROMPTS To detect ChatGPT texts from a certain language task with several interested topics, it is a realistic and practical scenario that the model trainer collects ChatGPT texts using certain prompts. However, they never know whether there are other unforeseen prompts used to obtain ChatGPT outputs during test. Thus, we aim to analyze how the detection models can generalize to unseen prompts. In detail, refer to Figure 1, we conduct experiment to train the model for multiple trials (in each individual task with the topics in HC-Var). For each task at each time, we train the model on ChatGPT generated texts from one prompt, and test the model on each of three prompts (which we designed in Section 3) individually. Besides, for each time of training, the human texts are randomly sampled to match the number of generated texts. In Figure 1, we report the F1 score of the trained classifiers. Notably, for these trained models, they have similar (close to 100%) accuracy on human texts (see Appendix B.1). Therefore, these F1-scores are majorly determined by their True Positive Rate, which measure their ability to correctly recognize ChatGPT texts. We report F1 score instead of AUROC, as AUROC considers all thresholds for decision making, which is impractical under unseen distribution shift. All experiments are conducted by 5 times, the average is reported. In this section, we study the ChatGPT detection generalization in terms of prompts and lengths under the same topic and domain. Note that we only report the result of a representative model, RoBERTa-base, and results for other models such as RoBERTa-large and T5 are in Appendix B. Observations. From Figure 1, we can observe a great disparity among models that trained and tested on different prompts. For example, under QA, the model trained on P1 or P2 has low F1 scores 0.64 and 0.79 on P3 respectively. While, the model trained on P3 has a better generalization, with F1 score 0.89 and 0.93 on P1 and P2 respectively. Thus, a natural question raises: Why does such disparity happen? Next, we unveil two potential reasons. Reason 1. Prompt Similarity: Intuitively, the generalization performance can be highly dependent on the “similarity” between the generated texts from two different prompts. In other words, if ChatGPT responds to two prompts in a similar way, it is very likely that the models trained on one prompt can also correctly recognize the texts from the other. Therefore, for two given prompts $P_i$ and $P_j$ (in the same task), we propose the concept of “prompt similarity”, denoted as $S(D^P_C, D^P_C)$, which refers to the similarity between the generated texts $D^P_C$ and $D^P_C$ from prompts $P_i$ and $P_j$. In this work, we calculate this similarity using MAUVE (Pillutla et al., 2021), which is a well-known similarity metric for text distribution, and we report every $S(D^P_C, D^P_C)$ in Figure 2. In Figure 3, we also visualize the texts from various prompts in the pen-ultimate layer of a pre-trained RoBERTa model. From figures, we can see that the “prompt similarity” has a great impact on generalization. Take QA as an example, the generated texts from P1 and P2 has a high MAUVE similarity 0.97, the representations of texts from P1 and P2 are also correspondingly close to each other. Meanwhile, in Figure 1d, the generalization between P1 and P2 is also high, showing that trained models on similar prompts can well generalize to each other. Reason 2. Human-ChatGPT Alignment: A more interesting study is about generalization between dissimilar prompts. In each task, there are cases where the training and test prompts are not similar but have a good generalization. For example, in review, P1 and P3 are not similar but the model trained on P3 has a high F1 score 0.99 on P1. It suggests that there are other reasons beyond prompt similarity that also affect the generalization performance. In this work, we find: for the training datasets which contain ChatGPT outputs closer to human written texts, the trained model has better generalization. We called this property as the “Human-ChatGPT (HC) alignment”, which refers to the similarity between $D^P_C$ and $D_H$, and denoted as $S(D^P_C, D_H)$. In Figure 4a, for each task, we measure HC-alignment for each prompt $P_i$, also using the MAUVE similarity. In Figure 4 (b)-(e), we re-organize the result in Figure 1 using bar plots to show the F1 score of the model trained and tested on each prompt. From the result, we note that the prompts with high “HC Alignment” have better generalization to other prompts. For prompts with low HC-Alignment, they have poorer generalization to other prompts unless they are tested on the prompts with high “prompt similarity” (which we give them a gray color in Figure 4(b)-(e)). Interestingly, the calculated HC-alignment also reflects our idea during prompt designing in data collection phase. Refer to Section 3.1 in “review” and “QA”, P3 is designed to guide the ChatGPT generate texts more conversational in QA and review. From Figure 4a, the HC alignment of P3 is also the highest. Insights. In practice, it is a realistic and reasonable setting to consider multiple and diverse prompts (which we don’t include in this discussion, since we only calculate HC-Alignment for each individual prompt). However, our studies draw key insights to bring cautions to the data collection during model training, which is the pitfall of only collecting samples far away from human data. To explain the impact of HC-Alignment on generalization, in Section 4.3, we construct a theoretical analysis to provide deeper understanding. In our discussion, we majorly claim that there can be two types of factors contributing to the HC Alignment. Specifically, for the ChatGPT data $D_C^{P_i}$ and human data $D_H$, they can differ in “ChatGPT Direction” and “Irrelevant Direction” (see Section 4.3 for more details). A larger difference in irrelevant direction can cause the ChatGPT generated texts have a lower HC-Alignment with human texts. Meanwhile, the detection models trained on datasets with low HC-alignment are likely to overfit to this irrelevant direction and suffer from poor generalization. In the next subsection, we provide an example study, to show that the length of the texts can be one possible “irrelevant direction” which affects generalization of the models. 4.2 Generalization to Length Shift Recall that in Section 3 when we design prompts to inquire ChatGPT outputs, we explicitly control the lengths of the generated texts. In this subsection, we show the impact of lengths on the model’s generalization. To have an overview on the length distribution of human and ChatGPT texts, in Figure 5a, we plot the density of human texts and ChatGPT texts in HC-Var in one language task “review”. Additionally, we include ChatGPT# to show the length distribution if we do not designate the lengths in the inquiries (i.e., by removing “in [50, 100, 200] words” in the prompts). From the Figure 5a, we can see the generated texts from ChatGPT# are much longer compared to human texts. Notably, previous studies (Guo et al., 2023) also find ChatGPT texts are longer than human in their collected QA dataset, HC-3. This suggests the length can be a commonly overlooked factor in previous studies during data collection. (See Appendix B.3 for length comparison in other tasks.) In our study, we find this difference in length will make a noticeable impact on the trained model’s performance. For example, in Figure 5b, we report the performance (TPR, 1-FPR) of the model trained on our dataset when it is tested on samples with various lengths. In Figure 5c, we conduct the same experiment, by replacing the ChatGPT texts in training set to ChatGPT# (without length designation). From the result, we can see the second model struggles on classifying short ChatGPT texts. In other words, the second model tends to predict short ChatGPT texts as human written. A likely reason is that this model is trained to heavily rely on the lengths of the texts for prediction. If a candidate text is short, the model will predict it as human-written. However, text lengths should be an “irrelevant feature” for detection, as ChatGPT can generate shorter or longer texts. In Figure 5b, this issue can be greatly alleviated under our dataset. It may be because our collected dataset HC-Var has a much slighter length difference between human and ChatGPT texts (see Figure 5a). This finding encourages us to collect ChatGPT texts to have similar lengths with human texts for training the detection models. It also demonstrates the pitfall if only collecting ChatGPT outputs that are very distinct from human texts. This conclusion echoes back to the discussions in Section 4.1. 4.3 THEORETICAL ANALYSIS In this section, we construct a theoretical model to understand our previous empirical results. We aim to show when the ChatGPT texts and human texts are not well-aligned, it is likely that the model has a poor generalization. To build the connection, our major argument is that the models tend to focus on “irrelevant directions” for detection when this alignment is low. In our study, we use a two-dimensional toy model with data samples from Gaussian distributions to illustrate our idea. Theoretical setup. We consider a simplified scenario that human texts and ChatGPT texts are lying in a two-dimensional data space. As illustrated in Figure 6, we define the $x_1$-axis refers to “ChatGPT Direction”, which includes principal features to decide whether a sample belongs to human or ChatGPT. For simplicity, we define the region to the right of line $x_1 = C (C > 0)$ as ChatGPT generated, and we define the left of $x_1 = H (H > 0)$ as human written. Orthogonal to the ChatGPT direction, we define the $x_2$-axis as “Irrelevant Direction”. This direction contains features that are irrelevant for ChatGPT detection. Previous discussion in Section 4.2 demonstrates that the length of the texts can be one source of irrelevant features. Under this data space, we define the human training data are sampled following a Gaussian distribution $D_H = \mathcal{N}(0, \sigma^2 I)$. For ChatGPT data, we also assume that they are sampled following a Gaussian distribution in the space $x_1 \geq C$. Recall the previous empirical studies, we find that using different prompts can generate texts with different HC-Alignedness. In our analysis, we aim to compare two data collection strategies, with different distances to human data (a.k.a HC Alignment in empirical studies). In detail, we compare the strategies to make samplings from $D_{C1}$ and $D_{C2}$: $$\begin{cases} D_{C1} = \mathcal{N}(\theta_1, \sigma^2 I), & \|\|\theta_1\|\|_2 = d, \\ D_{C2} = \mathcal{N}(\theta_2, \sigma^2 I), & \|\|\theta_2\|\|_2 = K \cdot d, \end{cases} \quad d \geq C, K > 1$$ (2) The key difference between the two data distributions is the existence of the term $K$, which decides their distance to the human data. For the centers $\theta_1$ and $\theta_2$, they are uniformly distributed in the ChatGPT region, as long as they have distances $d$ and $K \cdot d$ to the origin. Next, we will study the generalization performance for binary classification models trained on human and ChatGPT texts. Before that, we first define a necessary evaluation metric of model generalization. Definition 1 (False Negative Area). For a given model $f$, it could make errors in ChatGPT region under area surrounded by $f$, $x_1 = C$ and $x_2 = \pm T$, where $T > 0$ is a threshold value controlling the limitation of $x_2$. We define the False Negative Area (FNA) as the area of the enclosed region. As an illustration in Figure 6, $S_1$ and $S_2$ represent the corresponding FNA of $f_1$ and $f_2$, respectively. In our analysis, we denote the FNA of a model $f$ as $\Gamma(f)$. We use it to measure the models’ error rate on unforeseen ChatGPT generated data, which are not covered by the collected training data. Next, we formally state our main theory by analyzing the FNA of the models $f_1$ and $f_2$: Theorem 1. Given the human training data $D_H$, ChatGPT training data $D_{C1}$, $D_{C2}$. For two classifiers $f_1$ and $f_2$ which are trained to minimize the error under a class-balanced dataset: $$f_i = \arg \min_f \Pr(f(x) \neq y), \text{ where } \begin{cases} x \sim D_{Ci}, & \text{if } y = 1 \\ x \sim D_H, & \text{if } y = 0 \end{cases}$$ Suppose the maximal FNA that $f_1$ can achieve is denoted as $\sup \Gamma(f_1)$. Then, with probability at least $$\left(1 - \left(\frac{\pi}{2} - \frac{C}{d} + \Omega\left(\frac{C}{d}\right)^3\right)/\left(\frac{\pi}{2} - \frac{C}{Kd}\right)\right),$$ we have the relation: $$\left(\frac{\Gamma(f_2)}{\sup \Gamma(f_1)}\right)^2 \geq \left(1 + (K - 1) \cdot \frac{1}{1 + 2T \cdot \Omega(1/d)}\right) > 1.$$ (3) The proof is deferred to Appendix C. This theorem suggests that the FNA of $f_2$ is likely to be larger than the worst case of $f_2$ (with a moderate probability), since their FNA ratio is larger than 1. Moreover, both the probability term and the FNA ratio term (Eq 5) are monotonically increasing with the term $K$. It suggests the larger $K$ it is, the higher chance of $f_2$ can have a poorer generalization than $f_1$. Refer to Figure 6 compared with $f_1$, the model $f_2$ has a larger FNA, because its decision boundary has a smaller slope, which means $f_2$’s prediction is more relied on the irrelevant direction. 5 GENERALIZATION OF CHATGPT DETECTION ACROSS TOPICS & DOMAINS In this section, we discuss the circumstances that the models can face texts from unforeseen language tasks or topics. Under this setting, we find the trained models can also extract useful features to help the generalization to other unforeseen tasks or topics, which we called “transferable features”. We also validate one frequently applied strategy, transfer learning (Pan & Yang [2009]), can be benefited from this property. Notably, in this section, we only provide the results for task-level generalization, and we leave the topic-level study in Appendix B, where we can draw similar conclusions. ![Figure 7: Generalization of RoBERTa-base model across Various Language Tasks](image) (a) F1 Score (b) TPR (c) 1 - FPR Figure 7: Generalization of RoBERTa-base model across Various Language Tasks ![Figure 8: Representation space visualization on models trained on each task](image) (a) News (b) Review (c) Writing (d) QA Figure 8: Representation space visualization on models trained on each task 5.1 GENERALIZATION ACROSS TOPICS & DOMAINS In this subsection, we conduct experiments to test the RoBERTa-base classification method’s generalization across language tasks (and topics). In particular, in Figure 7, we train the model on the human and ChatGPT texts from each language task individually and we check whether it can correctly classify texts from other tasks. Since these tasks have different number of samples in HC-Var, we randomly sample 4,000 ChatGPT and 4,000 human samples for training in all experiments. In each training set, the ChatGPT texts will contain various topics (if exist) and various prompts. In the experiments, we report the evaluation metrics including F1-score, TPR and 1-FPR. Based our reported results in Figure 8, we can see that the trained models will have a performance drop on either human texts or ChatGPT texts. For example, the model trained on “writing” cannot effectively detect the ChatGPT generated texts in “QA”. Similarly, the models trained on “news” can hardly recognize human written texts in “writing”. This result shows that models could make errors on both human or ChatGPT texts. In Appendix B, we provide the results for topic-level generalization, where we can draw similar conclusions. In reality, due to the versatility of ChatGPT to handle various tasks, it is infeasible to collect texts from all possible tasks for model training. 5.2 FINE-TUNING WITH A FEW SAMPLES HELPS CROSS-DOMAIN / TOPIC DETECTION In this part, we identify a potential way to improve the ChatGPT detection in the unforeseen tasks (or topics). It is based on our finding that the models trained in each individual task can learn helpful features for other tasks. As an evidence, in Figure 8 we visualize the learned representations for various tasks rendered by the trained model in Section 5.1. From these figures, we note that the ChatGPT and human texts from unseen tasks during training are also well-separated in the representation space. It demonstrates the models can indeed learn useful features which are helpful to distinguish human and ChatGPT texts in other domains, so we call them “Transferable Features”. Table 4: Transfer Learning (Task-level) Performance via Linear Probing and Fine-Tuning \| Target \| news \| review \| writing \| QA \| \|--------------\|------\|--------\|---------\|----\| \| \| r→n \| w→n \| q→n \| n→r \| w→r \| q→r \| n→w \| r→w \| q→w \| n→q \| r→q \| w→q \| \| No Transfer \| 0.946\| 0.835 \| 0.927 \| 0.854\| 0.980\| 0.981\| 0.681\| 0.858\| 0.827\| 0.819\| 0.789\| 0.771\| \| LP-5 \| 0.991\| 0.990 \| 0.972 \| 0.901\| 0.958\| 0.987\| 0.901\| 0.967\| 0.902\| 0.772\| 0.860\| 0.849\| \| FT-5 \| 0.952\| 0.923 \| 0.932 \| 0.965\| 0.952\| 0.940\| 0.871\| 0.898\| 0.835\| 0.848\| 0.893\| 0.869\| \| LP-Scratch-5 \| 0.959±0.019\| \| 0.839±0.057\| \| 0.871±0.024\| \| 0.697±0.082\| \| \| FT-Scratch-5 \| 0.946±0.033\| \| 0.925±0.033\| \| 0.867±0.021\| \| 0.687±0.047\| \| \| LP-10 \| 0.990\| 0.978 \| 0.993 \| 0.932\| 0.986\| 0.984\| 0.916\| 0.956\| 0.934\| 0.839\| 0.880\| 0.859\| \| FT-10 \| 0.978\| 0.978 \| 0.983 \| 0.951\| 0.960\| 0.967\| 0.936\| 0.956\| 0.936\| 0.818\| 0.813\| 0.909\| \| LP-Scratch-10\| 0.979±0.005\| \| 0.934±0.013\| \| 0.906±0.023\| \| 0.764±0.071\| \| \| FT-Scratch-10\| 0.983±0.006\| \| 0.941±0.020\| \| 0.939±0.018\| \| 0.778±0.051\| \| We use the blue color to highlight the case that transfer learning outperforms training from scratch. To further verify the existence of transferable features, we conduct experiments to investigate transfer learning (Hendrycks et al., 2019) for domain adaption. In reality, if the model trainer encounters test samples from the language tasks (or topics) which are not involved in the training set, it is a practical and feasible solution for them to collect several samples in the same task as the test sample by themselves. Therefore, in our study, we consider two types of transfer learning strategies: Linear Probing (LP), which refers to the strategy that only the linear classifier (based on extracted features) is optimized; and Fine Tuning (FT) which refers to the strategy that all layers are optimized. In our experiment, we consider there are 5 and 10 more samples from both human data and ChatGPT texts are sampled for fine-tuning the models. In Table 4, we report the tuned models performance (F1 score) when tested on different targeted (downstream) tasks from various source models. For example, “r → n” means the model transferred from “review” for a downstream task “news”. Besides, we also include the original performance (from Figure 7), which is the performance before transfer learning (denoted as “No Transfer” in Table 4). For comparison, we report the result if these models are tuned from scratch (on pre-trained RoBERTa-base model without training for detection). From the result, we can see transfer learning can benefit the detection performance in general. For example, when compared with “No Transfer”, linear probing (LP) or Fine-tuning (FT) can improve the downstream task performance in most cases (except for w → r with 5 training samples). Moreover, when compared to the models training from scratch, the transferred models also achieve higher performance in all considered language tasks. It suggests that those pre-trained models can offer helpful features beyond the collected data samples for down-stream tuning. These results indeed show that there are shared features, a.k.a, transferable features, which are generally useful to distinguish human and ChatGPT texts in various domains. Remarkably, in Section 4.3, we introduce the notion “ChatGPT Direction”, which contains the fundamental and principal features to distinguish human and ChatGPT texts. Ideally, these features should be universally helpful for ChatGPT detection in all tasks and topics. However, it is hard to verify their existence in reality, because of the difficulty to consider all possible topics and tasks that ChatGPT can handle. Thus, we use “transferable features” to refer the shared features only in our studied topics and tasks. 6 CONCLUSION AND LIMITATION Conclusion: In this paper, we conduct a comprehensive analysis on the generalization behavior of training-based ChatGPT detection methods. Due to the limitation of existing datasets, we collect a new dataset HC-Var, with various types of ChatGPT generated texts and human texts. Our empirical and theoretical studies draw key findings on factors which affect the generalization. We provide insights on the data collection and domain adaption strategy for ChatGPT detection. Limitation: There also other factors which can influence the detection that are not discussed. For example, we have not investigate the scenario that the texts are composed by other language models, such as LLaMA2 (Touvron et al., 2023), or first generated by ChatGPT and then manipulated (i.e., rephrased) by other language models. It is also likely that a given candidate text is partially written by ChatGPT. Besides, as a foundation model, ChatGPT can achieve much more language tasks, such as programming code writing (Tian et al., 2023a). 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PFdjJiZjPj	Using self-generated code (instead of a placeholder) is not exactly a fair comparison when compared to previous work, as it samples the model 2x for each generation. Using the place-holder code was somewhat intended in the original technique so you only generate the code itself once.	THE PROGRAM TESTING ABILITY OF LARGE LANGUAGE MODELS FOR CODE Anonymous authors Paper under double-blind review ABSTRACT Recent development of large language models (LLMs) for code like CodeX and CodeT5+ demonstrates tremendous promise in achieving code intelligence. Their ability of synthesizing code that completes a program for performing a pre-defined task has been intensively tested and verified on benchmark datasets including HumanEval and MBPP. Yet, evaluation of these LLMs from more perspectives (than just program synthesis) is also anticipated, considering their broad scope of applications in software engineering. In this paper, we explore the ability of LLMs for testing programs/code. By performing thorough analyses of recent LLMs for code in program testing, we show a series of intriguing properties of these models and demonstrate how program testing ability of LLMs can be improved. Following recent work which utilizes generated test cases to enhance program synthesis, we further leverage our findings in improving the quality of the synthesized programs and show +11.77% and +4.22% higher code pass rates on HumanEval+ comparing with the GPT-3.5-turbo baseline and the recent state-of-the-art, respectively. 1 INTRODUCTION The community has witnessed a surge in the development of large language models (LLMs), which have achieved incredible ability in understanding and generating not only texts but also code. LLMs for code (CodeX [Chen et al., 2021], StarCoder [Li et al., 2023b], CodeT5+ [Wang et al., 2023b], etc) have been widely adopted to a variety of applications to achieve code intelligence. However, current evaluation of these LLMs mostly focuses on program completion/synthesis, despite the models can also be utilized in other applications. As the field continues to advance, evaluation of these models from more perspectives is anticipated, which could facilitate deeper understanding of the LLMs. The ability of automatically generating proper test suites is of great desire to software engineering, yet challenging. Being learning-based or not, current test generation efforts (e.g., fuzzing) primarily focus on creating diverse test inputs to identify faults in the code as much as possible via maximizing their coverage, e.g., line coverage and branch coverage [Fioraldi et al., 2020; Tufano et al., 2022; Dinella et al., 2022; Lemieux et al., 2023; Xia et al., 2023]. Although such test inputs try to verify the (non-)existence of crashes and hangs of the tested code, they lack the ability of determining whether the code adheres to the aim of the function which is represented by input-output relationships. Automatic test case generation for this aim not only requires an high coverage of the code being tested but also necessitates a correct understanding of the “true” desired input-output relationships in the tested code, leaving it a challenging open problem. Being capable of synthesizing correct code implementations given docstrings, LLMs for code seem capable of understanding the desired input-output relationship of a function described in natural language. This capability inspires applying these LLMs to generating test cases automatically [Chen et al., 2021]. However, the ability of these models for program testing has not been systematically evaluated. In this paper, we systematically compare the ability of recent LLMs for code in testing from two perspectives focusing on both the correctness and diversity of the test cases, considering that 1) program testing is of great interest in software engineering and software security as mentioned and 2) automatically generated test cases can further be adopted to improve the program synthesis performance [Chen et al., 2023]. Our analyses focus on algorithmic coding, based on the popular 164 problems from HumanEval+ [Liu et al., 2023a] and 427 sanitized problems from MBPP [Austin et al., 2021]. It is worth noting that the model may encounter various scenarios when generating test cases. It may generate test cases when provided with only natural language descriptions of the desire of the program, or it could generate test cases when given an “optimal” oracle implementation. In more complex situations, it may even need to test its own imperfect generated code or the code generated by other models. We consider 4 test-case generation settings (i.e., “self-generated” which uses each LLM to test code synthesized by the LLM itself, “all-generated” which lets all LLMs to test the same code synthesized by a group of four LLMs, “oracle” which tests an oracle implementation, and the “placeholder” in Figure1) and test a collection of 11 competitive LLMs for code. We conducted a variety of experiments, from which intriguing takeaway messages are delivered. As previously mentioned, several very recent papers (Shi et al., 2022; Li et al., 2023a; Chen et al., 2023) have shown that appropriate usage of generated test cases can improve the quality of program synthesis. Yet, the quality of generated test cases largely impacts the performance of such methods. Due to the lack of systematic evaluation of the testing ability of LLMs for code, it is unclear how to craft test cases that could be potentially more helpful to program synthesis. The studies in this paper also shed light on this. We will show that, substantially improved program synthesis performance can be gained by utilizing takeaway messages in our studies. Specifically, we can achieve +11.77% higher code pass rate on HumanEval+, in comparison with the GPT-3.5-turbo baseline. Compared with a very recent state-of-the-art called CodeT, our solution gains +4.22% higher code pass rate. 2 EVALUATION METRICS To make the evaluation more reliable and comprehensive, it is crucial to first design some suitable metrics, like BLEU (Papineni et al., 2002), ROUGE (Lin, 2004), and the pass rate (Chen et al., 2021) for evaluating machine translation, text summarization, and program synthesis, respectively. In this section, we specify two main evaluation metrics to evaluate the program testing ability of LLMs, from the perspective of correctness and diversity. Pass rate In software engineering, we expect test cases to represent some desired “ground-truth” functionality of the tested program/code. In practice, such “ground-truth” functionality can be described in the header comments of a function (i.e., docstrings of the function) and tested using the oracle implementation, as in HumanEval (Chen et al., 2021) and MBPP (Austin et al., 2021). The oracle program/code should be able to pass the test, if a generated test case is correct. Therefore, we leverage the pass rate as a measure to evaluate the correctness of the generated test cases. For a fair comparison, we instruct each model to generate three test cases in the prompt, and, when a model generates more than three test cases, we select the first three for evaluation. Assuming that there are in total $M$ programming problems in an experimental dataset and, for each problem, we have $N$ program/code implementations to be generated test cases for. Each model has only one chance to generate these test cases for each program/code. Then, we calculate the pass rate as: $$P = \frac{1}{MN} \sum_{i=1}^{M} \sum_{j=1}^{N} \frac{p_{ij}}{n_{ij}},$$ where $n_{ij}$ is the number of test cases in $Q_{ij}$ which includes no more than three test cases generated for the $j$-th program/code implementation of the $i$-th problem by the evaluated LLM at once, i.e., $Q_{ij} = \{(x_{ijk}, y_{ijk})\}_k$, and $p_{ij}$ is the number of test cases (in $Q_{ij}$) that do not fail the oracle. The pass rate defined in Eq. (1) measures correctness of the generated test cases. However, as can be seen in Figure1, the model can generate duplicate test cases that are less helpful, even though they are correct. To avoid such an evaluation bias, we further advocate deduplication in the set of test cases that are considered as correct, which leads to computation of a deduplicated pass rate defined as $P' = \frac{1}{MN} \sum \sum p'_{ij}/n'_{ij}$, where we use ‘ to denote the numbers of unique test cases. Coverage rate In addition to the above pass rates, we further consider coverage rate as a more fine-grained metric for evaluating the diversity of the generated test cases. According to its definition, converge rate computes the degree to which the code is executed, given a test case. Since, for each program/code, we keep no more than three test cases at once, we calculate how much percentage of the control structure is covered given these test cases. Similar to Eq. (1), we evaluate the performance of testing all programs/code over all $M \times N$ times of generation, i.e., we calculate $$C = \frac{1}{MN} \sum_{i=1}^{M} \sum_{j=1}^{N} c_{ij},$$ where $c_{ij}$ is the per-test-case branch coverage rate. We apply the `pytest` library to evaluate the branch coverage for all the three test cases for each code and average the results for all programs/code and all problems. Apparently, $C \leq 1$, and a higher $C$ shows better testing ability of an LLM, since we expect all parts of the programs/code to be executed to find our all potential bugs. 3 LARGE LANGUAGE MODELS FOR CODE In this section, we outline the evaluated models. We adopt some “small” models whose numbers of parameters are around 1B (to be more specific, from 770M to 1.3B in our choices) and some larger models that achieve state-of-the-art performance in the task of program synthesis. For the small models, we use InCoder (1.3B) (Fried et al., 2023), CodeGen2 (1B) (Nijkamp et al., 2023a), CodeT5+ (770M) (Wang et al., 2023b), and SantaCoder (1.1B) (Allal et al., 2023). InCoder is a unified generative model that can perform program/code synthesis as well as code editing, and it combines the strengths of causal language modeling and masked language modeling. The CodeGen2 model was trained on a deduplicated subset of the Stack v1.1 dataset (Kocetkov et al., 2023), and its training is formatted with a mixture of objectives for causal language modeling and span corruption. CodeT5+ is an encoder-decoder model trained on several pre-training tasks including span denoising and two variants of causal language modeling. SantaCoder was trained on the Python, Java, and JavaScript code in the Stack dataset. The pass rate (Chen et al., 2021) of programs generated by these models is compared in Table 1. When evaluating the (program) pass rate, we let the model generate 200 code implementations for each problem, and we set the temperature to 0.2, 0.6, and 0.8 for calculating pass@1, pass@10, and pass@100, respectively. As for larger models that achieve state-of-the-art program synthesis performance, we use CodeGen2 (16B) (Nijkamp et al., 2023a), CodeGen-Multi (16B) (Nijkamp et al., 2023b), CodeGen-Mono (16B) (Nijkamp et al., 2023b), StarCoder (15B) (Li et al., 2023b), WizardCoder (15B) (Luo et al., 2023), CodeGeeX2 (6B) (Zheng et al., 2023), and GPT-3.5-turbo. CodeGen-Multi and CodeGen-Mono are two large models from the first version of CodeGen. CodeGen-Multi was first trained on the pile dataset (Gao et al., 2020) and then trained on a subset of the publicly available BigQuery dataset which contains code written in C, C++, Go, Java, JavaScript, and Python. Based on the 16B CodeGen-Multi model, CodeGen-Mono (16B) was obtained by further tuning on a set of Python code collected from GitHub. Given a base model that was pre-trained on 1 trillion tokens from the Stack dataset, the 15B StarCoder model was obtained by training it on 35B tokens of Python code. WizardCoder further empowers StarCoder with instruction tuning, following a similar instruction evolution strategy as in WizardLM (Xu et al., 2023). CodeGeeX2, the second generation of a multilingual generative model for code, is implemented based on the ChatGLM2 architecture and trained on more code data. GPT-3.5-turbo is a very capable commercial LLM developed by OpenAI and we accessed it in August, 2023. For these large LLMs, we tested pass@1 of all models except GPT-3.5-turbo (whose result can be directly collected from Liu et al. (2023a)’s paper). By sorting their pass@1 from high to low, they are ranked as: GPT-3.5-turbo (61.7%), WizardCoder (46.23%, 15B), CodeGeeX2 (29.97%, 6B), StarCoder (27.9%, 15B), CodeGen-Mono (26.15%, 16B), CodeGen2 (19.33%, 16B), CodeGen-Multi (15.35%, 16B). The ranks on the MBPP dataset are similar. 4 CODE TO BE TESTED For evaluating the testing ability of LLMs, we need an oracle to express the ground-truth functionality of the tested code. Fortunately, current datasets for evaluating program synthesis performance often provide such oracles (see HumanEval (Chen et al., 2021) and MBPP (Austin et al., 2021)). In our experiments, we utilize an amended version of HumanEval called HumanEval+ (Liu et al., 2023a), together with MBPP (the sanitized version). These datasets are established to evaluate basic Python programming performance of LLMs, and they contain 164 and 427 problems, respectively. 4.1 IMPERFECT CODE IMPLEMENTATIONS In order to simulate real-world scenarios where the tested code is often buggy, we first adopt synthesized programs/code as the programs/code to be tested, considering that the synthesis of even --- 1https://pytest.org state-of-the-art LLMs is still imperfect. We evaluate the performance of each LLM in testing code that was generated by itself (which is denoted as “Self-generated”) and code in a set consisting of program completion results of several different LLMs (which is denoted by “All-generated”). That said, the compared LLMs take different code implementations when generating test cases for each programming problem in the self-generated setting. Whereas, in the all-generated setting, the same program/code implementations are given to different LLMs for generating test cases for comparison. In practice, we apply InCoder (1.3B), CodeGen2 (1B), CodeT5+ (770M), and SantaCoder (1.1B) to construct the all-generated program/code set, while, in the self-generated setting, each LLM first synthesizes code and complete a program to fulfill the requirement of each programming problem, and the LLM then generates test cases with the synthesized programs/code in its prompts. The temperature for all LLMs is uniformly set to 0.2 for synthesizing the programs/code in both settings. We obtain 100 program/code completions for each problem and we prompt each LLM to generate 3 test cases for every program/code implementation in the self-generated setting, and we sampled 100 implementations from the synthesis results of InCoder (1.3B), CodeGen2 (1B), CodeT5+ (770M), and SantaCoder (1.1B) to form the all-generated code set, i.e., we have $N = 100$ for these settings. We follow the same way of generating code as introduced in the papers of these LLMs. For model without instruction tuning, like InCoder and CodeT5+, we synthesize programs/code using the default prompt given by each programming problem in the test dataset, while, for models that have adopted instruction tuning, e.g., WizardCoder, we use the recommended prompt in their papers. ### 4.2 Optimal Code Implementations (Oracle) As a reference, we also report the performance of generating accurate and diverse test cases when the written code is perfectly correct, which is achieved by adopting the oracle as the programs/code to be tested (and such a setting is denoted by “Oracle”). Since Liu et al. (2023a) have reported that some oracle code in the HumanEval dataset can be incorrect, we adopt the amended oracle set in HumanEval+ in this setting. We further used the revised oracle code implementations instead of the original ones in evaluating the pass rate (i.e., $P'$) of the generated test cases. Considering that the public datasets often only provide one oracle implementation for each problem, and to keep the uncertainty of evaluation results consistent, we copy the oracle implementation by $100\times$ and we \| Model \| Size \| Pass@1 \| Pass@10 \| Pass@100 \| \|-------------\|--------\|--------\|---------\|----------\| \| InCoder \| 1.3B \| 6.95% \| 14.06% \| 23.76% \| \| CodeGen2 \| 1B \| 9.19% \| 17.50% \| 25.90% \| \| CodeT5+ \| 770M \| 12.95% \| 28.02% \| 37.56% \| \| SantaCoder \| 1.1B \| 15.21% \| 29.42% \| 43.80% \| Table 1: Program synthesis performance of the small LLMs (whose number of parameters is around 1 billion) evaluated on HumanEval+/MBPP (sanitized). prompt to generate 3 test cases for each of these copies. It can be regarded as letting $N = 100$, just like in the previous settings in Section 4.1. ### 4.3 No Implementation (Placeholder) In certain scenarios, we require test cases before the function/program has been fully implemented, hence we also evaluate in a setting where the main body of a tested function/program is merely a placeholder, as depicted in Figure 1(b). This scenario often occurs when the main code has not yet been implemented for a function/program or the test engineer does not want to introduce implementation bias to the LLM when generating test cases for a function/program. We denote such a setting as “Placeholder” in this paper. We also let $N = 100$, as in the oracle setting. ## 5 Test Case Generation In this section, we introduce how test cases can be generated, when the implementation of a function/program is given as described in Section 4. In this paper, a desired test case is a pair of input and its expected output for the function/program defined in the context. As an example, Figure 1 demonstrates some test cases for the programming problem of checking whether the two words satisfy a specific rotation pattern. To generate test cases, we use the LLMs introduced in Section 3. We wrote extra prompts to instruct the LLMs to generate three test cases for each given code which include docstrings that describe the purpose of this function, as depicted in Figure 1. Our instruction commands the LLMs (1) to “check the correctness of this function with three test” and (2) to start writing test code with an “assert” statement and the tested function, which specifies the format of the test cases as input-output pairs that can be parsed. For instance, given the example in Figure 1, the extra prompt should be “# Check the correctness of this function with three test cases \n assert cycpattern_check”. We then concatenate the extra prompt with the code and feed the concatenation into each LLM, for extracting test cases from the model output. The LLM will try to complete the given input by generating one or more “assert” statement(s), and we split the generation results into sub-strings, with “assert” as the separator. Each sub-string is then considered as a test statement, and we only take the first three statements if there exist more than three statements, as has been introduced in Section 2. Such a split can be considered as an effective post-processing operation which largely improves the quality of the generated test code, considering that some non-sense code pieces may be generated in the output of the LLMs. When using HumanEval+ and MBPP, we try removing test cases in the docstrings of the function, if there exist any, just to get rid of the broad hints from the docstrings (Chen et al., 2023). The temperature for generating test cases is kept as 0.2. Once obtained, the generated test cases are then compiled, and evaluated for their correctness and diversity to report the pass rate $P'$ and the coverage rate $C$. When calculating, for each problem and every set of completions generated, we create a temporary folder. ## 6 Main Results for Test Case Generation The experiment results of small and large LLMs on HumanEval+ can be found in Table 2 and Table 3 respectively. Table 4 shows the results on MBPP. There are several takeaways from these tables. - First, the test cases generated by LLMs can show a descent pass rate, and this pass rate is even higher than the code pass rate on HumanEval+, which holds for both large and small \| Model \| Size \| Oracle \| Self-generated \| All-generated \| Placeholder \| \|------------\|--------\|--------------\|----------------\|---------------\|-------------\| \| InCoder \| 1.3B \| 21.31% (61.43%) \| 23.37% (59.36%) \| 22.72% (61.10%) \| 25.19% (62.75%) \| \| CodeGen2 \| 1B \| 31.63% (71.55%) \| 30.62% (69.38%) \| 30.93% (69.70%) \| 30.69% (69.00%) \| \| CodeT5+ \| 770M \| 35.43% (71.45%) \| 32.34% (70.45%) \| 31.49% (69.75%) \| 32.67% (70.67%) \| \| SantaCoder \| 1.1B \| 30.97% (71.46%) \| 30.43% (70.81%) \| 30.13% (70.55%) \| 30.78% (71.24%) \| Table 2: The pass rates (and coverage rate) of the test cases generated on HumanEval+ in different settings for LLMs with around 1 billion parameters. \| Model \| Size \| Oracle \| Self-generated \| All-generated \| Placeholder \| \|----------------\|------\|------------\|----------------\|---------------\|-------------\| \| CodeGen-Multi \| 16B \| 43.88% (67.91%) \| 41.85% (69.30%) \| 40.38% (66.97%) \| 39.74% (68.28%) \| \| CodeGen2 \| 16B \| 46.34% (73.07%) \| 45.44% (73.17%) \| 42.00% (72.45%) \| 42.69% (72.86%) \| \| CodeGen-Mono \| 16B \| 49.03% (74.82%) \| 45.73% (73.74%) \| 43.91% (73.66%) \| 44.92% (73.63%) \| \| StarCoder \| 15B \| 55.07% (76.02%) \| 52.52% (72.45%) \| 48.20% (72.30%) \| 50.58% (74.52%) \| \| CodeGeeX2 \| 6B \| 57.03% (74.42%) \| 53.16% (73.55%) \| 49.28% (70.32%) \| 51.78% (73.08%) \| \| WizardCoder \| 15B \| 53.89% (77.87%) \| 55.47% (76.07%) \| 48.02% (75.27%) \| 49.89% (75.12%) \| \| GPT-3.5-turbo \| - \| 71.03% (77.85%) \| 72.45% (77.24%) \| 59.24% (74.99%) \| 66.28% (74.03%) \| Table 3: The pass rates (and coverage rate) of the test cases generated on HumanEval+ in different settings for LLMs whose parameters are obviously more than 1 billion. Figure 2: The correlation between code past rate and test pass rate in the “Oracle” setting. Figure 3: How the correctness of the test cases changes with their order when being generated. LLMs. Such a result is consistent with intuitions from previous work which rejects code that cannot pass the generated tests to improve the quality of program synthesis. • Second, the correctness of the generated test cases is positively correlated with the LLM’s ability of generating code (see Figure 2, where each red cross represents the performance of a model), which means an LLM showing the state-of-the-art program synthesis performance is possibly also the state-of-the-art LLM for program testing. As shown in Tables 2 and 3, GPT-3.5-turbo, which synthesizes programs/code with the highest correctness, provides test cases with the highest pass rate (71.03%) on HumanEval+. For an LLM, the more accurate it is capable of synthesizing programs/code on a dataset, the more powerful testing ability will probably be exhibited on the same dataset. There also exist a few exceptions, e.g., SantaCoder (1.1B) outperforms CodeT5+ (770M) and CodeGen2 (1B) in generating code, but it shows inferior performance in program testing on HumanEval+. By carefully examining the test cases yielded by SantaCoder on HumanEval+, we found that it tends to generate more complex and longer test cases than CodeT5+ for several problems on HumanEval+, which are often more desirable in program testing. This is also why the SantaCoder test cases show higher coverage rates in Table 2. To be concrete, in Problem 131 in HumanEval+, where the program is required to return the product of all digits with an odd position in a positive integer $ n $ (which is the input), the test input provided by CodeT5+ tends to be small for this problem, e.g., $ n = 2 $, while the SantaCoder test cases tend to have more digits (e.g., $ n = 12358 $), which is helpful in digging out hidden bugs. Yet, generating longer and more complex test cases is more challenging, and the correctness can be lower. • Third, as can be seen in Tables 3 and 4, generating test cases using large LLMs with their self-generated code (in the prompts) often leads to a higher level of correctness, compared with the placeholder results. This observation is in fact unsurprising, considering that generating code first and test case afterwards resembles the chain-of-thought prompting (Wei et al., 2022) (if adopting the placeholder is regarded as a plain prompting), which is beneficial to reasoning. Moreover, the self-generated performance of an LLM sometimes even outperforms its testing performance with an oracle, and we ascribe this to: 1) randomness in the style of the oracles which are few in number and/or 2) less distribution shift between self-generated code in prompt and the training code, for some powerful LLMs. \| Model \| Size \| Oracle \| Self-generated \| All-generated \| Placeholder \| \|---------------\|--------\|--------------\|----------------\|---------------\|-------------\| \| InCoder \| 1.3B \| 21.56% (46.81%) \| 17.98% (46.11%) \| 19.53% (46.45%) \| 22.58% (46.72%) \| \| CodeGen2 \| 1B \| 25.61% (54.26%) \| 21.85% (53.09%) \| 23.15% (50.43%) \| 22.81% (52.11%) \| \| CodeT+ \| 770M \| 29.02% (56.86%) \| 24.44% (52.31%) \| 24.84% (53.20%) \| 25.59% (55.81%) \| \| SantaCoder \| 1.1B \| 32.37% (55.68%) \| 26.40% (52.38%) \| 26.20% (52.83%) \| 26.53% (53.86%) \| \| CodeGen-Multi \| 16B \| 41.32% (60.63%) \| 35.96% (59.03%) \| 34.17% (58.09%) \| 34.84% (58.92%) \| \| CodeGen2 \| 16B \| 45.30% (62.15%) \| 38.67% (60.16%) \| 36.77% (58.59%) \| 37.27% (59.16%) \| \| CodeGen-Mono \| 16B \| 50.24% (64.39%) \| 43.94% (62.94%) \| 39.55% (61.99%) \| 42.41% (62.31%) \| \| StarCoder \| 15B \| 54.84% (65.10%) \| 46.77% (63.60%) \| 42.80% (61.95%) \| 45.35% (62.66%) \| \| CodeGeeX2 \| 6B \| 52.45% (64.64%) \| 44.52% (63.72%) \| 41.72% (60.48%) \| 43.86% (63.51%) \| \| WizardCoder \| 15B \| 57.85% (66.68%) \| 46.56% (64.86%) \| 41.62% (60.72%) \| 47.45% (64.54%) \| \| GPT-3.5-turbo \| - \| 74.30% (66.19%) \| 66.14% (65.30%) \| 49.56% (62.95%) \| 63.34% (64.72%) \| Table 4: The pass rates (and coverage rate) of the test cases generated on MBPP. • Fourth, with only a few exception, test cases obtained using the oracle code exhibit slightly higher code coverage, while the coverage rate achieved in the other settings (i.e., the self-generated, all-generated, and the placeholder settings) is often slightly lower. The above four takeaway messages can all be inferred from Tables 2, 3, and 4. In addition to all these results, we conduct more experiments to achieve the following takeaway messages. • Fifth, by analyzing the relationship between the quality of code in prompts and the correctness of test, we found that correct code implementation in the prompt often leads to higher quality of test code generation than the case when some incorrect code is given. We conducted an experiments where we first select programming problems in HumanEval+, where the code pass rate of an LLM is neither 0% or 100%. Then we separate self-generated programs/code of the model into two groups, with one group only contains programs/code that are considered as correct and the other only contains incorrect programs/code. In Table 5, we compare the performance of using these two sorts of code in the prompt, for generating test cases using the same LLM. Apparently, the quality of test cases obtained with correct programs/code is obviously higher. We further evaluate the overall testing performance of LLMs with only correct self-generated programs/code, if there exists any, in their prompts. Unlike in Table 5, where we do not take problems that can be 100% or 0% solved, we take all given problems in this evaluation, except, for every problem, we eliminate all incorrect self-generated programs/code if there exist at least one correct implementation synthesized by the evaluated LLM. By doing so, we can observe substantially improved program testing ability on HumanEval+ (i.e., 74.95% for GPT-3.5-turbo, 56.87% for WizardCoder, 54.33% for CodeGeeX2, and 53.24% for StarCoder), comparing with the original self-generated results in Table 5. The same on MBPP. • Sixth, by conducting an additional experiment, we further compare the quality of test cases collected from different positions in the generation results. For every set of the three generated test cases, we analyze the relationship between their correctness and the order when they are generated. The results are illustrated in Figure 3. As can be seen in the figure, the first generated test case often shows the best correctness and the latterly generated ones are more incorrect. This may be due to the fact that the model tends to first generate content with a high level of confidence (which is also more likely to be correct). 7 Improving Program Synthesis Using the Generated Test Cases High quality test cases are not only desired in program analyses, but also helpful to program synthesis. Previous methods have successfully used generated test cases to improve the performance of LLMs in synthesizing programs/code. For instance, [Li et al. (2023a)] designed a special prompt which involves the test cases as an preliminary, if they are available, for generating programs/code. One step further, [Chen et al. (2023)] proposed CodeT, which leverages the LLM to obtain test cases first and tests all synthesized programs/code with these test cases by performing a dual execution agreement, and it picks the code in the largest consensus set (i.e., the consensus set with the most code implementations and test cases) as output to obtain state-of-the-arts program synthesis performance. We encourage interested reader to read the original paper. \| Model \| Size \| w/ correct code \| w/ incorrect code \| #Problem \| \|---------------\|------\|-----------------\|-------------------\|----------\| \| InCoder \| 1.3B \| 28.55% \| 27.39% \| 27 \| \| CodeGen2 \| 1B \| 27.25% \| 25.74% \| 11 \| \| CodeT5+ \| 770M \| 40.19% \| 36.78% \| 27 \| \| SantaCoder \| 1.1B \| 37.45% \| 34.08% \| 24 \| \| CodeGen-Multi \| 16B \| 55.49% \| 50.06% \| 32 \| \| CodeGen2 \| 16B \| 43.56% \| 39.31% \| 29 \| \| CodeGen-Mono \| 16B \| 45.18% \| 42.86% \| 56 \| \| StarCoder \| 15B \| 58.16% \| 57.08% \| 68 \| \| CodeGeeX2 \| 6B \| 52.84% \| 48.63% \| 51 \| \| WizardCoder \| 15B \| 48.02% \| 45.12% \| 54 \| \| GPT-3.5-turbo \| - \| 75.39% \| 68.52% \| 126 \| Table 5: With the correct (self-generated) code, the LLMs show stronger ability of generating correct test cases on HumanEval+ (evaluated only on those problems that can neither be 0% solved nor 100% solved), than in the case where incorrect self-generated code is given in the prompts. Since most LLMs cannot generate any correct code for many hard problems while they often generate incorrect code even for easy problems, the number of tested problems in this experiment increases with the power of the tested LLM, as shown in the rightmost column. In the previous section, we have obtained results about many intriguing properties of the program testing performance of LLMs for code. In this section, we would like to drive the readers to think whether it is possible to utilize these results to improve the program synthesis performance, considering that the test cases (hand-crafted and given or automatically generated in particular) are widely and successfully used in program synthesis. We shall demonstrate that, by utilizing takeaway messages in Section 6, the program synthesis performance of previous methods can be improved significantly. Taking CodeT as an example of the previous state-of-the-art, the method uses a placeholder to generate test cases and treats all the test cases as equally correct as a prior. However, as discussed in our third takeaway message, using self-generated code helps to achieve more powerful ability in generating correct test cases. Moreover, if multiple test cases are provided in a single run of generation given an LLM, the correctness of the test cases decreases with their generation order, as shown in our fifth point. Hence, to obtain superior program synthesis performance, we introduce two simple modifications to it: 1) we employ the “self-generated” setting instead of the “placeholder” setting for generating test cases, which means we utilized synthesize programs in prompts when generating test cases for each program, 2) we assign different weights to the generated test cases based on their order in each generation result, which means we used the rank of each generated test case to re-weight its contribution to the consensus set it belongs to. We test the effectiveness of using 1) the prompt which involves self-generated (SG) code as the test cases generated in this setting show higher correctness than the baseline placeholder setting and 2) the rank-based re-weighted (RW) test cases, in improving program synthesis performance on HumanEval+. Following Chen et al. [2023], we used a temperature of 0.8 to generate code and self-generated test cases. After obtaining the consensus set, we re-weight test case by $p^{i-1}$ with $i$ being its order in the model output, and we let $p = 0.8$. That is, instead of directly using their counting numbers, we use the sum of $p^{i-1}$ and the final score of a consensus set is then the sum of a) $\sum p^{i-1}$ and b) the number of code implementations in the consensus set, and code implementations in the consensus set with the highest score are considered as the best solutions. Table 6 shows the results. We compare CodeT with CodeT+SG, CodeT+RW, and CodeT+SG+RW. For CodeT, we follow their official implementation and generate $100 \times 5$ test cases for each problem. For fair comparison, we ensure that our solutions with SR and/or RW generate the same numbers of program implementations and test cases as CodeT does. Hence, for each problem in HumanEval+, we synthesize a program together with its 5 test cases for 100 times when SR and/or RW are incorporated, i.e., we have $i \in \{1, 2, 3, 4, 5\}$. It can be seen from the table that both SG and WR improves the program synthesis performance considerably on most LLMs, except for Incoder, CodeGen2-1B, CodeT5+, and SantaCoder for which the test cases generated in the placeholder setting show similar or even higher correctness than in the self-generated setting and SG fails with them. For some LLMs, SG is more powerful, while, on the other models including SantaCoder and StarCoder, RW is more powerful. By combining SG and RW, the program synthesis performance of most powerful LLMs in Table 6 improves, comparing to only using one of the two. On GPT-3.5-turbo and WizardCoder, which are the best two models in synthesizing programs on HumanEval+, we achieve +4.22% and +3.04% performance gains for CodeT, respectively, with SG & RW. \| Model \| Size \| Baseline \| CodeT \| + SG \| + RW \| + SG & RW \| \|----------------\|-------\|----------\|-------\|------\|------\|-----------\| \| InCoder \| 1.3B \| 6.99% \| 9.85% \| 9.45%\| 10.26%\| 9.98% \| \| CodeGen2 \| 1B \| 9.19% \| 15.15%\| 14.89%\| 15.67%\| 15.35% \| \| CodeT5+ \| 770M \| 12.95% \| 16.57%\| 16.28%\| 17.19%\| 16.98% \| \| SantaCoder \| 1.1B \| 15.21% \| 18.43%\| 18.17%\| 18.75%\| 18.63% \| \| CodeGen-Multi \| 16B \| 15.35% \| 24.50%\| 25.71%\| 25.72%\| 26.95% \| \| CodeGen2 \| 16B \| 19.33% \| 27.56%\| 28.51%\| 28.43%\| 29.63% \| \| CodeGen-Mono \| 16B \| 26.15% \| 35.63%\| 36.69%\| 36.63%\| 37.95% \| \| StarCoder \| 15B \| 27.90% \| 40.46%\| 41.21%\| 42.12%\| 43.15% \| \| CodeGeeX2 \| 6B \| 29.97% \| 44.16%\| 45.23%\| 44.92%\| 46.32% \| \| WizardCoder \| 15B \| 46.23% \| 58.41%\| 60.13%\| 59.60%\| 61.45% \| \| GPT-3.5-turbo \| - \| 61.70% \| 69.25%\| 72.45%\| 70.75%\| 73.47% \| Table 6: Program synthesis performance (Pass@1) of LLMs can be significantly improved by using our takeaway messages in Section 6. The experiment is on HumanEval+. 8 RELATED WORK Test case generation via program analysis. Generating reasonable test cases for analyzing programs is a long-standing problem in the software engineering community. Various program analysis techniques, e.g., fuzzing, have been developed for achieving this goal. AFL++ (Fioraldi et al., 2020) is the most popular tool which incorporates many techniques in this category. A major weakness of these techniques is understandability of the generated test cases. Test case generation via deep learning. The invention of transformer and self-supervised pre-training have brought a breakthrough to programming language processing and program testing (Fioraldi et al., 2020; Tufano et al., 2022; Dinella et al., 2022). After being trained in a self-supervised manner on a large and diverse code corpus, LLMs have demonstrated remarkable abilities in understanding and synthesizing programs. We have also witnessed the adaptation of pre-trained LLMs (e.g., ChatGPT) to fuzzing (Xia et al., 2023) very recently. Similarly, Lemieux et al. (2023) utilized Codex to provide example test cases for under-covered functions, which prevents the coverage improvements stall. Nevertheless, there still lack and require in-depth analyses and intensive comparisons of different LLMs in program testing, considering that powerful LLMs emerge continuously. For instance, the recent WizardCoder (Luo et al., 2023) exhibits an obvious program synthesis superiority over other contemporary open-source LLMs. In our study, we focus on the analyses and comparison of the LLMs in writing test code and generating test cases. Evaluation of Large Language Model. Recently, large language models (LLMs) has incited substantial interest in both academia and industry. In order to evaluate the capabilities of large language models, a variety of effort have been devoted from the perspectives of natural/programming language processing accuracy, robustness, ethics, biases, and trustworthiness, etc. For instance, PromptBench (Zhu et al., 2023) demonstrates that current LLMs are sensitive to adversarial prompts, and careful prompt engineering is necessary for achieving descent performance with them. Another example, DecodingTrust (Wang et al., 2023a), offers a multifaceted exploration of trustworthiness of the GPT models, especially GPT-3.5 and GPT-4. The evaluation expands beyond the typical trustworthiness concerns to include several new critical aspects. Agentbench (Liu et al., 2023b) evaluates LLM as agents on challenging tasks in interactive environments. Their experimental results show that, while top commercial LLMs present a strong ability of acting as agents in complex environments, there is a significant disparity in performance between them and their open-source competitors. 9 CONCLUSION In this paper, we have performed thorough analyses of recent LLMs (mostly LLMs for code) in testing programs/code. Through comprehensive experiments with 11 LLMs on programming benchmark datasets including HumanEval+ and MBPP (the sanitized version), we have uncovered a range of intriguing characteristics of these LLMs for program/code testing. We have illustrated how the program testing capabilities of these LLMs can be enhanced in comparing intensive empirical results in four different settings. Based on our findings, we are also capable of improving the performance of state-of-the-art LLMs in synthesizing programs/code with test cases of higher quality. 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Dgc5RWZwTR	What is responsible for the efficiency of the proposed method? It is about 15 times faster than the closest method (Table 3) despite only changing the sampling scheme. Can the authors provide a breakdown of how other MTL techniques compute the influence matrix?	Efficient Training of Multi-task Combinatorial Neural Solver with Multi-armed Bandits Anonymous authors Paper under double-blind review Abstract Efficiently training a multi-task neural solver for various combinatorial optimization problems (COPs) has been less studied so far. In this paper, we propose a general and efficient training paradigm based on multi-armed bandits to deliver a unified combinatorial multi-task neural solver. To this end, we resort to the theoretical loss decomposition for multiple tasks under an encoder-decoder framework, which enables more efficient training via proper bandit task-sampling algorithms through an intra-task influence matrix. Our method achieves much higher overall performance with either limited training budgets or the same training epochs, compared to standard training schedules, which can be promising for advising efficient training of other multi-task large models. Additionally, the influence matrix can provide empirical evidence of some common practices in the area of learning to optimize, which in turn supports the validity of our approach. 1 Introduction Although a generic neural solver for multiple combinatorial optimization problems (COPs) is appealing, this problem is less studied in the literature, and training such a neural solver can be prohibitively expensive, especially in the era of large models. To relieve the training burden and better balance the resource allocation, in this paper, we propose a novel training paradigm via multi-armed bandits (MAB) from a multi-task learning (MTL) perspective, which can efficiently train a multi-task combinatorial neural solver under limited training budgets. To this end, we treat each COP with a specific problem scale as a task and manage to deliver a generic solver handling a set of tasks simultaneously. Different from a standard joint training in MTL, we employ MAB algorithms to select/sample one task in each training round, hence avoiding the complex balancing of losses from multiple tasks. To better guide the MAB algorithms, we employ a reasonable reward design derived from the theoretical loss decomposition for the widely adopted encoder-decoder architecture in MTL. This loss decomposition also brings about an influence matrix revealing the mutual impacts between tasks, which provides rich evidence to explain some common practices in the scope of COPs. To emphasize, our method is the first to consider training a generic neural solver for different kinds of COPs. This greatly differs from existing works focusing on either solution construction (Vinyals et al., 2015; Bello et al., 2017; Kool et al., 2019; Kwon et al., 2020) or heuristic improvement (Lu et al., 2020; Wu et al., 2021b; Agostinelli et al., 2021; Fu et al., 2021; Kool et al., 2022). Some recent works seek to generalize neural solvers to different scales (Hou et al., Li et al., 2021; Cheng et al., 2023; Wang et al., 2023) or varying distributions (Wang et al., 2021; Bi et al., 2022; Geisler et al., 2022), but with no ability to handle multiple types of COPs simultaneously. Experiments are conducted for 12 tasks: Four types of COPs, the Travelling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), the Orienteering Problem (OP) and the Knapsack Problem (KP), and each of them with three problem scales. We compare our approach with single-task training (STL) and extensive MTL baselines (Mao et al., 2021; Yu et al., 2020; Navon et al., 2022; Kendall et al., 2018; Liu et al., 2021a,b) under the cases of the same training budgets and same training epochs. Compared with STL, our approach needs no prior knowledge about tasks and can automatically focus on harder tasks so as to maximally utilize the training budget. What’s more, when comparing with STL under the same training epoch, our approach not only enjoys the cheaper training cost which is strictly smaller than that of the most expensive task, but also shows the generalization ability by providing a universal model to cover different types of COPs. Compared with the MTL methods, our method only picks the most impacting task to train at each time which improves the training efficiency without explicitly balancing the losses. In summary, our contributions can be concluded as follows: (1) We propose a novel framework for efficiently training a combinatorial neural solver for multiple COPs via MAB, which achieves prominent performance against standard training paradigms with limited training resources and can further advise efficient training of other large models; (2) We study the theoretical loss decomposition for the encoder-decoder architecture, leading to the influence matrix reflecting the inherent task relations and reasonable reward guiding the update of MAB algorithms.; (3) We verify several empirical observations for neural solvers from previous works [Kool et al., 2019; Joshi et al., 2021] by the influence matrix, demonstrating the validity and reasonableness of our approach. 2 RELATED WORK Neural solvers for COPs. Pointer Networks [Vinyals et al., 2015] pioneered the application of deep neural networks for solving combinatorial optimization problems. Subsequently, numerous neural solvers have been developed to address various COPs, such as routing problems [Bello et al., 2017; Kool et al., 2019; Lu et al., 2020; Wu et al., 2021b], knapsack problem [Bello et al., 2017; Kwon et al., 2020], job shop scheduling problem [Zhang et al., 2020], and others. There are two prevalent approaches to constructing neural solvers: solution construction [Vinyals et al., 2015; Bello et al., 2017; Kool et al., 2019; Kwon et al., 2020], which sequentially constructs a feasible solution, and heuristic improvement [Lu et al., 2020; Wu et al., 2021b; Agostinelli et al., 2021; Fu et al., 2021; Kool et al., 2022], which provides meaningful information to guide downstream classical heuristic methods. In addition to developing novel techniques, several works [Wang et al., 2021; Geisler et al., 2022; Bi et al., 2022; Wang et al., 2023] have been proposed to address generalization issues inherent in COPs. For a comprehensive review of the existing challenges in this area, we refer to the survey [Bengio et al., 2020]. Multi-task learning. Multi-Task Learning (MTL) aims to enhance the performance of multiple tasks by jointly training a single model to extract shared knowledge among them. Numerous works have emerged to address MTL from various perspectives, such as exploring the balance on the losses from different tasks [Mao et al., 2021; Yu et al., 2020; Navon et al., 2022; Kendall et al., 2018; Liu et al., 2021a,b] designing module-sharing mechanisms [Misra et al., 2016; Sun et al., 2020; Hu & Singh, 2021], improving MTL through multi-objective optimization [Sener & Kolotun, 2018; Lin et al., 2019; Momma et al., 2022], and meta-learning [Song et al., 2022]. To optimize MTL efficiency and mitigate the impact of negative transfer, some research focuses on task-grouping [Kumar & III, 2012; Zamir et al., 2018; Standley et al., 2020; Fifty et al., 2021], with the goal of identifying task relationships and learning within groups to alleviate negative transfer effects in conflicting tasks. On the application level, MTL has been extensively employed in various domains, including natural language processing [Collobert & Weston, 2008; Luong et al., 2016], computer vision [Zamir et al., 2018; Seong et al., 2019], bioinformatics [Xu et al., 2017], and many others. However, there are limited works on solving COPs using MTL. In this work, we highlight research on MTL for COPs and propose a learning framework to concurrently address various types of COPs. Multi-armed bandits. Multi-armed bandit (MAB) is a classical problem in decision theory and machine learning that addresses the exploration-exploitation trade-off. Several algorithms and strategies have been suggested to solve the MAB problem, such as the $\epsilon$-greedy, Upper Confidence Bound (UCB) family of algorithms [Lai et al., 1985; Auer et al., 2002], the Exp3 family [Littlestone & Warmuth, 1994; Auer et al., 1995; Gur et al., 2014], and the Thompson sampling [Thompson, 1933; Agrawal & Goyal, 2012; Chapelle & Li, 2011]. These methods differ in their balance of exploration and exploitation, and their resilience under distinct types of uncertainty. The MAB has been extensively studied in both theoretical and practical contexts, and comprehensive details can be found in [Slivkins et al., 2019; Lattimore & Szepesvári, 2020]. 3 METHOD We consider $K$ types of COPs, denoted as $T^i$ ($i = 1, 2, ..., K$), with $n_i$ different problem scales for each COP. Thus, the overall task set is $\mathcal{T} = \bigcup_{i=1}^{K} T^i := \{T^i_j \| j = 1, 2, ..., n_i, i = 1, 2, ..., K\}$. Figure 1: Pipeline of MAB for Solving COPs in view of MTL. We consider four types of COPs: TSP, CVRP, OP, and KP, each with a corresponding header and decoder. The encoder, which is common to all COPs, is also included. For each time step, we utilize the MAB algorithm to select a specific task for training, such as CVRP-100 depicted in the figure. We then obtain the loss for the selected task, perform loss decomposition as detailed in Section 3.1, and construct a reward using the methodology outlined in Section 3.2. Finally, we utilize the reward to update the MAB algorithm. Algorithm 1 MAB for Solving COPs in view of MTL Require: Combinatorial neural solver $S_\Theta$ with parameters $\Theta$, task set $\mathcal{T}$, MAB algorithm $\mathcal{A}(\mathcal{T})$, loss function $L(\Theta)$, number of training loops $L$, update frequency for MAB algorithm $freq$. 1: for $t = 1$ to $L$ do 2: Train $S_{\Theta(t)}$ on task $T^i_j$ selected by $\mathcal{A}(\mathcal{T})$ and store the gradient information $\nabla L^i_j(\Theta(t))$ 3: if $t \mod freq = 0$ then 4: Obtain reward $\vec{r}^i_j$ for each task $T^i_j$ using stored gradients $\{\nabla L^i_j(\Theta(t))\}_{t=t_1}^{t_2}$ following Section 3.2 5: Update $\mathcal{A}(\mathcal{T})$ with reward $\vec{r}^i_j$ for each task $T^i_j$ 6: Clear the record of the gradient information 7: end if 8: end for 9: return Well-trained neural solver $S_\Theta$ For each type of COP $T^i$, we consider a neural solver $S_{\Theta^i}(T^i_j) : T^i_j \rightarrow Y^i_j$, where $\Theta^i$ are the parameters for COP $T^i$, $T^i_j$ and $Y^i_j$ are the input instance the output space for COP $T^i$ with the problem scale of $n_j$ (termed as task $T^i_j$ in the sequel). The parameter vector $\Theta^i = (\theta^{\text{share}}, \theta^i)$ contains the shared and task-specific parameters for the COP $T^i$, and the complete set of parameters is denoted by $\Theta = \bigcup^K_{i=1} \Theta^i$. This parameter notation corresponds to the commonly used Encoder-Decoder framework in multi-task learning in Fig. 1, where $\theta^{\text{share}}$ represents the encoder - shared across all tasks, and $\theta^i$ represents the decoder - task-specific for each task. Given the task loss functions $L^i_j(\Theta^i)$ for COP $T^i$ with the problem scale of $n_j$, we investigate the widely used objective function: $$\min_\Theta L(\Theta) = \sum^K_{i=1} \sum^{n_i}_{j=1} L^i_j(\Theta^i).$$ \hspace{1cm} (1) We propose a general framework based on Multi-Armed Bandits (MAB) to dynamically select tasks during training rounds and a reasonable reward is constructed to guide the selection process. In particular, our approach establishes a comprehensive task relation by the obtained influence matrix, which has the potential to empirically validate several common deep learning practices while solving COPs. Overview. We aim to solve Eq. 1 using the MAB approach. Given the set of tasks $\mathcal{T} = \{T^i_j \| j = 1, 2, ..., n_i, i = 1, 2, ..., K\}$, we select an arm (i.e., task being trained) $a_t \in \mathcal{T}$ following an MAB algorithm, which yields a random reward signal $r_t$ that reflects the effect of the selection. The approximated expected reward is updated based on the received rewards. Essentially, our proposed 1 According to the Encoder-Decoder framework, encoder commonly refers to shared models, whereas decoder concerns task-specific modules. In this study, the decoder component comprises two modules: "Header" and "Decoder" as illustrated in Figure 1. method is applicable to any MAB algorithm. The general framework of MAB for solving COPs within the context of Multi-Task Learning (MTL) is outlined in Algorithm 1, and the overall pipeline is illustrated in Figure 1. 3.1 Loss Decomposition In the framework of MAB for solving COPs in view of MTL described in Algorithm 1, the way to design a reasonable reward to guide its update is crucial. In this part, we analytically drive a reasonable reward by decomposing the loss function for the Encoder-Decoder framework in Fig. 1. Following the previous notation, $ \Theta = \bigcup_{i=1}^{K} \Theta^i = \{\theta^{\text{share}}\} \cup \{\theta_i, i = 1, 2, ..., K\} $ are all trainable parameters. We suppose that a meaningful reward should satisfy the following two properties: (1) It can benefit our objective and reveal the intrinsic training signal; (2) When a task is selected, there always has positive effects on it in expectation. The difference on loss function is an ideal choice and previous work has used it to measure the task relationship (Fifty et al., 2021). However, such measurement is invalid in our context because there are no significant differences among tasks (see Appendix B), so using such information may mislead the bandit selection. What’s more, the computation cost of the “lookahead loss” in Fifty et al. (2021) is considerably expensive when frequent reward signals are needed. We instead propose a more fundamental way based on gradients to measure the impacts of training one task upon the others. To simplify the analysis, in Proposition 1, we assume the standard gradient descent (GD) is used to optimize Eq. 1 by training one task at each step $ t $, and then derive the loss decomposition under the encoder-decoder framework. Any other optimization method, e.g., Adam (Kingma & Ba, 2015), can also be used here with small modifications. We leave the detailed proofs for GD and Adam optimizer in Appendix B. Proposition 1 (Loss decomposition for GD). Using encoder-decoder framework with parameters $ \Theta = \bigcup_{i=1}^{K} \Theta^i = \{\theta^{\text{share}}\} \cup \{\theta_i, i = 1, 2, ..., K\} $ and updating parameters with standard gradient descent: $ \Theta(t+1) = \Theta(t) - \eta_t \nabla L(\Theta(t)) $, where $ \eta_t $ is the step size. Then the difference of the loss of task $ T_j^i $ from training step $ t_1 $ to $ t_2 $: $ \Delta L_j^i(t_1 \rightarrow t_2) = L_j^i(\Theta^i(t_2)) - L_j^i(\Theta^i(t_1)) $ can be decomposed to: \[ \Delta L_j^i(t_1 \rightarrow t_2) = - (\nabla^T L_j^i(\Psi^i(t_1))) \sum_{t=t_1}^{t_2} \mathbb{1}(a_t = T_j^i) \eta_t \nabla L_j^i(\Theta^i(t)) + \nabla^T L_j^i(\Psi^i(t_1)) \sum_{q \neq j} \sum_{t=t_1}^{t_2} \mathbb{1}(a_t = T_q^i) \eta_t \nabla L_q^i(\Theta^i(t)) \] \[ + \nabla^T \theta^{\text{share}} L_j^i(\Psi^i(t_1)) \sum_{p=1}^{K} \sum_{q=1}^{n_p} \sum_{t=t_1}^{t_2} \mathbb{1}(a_t = T_p^q) \eta_t \nabla \theta^{\text{share}} L_p^q(\Theta^p(t)), \] where $ \nabla L(\Theta) $ means taking gradient w.r.t. $ \Theta $ and $ \nabla L_\theta(\Theta) $ means taking gradient w.r.t. $ \theta \subseteq \Theta $, $ \Psi^i(t_1) $ is some vector between $ \Theta^i(t_1) $ and $ \Theta^i(t_2) $ and $ \mathbb{1}(a_t = T_j^i) $ is the indicator function. The idea behind Eq. 2 means the improvement on the loss for task $ T_j^i $ from $ t_1 $ to $ t_2 $ can be decomposed into three parts: (a) effects of training $ T_j^i $ itself w.r.t. $ \Theta^i $; (b) effects of training same kind of COP $ \{T_q^i, q \neq j\} $ w.r.t. $ \Theta^i $; and (c) effects of training other COPs $ \{T_p^q, p \neq i\} $ w.r.t. $ \theta^{\text{share}} $. Indeed, we quantify the impact of different tasks on $ T_j^i $ through this decomposition, which provides the intrinsic training signals for designing reasonable rewards. 3.2 Reward Design and Influence Matrix Construction In this part, we design the reward and construct the intra-task relations based on the loss decomposition introduced in Section 3.1. Though Eq. 2 reveals the signal during training, the inner products of gradients from different tasks can significantly differ at scale (see Appendix F). This will mislead the bandit’s update seriously since improvements may come from large gradient values even when they are almost orthogonal. To address this, we propose to use cosine metric to measure the influence between task pairs. Formally, for task $T_j^i$ from $t_1$ to $t_2$, the influence from training the same type of COP $T_q^i$ to $T_j^i$ is: $$m_q^i(t_1 \rightarrow t_2) = \frac{\nabla T L_j^i(\Psi^i(t_1)) \sum_{t=t_1}^{t_2} \eta_t 1(a_t = T_q^i) \nabla L_q^i(\Theta^i(t))}{\|\|\nabla T L_j^i(\Psi^i(t_1))\|\| \cdot \|\|\sum_{t=t_1}^{t_2} \eta_t 1(a_t = T_q^i) \nabla L_q^i(\Theta^i(t))\|\|},$$ (3) and the influence from training other types of COPs $T_p^q$ to $T_j^i$ is: $$m_p^q(t_1 \rightarrow t_2) = \frac{\nabla T_{\text{shared}} L_j^i(\Psi^i(t_1)) \sum_{t=t_1}^{t_2} \eta_t 1(a_t = T_p^q) \nabla L_p^q(\Theta^p(t))}{\|\|\nabla T_{\text{shared}} L_j^i(\Psi^i(t_1))\|\| \cdot \|\|\sum_{t=t_1}^{t_2} \eta_t 1(a_t = T_p^q) \nabla L_p^q(\Theta^p(t))\|\|}.$$ (4) Given Eq. [3,4], we denote the influence vector to $T_j^i$ as: $$m_j^i(t_1 \rightarrow t_2) = (\ldots, m_1^i(t_1 \rightarrow t_2), \ldots, m_j^i(t_1 \rightarrow t_2), \ldots, m_n^i(t_1 \rightarrow t_2), \ldots)^T$$ (5) Based on Eq. 5, an influence matrix $M(t_1 \rightarrow t_2) = (\ldots, m_j^i(t_1 \rightarrow t_2), \ldots)^T \in \mathbb{R}^{\sum_{k=1}^{K} n_k \times \sum_{k=1}^{K} n_k}$ can be constructed to reveal the relationship between tasks from time step $t_1$ to $t_2$. There are several properties about influence matrix $M(t_1 \rightarrow t_2)$: (1) $M(t_1 \rightarrow t_2)$ has blocks $M^i(t_1 \rightarrow t_2) \in \mathbb{R}^{n_i}$ in the diagonal position which is the sub-influence matrix of a same kind of COP with different problem scales; (2) $M(t_1 \rightarrow t_2)$ is asymmetry which is consistent with the general understanding in multi-task learning; (3) The row-sum of $M(t_1 \rightarrow t_2)$ are the total influences obtained from all tasks to one task; (4) The column-sum of $M(t_1 \rightarrow t_2)$ are the total influences from one task to all tasks. According to the implication of the elements in $M(t_1 \rightarrow t_2)$, the column-sum of $M(t_1 \rightarrow t_2)$: $$r(t_1 \rightarrow t_2) = 1^T \cdot M(t_1 \rightarrow t_2) \in \mathbb{R}^{1 \times \sum_{k=1}^{K} n_k}$$ (6) actually provides a meaningful reward signal for selecting tasks, which we can use to update the bandit algorithm. Moreover, we denote the update frequency of computing the influence matrix as $\Delta T$ and the overall training time is $n \Delta T$, then an average influence matrix $W$ can be constructed based on influence matrices $\{M(k \Delta T \rightarrow (k + 1) \Delta T), k = 0, 2, \ldots, n - 1\}$ collected during the training process: $$W = \frac{1}{n \Delta T} \sum_{k=1}^{n-1} M(k \Delta T \rightarrow (k + 1) \Delta T),$$ (7) revealing the overall task relations across the training process. When computing the bandit rewards, there remains an issue regarding the approximation of $\nabla T L_j^i(\Psi^i(t_1))$ in equations [3] and [4]. Moreover, there is a lack of theoretical works discussing this issue within the context of neural networks. We propose a heuristic method that relies on the widely accepted assumption in multi-task learning: Assumption 1. When using cosine metric on the gradients to measure the similarity between tasks, one task should have the similarity of 1 with itself (Wang et al., 2020; Yu et al., 2020). The training influences determined by Eq. [3] and [4] can be seen as the similarity between tasks measured by cosine metric, therefore we can determine: $$\nabla T L_j^i(\Psi^i(t_1)) = \sum_{t=t_1}^{t_2} \eta_t 1(a_t = T_j^i) \nabla L_j^i(\Theta^i(t))$$ (8) for Eq. [3] when $q = j$ in order to ensure that the self-task similarity $m_j^i(t_1 \rightarrow t_2)$ equals 1. 4 EXPERIMENTS In this section, we conduct a comparative analysis between our proposed method and both single-task training (STL) and extensive multi-task learning (MTL) methods to demonstrate the efficacy of our approach in addressing various COPs under different evaluation criteria. Specifically, we examine two distinct scenarios: (1) Under identical training budgets, we aim to showcase the convenience of our method in automatically obtaining a universal combinatorial neural solver for multiple COPs, circumventing the challenges of balancing loss in MTL and allocating time for each task in STL; (2) Given the same number of training epochs, we seek to illustrate that our method can derive a potent neural solver with excellent generalization capability. Furthermore, we employ the influence matrix to analyze the relationship between different COP types and the same COP type with varying problem scales. Experimental settings. We explore four types of COPs: the Travelling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), the Orienteering Problem (OP), and the Knapsack Problem (KP). Detailed descriptions can be found in Appendix A. Three problem scales are considered for each COP: 20, 50, and 100 for TSP, CVRP, and OP; and 50, 100, and 200 for KP. We employ the notation “COP-scale”, such as TSP-20, to denote a particular task, resulting in a total of 12 tasks. We emphasize that the derivation presented in Section 3.1 applies to a wide range of loss functions encompassing both supervised learning-based and reinforcement learning-based methods. In this study, we opt for reinforcement learning-based neural solvers, primarily because they do not necessitate manual labeling of high-quality solutions. As a representative method in this domain, we utilize the Attention Model (AM) (Kool et al., 2019) as the backbone and employ POMO (Kwon et al., 2020) to optimize its parameters. Concerning the bandit algorithm, we select Exp3 and the update frequency is set to 12 training batches. We discuss the selection of the MAB algorithms and update frequency in Appendix C, with details on training and configuration in Appendix E. 4.1 Comparison with Single Task Training and Multi Task Learning In this part, we explore the differences in performance between our method, MTL, and STL across various comparison criteria, highlighting our method’s superior efficiency and generalization ability. Comparison under same training budgets. We now consider a practical scenario with limited training resources available for neural solvers for all tasks. Our method addresses this challenge by concurrently training all tasks using an appropriate task sampling strategy. However, establishing a schedule for STL is difficult due to the lack of information regarding resource allocation for each task, and MTL methods are hindered by efficiency issues arising from joint task training. In this section, we compare our method with naive STL and MTL methods in terms of the optimality gap: \[ gap\% = \left\| \frac{\text{obj}}{\text{gt}} - 1 \right\| \times 100, \] averaged over 10,000 instances for each task under an identical training time budget. The total training time budget is designated as $ B $, with each type of COP receiving resources equitably for $ \frac{B}{T} $ within the STL framework. Two schedules are considered for the allocation of time across varying problem scales for the same category of COP: (1) Average allocation, denoted as STL$_{\text{avg}}$, indicating a uniform distribution of resources for each task; (2) Balanced allocation, denoted as STL$_{\text{bal}}$, signifying a size-dependent resource assignment with a 1:2:3 ratio from small to large problem scales, categorizing tasks into easy-median-hard levels. The first schedule is suitable for realistic scenarios where information regarding the tasks is unavailable, while the second is advantageous when prior knowledge is introduced. To mitigate the impact of extraneous computations, we calculate the time necessary to complete one epoch for each task and convert the training duration into the number of training epochs for STL. Utilizing the same device, the training time for each task with STL and MTL methods can be found in Table 1 and Table 3. We assess three distinct training budgets: (1) Small budget: the time required to complete 500 training epochs using our method, approximately 1.59 days in GPU hours; Table 1: Training time per epoch, represented in minutes. The COPs are classified into three scales: small, median, and large, which correspond to the sizes of 20, 50, and 100, respectively (50, 100, and 200 for KP). \| COP \| Small \| Median \| Large \| \|-----\|-------\|--------\|-------\| \| TSP \| 0.19 \| 0.39 \| 0.75 \| \| CVRP\| 0.27 \| 0.50 \| 0.90 \| \| OP \| 0.20 \| 0.41 \| 0.60 \| \| KP \| 0.34 \| 0.61 \| 1.10 \| Table 2: Comparison among our proposed method, multi-task learning (MTL), and single task training (STL) utilizing the same training budget. Specifically, STLavg. and STLbal. denote the allocation of resources, with an even distribution and a balanced allocation ratio of 1 : 2 : 3, respectively, among tasks with varying scales from small to large. The reported results depict the optimality gap (↓) in the main aspects. \| Method \| TSP20 \| TSP50 \| TSP100 \| CVRP20 \| CVRP50 \| CVRP100 \| OP20 \| OP50 \| OP100 \| KP50 \| KP100 \| KP200 \| Avg. Gap \| \|------------\|-------\|-------\|--------\|--------\|--------\|---------\|------\|------\|-------\|------\|-------\|-------\|----------\| \| STLavg. \| 0.009%\| 0.346%\| 3.934% \| 0.405% \| 2.292% \| 5.890% \| 1.075%\| 1.291%\| 5.674% \| 0.029%\| 0.015%\| 0.017%\| 1.573% \| \| STLbal. \| 0.019%\| 0.346%\| 2.967% \| 0.599% \| 2.292% \| 4.774% \| 1.073%\| 1.291%\| 4.771% \| 0.033%\| 0.015%\| 0.016%\| 1.346% \| \| Naive-MTL \| 0.029%\| 0.725%\| 3.427% \| 0.676% \| 2.455% \| 4.396% \| 0.445%\| 2.607%\| 5.564% \| 0.036%\| 0.014%\| 0.016%\| 1.624% \| \| Bandit-MTL \| 0.035%\| 0.401%\| 2.817% \| 0.717% \| 2.346% \| 4.460% \| 0.153%\| 1.148%\| 5.486% \| 0.036%\| 0.014%\| 0.016%\| 1.296% \| \| PCGrad \| 0.230%\| 0.762%\| 4.476% \| 1.051% \| 2.817% \| 5.606% \| 0.626%\| 2.773%\| 7.735% \| 0.041%\| 0.018%\| 0.022%\| 1.046% \| \| UW \| 0.036%\| 0.394%\| 1.905% \| 0.451% \| 1.667% \| 3.291% \| 0.562%\| 1.776%\| 3.989% \| 0.039%\| 0.016%\| 0.022%\| 1.085% \| \| CAGrad \| 0.634%\| 3.209%\| 8.433% \| 1.417% \| 4.631% \| 7.668% \| 0.536%\| 4.516%\| 8.232% \| 0.048%\| 0.024%\| 0.063%\| 3.284% \| \| IMTL \| 27.53%\| 53.71%\| 77.15% \| 175.3% \| 345.3% \| 560.3% \| 8.634%\| 31.43%\| 53.6% \| 71.8% \| 125.4%\| \| \| \| Nash-MTL \| 0.131%\| 0.280%\| 0.858% \| 0.466% \| 2.852% \| 1.471% \| 3.486%\| 7.412%\| 0.045%\| 0.016%\| 0.021%\| 0.206% \| \| Random \| 0.041%\| 0.402%\| 1.75% \| 0.489% \| 1.797% \| 3.298% \| 0.987%\| 0.794%\| 2.488% \| 0.032%\| 0.014%\| 0.015%\| 0.862% \| \| Ours \| 0.030%\| 0.297%\| 1.687% \| 0.422% \| 1.554% \| 2.861% \| 1.081%\| 0.533%\| 2.153% \| 0.031%\| 0.014%\| 0.014%\| 0.710% \| (2) Medium budget: 1000 training epochs, consuming 3.28 days in GPU hours; and (3) Large budget: 2000 training epochs, spanning 6.64 days in GPU hours. Extensive MTL baselines are considered here: Bandit-MTL (Mao et al., 2021), PCGrad (Yu et al., 2020), Nash-MTL (Navon et al., 2022), Uncertainty-Weighting (UW) (Kendall et al., 2018), CAGrad (Liu et al., 2021a) and IMTL (Liu et al., 2021b). We also involve the random policy which samples the task uniformly at each training slot, and the results are presented in Table 2. In general, our method outperforms MTL and STL methods in terms of average gap across all the budgets used. Specifically, our method yields consistent improvements for 10 out of 12 tasks under the small budget, 8 and 7 out of 12 tasks under the medium and large budget. Moreover, our approach demonstrates a stronger focus on more challenging problems, as it attains greater improvements for larger problem scales compared to smaller ones. What’s more, when comparing with all MTL methods, our method demonstrates two superior advantages: • Better performance on the solution quality and efficiency: In Table 2 typical MTL methods fail to obtain a powerful neural solver efficiently, and some of them even work worse than naive MTL and STL in limited budgets; • More resources-friendly: The computation complexity of typical MTL methods grows linearly w.r.t. the number of tasks, conducting these training methods still needs heavy training resources (High-performance GPU with quite large memories). The exact training time for one epoch w.r.t. GPU hour are listed in Table 3. Under the same training setting, intermediate termination of prolonged training epoch for typical MTL methods incurs wasted computation resources. However, our method trains only one task at each time slot, resulting in rapid epoch-wise training that facilitates flexible experimentation and iteration. Table 3: Time consumption for MTL methods w.r.t. the GPU hours for training one epoch in average. \| Method \| GPU Hours \| \|------------\|-----------\| \| Bandit-MTL \| 1.04 \| \| PCGrad \| 6.02 \| \| Nash-MTL \| 5.87 \| \| UW \| 1.00 \| \| IMTL \| 5.61 \| \| CAGrad \| 5.24 \| \| Ours \| 0.07 \| It’s also interesting to see that the random policy outperforms STL and the best-performing MTL baselines in our context, underscoring the positive effects of changing the training paradigm. Fur- 2Detailed analysis about the computation complexity of each MTL method is in Appendix D. Figure 2: A comparison between single task training (STL) and our method is showcased in this figure, with both methods utilizing the same number of training epochs (1000 in this case). While STL achieves superior performance, our method is capable of effectively tackling all tasks simultaneously, as evidenced by the strong mean results it produces. Furthermore, our proposed method surpasses the random policy, providing evidence of the additional improvements achieved through the integration of the bandit algorithm. As the training budgets increase, STL’s advantages become evident in easier tasks such as TSP, CVRP-20, OP-20, and KP-50. However, our method continues to deliver robust results for more difficult tasks like CVRP-100 and OP-100. Simultaneously, we observe a decrease in gain as the budget expands, aligning with our understanding that negative transfer exists among different tasks. In addition to performance gains, the most notable advantage of our approach is that it does not require prior knowledge of the tasks and is capable of dynamically allocating resources for each task, which is crucial in real-world scenarios. When implementing STL, biases are inevitably introduced with equal allocation. As demonstrated in Table 2, the performance of two distinct allocation schedules can differ significantly: STL-bal. consistently outperforms STL-avg. due to the introduction of appropriate priors for STL. Table 4: The comparison results are obtained by training our model for 1000 epochs and STL models for 100 epochs each, amounting to a total of 1200 epochs. \| \| TSP20 \| TSP50 \| TSP100 \| CVRP20 \| CVRP50 \| CVRP100 \| OP20 \| OP50 \| OP100 \| KP50 \| KP100 \| KP200 \| Avg. Gap \| \|-------\|-------\|-------\|--------\|--------\|--------\|---------\|------\|------\|-------\|------\|-------\|-------\|----------\| \| STL \| 0.011%\| 0.244%\| 1.578% \| 0.465% \| 1.706% \| 3.194% \| -1.133%\| 0.781%\| 2.898%\| 0.026%\| 0.013%\| 0.01237%\| 0.316% \| \| Ours \| 0.019%\| 0.202%\| 1.086% \| 0.348% \| 1.284% \| 2.362% \| -1.114%\| 0.224%\| 1.277%\| 0.030%\| 0.012%\| 0.01236%\| 0.478% \| Comparison under same training epochs. We conduct a comparison under the same number of training epochs by training our method on 12 tasks mentioned before for 1000 epochs in total, and comparing them with corresponding Single Task Learning (STL) neural solvers that are trained for 1000 epochs on each of their respective tasks. This is, by no means, a fair comparison, as our method dynamically chooses a task to train for 1000 epochs, resulting in a much smaller sample size than each task when using STL. Despite this, we choose this comparison as an intuitive way to demonstrate the superior generalization ability of our method under such extreme conditions. We present the results in Figure 2 and Table 4. Compared to individual tasks, shown in Table 4, our method (trained 1000 epochs) consistently outperforms STL (trained $100 \times 12 = 1200$ epochs) across most tasks, with exceptions noted in TSP20, OP20, and KP50. In most cases, our method’s performance is equivalent to that of using 100 to 300 epochs of STL. However, STL can only obtain one model in this context and lacks the ability to handle different types of COPs or to generalize well when presented with the same type of COP but with varying problem scales. As a result, our method demonstrates unparalleled superiority in three ways: (1) when considering the average performance on all problem scales for each type of COP, our method obtains the best results in CVRP, OP, and KP, and is equivalent to the results achieved by training TSP for about 500 epochs. This showcases our method’s excellent generalization ability for problem scales; (2) Our method can handle various types of COPs under the same number of training epochs, which is impossible for STL due to the existence of task-specific modules; (3) Our method’s training time is strictly shorter than the longest time-consuming task. 4.2 Study of the Influence Matrix Our approach has an additional advantage as it facilitates the identification of the task relationship through the influence matrix developed in Section 3.2. The influence matrix allows us to capture the inherent relationship among tasks. Additionally, we provide empirical evidence pertaining to the experience and observation in the learning to optimize community. We present a detailed view of the influence matrix in Figure 3, revealing significant observations: (1) Figure 3a highlights that the Figure 3: This figure provides a visual representation of the mutual influence between tasks. The left-hand side displays the average influence matrix, as defined in Eq. 7, which reveals significant mutual influences existing among the COPs of the same type. Meanwhile, the right-hand side illustrates the influence value, as defined in Eqs. 3-4, throughout the training process, further demonstrating the extensive mutual impacts among the COPs of the same type and the less pronounced interactions between COPs of different types. The influence matrix computed using Eq. 7 possesses a diagonal-like block structure. This phenomenon suggests a strong correlation between the same type of COP with different problem scales, which is not present within different types of COPs due to the corresponding elements being insignificant. Furthermore, within the same type of COP, we observe that the effect of training a task on other tasks lessens with the increase in the difference of problem scales. Hence, training combinatorial neural solvers on one problem scale leads to higher benefits on similar problem scales than on those that are further away. For instance, the influence of training TSP-20 on TSP-50 is 0.1007, which is higher than the influence on TSP-100, which is $-0.1196$. Similarly, training TSP-100 on TSP-50 has a larger influence than that on TSP-20, as can be observed from influences of $-0.0354$ and $-0.0978$, respectively; (2) Figure 3b presents a visualization of the influence resulting from Eq. 3-4 over the course of the training process. Each point in the chart represents the influence of a particular task on another task at a specific time step. Notably, tasks belonging to the same type of COP are highly influential towards each other due to the large variance of their influence values. Conversely, influences between different types of COPs are negligible, evident from the influence values being concentrated around 0. This striking observation showcases that the employed combinatorial neural solver and algorithm, AM (Kool et al., 2019) and POMO (Kwon et al., 2020), segregate the gradient space into distinct orthogonal subspaces, and each of these subspaces corresponds to a particular type of COP. Furthermore, this implies that the gradient of training each variant of COP is situated on a low-dimensional manifold. As a result, we are motivated to develop more parameter-efficient neural solver backbones and algorithms. 5 CONCLUSIONS In the era of large models, training a unified neural solver for multiple combinatorial tasks is in increasing demand, whereas such a training process can be prohibitively expensive. In this paper, given limited training budgets or resources, we propose an efficient training framework to boost the training of unified multi-task combinatorial neural solvers with a multi-armed bandit sampler. To achieve this, we perform the theoretical loss decomposition, resulting in the meaningful influence matrix that can reveal the intrinsic task relations among different COP tasks, providing evidence for several empirical observations in the area of learning to optimize. We believe that this framework can be powerful for multi-task learning in a broader sense, especially in scenarios where resources are limited, and generalization is crucial. It can also help analyze task relations in the absence of priors. Furthermore, the proposed framework is model-agnostic, which makes it applicable to any existing neural solvers. Different neural solvers may produce varying results on the influence matrix, and a perfect neural solver may gain mutual improvements even from different types of COPs. Therefore, there is an urgent need to study the unified backbone and representation method for solving COPs. REFERENCES Forest Agostinelli, Alexander Shmakov, Stephen McAleer, Roy Fox, and Pierre Baldi. 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o0oroLuPLZ	In equation 4, the set $S$ is not yet defined? I assume that $S$ is the feasible set for $x$. From the example in Equation 1, I presume that $S = \{ x \mid Ax \geq b.\}. I think it would be helpful if the authors could reiterate the downstream optimization task again in Section 3 for clarity.	SP-R-IP: A Decision-Focused Learning Strategy for Linear Programs that Avoids Overfitting Anonymous authors Paper under double-blind review Abstract For forecast-informed linear optimization problems, neural networks have shown to be effective tools for achieving robust out-of-sample performance. Various decision-focused learning paradigms have further refined those outcomes by integrating the downstream decision problem in the training pipeline. One of these strategies involves using a convex surrogate of the regret loss function to train the forecaster, called the SPO+ loss function. It allows for the training problem to be reformulated as a linear optimization program. However, this strategy has only been applied to linear forecasters, and is prone to overfitting. In this paper, we propose an extension of the SPO+ reformulation framework that solves the forecaster training procedure using an interior-point optimization method, and tracks the validation regret of intermediate results obtained for different weights of the barrier term. Additionally, we extend the reformulation framework to include the possibility of neural network forecasters with non-linear activation functions. On a real-life experiment of maximizing storage profits in a day-ahead electricity market using actual price data, we show that the proposed methodology effectively solves the problem of overfitting, and that it can outperform other decision-focused benchmarks including training the forecaster with implicit differentiation. 1 Introduction Linear programs (LPs) are ubiquitous in modern-day decision-making problems in operations research and finance. For many applications, the primary challenge of the problem: \[ \begin{align} \text{minimize} & \quad \hat{c}^T x \\ \text{subject to} & \quad Ax \geq b, \end{align} \] lies in representing the forecast $ \hat{c} $ of ground truth $ c $ as accurately as possible. This is the case for optimal scheduling of assets in energy markets, like storage systems, see e.g. Byrne et al. (2018); financial portfolio optimization, see e.g. Mansini et al. (2014); and deterministic inventory models, see e.g. Levi et al. (2004). The utilization of Machine Learning (ML) models has become common practice for modeling those forecasts. While the particular modeling strategy may differ among the various ML techniques, the main idea is always to fit a generic model’s parameters such that its output matches the ground truth given a set of inputs. The problem of training the forecaster, also referred to as the Empirical Risk Minimization (ERM) problem, involves minimizing a certain loss function over a train data set, which is in many cases solved with an implementation of the gradient descent algorithm. Within the above context, an aspect that is increasingly looked into is the loss function $ L $. The traditional approach is to apply a statistical error metric, e.g. Mean Squared Error (MSE), between the forecast and ground truth. This is a sensible approach when the downstream decision problem is unknown, resulting in a generic forecaster. In Decision-Focused Learning (DFL) or end-to-end learning, the downstream (optimization) problem is included in the forecasting pipeline. This typically includes adopting a task-aware loss function like the regret loss. For many downstream problems, including (mixed integer-)linear optimization, it is a well-known difficulty to minimize regret. with a gradient descent procedure. This is because of ill-defined gradients of the optimized output w.r.t. the forecast cost. Smoothing terms and perturbations with random noise have been proposed to overcome the issue, but such approaches involve approximating the downstream problem in the training phase and can as such distort the results. Another promising approach was proposed by Elmachtoub & Grigas (2022), involving a convex surrogate of the regret loss function, coined "SPO+" loss. They also show that the SPO+ loss allows for DFL with downstream LPs in two distinct ways. First, whereas the SPO+ loss is unsuitable for calculating exact gradients w.r.t. the forecasted cost, its convexity does allow for the subgradient method to be used for training the forecaster. Secondly, the training problem can be re-written to a single-level optimization program by applying duality theory. The latter technique is largely unexplored, and was only implemented for a linear forecasting model. In this paper, we build upon this idea by solving the single-level reformulation of the ERM with an Interior Point (IP) method. This approach, which is inspired by early stopping in traditional gradient descent training, allows for tracking the validation performance along the training iterations. Since IP methods can be used for non-convex optimization problems, this method also allows for extending the forecaster to a broader class of non-convex ML models, such as various types of Neural Networks (NN). The proposed approach comes at the cost of slower training compared to gradient-based methods. We mitigate this by adopting a two-stage procedure, where an initial large-scale forecaster is trained to minimize a statistical error metric. In the second stage, a smaller re-forecaster is trained by adopting the proposed DFL technique. 2 RELATED WORK AND CONTRIBUTIONS Decision-focused learning with gradient descent When solving DFL with gradient descent training, implicit differentiation of the KKT conditions can be used to compute gradients "through" an optimization program to update the parameter of the NN preceding the optimization. This line of research was catalyzed by the seminal work of Amos & Kolter (2017) and Donft et al. (2017), where such differentiation was achieved for quadratic programs. Agrawal et al. (2019) extended the method to convex optimization problems that can be written as disciplined parametric programs. The discontinuous nature of the output of linear and combinatorial programs is well-known to pose difficulties when trying to apply such methods. This sparked Wilder (2019) to add a quadratic smoothing term in the objective of the linear program, thereby enabling the calculation of gradients and facilitating the use of the above-mentioned methods. This idea was extended by Ferber et al. (2020) with a cutting planes approach to address mixed-integer linear programs as the downstream problem. In similar fashion and inspired by IP solution procedures, Mandi & Guns (2020) introduce a log-barrier smoothing term, rendering the objective function strictly convex. An approach that is conceptually similar to such smoothing terms is that of including a tunable noise term to the predicted cost, which was used in an additive, see Berthet et al. (2020), and multiplicative, see Dalle et al. (2022), fashion. For a comprehensive overview of DFL techniques, we refer the reader to Kotary et al. (2021) and Mandi et al. (2023). Interior point methods in machine learning Whereas gradient descent methods have dominated the field of ML recently because of their excellent scaling properties, other (albeit often slower) approaches of training ML models have been proposed, including IP solution strategies. This was first introduced by Trafalis et al. (1997) in a generic NN training setting. Kon et al. (2007) and Li & Liu (2022) show the effectiveness of IP-based solving of a logistic regression problem, which is widely used in the context of feature selection. Another sub-field of ML where IP methods gained significant traction is the training of Support Vector Machines (SVM). SVMs are mostly used for classification tasks, and their training problem can often be formulated as a quadratic program (QP). Ferris & Munson (2002), Woodsend & Gondzio (2011), and Gu et al. (2023) exploit the ability of interior point methods to efficiently solve for the global optimum of such QPs. It is noteworthy that none of these works discuss the possibility of tracking the validation performance along the iterations of the solution procedure, and rely solely on regularization terms to avoid overfitting. Contributions The scientific contribution of this work is threefold: (i) We develop an interior point-based neural network training algorithm that iteratively tracks the task-specific validation loss and dynamically updates the barrier term. (ii) We extend the original SPO+ reformulation ERM to accommodate neural network forecaster training. (iii) We demonstrate the effectiveness of the method by showcasing reduced out-of-sample regret values compared to state-of-the-art DFL models for an optimal scheduling problem of a storage system under price uncertainty using real-life data. 3 DECISION-FOCUSED LEARNING When training a forecaster $ f : \mathbb{R}^n \rightarrow \mathbb{R}^m $, mapping a vector of input features $ \alpha \in \mathbb{R}^n $ to the forecast $ \hat{c} \in \mathbb{R}^m $, the ERM problem can be written as: \[ \min_{\theta \in \Theta} \sum_{i \in I_{tr}} L(c_i, f(\alpha_i; \theta)) + \lambda \Omega(f), \] (2) with $ \theta $ the trainable parameters of the forecaster, $ I_{tr} $ the indices in the train data set, $ (\alpha_i, c_i) $ vector instances of the labeled trainset, $ L $ the chosen loss function and $ \Omega $ a regularization function that helps in avoiding to overfit on the train data. In DFL, it is acknowledged that the forecast is deployed in a downstream decision problem. A typical loss function that is chosen for DFL is the regret loss. When the downstream decision problem exhibits a linear objective function, the regret is defined as: \[ r(c, \hat{c}) = c^T x^(\hat{c}) - c^T x^(c), \] (3) being the difference in the ex-post downstream objective value of the optimal decisions $ x^* $ based on the forecasts, compared to that of the optimal decisions based on the ground truth. Setting the loss function in (2) to this regret function, the ERM becomes a bi-level optimization problem: \[ \min_{\theta \in \Theta} \left[ \sum_{i \in I_{tr}} c_i^T \cdot \arg \min_{x \in \{x \| Ax \geq b\}} (f(\alpha_i; \theta)^T x) \right], \] (4) which is well-known to be intractable for large-scale problems. In recent years, Neural Networks (NNs) have shown outstanding performance in many machine learning applications. When using nonlinear activation functions in the NN architecture, the ERM problem becomes highly non-convex regardless of the loss function, and is typically solved with gradient descent. Here, the NN parameters are iteratively updated by calculating the gradient of the loss function with respect to that parameter, and taking a step in the direction of steepest descent: \[ \theta \leftarrow \theta - \psi \frac{\partial L}{\partial \theta}, \] (5) where $ \psi $ represents the learning rate. When regret is used as the loss function in the ERM, the gradients can be calculated by using the chain rule: \[ \frac{\partial L}{\partial \theta} = \frac{\partial r}{\partial x^} \frac{\partial x^}{\partial \hat{c}} \frac{\partial \hat{c}}{\partial \theta}. \] (6) The first and third factors can be straightforwardly calculated with traditional techniques. The second factor, $ \frac{\partial x^}{\partial \hat{c}} $, being the gradient "through" the optimization program, can be calculated with implicit differentiation of the KKT optimality conditions, see Amos & Kolter (2017). Considering a downstream optimization problem of the form $ \min_{x \in S} g(x, \hat{c}) $, this leads to the following set of equations: \[ \begin{bmatrix} \frac{\partial^2 g}{\partial x^ \partial \hat{c}}(x^, \hat{c}) \\ 0 \end{bmatrix} + \begin{bmatrix} \frac{\partial^2 g}{\partial x^ \partial x^}(x^, \hat{c}) & -A^T \\ A & 0 \end{bmatrix} \begin{bmatrix} \frac{\partial x^}{\partial \hat{c}} \\ \frac{\partial \hat{c}}{\partial \theta} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}. \] (7) In order to solve for $ \frac{\partial x^}{\partial \hat{c}} $, the Hessian of the objective function should be non-zero, which is not the case for a linear optimization problem. To overcome this, smoothing terms have been proposed in the form of a quadratic term by Wilder (2019) and a log-barrier term by Mandi & Guns (2020). This indeed results in gradients that are readily computable. However, as the perturbed optimization problem is an approximation of the actual downstream problem, results may be suboptimal, as we show in Section 5. SPO+ reformulation Elmachtoub & Grigas (2022) propose an alternative approach and acknowledge that the regret function is non-convex in $ c $, which poses challenges in the training process. They re-write it to the "SPO+" loss: \[ l^{SPO+} = \max_{x \in S} \{c^T x - 2\hat{c}^T x\} + 2\hat{c}^T x^(c) - z^(c), \] and argue that when the underlying uncertainty distributions are well-behaved, minimizing this SPO+ loss corresponds to minimizing the regret function. In this formulation, $ z^(c) $ represents the optimal objective value. Interestingly, the SPO+ loss function is convex in $ \hat{c} $ and the authors provide an expression for a subgradient: \[ 2(x^(c) - x^(2\hat{c} - c)) \in \partial l^{SPO+}(c, \hat{c}). \] (8) This presents the opportunity to train a neural network to minimize this surrogate of the regret loss function using the subgradient method. However, our results in Appendix D.2 demonstrate that this expression of the subgradient can lead to suboptimal NN parameter updates in the training procedure. Elmachtoub & Grigas (2022) also provide a second, reformulation, approach: by substituting $ l^{SPO+} $ in (2), and leveraging duality theory, they find an ERM which is a linear optimization program when the forecaster is assumed to be a linear function of the input features, i.e. $ \hat{c} = B\alpha $. The ERM is given by: \[ \begin{align} \text{minimize}_{B,P} & \quad \sum_{i \in I_T} [-b^T p_i + 2(x^(c_i)\alpha_i^T) \bullet B - z^(c_i)] + \lambda \|B\| \\ \text{subject to} & \quad A^T p_i = 2B\alpha_i - c_i \quad \forall i \\ & \quad p_i \geq 0 \quad \forall i, \end{align} \] (9) where $ P $ represents the set of vectors $ p_i $, containing the dual variables associated with the constraints of the downstream optimization problem for every train sample $ i $. In the objective value, $ \bullet $ refers to the trace inner product. More details of this reformulation can be found in Appendix A. Thus, the bi-level ERM (4) is reformulated to a linear optimization program that can be solved with off-the-shelf solvers like CPLEX and Gurobi. No approximation of the mapping from forecasted parameter to decision was required. However, the limitation of this approach is threefold: (i) the training procedure has only been implemented for linear forecasters which restricts the predictive power of the forecaster, (ii) there is a significant risk of overfitting as the result gives the minimum objective value over the training set, and (iii) the ERM problem does not scale well, making it to a lesser extent applicable compared to a (sub)gradient-based approach when large amounts of data are required for the training procedure. ### 4 Method We aim to enhance the SPO+ reformulation framework by addressing two of the above-mentioned key limitations. The first extension is presented in Section 4.1, where we introduce the Sp-R-IP training method. This is an iterative interior point solution procedure to address the problem of overfitting. In section 4.2, we provide a second extension which consists of accommodating non-linear NN forecasters in the SPO+ framework, improving the predictive power of the trained forecast. Finally, Section 4.3 outlines a re-forecasting procedure designed to mitigate the scalability limitation of the IP-based solution method. #### 4.1 Sp-R-IP In training machine learning models, and especially NNs, overfitting to the train data is a well-known problem. The two main procedures to avoid overfitting are (i) regularization and (ii) early stopping. When deploying a regularization term in the ERM problem, high values in the parameters of the forecaster $f$ are penalized, which may lead to models which generalize better. Early stopping entails the procedure of tracking, with each iteration in the (gradient descent) parameter update procedure, the performance of the current value of $f$ on the validation set, i.e. data which is not included in calculating the gradient. When the forecaster stops improving the validation performance, the training procedure is terminated. Inspired by early stopping with gradient descent methods, we here propose a validation performance tracking procedure for the SPO+ reformulation. This requires an iterative approach to solving ERM (9). We observe that a simplex method will always explore vertices of the feasible region in its solution procedure. On the other hand, interior point methods can also be used to solve linear optimization problems and are arguably more suitable for this methodology. Indeed, the explored feasible points in the interior of the feasible region, which correspond to varying manifestations of the forecaster in problem (9), intuitively yield forecasters that generalize better than those derived from the extreme points of the feasible region. A second reason for using this IP-based solution procedure is that it accommodates the extension of ERM (9) to include NN forecasters, which renders the ERM non-linear, and therefore simplex methods inapplicable. The proposed method computes the validation performance of all forecasters obtained from the intermediate solutions accessed by an IP-based solver, and selects the forecaster with the best performance to be deployed on the test set. This SPO+ - Reformulation - Interior Point (Sp-R-IP) method is the core of our contribution. IP solution procedures involve the use of a barrier method as standard practice. Generalizing, we can write a non-linear optimization problem as: \[ \begin{align} \text{minimize} & \quad g(x) \\ \text{subject to} & \quad c(x) = 0, \\ & \quad x \geq 0. \end{align} \] The inequality constraint is replaced with a log-barrier term in the objective function: \[ \begin{align} \text{minimize} & \quad g(x) - \mu \ln(x) \\ \text{subject to} & \quad c(x) = 0, \end{align} \] with $\mu$ a positive number. The barrier term resulting from this nonzero $\mu$ penalizes solutions close to the boundary of the feasible region. As $\mu \to 0$, the barrier problem approaches the original one. The collection of optimal points for all possible values of $\mu$ is the central path: $CP = \{x^(\mu) \| \mu \in \mathbb{R}_+^\}$. The essence of interior point solvers is to iteratively decrease the value of $\mu$ and approximately following the central path toward the optimal solution. To find a solution of problem (11) for a specific value of $\mu$, one could invoke the KKT conditions and apply a Newton-Raphson procedure to iteratively update primal and dual variables towards intermediate solutions for a specific weight of the barrier term. Commercial and open-source solvers generally prioritize speed of obtaining the final (locally) optimal solution and design specialized algorithms for obtaining steps in the primal and dual variable space, and obtaining updated values of $\mu$, for that purpose. However, we argue that when the optimization program is an ERM for training a forecaster, the points on the central path should be regarded as actual intermediate solutions to be tested on the validation set. Indeed, these points constitute different realizations of the forecaster’s parameters, exhibiting decreasing regret as $\mu$ decreases and the optimal value is allowed closer to the edge of the feasible region. As such, the points on the central path are the direct equivalent of the intermediate realizations of a forecaster as it is updated with the gradient descent method. For that reason, we propose to prioritize obtaining optimal solutions of (11) for relevant values of $\mu$ over speed of getting to the optimal solution on the train set. To that end, we propose a dynamic update strategy of gradually decreasing $\mu$, adapting the rate of decrease based on the validation performance: \[ \mu_{n+1} = \frac{\mu_n}{d \cdot Z_{1,n} \cdot Z_{2,n}}, \] with \[ Z_{1,n} = \begin{cases} 1 - \epsilon_1, & \text{if } v_n < v_{n-1} \\ 1, & \text{otherwise} \end{cases} \] \[ Z_{2,n} = \begin{cases} 1 - \epsilon_2, & \text{if } v_n < v_i, \forall i < n \\ 1, & \text{otherwise}. \end{cases} \] In Eq. (12), $d$ represents a constant rate at which $\mu$ is decreased. In Eqs. (13)(14), $v_n$ denotes the validation performance, i.e. the value of the metric to be minimized, while $\epsilon_1$ and $\epsilon_2$ are small predetermined constants that modulate the rate at which $\mu$ decreases. Specifically, $\mu$ decreases more slowly when the latest validation performance improves compared to the iteration before (13) or sets a new best score (14). Adopting this dynamic update strategy ensures a more granular search in areas of high validation performance. Algorithm 1 depicts a high-level overview of the training procedure. Algorithm 1 Sp-R-IP algorithm* ``` 1: Input: $D_{tr} = \{\alpha_i, c_i \| i = 1, ..., N_{tr}\}, D_{val} = \{\alpha_j, c_j \| j = 1, ..., N_{val}\}$ ▷ train and validation data 2: Initialize $\mu$ ▷ barrier weight 3: Initialize $p_0, d_0$ ▷ primal and dual variables 4: for $n = 1, ..., \text{epochs}$ do 5: $p_n, d_n \leftarrow \text{SolveOpti}(p_{n-1}, d_{n-1}, \mu, D_{tr})$ ▷ using Problem (23) 6: Retrieve $\theta_n \in p_n$ 7: $v_n \leftarrow \text{ValPerfo}(f(\cdot; \theta_n), D_{val})$ 8: if $v_n < \min(\{v_i \| i = 1, ..., n - 1\})$ then 9: bestNet \leftarrow f(\cdot; \theta_n) 10: end if 11: $\mu \leftarrow \text{updateMu}(\mu, \{v_i \| i = 1, ..., n\})$ ▷ Via Eq. (12) 12: end for 13: Output: bestNet ``` ### 4.2 Neural Network in SPO+ Reformulation Existing implementations of the SPO+ reformulation ERM are limited to a linear forecaster. Here we propose to extend that framework to accommodate feedforward NN forecasters. Details of this derivation can be found in Appendix A. The ERM now reads: \[ \begin{align} \text{minimize}_{W^{(l)}, b^{(l)}, P} & \sum_{i \in I_{tr}} -b^T p_i + 2 \text{Tr}(x^(c_i)\hat{c}_i^T) + \lambda \sum_{(l)} \|W^{(l)}\| \\ \text{subject to} & A^T p_i = 2\hat{c}_i - c_i \quad \forall i \\ & p_i \geq 0 \quad \forall i \\ & \alpha_i^{(l)} = a^{(l)} \left(W^{(l)} \alpha_i^{(l-1)} + b^{(l)}\right) \quad \forall i, (l) = 1, ..., L - 1 \\ & \hat{c}_i = W^{(L)} \alpha_i^{(L-1)} + b^{(L)} \quad \forall i, \\ & c_{i,\text{init}} - \xi \|c_{i,\text{init}}\| \leq \hat{c}_i \leq c_{i,\text{init}} + \xi \|c_{i,\text{init}}\| \quad \forall i, \end{align} \] with $\text{Tr}()$ the trace operator, $L - 1$ the total amount of NN hidden layers, $W^{(l)}, b^{(l)}$ and $a^{(l)}$ the weights, biases and (nonlinear) activation function of layer $(l)$ respectively, and $\alpha^{(0)}$ the input features. This extension of the SPO+ ERM problem increases the predictive power of the forecaster, while rendering the ERM a nonlinear optimization program. As such, we loose the guarantee of finding a global optimum, and no longer have the option of solving the problem with simplex methods. In light of the discussion in the previous section, these drawbacks are arguably acceptable as the globally optimal solution is expected to be overfitted to the training set, and the interior point method is the option of choice for retrieving intermediate solutions that generalize well to unseen data. The last set of inequalities inhibits the NN to produce outputs that are very different from some initial forecast, $\hat{c}_{i,\text{init}}$ (see Section 4.3). $\xi$ becomes a new (nonnegative) hyperparameter of the problem, where $\xi \to \infty$ results in effectively removing the constraint, and $\xi \to 0$ forces the forecaster to produce the initial forecast, which may result in an infeasible optimization program, depending on the train data and the architecture of the neural network. The purpose of this constraint is to further reduce the ability of the NN to overfit on the train data. 4.3 Re-forecasting and Mini-batches As the proposed approach is a second-order method for training a NN, this is expected not to scale efficiently with the size of the forecaster or training dataset. Here, two heuristic procedures are proposed to mitigate this problem, being a re-forecasting methodology and a mini-batch implementation. Re-forecasting To mitigate the problem of high computational time and memory usage, we adopt a two-stage approach. In the first stage, an initial forecaster is trained to minimize the MSE of cost forecasts on an initial train data set. In the second stage, a refined set of input features, comprising a subset of the original features and the output of the initial forecaster, are used to train a secondary NN. This “re-forecaster” is trained to minimize the decision-focused loss function - being either the SPO+ loss function or the regret loss - on an auxiliary data set. As such, the dimensionality of the decision-focused learning problem is strongly reduced, which improves the computational tractability. This methodology bears a resemblance to the concept of Large Language Model (LLM) adapters, see e.g. [Hu et al., 2023]. In this framework, a second (adapter) NN is trained on top of the LLM, which is not updated in the auxiliary task-specific adapter training procedure. To avoid that the re-forecaster NN gets stuck in local minima exhibiting performance worse than that obtained by the initial forecaster, we implement a warm start methodology. Here, before solving the ERM problem, the re-forecaster is trained to yield outputs that are close to the output of the initial forecaster (by minimizing the MSE). This pretrained re-forecaster is used as a starting point for the ERM problem. The 2-stage forecasting procedure with optional warm start is visualized in Figure 1. Mini-batches Another extension of the SPO+ reformulation framework constitutes the use of mini-batches, which can improve the scalability of the method, as exemplified in Appendix E. We here develop a naive approach where the training set is split up in $M$ mini-batches. The procedure outlined in Algorithm 1 is executed independently for all $M$ mini-batches independently. This entails training $M$ distinct neural networks (NNs) and evaluating their performance on the complete validation dataset during the training process. While this leads to a loss of information for the separate NNs, this results in a greater variety of intermediate manifestations of the NN that potentially generalize better on unseen data. 5 Experiment We apply our DFL training approach to the day-ahead scheduling problem of an Energy Storage System (ESS) in the Belgian electricity market, with the objective of maximizing the profit based on forecasts of the day-ahead electricity price. The details of the linear optimization problem governing the ESS decisions are given in Appendix B. We evaluate the performance of the proposed methodology through a comparative analysis involving eight distinct models. The baseline model relies solely on the initial forecaster’s projections. The other methods deploy different DFL-oriented re-forecaster implementations on top of the initial forecast. We implement 2 methods using Implicit Differentiation (ID), one using the SPO+ loss function with SubGradient descent (Sp-SG), the baseline SPO+ reformulation method (Sp-R) and three variants of our proposed methodology, collectively referred to as Sp-R-IPx. For all DFL models, both a linear regression (-lin) and a single hidden layer NN variant with softplus activation function (-softplus) are implemented. The different models have different choices of hyperparameters. More details on the models and hyperparameters are provided in Appendix C. Figure 2 exemplifies the value of both the validation performance tracking procedure and controlling the barrier weight parameter. It displays in-sample training regret (Figure 2a) and out-of-sample validation regret (Figure 2b) of solutions to Problem (11) for decreasing values of the barrier weight $\mu$, comparing the Sp-R-IPx models. A first observation is that the results verify the viability of the SPO+ loss function. Indeed, the regret on the train set tends to decrease during the training procedure. Secondly, the figure serves as a compelling argument for the value of tracking the validation performance in the SPO+ reformulation training procedure. We remind the reader that the original reformulation approach, Sp-R, only considers the final optimal solution on the train set, represented by the (rightmost) point with the smallest value of $\mu$ in the figures. Whereas this in many cases corresponds to the optimal solution in terms of regret on the train set, Figure 2b clearly shows that this is not necessarily the case for the validation set, leading to the conclusion that the Sp-R method is prone to overfitting. In contrast, the Sp-R-IPs/d models capture forecasters with improved validation performance for intermediate values of $\mu$ compared to the last accessed point. This underscores the importance of our proposed validation tracking procedure. Finally, the figure shows the significance of controlling the $\mu$ update in the solution procedure. Indeed, the Sp-R-IP model, which produces the intermediate results of the barrier problem via IPOPT, explores a limited range of $\mu$ values. As such, it fails to find the intermediate solutions in the regions $\log(\mu) \approx -3$ and $\log(\mu) \approx -5$, where the other models find solutions with comparatively lower validation regret. When examining the Sp-R-IPd and Sp-R-IPs methods, we observe a slightly lower obtained validation regret when dynamically updating $\mu$ based on the validation performance, compared to the static update. However, this is less significant than the improvement achieved by controlled $\mu$ updates over the IPOPT solver output. Table 1 presents a comprehensive evaluation of the test performance across the different solution approaches. Absolute regret is calculated as the sum of the obtained regret over the days in test set, using standardized day-ahead prices. Relative regret measures this against the baseline of an ESS making decisions based on the output of the initial forecaster (initial FC in the table). As such, the relative regret is negative when the re-forecaster improves upon the performance of the initial forecaster. The first observation from the table is that the models from the proposed methodology (Sp-R-IPx) systematically outperform the models using a (sub)gradient descent method, and the original SPO+ reformulation approach, thus affirming the effectiveness of the proposed methodology. A second observation is that (sub)gradient-based models (ID-Q, ID-LB and Sp-SG) are unable to compete with the initial forecaster benchmark when the re-forecaster is cold-started. Deeper insights on these methods are provided in Appendix D. From those analyses, we conclude that the ID methods suffer from a difficult balancing exercise when choosing the value of the weight of the smoothing term, Table 1: Out-of-sample regret obtained on the test set, for both absolute ($r_{\text{abs}}$) and the relative improvement ($r_{\text{rel}}$) compared to the regret obtained by the initial forecaster. "Time" refers to the total time elapsed during the train procedure. \| Model \| Cold start \| Warm start \| \|---------------------\|------------\|------------\| \| \| $r_{\text{abs}}$ (€) \| $r_{\text{rel}}$ (%) \| Time (s) \| $r_{\text{abs}}$ (€) \| $r_{\text{rel}}$ (%) \| Time (s) \| \| Initial FC \| 0.095 \| - \| - \| 0.095 \| - \| - \| \| ID-Q-lin \| 0.278 \| +192 \| 407 \| 0.094 \| -1.3 \| 87 \| \| ID-Q-softplus \| 0.339 \| +257 \| 309 \| 0.089 \| -6.0 \| 157 \| \| ID-LB-lin \| 0.162 \| +70.5 \| 2,031 \| 0.094 \| -1.3 \| 1,960 \| \| ID-LB-softplus \| 0.191 \| +101 \| 1,309 \| 0.084 \| -11.6 \| 1,970 \| \| Sp-SG-lin \| 0.270 \| +184 \| 2,410 \| 0.101 \| +6.4 \| 2,320 \| \| Sp-SG-softplus \| 0.240 \| +153 \| 2,329 \| 0.096 \| +0.9 \| 2,309 \| \| Sp-R-lin \| 0.100 \| +5.1 \| 340 \| 0.100 \| +5.1 \| 738 \| \| Sp-R-softplus \| 0.086 \| -8.9 \| 2,051 \| 0.090 \| -5.4 \| 1,314 \| \| Sp-R-IP-lin \| 0.082 \| -14.0 \| 338 \| 0.082 \| -13.6 \| 738 \| \| Sp-R-IP-softplus \| 0.083 \| -12.8 \| 1,937 \| 0.094 \| -0.8 \| 1,076 \| \| Sp-R-IPs-lin \| 0.082 \| -13.9 \| 1,606 \| 0.080 \| -15.7 \| 1,295 \| \| Sp-R-IPs-softplus \| 0.079* \| -16.3 \| 14,731 \| 0.078 \| -17.2 \| 6,075 \| \| Sp-R-IPd-lin \| 0.082 \| -13.9 \| 2,053 \| 0.082 \| -13.9 \| 2,649 \| \| Sp-R-IPd-softplus \| 0.080 \| -15.5 \| 17,651 \| 0.083 \| -13.0 \| 13,771 \| and that the expression of the subgradient provided in [Elmachtoub & Grigas (2022)] does not always provide optimal parameter updates in the training process. Thirdly, for all the models, the difference in regret between a linear and a NN re-forecaster with a single hidden layer is limited. Even though this result is context-specific and should not be generalized, it underpins the utility of our two-stage re-forecasting procedure where the initial forecaster captures complex dynamics, and the second-stage re-forecaster finetunes the output for improved downstream performance. The final observation is that the Sp-R-IPs/d methods tend to have longer train times compared to the (sub)gradient-based methods. Most notably, whereas the (sub)gradient-based methods show similar train times for the linear regression and NN re-forecaster architectures, the train time dramatically increases for the IP-based methods. This increase is necessitated by the transition from the linear ERM formulation (9) to its non-linear counterpart (23). Even so, the two-stage re-forecasting procedure ensured a tractable training procedure, yielding superior performance for the proposed methods relative to benchmarks, all within an acceptable time frame for the specific application. 6 Conclusion and Future Work While implicit differentiation methods have gained prominence in decision-focused learning for non-linear convex problems, their application to downstream linear optimization necessitates the inclusion of smoothing terms. Our findings indicate that such approximations may not yield optimal results. On the other hand, we show that the SPO+ reformulation framework is prone to overfitting. To address this issue, we augment the SPO+ reformulation methodology to incorporate validation performance tracking across training iterations, employing an interior point solver, while also extending the method to include neural network forecasters. We have shown that this approach outperforms available decision-focused benchmarks for the optimal scheduling problem of an energy storage system. However, this comes at an increased computational cost. While our proposed mini-batch approach seems to alleviate this problem to some extent, future research could investigate how this can be implemented in a more rigorous way. 7 REPRODUCIBILITY STATEMENT In order to reproduce the results of this paper, readers can access the source code in the anonymous Github repository via this link: https://anonymous.4open.science/r/Sp-R-IP-55D2. This repository includes the data and scripts used for training the forecasters, as well as scripts for reproducing the figures.
drovOv7IKB	- Since the frequency-domain signal essentially is a one-to-one projection of the time-domain signal, how much does the proposed network differ from PatchTST (theoretically)? Specifically, do the two architectures share the same, or a subset of solution space? The performance gain over PatchTST seems very marginal in Table 3.	DIVIDE-AND-CONQUER TIME SERIES FORECASTING WITH AUTO-FREQUENCY-CORRELATION VIA CROSS-CHANNEL ATTENTION Anonymous authors Paper under double-blind review ABSTRACT To model various short-term temporal variations, we propose an effective design of Transformer-based, termed FreCoformer. FreCoformer is designed on top of the frequency domain and comprises three key designs: frequency patching operation and two independent observations of these patches. The patching process refines the frequency information, enhancing the locality. The subsequent observations extract the consistent representation within different channels by attention computation and summarize the relevant sub-frequencies to identify eventful frequency correlations for short-term variations. To improve the data fit for different time series scenarios, we propose a divide-and-conquer framework and introduce a simple linear projection-based module, incorporated into FreCoformer. These modules learn both long-term and short-term temporal variations of time series by observing their changes in the time and frequency domains. Extensive experiments show the effectiveness of our proposal can outperform other baselines in different real-world time series datasets. We further introduce a lightweight variant of FreCoformer with attention matrix approximation, which achieves comparable performance but with much fewer parameters and computation costs. The code is available: https://anonymous.4open.science/r/FreCoformer-6F2Z 1 INTRODUCTION Time series forecasting is an essential task in various applications and has recently witnessed great advancements powered by deep learning methods, especially Transformer (Zhou et al., 2021; Woo et al., 2022b; Nie et al., 2023; Wen et al., 2023). Such methods aim to discern consistent feature representations in historical observations and forecasting time series. Successful approaches usually involve learning representation in long-term temporal variations, e.g., trend and seasonality (Wen et al., 2020). These variations are typically extracted through time series decomposition (Woo et al., 2022a). Subsequently, they leverage the attention mechanism in Transformer to automatically learn temporal dependencies of these variations to yield consistent representations (Wen et al., 2023). Nevertheless, these approaches inevitably lead to information loss of short-term temporal variations in some complex scenarios (Liu et al., 2022c; Wu et al., 2023). Figure 1(a) illustrates an electricity case where modeling long-term variations mainly captures low-frequency features, neglecting many consistent mid-to-high frequency components. Such components manifest as short-term variations, such as fluctuations and periodicities over short durations, and are good guidances for several practical analyses (Crespo Cuaresma et al., 2004; Thompson & Wilson, 2016; Hammond et al., 2023). To end this, previous studies have leveraged frequency decomposition and spectrum information to assist Transformer in modeling temporal dependencies (Wu et al., 2021; Woo et al., 2022b). However, low-frequency components generally carry most of the energy in the spectrum and are dominant in real-world time series (Zhu & Shasha, 2002; Corripio et al., 2006). Influenced by such redundant low-frequency and noise, these approaches tend to prioritize long-term temporal variations (Figure 1(b)). Moreover, researchers directly deploy Transformers to the frequency domain to identify more eventfully relevant high-frequency components (Zhou et al., 2022). Despite enhancements in frequency attention, this approach relies on heuristic and empirical strategies, i.e., random or top-$K$ frequency selection, often capturing spurious correlations for forecasting (seen in Figure 1(c)). In this paper, we propose FreCoformer to represent various short-term temporal variations in complex time series automatically. It is designed on top of the frequency domain and comprises three key designs: frequency patching operation and two independent observations of these patches. The patching operation refines the frequency bands, providing an opportunity to learn representations from detailed views of frequency components. The first observation, a channel-wise attention mechanism, weighs channel-wise correlations for each independent sub-frequency component. These independent attentions share model parameters across all sub-frequency learning, preventing winner-take-all of redundancy low-frequency components. The second observation is channel-independent, which summarizes global frequency information (i.e., frequency-wise summarization) and eliminates channel correlations to facilitate multivariate time series forecasting. We further propose a 'divide-and-conquer' forecasting framework that integrates FreCoformer with a long-term modeling module, deploying in the time domain, to improve the data fit of time series scenarios. Additionally, we present a lightweight variant of FreCoformer to alleviate computational load, extending our proposal to various large-scale datasets. Our main contributions lie in three folds. 1) FreCoformer is a novel forecasting module designed for computing frequency correlation for representing short-term variations in time series. It can automatically identify the relevant and consistent frequency components in historical observations and forecast data points. Figure 1(d) illustrates our superiority to different previous methodologies in complex datasets. 2) The divided-and-conquer framework enhances data fit, and the ablation study shows the distinct contributions of each module under varying data scenarios. Extensive experimental results on eight benchmarks show the effectiveness of our proposal, achieving superior performance, with 41 top-1 and 21 top-2 cases out of 64 in total. 3) We incorporate the Nyström approximation to reduce the computational complexity of attention maps, achieving lightweight models with competitive performance. This opens new possibilities for efficient time series forecasting. Interestingly, results demonstrate that Nyström-FreCoformer can particularly enhance performance in datasets with a large number of channels. 2 RELATED WORKS Transformer for Time Series Forecasting. Forecasting is an important task in time series analysis (Alysha M. De Livera & Snyder, 2011; Hamilton, 2020). Transformer has recently achieved a progressive breakthrough in time series forecasting (Nie et al., 2023; Zhang & Yan, 2023; Jiang et al., 2023). Earlier attempts make efforts to improve the computational efficiency of Transformers to form them for time series forecasting tasks (Beltagy et al., 2020; Zhou et al., 2021; Liu et al., 2022a). Several works further apply Transformers to the time domain of time series to model inherent temporal dependencies (Li et al., 2019; Zhou et al., 2021; Liu et al., 2022b; Nie et al., 2023). Various studies have integrated frequency decomposition and spectrum analysis with Transformer in modeling temporal variations (Wu et al., 2021; Woo et al., 2022b), to improve the capacity of temporal-spatial representation. The work of (Zhou et al., 2022) designs the attention layers that directly function in the frequency domain to enhance spatial or frequency representation. Modeling Short-term Variation in Time Series. Short-term variations are intrinsic characteristics of time series data, playing a crucial role in effective forecasting (Crespo Cuaresma et al., 2004; Liu et al., 2022c). Numerous deep learning-based methods have been proposed to capture these transient patterns (Chung et al., 2014; Neil et al., 2016; Chang et al., 2018; Bai et al., 2018; Stoller et al., 2019; Wen et al., 2020; Wu et al., 2021; Woo et al., 2022a; Wang et al., 2022). Here, we summarize some works closely aligned with our proposal. Pyraformer ((Liu et al., 2022b) applies a pyramidal attention module with inter-scale and intra-scale connections to capture various temporal dependencies. FEDformer ((Zhou et al., 2022) incorporates the Fourier spectrum within the attention computation to identify pivotal frequency components. Beyond Transformers, TimesNet ((Wu et al., 2023) employs Inception blocks to capture both intra-period and inter-period variations. Channel-wise Correlation. Understanding the cross-channel correlation is also critical for time series forecasting. Several studies aim to capture intra-channel temporal variations and subsequently model the inter-channel correlations using Graph Neural Networks (GNNs) (Wu et al., 2020; Cao et al., 2021). Recently, Crossformer (Zhang & Yan, 2023) proposes a two-stage attention layer designed to simultaneously capture temporal variations and their cross-channel correlations. Extensive experimental results have demonstrated its effectiveness in multivariate time series forecasting. 3 Proposed Method Let $X = \{x^{(i)}_L\}_{m=1}^C$ denote a multivariate time-series consisting of $C$ channels, where each channel records an independent $L$ length historical observation. We aim to design an effective forecasting function $f_\theta(\cdot)$ that can accurately forecast $T$ data points for each channel, resulting in $\hat{X} \in \mathbb{R}^{C \times T}$. 3.1 FreCoformer. Forward Process. FreCoformer consists of four principal components: (1) a DFT-to-IDFT backbone, (2) frequency-wise patching, (3) channel-wise attention, and (4) frequency-wise summarization. An overview can be found in Figure 2(a). The DFT-to-IDFT backbone decomposes the input time series into its frequency components via DFT and learns a consistent representation of relevant frequency components (by frequency-wise patching, channel-wise attention, and frequency-wise summarization), enabling future time series generation through IDFT. Specifically, (i) The input $X$ is transformed to the real part $R \in \mathbb{R}^{C \times F}$ and imaginary part $I \in \mathbb{R}^{C \times F}$ of the frequency by DFT, where $F$ denotes the frequency bands. (ii) Along the $C$-axis, we segment these two matrices into a sequence of $N$ sub-frequency patches, i.e., $(R_1, ..., R_N)$ and $(I_1, ..., I_N)$, for all channels to refine the frequency information. (iii) Subsequently, cross-channel patches within the same sub-frequency are fed into the Transformer. Then, the Transformer sequentially and independently captures the channel-wise dependencies of each sub-frequency and, post-processing, concatenates all sub-frequencies. (iv) Along the $F$-axis, we further abstract the overall frequency information, resulting in two new real $R$ and imaginary $I$ parts. These two parts serve as IDFT for forecasting $X$. Frequency-wise Patching. Given the $R$ and $I$ matrices of DFT, a non-overlapping patching operation is performed on them. We segment the frequency entries of $(r^{(n)}_1, ..., r^{(n)}_F)$ and $(i^{(m)}_1, ..., i^{(m)}_F)$ of each channel into a set of sub-frequency patches with $P$ dimension, resulting in $(r^{(n)}_1, ..., r^{(n)}_N)$ and $(i^{(m)}_1, ..., i^{(m)}_N)$, where $N = F/P$ is the number of patches. Thus, the input $X$ will result in: Figure 2: System overview: (a) FreCoformer, (b) Divided-and-Conquer framework, and (c) Nyström-FreCoformer \[(R_1, \ldots, R_N), (I_1, \ldots, I_N) = \text{Patching}(DFT(X)), \quad R_{1:N}, I_{1:N} \in \mathbb{R}^{C \times P}\] The dimension of $P$ prevents information redundancy over fine-grained frequency bands, like neighboring 1Hz and 2Hz. This parameter is adjustable to real-world scenarios, e.g., an hourly sampling in daily recordings or alpha waveform typically occurring at 8–12 Hz (Adamantidis et al., 2019). Channel-wise Attention. We employ the Transformer encoder to learn the frequency-independent channel-wise correlation. For the $n$-th sub-frequency, where $n \in 1, 2, \ldots, N$, $r_n^{(m)}$ and $i_n^{(m)}$ are concatenated as the embedding for each channel, yielding all channel patches $W_n = \text{Concat}(R_n, I_n)$, where $W_n \in \mathbb{R}^{C \times 2P}$. These patches are then mapped to the Transformer latent space of dimension $D$ via a linear projection $E_n \in \mathbb{R}^{2P \times D}$, and a patch-wise normalization. This normalization is used to eliminate distributional differences across sub-frequency bands. Subsequently, we feed $C$ tokens of $W'_n = \text{PreNorm}(W_n, E_n)$, each at a time for self-attention computations, and this process is performed independently $N$ times for all sub-frequencies to obtain the complete representations. Therefore, the attention computation can be formalized as: \[A_n = \text{Attention}(Q_n, K_n, V_n) = \text{Softmax} \left( \frac{W'_n W'_n^T (W'_n W'_n)^T}{\sqrt{d}} \right) W'_n W'_n^T\] where $W'_n, W'_n^k, W'_n^v \in \mathbb{R}^{D \times M}$ are the weight matrices for generating the query matrix $Q_n$, key matrix $K_n$, and value matrix $V_n$. $\sqrt{d}$ denotes a scaling operation. The attention module also contains normalization and a feed-forward layer with residual connections (Dosovitskiy et al., 2021), and $A_n \in \mathbb{R}^{C \times M}$ weights the correlations among $C$ channels for the $n$-th sub-frequency band. Frequency-wise Summarization. We concatenate all independent attention maps $(A_1, \ldots, A_N)$, in sequence to form $A \in \mathbb{R}^{C \times (N \times M)}$. Given $A$ is derived from the $N$ times observation of independent frequency, we introduce a frequency-wise layer projection to summarize the overall frequency information, resulting in $A'$. Ultimately, two distinct linear layers are employed to generate the refined real and imaginary parts, serving IDFT for forecasting. \[\hat{X} = \text{IDFT}(\hat{R}, \hat{I}), \quad \text{where } \hat{R} = \text{Linear}(A'); \hat{I} = \text{Linear}(A')\] Notably, this frequency-wise summarization is channel-independent and shares the parameters of linear projection across all channels, i.e., $\mathbf{A}' = (\mathbf{A}'(1), ..., \mathbf{A}'(C)) = \text{Linear}(\mathbf{A}(1), ..., \mathbf{A}(C))$. This aims to mitigate channel correlations and enhance channel fits, referring to (Nie et al., 2023). ### 3.2 Divided-and-Conquer Framework Real-world time series exhibit variability across different scenarios. For instance, analyzing long-term variations in the data can reflect seasonal-trend patterns, such as differences between summer and winter and weekly changes in air quality (Vito, 2016; Karevan & Suykens, 2020). Conversely, areas like banking transactions, electricity consumption, and hospital foot traffic (Crespo Cuaresma et al., 2004; Lai et al., 2018b; Alysha M. De Livera & Snyder, 2011) require a focus on short-term variations. A successful forecasting function should adapt to various scenarios and capture eventful patterns to ensure precise forecasting. Therefore, we propose a ‘divide-and-conquer’ framework and introduce a simple linear projection-based module, incorporated into FreCoformer, to enhance adaptability to various types of time series data. Since FreCoformer is designed on top of the frequency domain, this new module, termed T-Net, operates in the time domain to further complement FreCoformer by improving the capability of modeling temporal dependencies. Given the input $\mathbf{X} \in \mathbb{R}^{C \times T}$, the first-order difference operation is applied independently to each univariate time series to remove the non-stationary variations and noise, yielding $(\tilde{\mathbf{X}}^1, ..., \tilde{\mathbf{X}}^C)$. Drawing inspiration from the works of (Zeng et al., 2023; Nie et al., 2023), for the $m$-th series, we also segment the time domain of $\tilde{\mathbf{X}}^m$ into a sequence of $N'$ temporal patches, where $N' = L/P'$ denotes the number of patches, each of dimension $P'$. We form two-stage linear projections: initially capturing the local temporal dependencies of each patch and subsequently learning the global temporal dependencies after concatenating all the learned patches. $$\hat{\mathbf{X}} = \text{Linear}^{\text{global}}(\text{Linear}^{\text{local}}(\tilde{\mathbf{X}}^1), ..., \text{Linear}^{\text{local}}(\tilde{\mathbf{X}}^C))$$ Notably, we intend for both FreCoformer and T-Net to independently learn different domain-based representations and each to have its own capacity to forecast the ground truth $\hat{\mathbf{X}}$. A summation is finally executed on the outputs of FreCoformer and T-Net without any additional operations. ### 3.3 Nyström-FreCoformer The $O(n^2)$ memory and time complexity of self-attention is the bottleneck for using longer historical time series for forecasting (Li et al., 2019; Zhou et al., 2021; Nie et al., 2023). With the patching operations in both the time and frequency domains, $O(LC^2)$ complexity has reduced to $O(\frac{L}{P}C^2)$. However, due to the channel-wise attention of FreCoformer, the computational cost increases proportionately with the number of channels, potentially leading to computational overloads when a large number of channels. We hence propose a lightweight Frecoformer inspired by NyströmFormer (Xiong et al., 2021) and conduct a matrix approximation for the attention map. Two main motivations drive our approach: First, employing the Nyström matrix approximation method allows us to further reduce our complexity to $O(\frac{L}{P}C)$ without modifying the feature extraction (attention computation) or the data stream structure within the Transformer, as opposed to previous methods (Zhou et al., 2021; Liu et al., 2022a; Wu et al., 2021; Zhou et al., 2022). Second, real-world time series data often exhibit redundancy across different dimensions due to consistent characteristics among similar variables, like the traffic volumes of neighboring locations Zhou et al. (2021). This redundancy can lead to unnecessary correlation computations in channel-wise attention processes. To compute the attention matrix $\mathbf{A}$, we first select $m$ landmark columns from the input $\mathbf{Q}_n$ and $\mathbf{K}_n$ matrices in each channel, denoted as $\mathbf{Q}_n$ and $\mathbf{Q}_n$, \| Methods \| Complexity \| \|------------------\|------------------\| \| Fedformer \| $O(LC)$ \| \| PatchTST \| $O(\frac{L}{P}C)$ \| \| Crossfromer \| $O(\frac{L^2}{P^2}C)$ \| \| Ours \| $O(\frac{L}{P}C^2)$ \| \| Ours(Nyström) \| $O(\frac{L}{P}C)$ \| Table 1: Computation complexity. $L$ is the input sequence length, $C$ is the channel count, and $P$ denotes the patch dimension. then compute: \[ \mathbf{F}_n = \text{softmax}\left(\frac{\mathbf{Q}_n \mathbf{K}_n^T}{\sqrt{d}}\right), \quad \mathbf{A}_n = \text{softmax}\left(\frac{\mathbf{Q}_n \mathbf{K}_n^T}{\sqrt{d}}\right)^+, \quad \mathbf{B}_n = \text{softmax}\left(\frac{\mathbf{Q}_n \mathbf{K}_n^T}{\sqrt{d}}\right) \] Where $ \mathbf{A}_n^+ $ is the Moore-Penrose inverse of $ \mathbf{A}_n $ (Xiong et al., 2021), and the Nyström approximation for $ n $-th channel-wise attention $ \mathbf{A}_n $ is: \[ \mathbf{A}_n \approx \tilde{\mathbf{A}}_n = \mathbf{F}_n \mathbf{A}_n \mathbf{B}_n. \] With the use of Nyström approximation on attention maps, the computational load has reduced from $ O(\frac{1}{P}C^2) $ to $ O(\frac{1}{P}C) $. Detailed derivations and proofs can be found in Appendix A.1. 4 EXPERIMENTS 4.1 PROTOCOLS Table 2: Benchmark datasets summary \| Datasets \| Weather \| Electricity \| ETTh1 \| ETTh2 \| ETTm1 \| ETTm2 \| Air \| Traffic \| \|----------\|---------\|-------------\|-------\|-------\|-------\|-------\|-----\|--------\| \| #channel \| 21 \| 321 \| 7 \| 7 \| 7 \| 7 \| 12 \| 862 \| \| #timesteps \| 52969 \| 26304 \| 17420 \| 17420 \| 69680 \| 69680 \| 6941 \| 17544 \| Datasets. We conducted extensive experiments on eight real-world benchmark datasets: Weather, four ETT datasets (ETTh1, ETTh2, ETTm1, ETTm2), Electricity, Traffic, and Air $ ^1 $ (Vito, 2016), where the former seven datasets are available in the work (Wu et al., 2021) $ ^2 $. A summary of the datasets is presented in Table 2 and details can be found in Appendix A.2. Baselines. We selected some state-of-the-art (SOTA) time series forecasting works as our baselines: PatchTST (Nie et al., 2023), TimesNet (Wu et al., 2023), Fedformer (Zhou et al., 2022) and Pyraformer (Liu et al., 2022a). PatchTST represents a new SOTA, outperforming several authoritative early works, including Autoformer (Wu et al., 2021), Informer (Zhou et al., 2021), and DLinear (Zeng et al., 2023). Other baselines with differing architectures are designed to capture short-term temporal variations. Besides, we compared our proposal to a multi-channel modeling SOTA, i.e., Crossformer (Zhang & Yan, 2023). Setup. All baselines adhere to the same prediction length with $ T \in \{24, 36, 48, 60\} $ for the Air dataset and $ T \in \{96, 192, 336, 720\} $ for other datasets. The look-back window $ L = 336 $ was used in our setting for fair comparisons, referring to (Nie et al., 2023). Besides, we further explored the impact of an extended look-back window by evaluating with $ L = 512 $. - For the Air dataset, we tested our model and all baselines with a look-back window $ L = 104 $, based on the settings recommended for small datasets in (Zhou et al., 2021). - For other datasets, we collected all available results of PatchTST, Fedformer, and Pyraformer from (Nie et al., 2023). Results for Crossformer with prediction lengths $ T \in \{336, 720\} $ were collected from (Zhang & Yan, 2023). For unavailable $ T \in \{96, 192\} $, we implemented Crossformer to obtain the results. We collected results of TimesNet from (Wu et al., 2023) with the default $ L = 96 $ and implemented TimesNet with our default $ L = 336 $ to select the best outcomes for a fair comparison. 4.2 RESULTS 4.2.1 MAIN RESULTS Table 3 shows the main results for multivariate long-term forecasting. Overall, with default a look-back window of $ L = 336 $, our proposal shows leading performance on most datasets, as well as on different prediction length settings, with 27 top-1 and 34 top-2 cases out of 64 in total. When the look-back window is extended to $ L = 512 $, our framework demonstrates superior performance, achieving 41 top-1 and 21 top-2 rankings out of 64 cases. Considering both look-back Twindow settings, our framework achieves top-1 rankings in 63 out of 64 cases. $ ^1 $https://archive.ics.uci.edu/dataset/360/air-quality $ ^2 $https://drive.google.com/drive/folders/1ZOYpTUa82_jCcxdTmyrt0LXQfvaM9vIy Table 3: Multivariate long-term forecasting results with MSE/MAE. Bold/underline indicates the best/second results. The asterisk* denotes the results are implemented by us; Other results are from original papers (Nie et al., 2023; Zhang & Yan, 2023; Wu et al., 2023). \| Models \| Ours 512 \| Ours 336 \| PatchTST \| Crossformer \| TimesNet \| Fedformer \| Pyraformer \| \|--------\|----------\|----------\|-----------\|-------------\|----------\|-----------\|------------\| \| Metric \| MSE \| MAE \| MSE \| MAE \| MSE \| MAE \| MSE \| \| Weather \| \| \| \| \| \| \| \| \| 96 \| 0.146 \| 0.194 \| 0.149 \| 0.196 \| 0.152 \| 0.199 \| 0.166* \| \| 192 \| 0.190 \| 0.240 \| 0.193 \| 0.238 \| 0.197 \| 0.243 \| 0.235* \| \| 336 \| 0.242 \| 0.290 \| 0.245 \| 0.295 \| 0.250 \| 0.298 \| 0.296* \| \| 720 \| 0.315 \| 0.334 \| 0.317 \| 0.372 \| 0.320 \| 0.335 \| 0.353* \| \| Electicity \| \| \| \| \| \| \| \| \| 96 \| 0.128 \| 0.224 \| 0.129 \| 0.225 \| 0.130 \| 0.222 \| 0.198* \| \| 192 \| 0.145 \| 0.239 \| 0.146 \| 0.240 \| 0.148 \| 0.240 \| 0.239* \| \| 336 \| 0.162 \| 0.259 \| 0.165 \| 0.259 \| 0.167 \| 0.261 \| 0.259* \| \| 720 \| 0.197 \| 0.290 \| 0.202 \| 0.294 \| 0.205 \| 0.291 \| 0.433* \| \| ETTh1 \| \| \| \| \| \| \| \| \| 96 \| 0.359 \| 0.390 \| 0.362 \| 0.390 \| 0.375 \| 0.390 \| 0.428* \| \| 192 \| 0.401 \| 0.418 \| 0.406 \| 0.415 \| 0.411 \| 0.421 \| 0.422* \| \| 336 \| 0.401 \| 0.418 \| 0.406 \| 0.415 \| 0.431 \| 0.436 \| 0.440* \| \| 720 \| 0.436 \| 0.459 \| 0.433 \| 0.452 \| 0.449 \| 0.466 \| 0.519* \| \| ETTh2 \| \| \| \| \| \| \| \| \| 96 \| 0.268 \| 0.339 \| 0.273 \| 0.335 \| 0.274 \| 0.336 \| 0.801* \| \| 192 \| 0.328 \| 0.373 \| 0.337 \| 0.378 \| 0.339 \| 0.379 \| 0.854* \| \| 336 \| 0.342 \| 0.382 \| 0.345 \| 0.385 \| 0.341 \| 0.385 \| 0.903* \| \| 720 \| 0.372 \| 0.421 \| 0.374 \| 0.419 \| 0.379 \| 0.422 \| 1.146* \| \| ETThm1 \| \| \| \| \| \| \| \| \| 96 \| 0.286 \| 0.340 \| 0.285 \| 0.338 \| 0.290 \| 0.342 \| 0.378* \| \| 192 \| 0.326 \| 0.366 \| 0.322 \| 0.362 \| 0.332 \| 0.369 \| 0.394* \| \| 336 \| 0.356 \| 0.384 \| 0.353 \| 0.385 \| 0.366 \| 0.392 \| 0.404* \| \| 720 \| 0.372 \| 0.409 \| 0.370 \| 0.420 \| 0.424 \| 0.424 \| 0.550* \| \| ETThm2 \| \| \| \| \| \| \| \| \| 96 \| 0.165 \| 0.257 \| 0.164 \| 0.254 \| 0.165 \| 0.257 \| 0.511* \| \| 192 \| 0.230 \| 0.295 \| 0.218 \| 0.291 \| 0.227 \| 0.292 \| 0.553* \| \| 336 \| 0.270 \| 0.327 \| 0.270 \| 0.276 \| 0.278 \| 0.292 \| 1.556* \| \| 720 \| 0.358 \| 0.383 \| 0.361 \| 0.381 \| 0.367 \| 0.385 \| 1.566* \| \| Air* \| \| \| \| \| \| \| \| \| 24 \| 0.577 \| 0.568 \| 0.572 \| 0.562 \| 0.607* \| 0.582* \| 0.574* \| \| 36 \| 0.665 \| 0.675 \| 0.665 \| 0.661 \| 0.661 \| 0.661 \| 0.746* \| \| 48 \| 0.702 \| 0.705 \| 0.685 \| 0.700 \| 0.722* \| 0.744* \| 0.888* \| \| 60 \| 0.720 \| 0.645 \| 0.738 \| 0.653 \| 0.766* \| 0.667* \| 0.815* \| \| Traffic \| \| \| \| \| \| \| \| \| 96 \| 0.356 \| 0.248 \| 0.358 \| 0.250 \| 0.367 \| 0.251 \| 0.502* \| \| 192 \| 0.378 \| 0.257 \| 0.378 \| 0.259 \| 0.507* \| 0.287* \| 0.513* \| \| 336 \| 0.384 \| 0.260 \| 0.391 \| 0.264 \| 0.398 \| 0.265 \| 0.513* \| \| 720 \| 0.424 \| 0.283 \| 0.322 \| 0.287 \| 0.530 \| 0.300 \| 0.640* \| Figure 3: (a) Visualized predictions from our model and baselines on the ETTh1 dataset. The X-axis denotes time steps, Y-axis is the amplitude of the time series. (b) Heatmaps of the input and output matrixes of FreCoformer’s Transformer encoder on ETTh1. We showed 3 samples from different channels. These output matrixes will be used to generate forecasting. The X-axis denotes frequency components, Y-axis is the dimension of the feature vector. These heat maps show the energy distribution in the frequency domain. 4.2.2 MODEL ANALYSIS Figure 1 already illustrates the ability of our proposal to accurately capture mid-to-high frequency components, demonstrating superiority over time-domain modeling methods (PatchTST), frequency decomposition-assisted temporal modeling methods (Autoformer), and frequency attention methods (Fedformer). We further visualize the time domain representation with more advanced baselines in... Table 4: Left part: Module ablation of our framework, FreCoformer only, and T-Net only, where bold/underline indicates the best/second results. Right part: Ablation study of channel-wise attention and frequency patching, where * denotes the better forecasting performance. ‘Non-CW’ denotes the removal of channel attention, replaced by an alternative linear projection; ‘Non-FP’ indicates that the entire frequency bands are used as tokens for channel-wise attention. \| Setting \| Complete \| FreCoformer \| T-Net \| Non-CW \| Non-FP \| \|---------\|----------\|-------------\|-------\|--------\|--------\| \| Dataset \| Metric \| MSE MAE \| MSE MAE \| MSE MAE \| MSE MAE \| \| ETTh1 \| 96 \| 0.362 0.391 \| 0.364 0.391 \| 0.371 0.399 \| 0.372* 0.398* \| 0.373 0.400 \| \| \| 192 \| 0.403 0.411 \| 0.403 0.412 \| 0.411 0.421 \| 0.405* 0.414* \| 0.410 0.419 \| \| \| 336 \| 0.406 0.415 \| 0.416 0.423 \| 0.420 0.439 \| 0.419* 0.424* \| 0.423 0.429 \| \| \| 720 \| 0.433 0.452 \| 0.434 0.452 \| 0.446 0.464 \| 0.435* 0.453* \| 0.458 0.469 \| \| Weather \| 96 \| 0.149 0.196 \| 0.173 0.225 \| 0.150 0.197 \| 0.176 0.227 \| 0.174* 0.225* \| \| \| 192 \| 0.193 0.238 \| 0.216 0.262 \| 0.194 0.239 \| 0.218 0.262* \| 0.217* 0.262* \| \| \| 336 \| 0.245 0.279 \| 0.263 0.295 \| 0.246 0.280 \| 0.265* 0.295* \| 0.266 0.298 \| \| \| 720 \| 0.318 0.332 \| 0.328 0.342 \| 0.319 0.333 \| 0.332* 0.343* \| 0.333 0.347 \| Figure 3(a). Both input and output are from the ETTh1 dataset, and the length is 336. Fedformer and TimesNet fail to accurately capture both long-term and short-term patterns. Compared to the best-performing PatchTST, our model exhibits an advantage in identifying short-term variations, resulting in detailed fluctuations in periodicity variation. More results can be seen in Appendix A.3. To demonstrate the efficacy of the core design—channel-wise attention in FreCoformer, we visualized the heatmaps of the input and output DFT matrices of the Transformer encoder in FreCoformer in Figure 3(b). The energy of the original data is primarily concentrated in the low-frequency range, leading to a potential imbalance in energy distribution. In the output of the transformer encoder, there is a balanced energy distribution between low-frequency and mid-to-high-frequency components. This balance likely enables our method to efficiently extract pivotal frequency features across the entire frequency spectrum and various temporal variations, enhancing prediction outcomes. ### 4.2.3 Ablation Study Module Ablation Study. We investigate the efficiency of our framework and its modules by using the ETTh1 and Weather datasets. The ETTh1 dataset contains more intricate mid-to-high-frequency information, while the Weather dataset primarily focuses on low-frequency data. We independently implement two modules for forecasting these datasets and compare their results to the complete framework (in Table 4 (Left)). It shows on datasets like ETTh1, which are rich in complex high-frequency information, FreCoformer consistently has better performance. Conversely, on datasets like Weather, where long-term variations (low frequency) are dominant, using solely the time domain modeling has better outcomes, but combining both has superior results. These observations imply that frequency modeling has more contributions to our framework in intricate datasets. Also, it does not bring redundancy to time domain modeling in simple and stationary time series. Channel-wise Attention and Frequency Patching Ablations We further investigate the impact of channel-wise attention and frequency patching (refinement) on forecasting accuracy. As shown in Table 4 (Right), our framework consistently achieved superior accuracy in all experiments. In datasets like ETTh1, characterized by more complex frequency information, channel-wise attention achieves better performance in forecasting than frequency patching, emphasizing the significance of our fundamental design of channel-wise attention. ### 4.2.4 Nyström-FreCoformer We conduct comparative experiments to evaluate forecasting accuracy and computational complexity against various baseline methods. In Figure 4, the X-axis denotes GPU memory usage, while the Y-axis indicates prediction accuracy. Obviously, our framework outperforms in terms of accuracy across both datasets. In datasets with fewer channels, like ETTh1 (7 channels), our model excels in both accuracy and computational efficiency. In contrast, when dealing with datasets having a larger number of channels, like the Weather (21 channels), our original method still retains Figure 4: Visualization of prediction accuracy and computational complexity comparing various baselines, FreCoformer, and Nyström-FreCoformer. the highest accuracy, though with a slight increase in computational load (as denoted by 'Ours' in Figure 4 (Right)). We further maintain constant parameters, modifying only the computational method for self-attention by employing Nyström, which allowed for a substantial reduction in computational demand without sacrificing accuracy (Ours(N)). Moreover, refining the parameters in the Nyström variant allowed us to realize further computational efficiencies without compromising accuracy (Nyström-freCoformer). Consequently, our model demonstrates superiority in both computational cost and accuracy in this setup. Table 5: A more comprehensive comparison, taking into account MSE and GPU memory usage. We used the ETTh1 and Weather, Electricity, Traffic dataset with a look-back window of length 336. For ETTh1, the prediction length is 96, and for Weather, Electricity, and Traffic, it’s 720 time steps. We evaluated the effectiveness of various methods based on these metrics and also considered runtime memory consumption. The best results are in bold and the second best results are in _underlined_. \| Dataset(channels) \| Metric \| Ours \| Ours (Nyström) \| PatchTST \| Crossformer \| TimesNet \| Fedformer \| \|------------------\|--------\|------\|----------------\|----------\|-------------\|----------\|-----------\| \| ETTh1(7) \| MSE \| 0.362\| - \| 0.375 \| 0.424 \| 0.384 \| 0.376 \| \| \| O \| 1661 \| - \| 1683 \| 3096 \| 2015 \| 3989 \| \| Weather(21) \| MSE \| 0.318\| 0.319 \| 0.320 \| 0.353 \| 0.341 \| 0.389 \| \| \| O \| 6029 \| 2422 \| 5775 \| 2635 \| 3943 \| 6115 \| \| Electricity(321) \| MSE \| 0.129\| 0.129 \| 0.130 \| 0.198 \| 0.168 \| 0.186 \| \| \| O \| 10113\| 6261 \| 17379 \| 15375 \| 15593 \| 7285 \| \| Traffic(862) \| MSE \| 0.431\| 0.426 \| 0.434 \| 0.530 \| 0.640 \| 0.621 \| \| \| O \| 22317\| 13357 \| 33823 \| 39387 \| 25689 \| 17023 \| 5 CONCLUSION This paper proposes an effective design of Transformer-based models for modeling short-term temporal variation through frequency modeling, termed FreCoformer. FreCoformer is built upon the Transformer model and has three key components: (i) frequency refinement, (ii) channel-wise attention to independent frequency bands, and (iii) frequency-wise summarization. Compared to the previous works, FreCoformer locally and globally learns the frequency correlations of various short-term variations in time series. We further propose a divide-and-conquer framework and introduce a simple linear projection-based module incorporated into FreCoformer, to enhance adaptability to various types of time series data. Extensive experiments show the effectiveness of our proposal can outperform other baselines in different real-world time series datasets. The ablation shows the success of our FreCoformer design. We further incorporate Nyström approximation to reduce the computational complexity of attention maps, achieving lightweight with competitive forecasting performance. This introduces a new perspective for effective time series forecasting. 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w3YZ9MSlBu	What are the typical sources for mining? Youtube, streaming services, Freesound, or something else? What is the typical audio quality? Are they copyrighted, or not? Do you keep the audio or only the relevant features (MFCC, CQT?)	MERT: Acoustic Music Understanding Model with Large-Scale Self-supervised Training Yizhi Li1,2,4 • Ruibin Yuan3,4,6,* • Ge Zhang3,4,5,6,* • Yinghao Ma8,7,* • Xingran Chen9 • Hanzhi Yin8,3 • Chenghao Xiao8,8 Chenghua Lin1,2,† • Anton Ragni2 • Emmanouil Benetos7 • Norbert Gvengc2 • Roger Dannenberg3 • Ruibo Liu9 Wenhui Chen8,5 • Gus Xia10,11 • Yemin Shi8,6,12 • Wenhao Huang2,6 • Zili Wang8 • Yike Guo4 • Jie Fu3,4,6 † ∗ m-a-p.ai • University of Manchester • University of Sheffield • Carnegie Mellon University † Hong Kong University of Science and Technology • University of Waterloo • Beijing Academy of Artificial Intelligence ‡ Queen Mary University of London • Durham University • Dartmouth College • MBZUAI • New York University • linksoul.ai Abstract Self-supervised learning (SSL) has recently emerged as a promising paradigm for training generalisable models on large-scale data in the fields of vision, text, and speech. Although SSL has been proven effective in speech and audio, its application to music audio has yet to be thoroughly explored. This is partially due to the distinctive challenges associated with modelling musical knowledge, particularly tonal and pitched characteristics of music. To address this research gap, we propose an acoustic Music undERstanding model with large-scale self-supervised Training (MERT), which incorporates teacher models to provide pseudo labels in the masked language modelling (MLM) style acoustic pre-training. In our exploration, we identified an effective combination of teacher models, which outperforms conventional speech and audio approaches in terms of performance. This combination includes an acoustic teacher based on Residual Vector Quantisation - Variational AutoEncoder (RVQ-VAE) and a musical teacher based on the Constant-Q Transform (CQT). Furthermore, we explore a wide range of settings to overcome the instability in acoustic language model pre-training, which allows our designed paradigm to scale from 95M to 330M parameters. Experimental results indicate that our model can generalise and perform well on 14 music understanding tasks and attain state-of-the-art (SOTA) overall scores. 1 Introduction Pre-trained language models (PLMs) can learn generalisable representations of data without human annotated labels in a self-supervised learning (SSL) style, leading to remarkable performance improvement in natural language processing and related fields (Brown et al., 2020; Fang et al., 2022; Chen et al., 2021a). Music is widely recognised as a special language that can be used to communicate across different cultures (Mehr et al., 2019). The internal similarity between music and language as a communication interface lays a promising foundation for adapting PLM-based methods to model music sequences. We argue that the benefit is twofold. First, PLMs can potentially pave the way to unify the modelling of a wide range of music understanding, or the so-called Music Information Retrieval (MIR) tasks, including but not limited to music tagging, beat tracking, music transcription, source separation, etc., so that different tasks no longer need task-specific models or features. Second, releasing a PLM for acoustic music understanding allows the redistribution of the musical knowledge rather than the data itself, which saves the costs of manual annotation and copyright law restrictions. Unfortunately, we are yet to see a general-purpose and cost-effective open-source PLM on acoustic music understanding. Most existing studies are designed to solely address music tagging problems (Pons and Serra, 2019; Spijkervet and Burgoyne, 2021; McCallum et al., 2022; Huang et al., 2022; Zhu et al., 2021; Zhao and Guo, 2021), and many of them do not provide open-source code bases or checkpoints for further evaluation. A promising model is JukeMIR (Castellon et al., 2021), which is based on Jukebox (Dhariwal et al., 2020) and provides a comprehensive evaluation on MIR. The authors contributed equally to this work. †Corresponding authors. downstream tasks. However, this foundation model uses cumbersome hierarchical auto-regressive transformer decoders containing billions of parameters to model music audio, leading to significant inefficiency for conducting general music understanding tasks (e.g., it takes weeks to inference on datasets like MTG (Bogdanov et al., 2019) with a consumer-grade 3090 GPU). The aforementioned research gap has urged us to design and open-source a generalisable and computationally affordable pre-trained acoustic music model. In this paper, we propose an acoustic Music undERstanding model with large-scale self-supervised Training (MERT). MERT inherits a speech SSL paradigm, employing teacher models to generate pseudo targets for sequential audio clips. Specifically, to capture the distinctive pitched and tonal characteristics in music, MERT incorporates a multi-task paradigm to balance the acoustic and musical representation learning as demonstrated in Fig. 1. In the proposed design, an Residual Vector Quantisation - Variational Autoencoder (RVQ-VAE) (Défossez et al., 2022) is used as the acoustic teacher to provide discretised acoustic-level summarisation of the music signal. The Constant-Q Transformation (CQT) (Brown, 1991) model is further introduced as the music teacher for capturing pitch and harmonic inductive bias. Regarding the context dependencies and music hierarchies, as indicated in Borsos et al. (2022), we leave the task of modelling high-level and abstract patterns to the stacked layers of self-attentions in the transformer. We also explore a wide range of settings for the transformer and 1D convolution encoder to overcome the instability in acoustic model pre-training, which permits effective scaling up of MERT from 95M to 330M size when blending acoustic and musical knowledge. By scaling up to 330M size (only 7% the size of JukeBox), MERT achieves overall state-of-the-art (SOTA) results on various MIR tasks, which demonstrates a strong generalisability on music understanding. Last but not least, we analyse multiple pre-trained settings considering the teachers and share our decision routes § 5.2 and § 5.3, which could potentially guide future acoustic music understanding pre-training research. To summarise, our contributions are: • proposing a multi-task style predictive acoustic self-supervised learning paradigm, which achieves SOTA performance on various MIR tasks, including important yet unexplored tasks for pre-training such as pitch detection, beat tracking and source separation applications; • conducting a broad range of analysis based on ablation study of the proposed MERT pre-training paradigm; • exploring robust and stable strategies for acoustic music model training to overcome training instability and frequent crashes when scaling up the pre-training on model size; • providing an open-source, generalisable and computationally affordable acoustic music pre-trained model, which addresses the needs of both industry and research communities. 2 RELATED WORK PLMs for Acoustic Music The field of music information retrieval (MIR) has long been facing challenges in data availability due to the costs associated with music audio annotation and country-specific copyright laws (Chen et al., 2019; Castellon et al., 2021). To address this challenge, pre-trained language models (PLMs) for acoustic music have been proposed to provide reusable learned representations, enabling transfer learning for various downstream MIR tasks without the need for extensive data annotation (Castellon et al., 2021). However, current acoustic music pre-trained models still have room for improvement in terms of providing open-source, generalisable, and lightweight learned representations suitable for both industrial and research applications (McCallum et al., 2022). Existing acoustic music pre-trained models primarily focus on tagging tasks and rely on supervised tagging labels for pre-training (Pons and Serra, 2019; Spijkervet and Burgoyne, 2021; McCallum et al., 2022; Huang et al., 2022). While some studies have explored contrastive learning for acoustic music pre-training, they face limitations in training data and model size, hampering the performance improvements (Choi et al., 2017; Li et al., 2022). Additionally, several models trained on inaccessible datasets or without publicly available codes and model weights make it difficult to reproduce or extend their approaches (McCallum et al., 2022; Castellon et al., 2021; Li et al., 2022; Zhu et al., 2021; Zhao and Guo, 2021). Although some general-purpose audio representation models show potential for music audio representation learning, their performance is mostly evaluated on limited MIR downstream tasks (Saeed et al., 2021; Borsos et al., 2022; Wang et al., 2023). This lack of comprehensive evaluation hinders further studies and a thorough understanding of the pros and cons of existing models. Self-Supervised Speech Processing Music and speech processing are closely related (Jasmin et al., 2020) since they usually use the same audio data formats. Additionally, both acoustic music and speech processing models need to deal with the cocktail party problem (Brown and Bidelman, 2022; Petermann et al., 2022) since good source separation capabilities help both separating noises and background sounds with speech and processing polyphonic music audio. These common grounds between music and speech processing inspire us to adapt SOTA speech pre-trained models and tailor them specifically for music audio processing tasks. For instance, existing research work targeting general-purpose audio representations (Saeed et al., 2021; Borsos et al., 2022; Wang et al., 2023) has verified that self-supervised speech processing models can be extended beyond speech to downstream entry-level music tasks, including generating mono piano music and music reconstruction. Audio Representation with Language Modelling Mask strategy-based large-scale language models have been applied to a wide range of domains (Lample and Charton, 2019; Chen et al., 2021a; Fang et al., 2022), but still remain under-explored in acoustic music understanding. For audio, Dhariwal et al. (2020) investigates generating hierarchical tokens which can be further employed to reconstruct music, inspiring subsequent research to understand and generate acoustic music based on extracted discrete tokens from continuous features. Baevski and Mohamed (2020) introduce a pre-trained VQ-VAE (Baevski et al., 2019) to provide prediction targets to conduct speech representation learning with MLM. While introducing K-means to provide discrete token codebooks and pre-training the model to detect sound units, Hsu et al. (2021) claim that a better teacher model in SSL could lead to better downstream task performance. Additionally, recent speech processing pre-trained models (Borsos et al., 2022; Wang et al., 2023) propose to train or adopt separately trained codecs (Zeghidour et al., 2021; Défossez et al., 2022) for discrete token extraction. Based on the conclusion from previous studies, the recently released RVQ-VAEs (Zeghidour et al., 2021; Défossez et al., 2022), achieving good results in music reconstruction, could be adopted as teacher models for music understanding pre-training and provide acoustic information guidance. Yet some of the uniqueness of music processing such as timbre and harmony remains unexplored. We thus propose to incorporate a corresponding musical teacher model in MERT to fill such an important gap. 3 METHODOLOGY This section introduces the pre-training paradigm and architecture of our models. It includes prediction to acoustic teachers such as k-means or deep music features, and reconstruction to music teachers such as CQT spectrum, both based on the well-established masked language modelling (MLM). 3.1 PRE-TRAINING WITH MLM Supervised Learning requires a labelled dataset $D_t = \{x_i^{(t)}, y_i^{(t)}\}_{i=1}^N$. Here, $N$ is the number of data samples, $x_i^{(t)}$ is the $i^{th}$ data sample in the dataset, and $y_i^{(t)}$ is the corresponding label. From $D_t$, we can train a machine learning algorithm $f_\theta(\cdot)$ parameterised with $\theta$ that makes label predictions on each data sample. Unsupervised learning, in contrast, learns an algorithm based on an unlabelled dataset $ D = \{x_i\}_{i=1}^M $, with SSL being a specific type of this class. For each data sample $ x_i $, SSL derives a new data $ x'_i $ with a pseudo label $ y'_i $. The training process is to minimise the loss between each pseudo label $ y'_i $ and the prediction based on new data $ \hat{y}_i = f_\theta(x'_i) $ as denoted in Eq.1. \[ \theta^ = \arg\min_\theta \sum_{x'_i \in D} L(f_\theta(x'_i(t)), y'_i(t)). \] (1) MLM is a famous example of pseudo-label generation. Let $ x_i = [x^{(1)}_i, x^{(2)}_i, \ldots, x^{(L)}_i] $ be the $ i $th data sample in a sequential dataset with length $ L $, and $ M \subset [L] $ is a subset of indices randomly chosen from 1 to $ L $. Then, the new data is defined by the following equation \[ x'_i = [1_{[L]\setminus M}(1) \cdot x^{(1)}_i, 1_{[L]\setminus M}(2) \cdot x^{(2)}_i, \ldots, 1_{[L]\setminus M}(L) \cdot x^{(L)}_i] \] (2) where $ 1_{[L]\setminus M}(x) $ denotes the indicator function, that is, $ 1_{[L]\setminus M}(x) = 1 $ if and only if $ x $ is outside the masked indices set $ M $. The pseudo-label that needs to be learned is typically $ y'_i = x_i - x'_i $, i.e., the masked data. However, reconstructing masked data $ y' $ for raw audio tasks as pseudo-label is hard to train. HuBERT (Vaswani et al., 2017; Hsu et al., 2021) uses a dimension-reduced feature $ z' $ derived from $ y' $ with phonetic acoustic information, which forms the design basis of our pre-training strategy. As a speech SSL system, HuBERT utilises offline clustering to acquire pseudo labels for a BERT-like prediction loss. Specifically, it uses Mel-frequency cepstral coefficients (MFCCs), a widely-used traditional feature in speech-related tasks, as acoustic features for clustering. The obtained results are then utilised as pseudo labels in the first iteration of pre-training. It then uses the learned representation for clustering to get a pseudo label for the second iteration pre-training. Such a pseudo label includes acoustic information in human speech and can be aligned to phonemes. The loss functions of HuBERT are formulated as follows: \[ L_H(f; x, M, Z) = \sum_{t \in M} \log p_f(z_t \| x', t) \] (3) where $ \log p_f(\cdot \| x', t) $ is the log-likelihood function on clustering results given the masked input $ x' $ and position $ t $ derived from $ f $; likelihood function $ p_f $ is the Noise Contrastive Estimation (NCE) loss which is defined as \[ p_f(c \| x', t) = \frac{\exp(\text{sim}(T(o_t), e_c)/\tau)}{\sum_{c'=1}^C \exp(\text{sim}(T(o_t), e_{c'})/\tau)}, \] (4) Here, $ c \in [C] $ is a codeword of the clustering results and $ e_c $ represents its embedding; sim is the cosine similarity; $ o_t $ is the output of the model at timestep $ t $; and $ T(o_t) $ is the linear transformation of $ o_t $, making it have the same dimension as $ e_c $. Besides, $ \tau $ scales the logit which is set to 0.1 in HuBERT. The linear transformation $ T $, the model to generate outputs, and the embedding of all the clustering results are all learnable. Overall, we use the same model as HuBERT but introduce several notable variations tailored to music. Specifically, we designed a better hidden-unit $ z $ as pseudo tags for pre-training with multiple music acoustic features. In addition, we added a reconstruction loss to music features and employed additional music augmentation tricks. ### 3.2 Modelling Acoustic Information The MFCC features are only good at modelling acoustic and timbre information for single-pitch signals, and therefore, the clustering results do not provide much timbre information in music recording. We proposed two potential approaches as the teacher on acoustic information: one based on traditional features, and the other based on deep learning. The first method uses k-means on the log-Mel spectrum and Chroma features for timbre and harmonic acoustic information, respectively. In the case of music representation, each frame contains more information compared to speech, necessitating a larger number of classes for k-means clustering. The complexity of the k-means algorithm is linear with the number of centroids (clustering centres), leading to a time-consuming k-means for the music feature. To tackle this problem, we employ 300-means for the log-Mel spectrum with dimension 229, and 200-means for Chroma features with dimension 264, resulting in a total of 60,000 classes (200 centroids for Chroma features multiplied by 300 centroids for the log-Mel spectrum). Despite the increased number of classes, the computational complexity remains comparable to that of HuBERT. The disadvantage of k-means is that it is difficult to scale up to a larger number of classes and larger datasets, and the results are sensitive to initialisation. The second choice for our acoustic teacher is EnCodec (Défossez et al., 2022), a recent learnable feature with 8-layer residual Vector Quantized-Variational AutoEncoder (VQ-VAE). Each acoustic feature, denoted as $z_{enc} \in [C]^{L \times 8}$, is a 2-dimensional auditory code matrix, and $L$ is the length of the recording. The row vector of each matrix $z_{enc}[t,:]$ represents the results of 8 different clusterings for frame $t$, and the column vector of each matrix $z_{enc}[:,j]$ represents the results from the $j^{th}$ codebook of the audio sequence, where $j \in \{1,\ldots,8\}$. EnCodec converts 24kHz input waveforms to 8 different embeddings at 75Hz with a 320-fold reduction, and the quantizer has 1024 dimensions. In this setting, for each 5-second waveform, the discrete acoustic feature is a matrix with $375 \times 8$ entries, representing 375 frames (75Hz $\times$ 5s) and 8 deep acoustic features. With these embeddings, the decoder of EnCodec can reconstruct the waveform at 24 kHz with authentic information in timbre. ### 3.3 Modelling Musical Information Apart from acoustic information, we added a new reconstruction loss to the Constant-Q transform (CQT) spectrogram to emphasise pitch-level information. The CQT is a type of frequency transform that is widely used in various MIR tasks, such as pitch detection, chord recognition, and music transcription. It is similar to the Fourier transform, but bin widths are proportional to frequency rather than equal, giving each octave the same number of bins, resulting in a better time-frequency trade-off for music audio where multiple pitches occur in multiple octaves. We utilize mean squared error (MSE) loss to reconstruct the CQT spectrum $z_{cqt}$ from the masked input audio $x'$. That is, $$L_{CQT}(f_{cqt}; x, M, z_{cqt}) = \sum_{t \in M} \\|z_{cqt,t} - f_{cqt}(x')_t\\|_2$$ And the final loss function $L$ is a linear combination of both the acoustic loss function $L_H$ and the musical-pitch loss function $L_{CQT}$: $$L = \alpha \cdot L_H + L_{CQT}$$ ### 4 Experiments #### 4.1 Evaluation Protocol Downstream Tasks We evaluate our method and compare it with baseline models on 14 downstream tasks, including frame-level classification or regression tasks like music tagging, key detection, genre classification, emotion score regression, instrument classification, pitch classification, vocal technique detection, and singer identification; and sequential tasks like beat tracking and source separation. For instrument classification, we use the Nsynth (Engel et al., 2017) and MTG-instrument datasets, with receiver operating characteristic (ROC), and average precision (AP). The NSynth dataset is also used for pitch classification, with accuracy (ACC) as the evaluation metric. Vocal technique detection and singer identification based on the VocalSet dataset (Wilkins et al., 2018), with accuracy as the metric. For music tagging, we utilise the MagnaTagATune (MTT) (Law et al., 2009) and MTG-Jamendo (Bogdanov et al., 2019) datasets, averaging multiple embeddings for long audio recordings. Key detection is accomplished using the Giantsteps and Giantsteps-MTG-keys datasets (Knees et al., 2015; Korzeniowski and Widmer, 2017), with a refined accuracy ($ACC_{refined}$) metric. Genre classification is performed using the GTZAN (Tzanetakis and Cook, 2002) and MTG-Genre datasets, with ROC, and AP metrics. Emotion score regression is conducted on the Emomusic dataset (Soleymani et al., 2013), with the coefficient of determination (R2 score) of arousal and valence as evaluation metrics. Beat tracking is conducted on the GTZAN Rhythm dataset (Marchand and Peeters, 2015), using the F-measure (F1). Finally, source separation is accomplished using the MUSDB18 dataset (Raffi et al., 2017), with the Source-to-Distortion Ratio (SDR) as the evaluation metric. The full descriptions of the datasets and tasks can be found in Appendix B.1. Probing Protocol Following Castellon et al. (2021); Yang et al. (2021), we restrict the testing protocol with probing rather than fine-tuning, i.e. freezing the backbone pre-trained models as deep feature extractor and only train a simple downstream structure, typically a multilayer perceptron (MLP) for frame-level tasks. For a fair comparison, we also limit the space for hyper-parameters searching. For full details please refer to Appendix B.2. 4.2 Baseline Methods We select models pre-trained with various paradigms from both music and speech domains as our baselines to provide a comprehensive evaluation of the generalisation ability of the designs. Mu-siCNN (Pons and Serra, 2019) is selected as a representative supervised method, which is pre-trained with supervision from the Million Song Dataset tags (Bertin-Mahieux et al., 2011). CLMR (Spijkervet and Burgoyne, 2021) and MULE (McCallum et al., 2022) are selected as representatives of SOTA music representations trained with contrastive learning. Jukebox (Dhariwal et al., 2020) and the corresponding transfer learning method, JukeMIR (Castellon et al., 2021) is selected as the representative of transfer learning from a large-scale generative pre-trained musical representation. We also select the recently proposed strong speech SSL models, HuBERT (Hsu et al., 2021) and data2vec (Baevski et al., 2022), as our baselines since they share the same MLM pre-training paradigm with MERT. While HuBERT reconstructs the masked discrete tokens provided by the K-means teacher, data2vec uses the student model updated with an exponential moving average gradient to produce continuous representations for MLM prediction. In order to reveal the effectiveness of the pre-training paradigm itself rather than the training data distribution, we re-train the speech models and denote them by HuBERT\textsuperscript{music} and data2vec\textsuperscript{music}. Additionally, we present the current SOTA for each task including results from both supervised and self-supervised methods. 4.3 Implementation Details Training Settings We deploy the proposed SSL architecture in the training of various model sizes with matched scales of data. We mined 160K hours of music recordings from the Internet to build a large-scale music dataset. Accordingly, the base size models (95M) are trained with a 1K hours subset whereas the whole dataset is used for the large model (330M). Specifically, we provide a special edition of the base model, MERT\textsuperscript{−95M-public}, that is trained on a totally publicly available music dataset, music4all (Santana et al., 2020), with a data size of 910 hours. In the context of self-attention, the computational complexity scales quadratically with the sequence length. Therefore, when dealing with limited computational resources, there exists a trade-off between the batch size and the sequence length. In our preliminary experiments, we have observed that increasing the batch size provides greater performance improvements compared to extending the context length. To allow a larger batch size under the computational limitation, we adopt a strategy of randomly truncating audio clips into 5-second segments following Ma et al. (2023). This duration roughly corresponds to a 2-bar context in music. It is worth noting that our model utilises a convolutional relative positional embedding, similar to the approach introduced by Baevski et al. (2020) in Wav2Vec, enabling it to operate effectively in longer contexts, if required. The effective batch sizes and learning rates for the base model and large model are set to 1.5 and 5.5 hours, and their learning rates are set to $5e^{-4}$, $1.5e^{-3}$, respectively. Pre-training is carried out with the fairseq\footnote{https://github.com/facebookresearch/fairseq} framework. Models are trained with 64 A100-40GB GPUs with fp16. We also implement a data augmentation of randomly adding short segments to improve the representation robustness, and describe the details in Appendix A.1 Training Stability In our empirical findings, we observe that when scaling up acoustic encoder-only models, they tend to exhibit a higher susceptibility to training instability compared to models of similar size in text or image domains. Such instability can result in decreased performance or, in extreme cases, even lead to crashes in model training. During experimentation with scaling up to the MERT\textsuperscript{−330M} model, we encounter notable instability manifested by constant gradient clipping and sporadic spikes in losses. This instability has a detrimental effect on the accuracy of MLM predictions and results in decreased performance on downstream tasks. Our attempts to resume training from previously saved checkpoints and data batches are proved unsuccessful in mitigating the instability issue. Furthermore, we observe that reducing the learning rate in this context not only fails to address the issue but also leads to a decline in performance and hindered the training convergence. We further explore the effectiveness of a seemingly-powerful method DeepNorm (Wang et al., 2022a) in stabilising acoustic language model pre-training, but find it to be ineffective. Eventually, we discover that incorporating attention relaxation techniques (Chen et al., 2021b) is beneficial in addressing the instability challenges. We also found that transitioning from post-layer normalisation (Post-LN) to pre-layer normalisation (Pre-LN) offers a potential solution of allowing training to continue. More information can be found in Appendix B.3. Table 1: Experimental Performances of MERT and Baselines on Downstream Tasks (1/2). The baselines are grouped by supervised and unsupervised pre-training paradigms. The superscripts denote the category of the acoustic teacher used by MERT models. “public” refers to the MERT model trained with only open-source dataset. Results with star* are claimed in the references. \| Dataset Task \| Metrics \| ROC \| AP \| Acc<sub>best</sub> \| Acc \| F1<sup>std</sup> \| R2<sup>V</sup> \| R2<sup>A</sup> \| Acc \| Acc \| Acc \| Acc \| \|--------------\|---------\|-----\|----\|-----------------\|-----\|-------------\|--------\|--------\|-----\|-----\|-----\|-----\| \| MERT-95M<sup>k-means</sup> \| 90.6 \| 38.4 \| 65.0 \| 78.6 \| 88.3 \| 52.9 \| 69.9 \| 71.3 \| 92.3 \| 74.6 \| 77.2 \| \| MERT-95M-public<sup>k-means</sup> \| 90.7 \| 38.4 \| 67.3 \| 72.8 \| 88.1 \| 59.7 \| 72.5 \| 70.4 \| 92.3 \| 75.6 \| 78.0 \| \| MERT-95M<sup>RVQ-VAE</sup> \| 91.0 \| 39.3 \| 63.5 \| 78.6 \| 88.3 \| 60.0 \| 76.4 \| 70.7 \| 92.6 \| 74.2 \| 83.7 \| \| MERT-330M<sup>RVQ-VAE</sup> \| 91.3 \| 40.2 \| 65.6 \| 79.3 \| 87.9 \| 61.2 \| 74.7 \| 72.6 \| 94.4 \| 76.9 \| 87.1 \| (Previous) SOTA \| 92.0 [26] \| 41.4 [15] \| 74.3 [30] \| 83.5 [36] \| 80.6 [24] \| 61.7 \| 72.1 [15] \| 78.2 [53] \| 89.2 [36] \| 65.6 [55] \| 80.3 [39] \| Table 2: Experimental Performances of MERT and Baselines on Downstream Tasks (2/2). Average scores across task are calculated on the SOTA results and models applicable to all the tasks. \| Dataset Task \| Metrics \| ROC \| AP \| ROC \| AP \| ROC \| AP \| ROC \| AP \| SDR<sup>mix</sup> \| SDR<sup>drums</sup> \| SDR<sup>mix</sup> \| SDR<sup>other</sup> \| \|--------------\|---------\|-----\|----\|-----\|----\|-----\|----\|-----\|----\|-----------------\|-----------------\|-----------------\|-----------------\| \| MERT-95M<sup>k-means</sup> \| 77.2 \| 19.6 \| 75.9 \| 13.7 \| 87.0 \| 18.6 \| 82.8 \| 29.4 \| 5.6 \| 5.6 \| 4.0 \| 3.0 \| 62.9 \| \| MERT-95M-public<sup>k-means</sup> \| 77.5 \| 19.6 \| 76.2 \| 13.3 \| 87.2 \| 18.8 \| 83.0 \| 28.9 \| 5.5 \| 5.5 \| 3.7 \| 3.0 \| 63.0 \| \| MERT-95M<sup>RVQ-VAE</sup> \| 77.5 \| 19.4 \| 76.4 \| 13.4 \| 87.1 \| 18.8 \| 83.0 \| 28.9 \| 5.5 \| 5.5 \| 3.8 \| 3.1 \| 63.7 \| \| MERT-330M<sup>RVQ-VAE</sup> \| 78.1 \| 19.8 \| 76.5 \| 14.0 \| 86.7 \| 18.6 \| 83.4 \| 29.9 \| 5.3 \| 5.6 \| 3.6 \| 3.0 \| 64.7 \| (Previous) SOTA \| 78.8 \| 20.2 [11] \| 78.6 \| 16.1 [36] \| 87.7 \| 20.3 [1] \| 84.3 \| 32.1 [36] \| 9.3 \| 10.8 \| 10.4 \| 6.4 [44] \| 64.5 \| 5 RESULTS ANALYSIS 5.1 PERFORMANCE & EFFICIENCY OF MERT MODELS The results on all the downstream tasks are provided in Tab. 1 and Tab. 2. As suggested by the average scores in Tab. 2, MERT-330M<sup>RVQ-VAE</sup> achieves the same score as the combination of the previous SOTAs (from 10 different models even including supervised methods) and becomes the new SOTA on 4 metrics. It is also noteworthy that the other smaller MERT-95Ms still have comparable performance. Generally, MERTs perform well on tasks focusing on local-level musical information such as beat, pitch and local timbre such as singer information, and remain competitive on the other tasks requiring more global-level information, such as music tagging, key detection, and genre classification. This indicates the blending of acoustic and musical teachers could provide comprehensive guidance for the understanding of music recordings, though pre-trained in only a 5-second context length. Nevertheless, the performances of our models in tasks with more global music information are close to strong baselines, suggesting MERT models are capable of recognising global patterns well, thanks to the relative position embeddings and the contextualisation of the transformers. In addition, our models demonstrate good results with limited data, even when training with public data that may lack enough diversity. MERT-95M-public and MERT-95M are both trained on a ~1k hour dataset and give competitive performance compared with the SOTA and MERT-330M, proving that MERT can converge effectively and learns generalisable music representations with limited training data. Moreover, the MERT-95M-public is trained with Music4ALL (Santana et al., 2020), a 910-hours public music dataset with mainly pop music and lack of diversity in music style, and shows comparable performance to other settings. In particular, its performance does not have a significant difference besides genre classification on GTZAN compared to MERT-95M. We evaluate the performance of the MERT-RVQ-VAE model with a parameter size of 95M and 330M, given the use of the EnCodec feature enables us to scale up the dataset compared to the K-means. The results demonstrate that increasing the model size to 330M yields improved performance or maintains similar performance compared to MERT-95M-RVQ-VAE (less than 0.1%) on most of the tasks besides beat tracking. More importantly, the lightweight sizes of MERTs open up new possibilities for transferring one general understanding model for large-scale classification or sequence labelling MIR tasks. MERT series models achieve better or comparable performance with only 1.9% (95M) and 6.6% (330M) parameters compared to the self-supervised baseline Jukebox-5B (Dhariwal et al., 2020). Even when our evaluation is in probing setting, most models could not be trained on sequence labelling tasks like beat tracking or source separation with affordable computational costs except for MERT and baseline models with similar architecture (Hsu et al., 2021; Baevski et al., 2022). Table 3: Evaluation Results from Models Trained with Different Teacher Settings. Models labeled with △2 and ▲2 suggest that the K-means teachers are trained with the features from △1 and ▲1 models. All the listed models are sized in (95M) and not augmented with the in-batch noise mixture. \| Acoustic Teacher \| Acoustic Target Class \| Musical Teacher \| MTT Tagging ROC \| AP \| GS Key AccRefined \| GTZAN Genre Acc \| R2Y \| R2A \| Avg. \| \|------------------\|----------------------\|----------------\|----------------\|----\|------------------\|-----------------\|-----\|-----\|------\| \| K-means, MFCC \| 100 \| \| 89.8 \| 36.3 \| 15.1 \| 66.2 \| 39.6 \| 67 \| 49.4 \| \| K-means, MFCC \| 500 \| \| 90.3 \| 38 \| 17 \| 70 \| 40.6 \| 67.5 \| 51.3 \| \| K-means, MFCC \| 2000△1 \| \| 90.2 \| 37.6 \| 15.6 \| 70 \| 44.3 \| 67.6 \| 51.4 \| \| K-means, Logmel+Chroma \| 300 + 200 ▲1 \| N/A \| 90.5 \| 37.6 \| 55.1 \| 75.2 \| 40.1 \| 68.2 \| 62.1 \| \| K-means, MFCC \| 2000△2 \| \| 90.4 \| 37.5 \| 16.1 \| 68.3 \| 43.9 \| 67.7 \| 51.0 \| \| K-means, Logmel+Chroma \| 500 ▲2 \| \| 90.4 \| 37.7 \| 49.2 \| 72.8 \| 46.5 \| 66.9 \| 60.7 \| \| K-means, MFCC+CQT \| 300+200 \| \| 89.4 \| 35.3 \| 53.2 \| 69.0 \| 45.8 \| 66.8 \| 60.2 \| \| K-means, Logmel+Chroma \| 300+200 CQT \| \| 90.6 \| 38.4 \| 65.0 \| 78.6 \| 53.1 \| 68.7 \| 67.3 \| \| 1024×8 all codebook \| N/A \| \| 90.7 \| 38.7 \| 60.5 \| 72.8 \| 55.3 \| 69.0 \| 65.0 \| \| 1024×8 all codebook \| N/A \| \| 90.5 \| 38.4 \| 63.2 \| 77.2 \| 53.2 \| 72.3 \| 66.9 \| \| RVQ-VAE \| 1024 codebook7 \| \| 88.6 \| 34.4 \| 63.5 \| 62.1 \| 33.3 \| 53.2 \| 57.6 \| \| RVQ-VAE \| 1024 codebook0 \| \| 90 \| 36.7 \| 59.4 \| 67.2 \| 39.7 \| 64.5 \| 60.5 \| \| RVQ-VAE \| 1024×8 random codebook \| \| 90.6 \| 38.1 \| 66.8 \| 73.8 \| 48.1 \| 68.6 \| 65.8 \| 5.2 THE EFFECTIVENESS OF ACOUSTIC & MUSICAL TEACHER As demonstrated in Tab. 3, we explore optimal combinations and selections of the teacher models in the MERT paradigm with a subset of downstream tasks following Castellon et al. (2021), including auto-tagging, key detection, genre classification, and emotion recognition. We reproduce the original HuBERT (Hsu et al., 2021) setting on music datasets with the acoustic teacher K-meansMFCC△1 and the teacher K-meansMFCC▲2 trained on features produced by HuBERT model from the first stage, similar to DeepCluster (Caron et al., 2018). We observe that such models perform poorly on the key detection and emotion recognition tasks even we increase the dimension of the MFCC features from 100 to 2000. As the re-clustering K-means does not bring significant improvement in the second stage pre-training, we stick to the ordinary one stage pre-training to study the influence brought by the teachers with less computational cost. Given that the key information is highly related to the pitch classes of the audio, we then introduce such inductive bias by providing the K-means acoustic teacher with additional Chroma or CQT features, denoted as K-meansLogmel+Chroma△1 and K-meansMFCC+CQT▲2. The additional pitch information indirectly brought by the Chroma and CQT features immediately endow the model a certain level of key detection ability and raise the accuracy from 15.6 to 55.1 and 53.2 while keeping or increasing performances on other tasks. This confirms that the potentials of transformer models can be better excavated from more dimensions by introducing extra pseudo prediction targets in the MLM scheme. Following such an intuition, it could be further assumed that designing a proper multi-task learning pre-training paradigm can guide the model to produce more general representations for various music... understanding tasks. We thus propose leveraging the CQT explicitly as a musical teacher to introduce harmonic inductive bias during the pre-training. Compared to models trained with only the acoustic teacher $ \text{MFCC}_\Delta^1 $ or K-means$^{\text{Logmel+Chroma}}_1$, MERT models trained with the newly proposed CQT musical teacher that are naturally more aligned to music audio can achieve significant performance gains on not only the key detection task but also the tasks requiring the high-level information like genre classification and emotion recognition. However, given that K-means models are difficult to scale up on large datasets due to memory and computational requirements, we use the RVQ-VAE model EnCodec (Défossez et al., 2022) as the final version of acoustic teacher without searching for the immeasurable hyper-parameter $ K $. The EnCodec could intuitively provide more comprehensive acoustic information since the audio can be greatly recovered from the intermediate discrete codecs from the encoder by a neural decoder. We observe that leveraging only one top ($ 1024^{\text{codebook}}_7 $) or bottom layer ($ 1024^{\text{codebook}}_0 $) of the residual codebooks in RVQ-VAE already provide substantial information in pre-training, and the use of all layers in the codebooks allows the student models to learn richer acoustic patterns. Although the strategy of randomly accessing one of the codebooks for each batch can alleviate the use of GPU memory and lead to similar performance compared to using all of them at a time, the setting of predicting 8 codebooks together is adopted for faster convergence in the finalised design. By replacing the acoustic teacher with RVQ-VAE, MERT achieves an average score of 66.9, similar to that of the K-means$^{\text{Logmel+Chroma}}_1$ version (i.e., 67.3) while largely reducing the cost of scaling up K-means. ### 5.3 Study on Musical Loss Table 4: Evaluation Results for Musical Loss Study. The listed models are not augmented with the in-batch noise mixture. \| Parameter Size \| Acoustic Teacher Model \| Acoustic Target Class \| Musical Loss Weight \| MTT Tagging ROC \| AP \| GS Key Acc Refined \| GTZAN Genre Acc \| EMO Emotion R2^V \| Avg. R2^A \| \|----------------\|------------------------\|-----------------------\|---------------------\|----------------\|----\|-------------------\|----------------\|------------------\|---------\| \| 95M \| K-means$^{\text{Logmel+Chroma}}$ \| 300 + 200 \| N/A \| 90.5 \| 37.6 \| 55.1 \| 75.2 \| 40.1 \| 68.2 \| 62.1 \| \| \| \| \| \| 1 \| 90.6 \| 38.4 \| 65.0 \| 78.6 \| 53.1 \| 68.7 \| 67.3 \| \| \| \| \| \| 2 \| 90.6 \| 38.1 \| 62.7 \| 66.9 \| 45.5 \| 67.9 \| 62.7 \| \| \| \| \| \| 5 \| 90.4 \| 37.3 \| 65.3 \| 70.3 \| 45.7 \| 68.3 \| 64.1 \| \| 95M \| RVQ-VAE \| \| 1024 × 8 all codebook \| N/A \| 90.7 \| 38.7 \| 60.5 \| 72.8 \| 55.3 \| 69.0 \| 65.0 \| \| \| \| \| 1024 × 8 all codebook \| 1 \| 90.5 \| 38.4 \| 63.2 \| 77.2 \| 53.2 \| 72.3 \| 66.9 \| We conducted a hyperparameter search to determine the optimal weight for the musical loss applied to masked audios in the k-means setting. In Table 4, we present the results of the varying musical loss weights, which uses the same evaluation setting in § 5.2. By adjusting the weight, we find that a weight of 1 yielded the best overall performance for the base model. We observe that when switching the acoustic teacher to RVA-VAE, the models performs slightly worse on GS than those with K-means. Overall, our study provides valuable insights into the impact of musical loss and different acoustic models on the performance of the acoustic language model. These findings can inform the future development of more effective and efficient models in the domain of acoustic processing. ### 6 Conclusion In conclusion, our work underscores the potential of SSL for modelling raw music audio and the efficacy of our approach, MERT, in pre-training sizeable models. We present a novel paradigm that integrates RVQ-VAE and CQT teacher models, providing a unique blend of acoustic and musical information necessary for MLM-based pre-training for music understanding. This integration, bolstered by the application of an in-batch noise mixup data augmentation and Pre-LN, enables the learning of robust music representations with further training stability. The performance of the MERT model surpasses previous SSL baselines, achieving SOTA or comparable results across a wide range of MIR tasks while using significantly smaller parameter size. We anticipate that our method and the forthcoming public release of our codes and models will catalyse further research into the application of SSL in music audio, thereby broadening the scope and depth of human understanding of music. Despite being capable of handling longer sequences with relative positional embedding, our models are limited by the short 5-second training context, so our approach could be further improved for tasks requiring understanding extended musical contexts if trained on longer sequences. LIMITATION AND FUTURE WORK Our models are trained using only 5-second audio signals due to constraints in computational resources and the extended length of audio signals. Despite these models being capable of handling longer sequences thanks to relative positional embedding, this approach could potentially limit their performance in tasks requiring a comprehensive understanding of extended musical contexts. We plan to continue training our models on a longer context once gaining access to more computing resources. Moreover, although we propose several techniques to improve the training stability for the acoustic pre-training, we still suffer from the gradient exploding issues with the half-precision training for settings with larger batch sizes and model sizes. In addition, we observe inverse-scaling effect in specific tasks while scaling-up to 330M, which indicates that our design could be further improved by stabilising the training. ACKNOWLEDGEMENT This paper is a tribute to our talented friend Anqiao Yang, for his friendship and valuable advice to this work. Yizhi Li is a Ph.D. student fully funded by the Department of Computer Science, University of Manchester, UK. This work is partially funded by Theme based Research Scheme (T45-205/21-N), Research Grants Council of Hong Kong. Yinghao Ma is a research student at the UKRI Centre for Doctoral Training in Artificial Intelligence and Music, supported by UK Research and Innovation [grant number EP/S022694/1]. Emmanouil Benetos is supported by a RAEng/Leverhulme Trust Research Fellowship [grant number LTRF2223-19-106]. We acknowledge IT Services at The University of Sheffield for the provision of services for High Performance Computing. REFERENCES [1] Alonso-Jiménez, P., Serra, X., and Bogdanov, D. (2022). Music representation learning based on editorial metadata from discogs. International Society for Music Information Retrieval (ISMIR). [2] Baevski, A., Hsu, W.-N., Xu, Q., Babu, A., Gu, J., and Auli, M. (2022). Data2vec: A general framework for self-supervised learning in speech, vision and language. In International Conference on Machine Learning, pages 1298–1312. PMLR. [3] Baevski, A. and Mohamed, A. (2020). Effectiveness of self-supervised pre-training for asr. In ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7694–7698. [4] Baevski, A., Schneider, S., and Auli, M. (2019). vq-wav2vec: Self-supervised learning of discrete speech representations. arXiv preprint arXiv:1910.05453. [5] Baevski, A., Zhou, Y., Mohamed, A., and Auli, M. (2020). wav2vec 2.0: A framework for self-supervised learning of speech representations. Advances in neural information processing systems, 33:12449–12460. [6] Bertin-Mahieux, T., Ellis, D. P., Whitman, B., and Lamere, P. (2011). The million song dataset. Proceedings of the 12th International Conference on Music Information Retrieval (ISMIR). [7] Böck, S., Korzeniowski, F., Schlüter, J., Krebs, F., and Widmer, G. (2016a). madmom: a new Python Audio and Music Signal Processing Library. In Proceedings of the 24th ACM International Conference on Multimedia, pages 1174–1178, Amsterdam, The Netherlands. [8] Böck, S., Krebs, F., and Widmer, G. (2016b). Joint beat and downbeat tracking with recurrent neural networks. In ISMIR, pages 255–261. New York City. [9] Bogdanov, D., Won, M., Tovstogan, P., Porter, A., and Serra, X. (2019). The mtg-jamendo dataset for automatic music tagging. Machine Learning for Music Discovery Workshop, International Conference on Machine Learning (ICML 2019). [10] Borsos, Z., Marinier, R., Vincent, D., Kharitonov, E., Pietquin, O., Sharifi, M., Teboul, O., Grangier, D., Tagliasacchi, M., and Zeghidour, N. (2022). Audiolm: a language modeling approach to audio generation. arXiv preprint arXiv:2209.03143.
lwT5CRq1PO	- For the FedDF method, the shared images for KD are only saved on the server side and it is unlabeled. But here in Figure 1, we can see that patches are sent from the server to the clients, but the paper doesn't state why the image needs to be shared with the client.	Federated Learning with a Single Shared Image Anonymous authors Paper under double-blind review Figure 1: Schematic illustration of our federated learning algorithm using single images. Our algorithm works on the principle of generating a common distillation dataset from only one shared single image using deterministic augmentations. To this end, our method dynamically selects the best patches for the training of the global model in the next round using knowledge distillation. Abstract Federated Learning (FL) enables multiple machines to collaboratively train a machine learning model without sharing of private training data. Yet, especially for heterogeneous models, a key bottleneck remains the transfer of knowledge gained from each client model with the server. One popular method, FedDF, uses distillation to tackle this task with the use of a common, shared dataset on which predictions are exchanged. However, in many contexts such a dataset might be difficult to acquire due to privacy and the clients might not allow for storage of a large shared dataset. To this end, in this paper, we introduce a new method that improves this knowledge distillation method to only rely on a single shared image between clients and server. In particular, we propose a novel adaptive dataset pruning algorithm that selects the most informative crops generated from only a single image. With this, we show that federated learning with distillation under a limited shared dataset budget works better by using a single image compared to multiple individual ones. Finally, we extend our approach to allow for training heterogeneous client architectures by incorporating a non-uniform distillation schedule and client-model-mirroring on the server-side. 1 Introduction Federated Learning (FL) is a paradigm in the field of distributed machine learning which enables multiple clients to collaboratively train powerful predictive models without the need of centralising the training data (Zhang et al., 2021). It comes with its own set of key challenges in terms of skewed non-IID distribution of data between the participating clients (Zhu et al., 2021a; Li et al., 2020; Chai et al., 2019; Hsu et al., 2019; Lin et al., 2020) and communication efficiency during training (Konečný et al., 2016; Lin et al., 2020) among others. These challenges are not directly answered with the classical approaches such as FedAvg (McMahan et al., 2023), which rely primarily on a naive client network parameter sharing approach. Since the inclusion of clients with different data... distributions has a factor of heterogeneity involved (Zhu et al., 2021a; Hsu et al., 2019), another well-known work (Li et al., 2020) counteracts this heterogeneity directly during the client training. This tries to solve one challenge related to non-iidness in private data distribution, but other key challenges related to network parameter sharing remain including concerns with privacy leakage during parameter sharing (Wang et al., 2019; Sun et al., 2021), heterogeneity of client architectures (Lin et al., 2020; Chai et al., 2019) and high bandwidth cost of parameter sharing (Konečný et al., 2016). To this end, along a second line of thought implementing a server-side training regime, approaches suggested in (Lin et al., 2020; Li & Wang, 2019; Zhu et al., 2021b; Sui et al., 2020) make use of the process of knowledge distillation (KD) (Hinton et al., 2015; Gou et al., 2021) to overcome these challenges without the exclusive need of network parameter sharing. To facilitate central network training with the help of KD, the sharing of public data is needed between the clients and the server. In this work, we propose a novel approach of making use of a single datum source to act as the shared distillation dataset in ensembled distillation-based federated learning strategies. Our approach makes use of a novel adaptive dataset pruning algorithm on top of generating the distillation data from a single source image during the training. This combination of shared data generation and instance selection process not only allows us to train the central model effectively but also outperforms the other approaches which make use of multiple small-sized images in place of a single image under a limited shared dataset budget. The use of a single datum source has added benefits in domains, where publicly available data and client resources (e.g., network bandwidth and connectivity) are limited in nature. The use of a single datum source has been explored (Asano et al., 2020; Asano & Saeed, 2023) under the settings of self-supervised learning and understanding extrapolation capabilities of neural networks with knowledge distillation, but it has not yet been explored in federated setting for facilitating model training. ![Figure 2](image.png) Figure 2: Comparison of test performance in federated setting using a single image with patch selection compared to the equivalent size of multiple independent training samples from a labelled dataset as shared distillation dataset. We use different rates of FedAvg. initialisations to emulate different network bandwidth conditions. Detailed result in Table 4. We perform a series of experiments to examine the viability of our proposed algorithm under varying conditions of heterogeneity in private client data, client-server model architectures, rate of pre-training network initialisations before distillation, shared dataset storage budget and real-world domain of the single images. We also extend our experiments to a use-case of heterogeneous client architectures involved during a single federated training with the help of client-model mirroring on the server side. To facilitate this, we keep one copy of the client model of each type on the server end, which acts as a global model for the clients that have the same network architecture. The global models are improved with knowledge distillation after each round of local client training with the help of shared logits over the single image dataset. The results we obtain during the aforementioned experiments demonstrate positive reinforcement towards reaching our goal of efficient federated training using Knowledge Distillation under a limited shared dataset budget. The primary contributions of this work are: 1. Demonstrating the efficiency of a single image as a powerful medium for knowledge transfer in a federated learning setting using knowledge distillation. 2. Novel algorithm for dynamic data pruning which evolves with the current global model during federated learning. 3. Extensive evaluation of our proposed methods under a variety of conditions in a federated setting. 2 RELATED WORK Federated Learning using Knowledge Distillation Knowledge Distillation (KD) (Hinton et al., 2015) has been shown to successfully transfer the knowledge of an ensemble of neural networks into a single network with the means of output logits over a shared dataset. KD has also been leveraged in federated setting, such as Federated Distillation Fusion (FedDF) (Lin et al., 2020) and Federated Ensemble Distillation (FedED) (Sui et al., 2020), where the respective authors make use of KD to allow robust and faster convergence on top of using other ensembling methods such as the ones suggested in Federated Averaging (McMahan et al., 2023) for initialising the central network before the distillation training of the global model. On the other hand, authors of works such as Federated Model Distillation (FedMD) (Li & Wang, 2019) have also successfully shown that KD can be used for knowledge transfer in a federated setting for the purpose of client model personalisation. However, the application of algorithms such as FedMD is targetted for personalisation by client-side knowledge distillation rather than improvement of a central model, hence we have not delved into it in the scope of our research. In the case of ensembling methods, it has been shown in (Lin et al., 2020) that in the absence of an ensemble of local parameters before distillation training, the final test performance of the central network tends to suffer. As a result, these methods have been shown by the authors to significantly rely on parameter exchange every round similar to naive parameter exchange-based algorithms such as FedAvg (McMahan et al., 2023) for robust performance on top of KD. Since the aforementioned KD-based federated algorithms also require significant regular ensembling using network parameter exchange, our approach focusses on improving this aspect by relying significantly on knowledge distillation with the help of data pruning and augmentations on the shared public dataset, which has not yet been explored in these works. Communication Efficient Federated Learning To solve the high bandwidth costs related to parameter sharing, authors of (Caldas et al., 2018; Konečnỳ et al., 2016) have shown that quantisation of network parameters before the transfer can significantly reduce the bandwidth costs incurred during their transfer. However, with the application of the same low-bit quantization methods with the KD-based federated learning methods in (Lin et al., 2020), the authors have also shown a significant decrease in the overall test performance of models compared to their non-quantized counterparts. On the other hand to not rely on public data sources, authors of the work (Zhu et al., 2021b) have successfully shown that data-free approaches using a centrally trained generative network for producing the public shared dataset works robustly. However, this also requires an additional exchange of the generative network parameters before each round, which leads to an increase in the network bandwidth usage itself. In pursuit of reducing the bandwidth costs pertaining to network parameter exchanges as well as shared dataset sharing, these works have not yet made an attempt to make use of a storage-efficient single data source, which can simultaneously generate a public distillation dataset alongside being used for dynamic selection without added bandwidth costs. We explore this in our work. Single Image Representation Learning In (Asano et al., 2020), the authors have successfully made use of a single image to produce augmented patches for facilitating self-supervised learning of neural networks required for solving various downstream tasks. However, the focus of our work is not on the process of solving tasks with the help of self-supervised learning, but on the implications of making use of the single image patches in a federated setting as a medium of knowledge transfer for training robust classifiers. To this end, in a closely resembling work to our target task, the authors in (Asano & Saeed, 2023) have shown to be able to successfully use KD with single image patches to transfer the knowledge between a pre-trained network and an untrained network to solve the classification task of ImageNet-1k. However, the experiments by the authors were all based in a non-federated setting. In our work, we explore the usage of single image patches in a federated setting as the shared public distillation dataset and its implications in limited shared dataset budget settings. 3 METHOD Our method focuses on a dynamic procedure to utilize a single image to act as a proxy dataset for distillation in addition to a federated learning setup which is similar to existing ensemble-based knowledge distillation-based methods such as FedDF (Lin et al., 2020). Alongside the generation of a distillation dataset from a single data source, we take care of dynamically selecting the best patches every round to improve the training. The two important parts of our federated strategy are described in the following sections: 3.1 Distillation Set Generation and 3.2 Patch Subset Selection. 3.1 DISTILLATION DATASET GENERATION For generating meaningful image representations out of a single image, we make use of the patchification technique. Using this technique, we generate a large number of small-size crops from a big image by making combined use of augmentations such as 35-degree rotations, random horizontal flipping, color transformations etc., similar to the ones used in (Asano & Saeed, 2023) for knowledge distillation based learning. The image generation procedure can be controlled by a seed, which allows all the clients to be able to generate the same set of patches using the same augmentations from a single image. This provides us with the means of reducing the bandwidth usage pertaining to the transfer of the distillation proxy set to the clients required for improving the global model using Knowledge Distillation. Due to the flexibility provided by augmentations in combination with the subset selection procedure described in Section 3.2, one can make use of a single image to produce varying desired number of patches for the fixed amount of single image data. 3.2 PATCH SUBSET SELECTION After we have an initial dataset for distillation using the method described in Section 3.1, we apply dataset pruning methods on this dataset to ensure the selection of information-rich patches for assisting the current round of federated training. The dataset generation procedure is based on the whole image, due to which it has the ability to produce bad representation patches such as: containing no entities, overlapping with others, being dissimilar to the target domain, and similar problems arising due to heavy augmentations and information less regions of the single image. To prune the bad patches, we make use of the following two mechanisms: KMeans Balancing (3.2) and Entropy Selection (3.2). These mechanisms depend on the current global model for their operation, which makes them dynamic in nature and improves their data-pruning ability with the improvement in the global model. As a result, better representations are selected with better global models. Entropy Selection Entropy Selection is based on the use of randomness present in the output logits of the distillation training examples to prune their dataset. To achieve this, we examine the maximum softmax values of the logits obtained for each distillation training example using the current global model. On the basis of a removal heuristic $H^E \in \{\text{Top, Bottom, Random}\}$, we remove the top k percent of examples from each group (grouped on the basis of their predicted class using current global model). Top removes training examples with high softmax values while Bottom removes the ones with low softmax values. The algorithm has been described in detail in Alg. 1. KMeans Balancing KMeans Balancing is based on the use of unsupervised KMeans clustering on the embedding layer representations of the training examples. To accomplish this, we establish a KMeans clustering model (based \begin{algorithm} \caption{Entropy Selection} \begin{algorithmic}[1] \Require Distillation Training Dataset ($X$), Current Global Model ($M^G$) \Parameters Percentage of Examples to Prune ($k$), Removal Heuristic ($H^E$) \Ensure Pruned Distillation Training Dataset with Entropy Selection ($X^E$) \begin{itemize} \item For all $x_n \in X : n \in [1..S]$ where $S = \text{size of } X$, find $Y = \{y_n : y_n = \text{Max}(Softmax(Classifier Output } (M^G, x_n))\}$. \item Select indices of the training examples using the removal heuristic $H^E \in \{\text{Top, Bottom, Random}\}$ with their corresponding values in $Y$. \item Push the selected indices in the new dataset ($X^E$). \end{itemize} \end{algorithmic} \end{algorithm} on Euclidean distance) with K cluster centers and try to fit the embedding representations (using the current global model) of the distillation training examples on it. Using a selection heuristic $ H^K \in \{ \text{Easy, Hard, Mixed} \} $ on their calculated cluster distances $ D $, we can select the training examples for the next round of training. Easy prefers examples with low cluster distance values while Hard prefers high cluster distance values. A class balancing factor $ F^K \in [0.0, 1.0] $ ensures that there’s a fixed lower bound for selecting a minimum number of training examples from each of the predicted classes (using the current global model) on the distillation training set. The algorithm has been described in detail in Alg. 2. Algorithm 2: K-Means Balancing Input: Distillation Training Dataset ($ X $), Current Global Model ($ M^G $) Parameters: Number of Clusters ($ K $), Size of New Dataset ($ s $), Balancing Factor ($ F^K $), Selection Heuristic ($ H^K $) Output: Pruned Distillation Training Dataset with KMeans Selection ($ X^K $) begin 1. For all $ x_n \in X : n \in [1..S] $ where $ S = \text{size of } X $, find \[ Z = \{ z_n : z_n = \text{Embedding Representation} (M^G, x_n) \} \] \[ Y = \{ y_n : y_n = \text{Max-Index Classifier Output} (M^G, x_n) \}. \] 2. Define $ C^P = \{\text{Set of unique classes in } Y\} $ and Number of unique classes $ C = \|C^P\| $. 3. Initialise an independent unsupervised KMeans Clustering Model ($ M^C $) using $ K $ number of cluster centers. Fit $ M^C $ on $ Z $ and find $ D = \{ d_n : d_n = \{\text{Shortest euclidean distance of } z_n \text{ to its cluster center}\} $. 4. Define the minimum number of examples (balancing lower bound) to be selected from each class $ c_i \in C^P $, as $ LB = \lceil \frac{s}{C} \times F^K \rceil $. 5. forall $ c_i \in C^P : i \in [1..C] $ do - Find the indices of examples belonging to the $ c_i $ using $ y_n \in Y : y_n = c_i $. - Select indices of the new training examples on the basis of selection heuristic $ H^K \in \{\text{Easy,Hard,Mixed}\} $ with their corresponding cluster distance values $ D $. - Push the training examples from $ X $ with selected indices in the new dataset ($ X^K $). - Remove the selected training examples from $ X $ and $ D $. 6. Remaining number of examples to be selected can be calculated as given by : $ s - \text{size of } X^K $. 7. Using selection heuristic $ = H^K $ on the cluster distance values in $ D $, find the indices of the remaining examples to be selected. Push the training examples with selected indices in the new dataset ($ X^K $) end ### 4 EXPERIMENTS #### 4.1 EXPERIMENTAL SETUP Our experimental setup for federated training using our algorithm has been shown schematically in Fig. 1. Dataset We do our experiments across the following publically available datasets: CIFAR10/100 (Krizhevsky et al., 2009) and MedMNIST (PathMNIST) (Yang et al., 2023). For the distribution of private data among the clients from the collective training data, we use a similar strategy to the one suggested in (Hsu et al., 2019), which allows us to control the degree of heterogeneity using the parameter $ \alpha $ (lower $ \alpha $ = higher degree of non-iidness and vice-versa). We use the full test sets corresponding to the private client datasets as evaluation sets for the global model (testing is only server side). 10% of the training examples are held as a validation dataset. For the shared public dataset, we generate patches out of a single image for all the experiments with our method. For the FedDF experiments, we have made use of CIFAR100 training set for CIFAR10 experiments, unless mentioned otherwise. The single images have been visualised in Appendix A alongside the patches and t-SNE visualisations during training. Server-Client Model Architecture ResNets (trained from scratch) have been used for most of our experiments as the model of choice for the server and clients (He et al., 2016). WideResNets have also been used for some of the experiments (Zagoruyko & Komodakis, 2016). The models have been explicitly defined in the table descriptions for unambiguity. Hyper-parameter Configuration The values of the learning rate (local and global) have been motivated by the experiments described in Appendix C. We use a client learning rate of 0.01 for ResNet and WResNet, while the distillation training learning rate is 0.005. For KMeans Balancing, we use a KMeans model with 1000 clusters, a class balancing factor of 1.0, and the ‘Hard’ selection heuristic. For Entropy selection, we remove 90% of the training examples using the ‘Top’ removal heuristic (Appendix B). For the experiment in Table 2, we do local client training for 10 epochs and server-side distillation for 250 steps, while 40 epochs and 500 distillation steps have been our choice for other experiments unless mentioned otherwise. We prefer to keep the use of FedAvg initialisations to 20% in our experiments unless mentioned otherwise. For all the experiments, we simulate 20 private clients, with a selection probability (C) of 0.4 per training round. 4.2 Selecting the Best Image for Domain of Task We conduct cross-dataset single-image testing using our algorithm across 3 private training datasets and 3 images, with two of them corresponding to one of the dataset domains and the third one being a random noise. The results in Table 1 exhibit that it is necessary to use a single image that is similar to the domain of the target task for optimal performance. In the case of using a single random noise image as the distillation proxy, we get the lowest test performance as it is hard for random augmentations to convert random noise patches into a knowledge transfer medium. Hence, care must be taken in choosing a single image with similar patch representations as the target task for optimal learning with our algorithm. There can be an interesting area to explore with more augmentation experiments and generative algorithms, if it is possible to use a random noise image viably as a single image with our method. We leave this as future work. \| Image \| Dataset \| \|------------------------------\|---------------\| \| \| CIFAR10 \| CIFAR100 \| PathMNIST \| \| City Street \| 75.3 \| 32.0 \| 69.7 \| \| Multi Colon Pathology Samples\| 69.0 \| 12.0 \| 71.6 \| \| Random Noise \| 39.4 \| 6.8 \| 33.0 \| Table 1: Best test performance during 30 rounds of training using our federated method with varying Pvt. Datasets (Distribution α = 100.0) and 5k Single Image Patches (Distillation Proxy Set) on ResNet-8 architecture with 20% rate of FedAvg. initialisation. 4.3 Ablation Studies with Patch Selection Mechanism Finding the Best Patch Subselection Strategies across Varying Pvt. Dataset To find the effectiveness of patch subset selection mechanisms, we test it under different private datasets from different real-world domains (General and Medical). Through Table 2, it is evident that the single image patches work best in the presence of a selection strategy in our federated algorithm. On their own, both KMeans Balancing (3) and Entropy Selection (3.2) strategy works better than employing no selection for the same number of patches. Together, they perform best across all the datasets we have tested which is what we use in our other experiments in this work. Both of the selection strategies and their combination significantly impact the final performance. We have done our primitive analysis with them in light of this work to find an optimal setting (Appendix B, but there might be a correlation between their settings which we have not delved into. We can propose this detailed analysis of their combinative work as future work for improving the test performance of our federated strategy with the means of better data pruning. Through the T-SNE visualisation in Fig. 3 during different phases of federated training with a single image and our data pruning method, we observe the formation of identifiable boundary structures among the selected patches as the global model accuracy improves. This provides a visual qualitative \| Selection Strategy \| Private Dataset \| \|-------------------\|-----------------\| \| \| CIFAR10 \| CIFAR100 \| PathMNIST \| \| No Selection \| 63.4 ± 1.4 \| 24.2 ± 1.1 \| 64.5 ± 4.7 \| \| KMeans \| 66.2 ± 0.8 \| 21.8 ± 2.1 \| 67.9 ± 8.4 \| \| Entropy \| 65.9 ± 1.0 \| 26.3 ± 1.0 \| 76.4 ± 2.8 \| \| KMeans + Entropy \| 67.0 ± 1.1 \| 26.4 ± 1.2 \| 77.1 ± 3.0 \| Table 2: Best test performance achieved during 30 rounds of training with different selection mechanisms (Distillation Set Size = 5000 patches) across different private datasets ($\alpha = 1.0$) using our federated strategy with ResNet-8 while using 20% rate of FedAvg. initialisations. (2 seeds) assessment of our design claims regarding the positive correlation of the effectiveness of our patch subset selection algorithm with the evolution of the global model. ![Scatter plot of TSNE embeddings](image) (a) Global Model Accuracy = 52.8 (b) Global Model Accuracy = 76.7 Figure 3: Scatter plot of TSNE embeddings of single image patches during different phases of training, using our method with FedAvg and ResNet-8 on CIFAR10. Testing the Impact of Selection Mechanism with Manually Labelled Distillation Set We test the viability of our selection mechanism in case of extending it to the use cases where we already have a shared public dataset at hand in Table 3. During the regular exchange of initialisation parameters, the application of our selection mechanism exhibits no advantage. However, when we reduce the exchange of initialisation parameters to emulate low bandwidth conditions, it shows significant gains. This shows that even with the availability of standard distillation sets at hand in ensembled distillation methods, the subset selection mechanism can play an important role in low bandwidth cost federated training. \| Selection Mechanism Applied \| FedAvg Initialisation Rate (in %) \| \|-----------------------------\|----------------------------------\| \| \| 100 \| 50 \| 20 \| \| $\times$ \| 75.0 ± 0.5 \| 73.1 ± 1.1 \| 67.4 ± 0.5 \| \| $\checkmark$ \| 73.8 ± 1.9 \| 72.3 ± 0.6 \| 70.7 ± 1.2 \| Table 3: Comparison of best test performance during 30 rounds of training with CIFAR10 Pvt. Data with Distribution $\alpha = 1.0$ using FedDF (with ResNet-8) between use/non-use of selection mechanism across varying rate of using FedAvg initialisation. 1000 samples from CIFAR100 train split make the distillation proxy dataset. ### 4.4 Ablation Studies with Varying Network and Storage Conditions Comparing Performance of a Single Image in Limited Shared Dataset Budget Settings This is our most significant experiment in terms of exhibiting the viability of federated learning under limited shared dataset budget settings using a single image. Going through the results in Table 4, we see that for the same amount of storage budget, a single image with patch selection outperforms similarly sized individual samples. If we also lower the network budget and the rate of exchange of initialisation parameters, it is able to hold at par with individual training samples 10 times its size. This shows a promising future for our work in the scenario where there is limited availability of publicly shared datasets as well as storage budget would be low on participating clients. \| Distillation Dataset \| No. of Pixels \| FedAvg Initialisation Rate (in %) \| \|---------------------------\|--------------\|----------------------------------\| \| \| \| 100 \| 50 \| 20 \| \| 5K CIF100 Samples \| 5M \| 76.4 ± 1.4 \| 74.1 ± 1.6 \| 68.9 ± 1.4 \| \| Single Image with Patch Selection \| 0.5M \| 74.8 ± 2.6 \| 73.2 ± 3.2 \| 68.6 ± 0.8 \| \| 500 CIF100 Samples \| 0.5M \| 73.2 ± 1.7 \| 71.3 ± 2.0 \| 66.5 ± 0.9 \| Table 4: Best test performance during 30 rounds of training with CIFAR10 Pvt. Data with Distribution $ \alpha = 1.0 $ using ResNet-8 with different distillation datasets and rate of using FedAvg initialisation. Testing Performance in Limited Network Bandwidth Settings against Heterogeneous Data Distributions To test the impact of high data distribution heterogeneity on our FL strategy against an existing SOTA federated learning strategy based on knowledge distillation, we show the performance gains in Table 5. We also vary the network initialisation rate to test our method in high and low-bandwidth situations. We notice that with the help of patch subset selection, our methods outperform the fed strategy which doesn’t make use of this process. This trend is constant across all bandwidth scenarios and local client training expenditures. We have also extended our approach to incorporate FedProx local client training regime, which shows better results than naive local client training. This extendability makes our method unique and viable to more approaches than just one kind of local training which can have added performance benefits with our algorithm. \| Strategy \| Local Epochs \| FedAvg Initialisation Rate (in %) \| \|-------------------\|--------------\|----------------------------------\| \| \| \| $ \alpha = 1.0 $ \| $ \alpha = 0.1 $ \| $ \alpha = 1.0 $ \| $ \alpha = 0.1 $ \| $ \alpha = 1.0 $ \| $ \alpha = 0.1 $ \| \| FedDF \| 20 \| 75.7 ± 1.2 \| 48.2 ± 2.6 \| 73.9 ± 0.8 \| 47.3 ± 5.2 \| 71.1 ± 0.5 \| 42.2 ± 9.4 \| \| \| 40 \| 75.7 ± 0.9 \| 49.5 ± 3.1 \| 74.9 ± 1.9 \| 49.3 ± 1.1 \| 72.5 ± 0.5 \| 46.1 ± 6.6 \| \| Ours w/ FedAvg \| 20 \| 76.9 ± 0.6 \| 47.8 ± 5.3 \| 75.8 ± 0.3 \| 47.3 ± 5.5 \| 73.7 ± 1.0 \| 45.5 ± 5.1 \| \| \| 40 \| 77.0 ± 0.6 \| 47.8 ± 5.4 \| 76.2 ± 1.4 \| 49.5 ± 2.2 \| 74.3 ± 0.6 \| 46.6 ± 6.7 \| \| Ours w/ FedProx \| 20 \| 77.2 ± 0.8 \| 47.2 ± 7.0 \| 74.5 ± 1.3 \| 44.6 ± 7.9 \| 73.1 ± 0.2 \| 46.9 ± 4.5 \| \| \| 40 \| 77.7 ± 0.8 \| 47.7 ± 3.8 \| 76.3 ± 0.4 \| 46.0 ± 5.3 \| 74.3 ± 1.1 \| 45.1 ± 6.0 \| Table 5: Comparison of best test performance under different settings (FedAvg Initialisation Rate, Degree of Heterogenity ($ \alpha $), Local Training Epochs) using different federated learning strategies with ResNet-8 on CIFAR10 during 30 rounds of training (2 seeds). 5000 single image patches have been used as distillation proxy set (w/o selection mechanism for FedDF). ### 4.5 Ablation Studies with Varying Client-Server Neural Network Architectures Testing our Strategy under Homogeneous Network Architecture Settings We perform all the experiments in the earlier sections using ResNet-8 as the client and server models. To make sure our federated strategy works equally well among other homogenous network distributions, we put it to the test against FedDF using ResNet-20 as well as W-ResNet-16-4 in Table 6. We see that under the same distillation set storage budget, our method works better under all the tested network architectures. As per nominal expectations, network architectures with more parameters show better results than the ones with less number of parameters which enables us to achieve better test performance with more complex networks. Irrespective of the network architecture, the trend is constant when it comes to our FL strategy outperforming other strategies making use of a labelled distillation dataset in a limited storage budget scenario. Testing our Strategy under Heterogeneous Network Architecture Settings In the final experimental section, we test our federated strategy in the presence of heterogeneity in the client model... \| Fed Strategy \| Network Architecture \| \|--------------\|----------------------\| \| \| ResNet-8 \| ResNet-20 \| W-ResNet-16-4 \| \| FedDF \| 67.3 ± 1.9 \| 73.0 ± 0.6 \| 75.3 ± 1.2 \| \| Ours \| 70.2 ± 0.8 \| 74.1 ± 0.9 \| 75.7 ± 0.9 \| Table 6: Best test performance during 30 rounds of training using CIF10 Pvt. Data with Distribution $\alpha = 1.0$ using different Fed strategies and homogeneous client-server network architectures with 20% rate of FedAvg. initialisation. FedDF uses 500 CIF100 samples as distillation proxy, while our method makes use of a single image of equivalent size with patch subset selection. architectures. The results present in Table 7 show the success of our method in training the global models when pitted against a strategy not utilising a single image. It also exhibits the importance of constant distillation training for the success of our methods, as our non-uniform approach gives subpar results with less training time. However, when going from 15k to 11k steps, we also save about 1/3 of the training time and computation resources used on the server side. It can be an interesting point of extension to our work to improve upon this non-uniform scheduling to allow for more robust training of heterogeneous models with less computation time. \| Fed Strategy \| Total Distillation Steps \| Macro-Avg Accuracy (Server Models) \| \|--------------\|--------------------------\|-----------------------------------\| \| FedDF \| 15K \| 67.4 ± 0.6 \| \| Ours \| 15K \| 68.5 ± 1.1 \| \| Ours w/ Scheduling \| 11.3K \| 65.2 ± 1.3 \| Table 7: Best test performance across during 30 rounds of training using CIF10 Pvt. Data with Distribution $\alpha = 1.0$ using different Fed strategies and distillation step scheduling, under a heterogenous client distribution (6 ResNet-8, 7 ResNet-20, 7 W-ResNet-16-4) with 20% rate of FedAvg. Initialisation. 500 CIF100 samples have been used as distillation proxy for FedDF, while our method makes use of a Single Image of equivalent size with patch selection. 5 CONCLUSION Through this work, we present a novel approach for federated learning using ensembled knowledge distillation with the use of augmented image patches from a single image with patch subset selection. We successfully exhibit the performance gains with our approach in a limited shared dataset budget scenario as well as low network bandwidth requiring scenarios with less exchange of network parameters. Alongside low resource usage, the use of a single image also enables our federated strategy to be applicable to scenarios where we have a lack of public datasets required during federated training of multiple clients. Prospective Future of our Work We mention a few specialised avenues of extension to our work during the discussion of results in Section 4. Some of the key points that were not mentioned in it include: Application of the single datum based federated learning to other modalities and machine learning tasks; Application of our work to other knowledge distillation-based algorithms in federated learning other than ensembled methods, such as FedMD (Li & Wang, 2019); Analysis of different kind of augmentations to improve the robustness of our method. With the aforementioned points, significant work can be done to improve the viability of our novel approach presented in this work to incorporate more real-world challenges. REFERENCES Yuki M. Asano and Aaqib Saeed. 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v1VvCWJAL8	Do we have any ideas on the identifiability of the model? This is an important question because we discuss causality. Although the theory in the paper converts IASCMs on a set of domains into an easy-to-deal-with form that is the canonical ILD. There are no discussions on the identifiability of canonical ILDs, that is, when we really try to learn the canonical ILDs, can we identify the single eq class of canonical ILDs, which contains the true one? The practical application of the proposed idea depends critically on this question.	Towards Characterizing Domain Counterfactuals for Invertible Latent Causal Models Zeyu Zhou, Ruqi Bai, Sean Kulinski, Murat Kocaoglu, David I. Inouye Elmore Family School of Electrical and Computer Engineering Purdue University {zhou1059, bai116, skulinsk, mkocaoglu, dinouye}@purdue.edu Abstract Answering counterfactual queries has important applications such as explainability, robustness, and fairness but is challenging when the causal variables are unobserved and the observations are non-linear mixtures of these latent variables, such as pixels in images. One approach is to recover the latent Structural Causal Model (SCM), which may be infeasible in practice due to requiring strong assumptions, e.g., linearity of the causal mechanisms or perfect atomic interventions. Meanwhile, more practical ML-based approaches using naïve domain translation models to generate counterfactual samples lack theoretical grounding and may construct invalid counterfactuals. In this work, we strive to strike a balance between practicality and theoretical guarantees by analyzing a specific type of causal query called domain counterfactuals, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). We show that recovering the latent SCM is unnecessary for estimating domain counterfactuals, thereby sidestepping some of the theoretic challenges. By assuming invertibility and sparsity of intervention, we prove domain counterfactual estimation error can be bounded by a data fit term and intervention sparsity term. Building upon our theoretical results, we develop a theoretically grounded practical algorithm that simplifies the modeling process to generative model estimation under autoregressive and shared parameter constraints that enforce intervention sparsity. Finally, we show an improvement in counterfactual estimation over baseline methods through extensive simulated and image-based experiments. 1 Introduction Causal reasoning and machine learning, two fields which historically evolved disconnected from each other, have recently started to merge with several recent results leveraging the available causal knowledge to develop better ML solutions (Kusner et al., 2017; Moraffah et al., 2020; Nemirovsky et al., 2022; Calderon et al., 2022). One such setting is causal representation learning (Schölkopf et al., 2021; Brehmer et al., 2022), which aims to take data from a complex observed space (e.g., images) and learn the latent causal factors that generate the data. A common scenario is when we have access to diverse datasets from different domains, where from a causal perspective, each domain is generated via an unknown intervention on some domain-specific latent causal mechanisms. With this in mind, we focus on a specific causal query called a domain counterfactual (DCF), which hypothesizes: “What would this sample look like if it had been generated in a different domain (or environment)?” For example, given a patient’s medical imaging from Hospital A, what would it look like if it had been taken at Hospital B? Answering this DCF query could have applications in fairness, explainability, and model robustness. A naïve ML approach to answering this query is to simply train generative models to map between the two distributions without any causal assumptions or causal constraints (e.g., Kulinski and Inouye (2023)); however, this lacks theoretic grounding and may produce invalid counterfactuals. One common causal approach for answering such a counterfactual query would be a two-stage method of first recovering the causal structure and then estimating the counterfactual examples (Kocaoglu Equal contribution. Listing order is random. Table 1: This table of related causal representation learning works, focuses mostly on works that study learning a latent SCM, shows that most prior works in this area aim for identifiability of the (latent) SCM, and thus require strong technical assumptions which may not hold in real-world scenarios (e.g., perfect single-node interventions for each variable). \| SCM type \| Observ. Function \| Other Assumptions \| Observ. Function Identifiability \| Characterization of Counterfactual Equiv. \| \|-------------------\|------------------\|-------------------\|---------------------------------\|------------------------------------------\| \| Nasr-Esfahany et al. (2023) \| Invertible observed \| N/A \| 1) Access to ground-truth DAG \| N/A \| Single mechanism counterfactuals under specific contexts \| \| Brehmer et al. (2022) \| Invertible latent \| Invertible \| 1) Atomic stochastic hard interv. 2) Training set is counterfactuals pairs \| Mixing and elementwise \| N/A - Counterfactuals as input \| \| Squires et al. (2023) \| Linear latent \| Linear \| 1) Atomic hard interv. \| Scaling \| No \| \| Liu et al. (2022a) \| Linear latent \| Non-linear \| 1) Significant causal weights variation \| Mixing and scaling \| No \| \| Varici et al. (2023) \| Latent non-linear \| Linear \| 1) Atomic stochastic hard interv. \| Mixing or scaling \| No \| \| Khemakhem et al. (2021) \| Invertible observed (implicit) \| Affine \| 1) Bivariate requirement for identifiability \| Full (bivariate only) \| No \| \| Ours \| Invertible latent \| Invertible \| 1) Access to domain labels \| No \| Domain counterfactual \| et al., 2018; Sauer and Geiger, 2021; Nemirovsky et al., 2022). However, most of the existing methods for causal structure learning either assume the causal variable to be observed (as opposed to our setting where the causal variables are latent) or require restrictive assumptions for recovering the latent causal structure, such as atomic interventions (Brehmer et al., 2022; Squires et al., 2023; Varici et al., 2023), or access to counterfactual pairs (Brehmer et al., 2022), or assume model structures like linearity or polynomial (Khemakhem et al., 2021; Squires et al., 2023), which often do not hold in practice. A summary of existing works can be found in Table 1. In this paper, we strive to balance practicality and theoretical guarantees by answering the question: “Can we theoretically and practically estimate domain counterfactuals without the need to recover the ground-truth causal structure?” With weak assumptions about the true causal model and available data, we analyze invertible latent causal models and show that it is possible to estimate domain counterfactuals both theoretically and practically, where the estimation error depends on the intervention sparsity. We summarize our contributions as follows: C1 For a class of invertible latent domain causal models (ILD), we show that recovering the true ILD model is unnecessary for estimating domain counterfactuals by proving a necessary and sufficient characterization of domain counterfactual equivalence. C2 We prove a bound on the domain counterfactual estimation error which decomposes into a data fit term and intervention sparsity term. If the true intervention sparsity is small, this bound suggests adding a sparsity constraint for DCF estimation. C3 Towards practical implementation, we prove that any ILD model with intervention sparsity $k$ can be written in a canonical form where only the last $k$ variables are intervened. This significantly reduces the modeling search space from $\binom{m}{k}$ causal structures to only one. C4 In light of these theoretic results, we propose an algorithm for estimating domain counterfactuals by searching over canonical ILD models while restricting intervention sparsity (inspired by C2 and C3). We validate our algorithm on both simulated and image-based experiments. Notation We denote function equality between two functions $f : X \rightarrow Y$ and $f' : X \rightarrow Y$ as simply $f = f'$, which more formally can be stated as $\forall x \in X, f(x) = f'(x)$. Similarly, $f \neq f'$ means that there exists $x \in X, f(x) \neq f'(x)$. We use $\circ$ to denote function composition, e.g., $g(f(x)) = g \circ f(x)$ or simply $h = g \circ f$. We use subscripts to denote particular indices (e.g., $x_j \in \mathbb{R}$ is the $j$-th value of the vector $x$ and $x_{<j} \in \mathbb{R}^{j-1}$ is the subvector corresponding to the indices 1 to $j - 1$). For function outputs, we use bracket notation to select a single item (e.g., $[f(x)]_j \in \mathbb{R}$ refers to the $j$-th output of $f(x)$) or subvector (e.g., $[f(x)]_{<j} \in \mathbb{R}^j$ refers to the subvector for indices 1 to $j$ inclusive). Similarly, for (unbound) functions, let $[f]_j : \mathbb{R}^m \rightarrow \mathbb{R}$ refer to the scalar function corresponding to the $j$-th output or $[f]_{<j} : \mathbb{R}^m \rightarrow \mathbb{R}^j$ refer to the vector function corresponding to first $j$ outputs. For any positive integer $m$, we define $[m] \triangleq \{1, \ldots, m\}$. We denote $N_d$ as number of domains in the ILD model. --- 1 Code can be found in https://github.com/inouye-lab/ild-domain-counterfactuals. 2 Domain Counterfactuals with Invertible Latent Domain Causal Models Given a set of domains (or environments), a domain counterfactual (DCF) asks the question: “What would a sample from one domain look like if it had (counterfactually) been generated from a different domain?” Each domain represents different causal model on the same set of causal variables, i.e., they can be viewed as interventions of a baseline causal model. If we let $ D $ be an auxiliary indicator variable denoting the domain, a DCF can be formalized as the counterfactual query $ p(X_{D=d'} \| X = x, D = d) $, where $ x $ is the observed evidence, $ d $ is the original domain, and $ X_{D=d'} $ is the counterfactual random variable when forcing the domain to be $ d' $. In this work, we aim to find DCF for a class of invertible models (which we define in Section 2.1) and we will assume that the causal variables are unobserved (i.e., latent). To compare, Causal Representation Learning (CRL) has a similar latent causal model setup (Schölkopf et al., 2021). However, most CRL methods aim for identifiability of the latent representations, which is unsurprisingly very challenging. In contrast, we show that estimating DCFs is easier than estimating the latent causal representations and may require fewer assumptions in Section 2.2. 2.1 ILD Model We now define the causal model based primarily on the assumption of invertibility. First, we assume that the observation function (or mixing function) shared between all domains is invertible (as in Liu et al. (2022a); Zhang et al. (2023); von Kügelgen et al. (2023)). This means that the latent causal variables are invertible functions of the observed variables. Second, we assume that the latent SCMs for each domain are also invertible with univariate exogenous noise terms per causal variable. We assume the standard Directed Acyclic Graph (DAG) constraint on the SCMs. For notational simplicity, we will assume w.l.o.g. that the DAG is a complete graph (i.e., it includes all possible edges), but some edges could represent a zero dependency which is functionally equivalent to the edge being missing. Given the topological ordering respecting the complete DAG, we prove that an invertible SCM can be written as a unique autoregressive invertible function that maps from all the exogenous noises to the latent endogenous causal variables (See Appendix B.1). Note that the SCM invertibility assumption excludes causal models where causal variables have multivariate exogenous noise. Given all this, we now define our ILD model class that joins together the shared mixing function and the latent SCMs for each domain. Definition 1 (Invertible Latent Domain Causal Model). An invertible latent domain causal model (ILD), denoted by $(g, F)$, combines a shared invertible mixing function $ g : Z \rightarrow X $ with a set of $ N_d $ domain-specific latent SCMs $ F \triangleq \{ f_d : \mathbb{R}^m \rightarrow Z \}_{d=1}^{N_d} $, where $ f_d $ are invertible and autoregressive. The exogenous noise is assumed to have a standard normal distribution, i.e., $ \epsilon \sim \mathcal{N}(0, I_m) $. While we discuss the model in depth in Appendix A, we first briefly discuss why the autoregressive and standard normal exogenous noise assumptions are not restrictive. For any model that violates the topological ordering, an equivalent ILD model can be constructed by merging the original mixing function with a variable permutation. Similarly, for any continuous exogenous distribution, we can construct an equivalent Gaussian noise-based ILD model via merging the original SCM with the Rosenblatt transform (Rosenblatt, 1952) and inverse element-wise normal CDF transformation. Moreover, we prove in the appendix that for any observed domain distributions, there exists an ILD model that could match these domain distributions. Therefore, these two assumptions are not critical but will simplify theoretical analysis. Given our definition, we note that interventions between two ILDs are implicitly defined by the difference between two domain-specific causal models and the intervention set is denoted by $ I(f_d, f_{d'}) \subseteq [m] $, which is the set of the intervened causal variables’ indices. In Appendix B.3 in the appendix, we prove that the standard notion of causal intervention is equivalent to checking if the inverse subfunctions are equal, i.e., $ j \in I(f_d, f_{d'}) \iff [f_d^{-1}]_j \neq [f_{d'}^{-1}]_j $. We further define the ILD intervention set as the union over all pairs of domains, i.e., $ I(F) \triangleq \bigcup_{f_d, f_{d'} \in F} I(f_d, f_{d'}) = \bigcup_{d \leq N_d} I(f_1, f_d) $. These implicit ILD interventions could be a hard intervention (i.e., remove dependence on parents) or a soft intervention (i.e., merely change the dependence structure with parents). Because any intervened causal mechanism is invertible by our definition, ILD interventions must be stochastic rather than do-style interventions, which would break the invertibility of the latent SCM. Finally, we define a notion of two ILD models being equivalent with respect to their observed distributions based on the change of variables formula. This notion, which is a true equivalence relation because the equation in (2) has the properties of reflexivity, symmetry, and transitivity by the properties of the equality of measures, will be important for defining an upper bound on DCF estimation in Section 3.1 and for developing practical algorithms that minimize the divergence between the ILD observed distribution and the training data in Section 3.3. Definition 2 (Distribution Equivalence). Two ILDs $(g, \mathcal{F})$ and $(g', \mathcal{F'})$ are distributionally equivalent, denoted by $(g, \mathcal{F}) \simeq_D (g', \mathcal{F'})$, if the induced domain distributions are equal, i.e., \[ \forall d, \quad p_N(f_d^{-1} \circ g^{-1}(x)) \|J_{f_d^{-1} \circ g^{-1}}(x)\| = p_N(f_d'^{-1} \circ g'^{-1}(x)) \|J_{f_d'^{-1} \circ g'^{-1}}(x)\|. \] ### 2.2 ILD Domain Counterfactuals With our ILD model defined, we now formalize a DCF query for our ILD model. For that, we remember the three steps for computing (domain) counterfactuals (Pearl, 2009, Chapter 1.4.4): abduction, action, and prediction. The first step is to infer the exogenous noise from the evidence. For ILD models, this simplifies to a deterministic function that inverts the mixing function and latent SCM, i.e., $e = f_d^{-1} \circ g^{-1}(x)$. The second step and third steps are to perform the target intervention and run the exogenous noise through the intervened mechanisms. For ILD, this is simply applying the other domain’s causal model and the shared mixing function, i.e., $x_{d \rightarrow d'} = g \circ f_{d'}(e)$. Combining these steps yields the simple form of a DCF for ILD models: \[ x_{d \rightarrow d'} \triangleq g \circ f_{d'} \circ f_d^{-1} \circ g^{-1}(x), \text{ where } f_d, f_{d'} \in \mathcal{F}. \] DCF for ILD models are deterministic counterfactuals (de Lara et al., 2023) since they have a unique mapping, i.e., given the evidence $x$ from $d$, the counterfactual $x_{d \rightarrow d'}$ is deterministic. We now provide a notion that will define which ILDs have the same DCFs (see Appendix B.4 for the equivalence relation proof). Definition 3 (Domain Counterfactual Equivalence). Two ILDs $(g, \mathcal{F})$ and $(g', \mathcal{F'})$ are domain counterfactually equivalent, denoted by $(g, \mathcal{F}) \simeq_C (g', \mathcal{F'})$, if all domain counterfactuals are equal, i.e., \[ \forall d, d' : g \circ f_{d'} \circ f_d^{-1} \circ g^{-1} = g' \circ f_{d'} \circ f_d^{-1} \circ g'^{-1}. \] While Definition 3 succinctly defines the equivalence classes of ILDs, it does not give much insight into the structure of the equivalence classes. To fill this gap, we now present one of our main theoretic results which characterizes a necessary and sufficient condition for being domain counterfactually equivalent and relates proves that their intervention set size must be equal. Theorem 1 (Characterization of Counterfactual Equivalence). Two ILDs are domain counterfactually equivalent, i.e., $(g, \mathcal{F}) \simeq_C (g', \mathcal{F'})$ if and only if: \[ \exists h_1, h_2 \in \mathcal{F}_I \text{ s.t. } g' = g \circ h_1^{-1} \in \mathcal{F}_I \text{ and } f_d = h_1 \circ f_d \circ h_2 \in \mathcal{F}_A, \forall d, \] and moreover, counterfactually equivalent models share the same intervention set size, i.e., if $(g, \mathcal{F}) \simeq_C (g', \mathcal{F'})$, then $\|\mathcal{I}(\mathcal{F})\| = \|\mathcal{I}(\mathcal{F'})\|$. See Appendix B.5 for proofs. Importantly, Theorem 1 can be used to construct domain counterfactually equivalent models and verify if two models are domain counterfactually equivalent (or determine they are not equivalent). In fact, for any two invertible functions $h_1$ and $h_2$ that satisfy the implicit autoregressive constraint, i.e., for all $d, h_1 \circ f_d \circ h_2 \in \mathcal{F}_A$, we can construct a counterfactually equivalent model—which can have arbitrarily different latent representations defined by $g' = g \circ h_1^{-1}$ since $h_1$ can be an arbitrary invertible function. Ultimately, this result implies that to estimate domain counterfactuals, we indeed do not require the recovery of the latent representations or the full causal model. ### 3 Estimating ILD Domain Counterfactuals in Practice While the previous section proved that recovering the latent causal representations is not necessary for DCFs, this section seeks to design a practical method for estimating DCFs. Since we only assume access to i.i.d. data from each domain, one natural idea is to fit an ILD model that is distributionally equivalent to the observed domain distributions. Yet, distribution equivalence is only a distribution-level property while counterfactual equivalence is a point-wise property, i.e., the domain distributions can match while the counterfactuals could be different. Indeed, we show in Theorem 2 that even under the constraint of distribution equivalence, the counterfactual error can be very large. To mitigate this issue, we choose a relatively weak assumption called the Sparse Mechanism Shift (SMS) hypothesis (Schölkopf et al., 2021), which states that the differences between domain distributions are caused by a small number of intervened variables. Given this assumption about the true ILD model, it is natural to impose this intervention sparsity on the estimated ILD model. Therefore, we now have two components to ILD estimation: a distribution equivalence term and a sparsity constraint which are based on the dataset and our assumption respectively. We first prove that both of these components are important for DCF estimation by providing a bound on the counterfactual error (defined below). Then, we prove that the sparsity constraint can be enforced by only optimizing over a canonical version of ILD models, which have all intervened variables last in a topological ordering. This greatly simplifies the practical optimization algorithm since only one sparsity structure is needed than the potentially $ \binom{m}{k} $ different sparsity structures, where $ k $ is the sparsity level. Finally, we bring all of this together to form a practical optimization objective with sparsity constraints. ### 3.1 Domain Counterfactual Error Bound In this section, we will prove a bound on counterfactual error that depends on both distribution equivalence and intervention sparsity. Towards this end, let us first define a counterfactual pseudo-metric between ILD models via RMSE (proof of pseudo-metric in Lemma 6 in the appendix). Definition 4 (Counterfactual Pseudo-Metric for ILD Models). Given a joint distribution $ p(x, d) $, the counterfactual pseudo metric between two ILDs $ (g, F) $ and $ (g', F') $ is defined as the RMSE over all counterfactuals, i.e., \[ d_C((g, F), (g', F')) = \sqrt{\mathbb{E}_{p(x, d)p(d')}[\\| g \circ f_{d'} \circ f_d^{-1} \circ g^{-1}(x) - g' \circ f_{d'} \circ f_d^{-1} \circ g'^{-1}(x) \\|_2^2]}, \] where $ p(d') = p(d) $ is the marginal distribution of the domain labels. Given this pseudo-metric, we can now derive a bound on the counterfactual error between an estimated ILD $ (\hat{g}, \tilde{F}) $ and the true ILD $ (g^, F^) $ defined as $ \varepsilon(\hat{g}, \tilde{F}) = d_C((\hat{g}, \tilde{F}), (g^, F^)) $. Theorem 2 (Counterfactual Error Bound Decomposition). Given a max intervention sparsity $ k \geq 0 $ and letting $ M(k) = \{(g, F) : (g, F) \simeq_D (g^, F^), \|I(F)\| \leq \max\{k, \|I(F^)\|\}\} $, the counterfactual error can be upper bounded as follows: \[ \varepsilon(\hat{g}, \tilde{F}) \leq \min_{(g', F') \in M(k)} d_C((\hat{g}, \tilde{F}), (g', F')) + \max_{(\hat{g}, \tilde{F}) \in M(k)} d_C((\hat{g}, \tilde{F}), (g^, F^)) . \] Furthermore, if we assume that the ILD mixing functions are Lipschitz continuous, we can bound the worst-case error $ B $ as follows: \[ (B) \leq \left[ \max_{(\hat{g}, \tilde{F}) \in M(k)} \tilde{k} L_g^2 \max_{i \in [m]} \mathbb{E}\left[ \\| \tilde{f}_d(\epsilon) - \tilde{f}_{d'}(\epsilon) \\|_2^2 \right] + k^ L_{g^}^2 \max_{i \in [m]} \mathbb{E}\left[ \\| f_{d^}^(\epsilon) - f_{d'^}^(\epsilon) \\|_2^2 \right] \right]^{1/2}, \] where $ \tilde{k} = \|I(\tilde{F})\| $ and $ k^ = \|I(F^)\| $, $ L_g $ is the Lipchitz constant of $ g $, and the expectation is over $ p(d, d', \epsilon) = p(d)p(d')p(\epsilon) $. Please check proof in Appendix B.6. The first term (A) corresponds to a data fit term and could be reduced by minimizing the divergence between the ILD model and the observed distributions. If the estimated ILD already matches the ground truth distribution, then this term would be zero. The second term (B), however, does not involve the data distribution and cannot be explicitly reduced. Yet, the bound on this second error term shows that it can be implicitly controlled by constraining the target intervention sparsity $ k $ of the estimated model. Informally, the (B) term depends on the intervention sparsity, Lipschitz constant, and a term that corresponds to the largest feature difference between domain SCMs. This last term can be interpreted as the worst case single-feature difference between latent counterfactuals. We do not claim this bound is tight, but rather simply aim to show that the domain counterfactual error depends on the target intervention sparsity $ k $ such that reducing $ k $ (as long as $ k \geq k^ $) can improve DCF estimation. Therefore, our error bound elucidates that both data fit and intervention sparsity are needed for DCF estimation. 3.2 Canonical ILD Model While the last section showed that imposing intervention sparsity helps control the counterfactual error, imposing this sparsity constraint can be challenging. In particular, the ground truth sparsity pattern, i.e., which of $ k $ causal mechanisms are intervened, is unknown. A naïve solution would be to optimize an ILD model for all possible $ \binom{k}{m} $ sparsity patterns. In this section, we prove that we only need to optimize one sparsity pattern without loss of generality. In particular, we can assume that all intervened mechanisms are on the last $ k $ variables. We refer to such a model as a canonical ILD model which we formalize next. Definition 5 (Canonical Domain Counterfactual Model). An ILD $(g, F)$ is a canonical domain counterfactual model (canonical ILD), denoted by $(g, F) \in C$, if and only if the last variables are intervened, i.e., $(g, F) \in C \iff I(F) = \{ m - j : 0 \leq j < \|I(F)\| \}$. While this definition may seem quite restrictive, we prove that (perhaps surprisingly) any ILD can be transformed to an equivalent canonical ILD. Theorem 3 (Existence of Equivalent Canonical ILD). Given an ILD $(g, F)$, there exists a canonical ILD that is both counterfactually and distributionally equivalent to $(g, F)$ while maintaining the size of the intervention set, i.e., $ \forall (g, F), \exists (g', F') \in C \ s.t. \ (g', F') \simeq_{C,D} (g, F) $ and $\|I(F)\| = \|I(F')\|$. See Appendix B.7 for full proof and Example 1 in the appendix for a toy example. This result is helpful for theoretic analysis and, more importantly, it has great practical significance as now we can merely search over canonical ILD models. 3.3 Proposed ILD Estimation Algorithm Given the error bound in Theorem 2, the natural approach is to minimize the divergence between the observed domain distributions (represented by the training data) and the model’s induced distributions while constraining to $ k $ interventions. From Theorem 3, we can simply optimize over canonical ILD models without loss of generality. Therefore, we optimize the following constrained objective given a target intervention size $ k $: \[ \min_{g,F} \mathbb{E}_{p(x,d)}[-\log q_{g,F}(x,d)] \quad \text{s.t.} \quad [f_d]_{\leq m-k} = [f_{d'}]_{\leq m-k}, \forall d \neq d'. \] Concretely, the practical algorithm means training a normalizing flow for each domain while sharing most (but not all) parameters and enforcing autoregressiveness for part of the model. The non-shared domain-specific parameters correspond to the intervened variable(s). For higher dimensional data, we also relax the strict invertibility constraint and implement this design using VAEs. 4 Related Work Causal Representation Learning Causal representation learning is a rapidly developing field that aims to discover the underlying causal mechanisms that drive observed patterns in data and learn representations of data that are causally informative (Schölkopf et al., 2021). This is in contrast to traditional representation learning, which does not consider the causal relationships between variables. As this is a highly difficult task, most works make assumptions on the problem structure, such as access to atomic interventions, the graph structure (e.g., pure children assumptions), or model structure (e.g., linearity) (Yang et al., 2022; Huang et al., 2022; Xie et al., 2022; Squires et al., 2023; Zhang et al., 2023; Sturma et al., 2023; Jiang and Aragam, 2023; Liu et al., 2022a). Other works such as (Brehmer et al., 2022; Ahuja et al., 2022; Von Kügelgen et al., 2021) assume a weakly-supervised setting where one can train on counterfactual pairs $(x, \tilde{x})$ during training. In our work, we aim to maximize the practicality of our assumptions while still maintaining our theoretical goal of equivalent domain counterfactuals (as seen in Table 1). Counterfactual Generation A line of works focus on the identifiability of counterfactual queries (Shpitser and Pearl, 2008; Shah et al., 2022). For example, given knowledge of the ground-truth causal structure, Nasr-Esfahany et al. (2023) are able to recover the structural causal models up to equivalence. However, they do not consider the latent causal setting and assume some prior knowledge of underlying causal structures such as the backdoor criterion. There is a weaker form of counterfactual generation without explicit causal reasoning but instead using generative models Zhu et al. (2017); Nemirovsky et al. (2022). These typically involve training a generative model with a meaningful latent representation that can be intervened on to guide a counterfactual generation (Ilse et al., 2020). As these works do not directly incorporate causal learning in their frameworks, we consider them out of scope for this paper. Another branch of works estimate causal effect without trying to learn the underlying causal structure, which typically assume all variables are observable (Louizos et al., 2017). An expanded related work section is in Appendix F. 5 EXPERIMENTS We have shown theoretically the benefit of our canonical ILD characterization and restriction of intervention sparsity. In this section, we empirically test whether our theory could guide us to design better models for producing domain counterfactuals while only having access to observational data $ x $ and the corresponding domain label $ d $. In our simulated experiment, under the scenario where all of our modeling assumptions hold, we try to answer the following questions: (1) When we know the ground truth sparsity, does sparse canonical ILD lead to better domain counterfactual generation over naïve ML approaches (dense models)? (2) What would happen if there is a mismatch of sparsity between the dataset and modeling and what is a good model design strategy in practice? After this simulated experiment, we perform experiments on image datasets to determine if sparse canonical models are still advantageous in this more realistic setting. In this case, we assume the latent causal model lies in a lower dimensional space than the observed space and thus we use autoencoders to approximate an observation function that is invertible on a lower-dimensional manifold. 5.1 SIMULATED DATASET Experiment Setup To extensively address our questions against diverse causal mechanism settings, for each experiment, we generate 10 distinct ground truth ILDs. The ground truth latent SCM $ f^_d \in \mathcal{F}_{IA} $ takes the form $ f^_d(\epsilon) = F^_d \epsilon + b^_d I_T $ where $ F^_d = (I - L^_d)^{-1}, L^_d \in \mathbb{R}^{m \times m} $ is a domain-specific lower triangular matrix that satisfies the sparsity constraint, $ b^_d \in \mathbb{R} $ is a domain-specific bias, $ I_T $ is an indicator vector where entries corresponding to the intervention set are 1, and $ L^_d $ and $ b^_d $ are randomly generated for each experiment. The observation function takes the form $ g^(x) = G^ \text{LeakyReLU}(x) $ where $ G^* \in \mathbb{R}^{m \times m} $ and the slope of LeakyReLU is 0.5. We use maximum likelihood estimation to train two ILDs (like training of a normalizing flow): ILD-Can as introduced in Section 3.2 and a baseline model, ILD-Dense, which has no sparsity restrictions on its latent SCM. To evaluate the models, we compute the mean square error between the estimated counterfactual and ground truth counterfactual. More details on datasets and models, and illustrating figures of the models can be found in Appendix C.1. Result To answer whether sparse canonical ILD provides any benefit in domain counterfactual generation, we first look at the simplest case where the latent causal structure of the dataset and our model exactly match. In Figure 1a, we notice that when the growth truth intervention set $ T^* $ is \{5, 6\} (i.e. the last two nodes), ILD-Can significantly outperforms ILD-Dense. Then we create a few harder and more practical tasks where the intervention set size is still 2 but not constrained to the last few nodes. Again, in Figure 1a, we observe that no matter which two nodes are intervened on, ILD-Can performs much better than the naïve ML approach ILD-Dense. This first checks that restricting model structure to the specific canonical form does not harm the optimization even though the ground truth structure is different. Furthermore, it validates the benefit of our model design for domain counterfactual generation. More results with different number of domains and latent dimensions can be found in Appendix C.2, which all show that ILD-Can consistently perform better than ILD-Dense. We also include an illustrating figure visualizing how ILD-Can achieves lower counterfactual error. We then transition to the more practical scenario where the true sparsity $ \|T^\| $ is unknown. In Figure 1b, at first glance, we observe a trend of the decrease in counterfactual error as we decrease $ \|T\| $. For the case where $ \|T\| \geq \|T^\| $ (i.e. when $ \|T\| = 2, 3, 4 $), this aligns with our intuition that the smaller search space of ILD-Can leads to a higher chance of finding model with low counterfactual error. For the case where $ \|T\| = 1 $, we notice that it performs better than the canonical model that matches the true sparsity. Though it cannot reach distribution equivalence, the reduction in worst-case error (see Theorem 2) seems to be enough to enable comparable or better counterfactuals on average. We further check the performance of the data fitting and see a significant decrease in the fit of ILD-Can once $ \|T\| < \|T^\| $, which supports that the performance in data (a) With knowledge of $\|\mathcal{I}^\|$ and $\|\mathcal{I}^\| = \|\mathcal{I}\| = 2$. (b) Without knowledge of $\|\mathcal{I}^\|$ and $\mathcal{I}^* = \{5, 6\}$ Figure 1: Simulated experiment results ($N_d = 3$) averaged over 10 runs with different ground truth SCMs and the error bar represents the standard error. (a) This shows ILD-Can is consistently better than ILD-Dense regardless of intervened nodes in the dataset. (b) Here we test varying $\|\mathcal{I}\|$ while holding $\mathcal{I}^$ fixed. The performance of ILD-Can approaches to that of ILD-Dense as we increase $\|\mathcal{I}\|$. An unexpected result is that ILD-Can performs best when $\|\mathcal{I}\| = 1$ and that results from a worse data fitting which is more carefully investigated in Appendix C.2. fitting can be used as an indicator for whether we found the appropriate $\|\mathcal{I}\|$. Additional results on data fitting performance and experiments with different setups, including more complex $g$ based on normalizing flows and VAEs, can be found in Appendix C, and they all lead to the conclusion that ILD-Can produces better counterfactuals than ILD-Dense even though we do not know $\|\mathcal{I}^\|$. 5.2 IMAGE-BASED COUNTERFACTUAL EXPERIMENTS Here we seek to learn domain counterfactuals in the more realistic image regime. Following the manifold hypothesis (Gorban and Tyukin, 2018; Schölkopf et al., 2021), we assume that the causal interactions in this regime happen through lower-dimensional semantic latent factors as opposed to high-dimensional pixel-level interactions. To allow for learning of the lower dimensional latent space, we relax the invertibility constraint of our image-based ILD to only require pseudoinvertibility and test our models in this practical setting. High-dim ILD Modeling We modify the ILD models from Section 5.1 to fit a VAE (Kingma and Welling, 2013) structure where the variational encoder, $(g^+, F^+)$, first projects to the latent space via $g^+$ to produce the latent encoding $z$, which is then passed to two domain-specific latent causal models $f_{d,\mu}^+, f_{d,\sigma}^+$ which produce the parameters of posterior noise distribution. The decoder, $(g, F)$, follows the typical ILD structure: $g \circ f_d$, where, $g$ and $f_d$ can be viewed as pseudoinverse of $f_{d,\mu}^+$ and $g^+$. A detailed description and diagram of the models can be found in Figure 19, but informally, these modified ILD models can be seen as training a VAE per domain with the restriction that each VAE shares parameters for its initial encoder and final decoder layers (i.e. $g$ is shared). As an additional baseline, we compare against the naïve setup, which we call ILD-Independent, where each VAE has no shared parameters (i.e. a separate $g$ is learned for each domain). These models were trained using the $\beta$-VAE framework (Higgins et al., 2017). Further details can be found in the Appendix D.4. After training, we can perform domain counterfactuals as described in Section 2.2. Dataset We apply our methods to five image-based datasets: Rotated MNIST (RMNIST), Rotated FashionMNIST (RFMNIST) (Xiao et al., 2017), Colored Rotated MNIST (CRMNIST), 3D Shapes (Burgess and Kim, 2018) and Causal3DIdent (Von Kügelgen et al., 2021), which all have both domain information (e.g., the rotation of the MNIST digit) and class information (e.g., the digit number). For each dataset, we split the data into disjoint domains (e.g., each rotation in CRMNIST constitutes a different domain) and define class variables which are generated independently of domains (e.g., digit class in CRMNIST), to evaluate our model’s capability of generating domain counterfactuals. Specifically, for RMNIST, RFMNIST and 3D Shapes, all latent variables are independently generated, and for CRMNIST and Causal3DIdent, there is a more complicated causal graph containing the domain, class and other latent variables. Further details on each dataset and (assumed) ground-truth latent causal graphs could be found in Appendix D.1 and Appendix D.3. Metrics Inspired by the work in Monteiro et al. (2023), we evaluate the image-based counterfactuals with latent SCMs via the following metrics, where $h_{domain}$ and $h_{class}$ represents pretrained domain classifier and class classifier respectively: (1) Effectiveness - whether the counterfactual truly changes the domain defined as $\mathbb{P}(h_{domain}(\hat{x}_{d \rightarrow d'}) = d')$; (2) Preservation - whether the domain counterfactual only changes domain-specific information defined as $\mathbb{P}(h_{class}(\hat{x}_{d \rightarrow d'}) = y)$; (3) Composition - whether the counterfactual model is invertible defined as $\mathbb{P}(h_{class}(\hat{x}_{d \rightarrow d}) = y)$; and (4) Table 2: Quantitative result for Composition (Comp.), Reversibility (Rev.), Preservation (Pre.), and Effectiveness (Eff.), where higher is better. CRMNIST, 3D Shapes, Causal3DIdent are averaged 20, 5, 10 runs respectively. Best models are bold (within 1 standard deviation) and due to space constraints, expanded tables with additional datasets and standard deviation are in Appendix D.5. \| \| CRMNIST \| \| \| \| 3D Shapes \| \| \| \| Causal3DIdent \| \|----------------\|---------\|----------\|----------\|----------\|-----------\|----------\|----------\|----------\|--------------\| \| \| Comp. \| Rev. \| Eff. \| Pre. \| Comp. \| Rev. \| Eff. \| Pre. \| Comp. \| \| ILD-Independent \| 87.24 \| 59.38 \| 64.65 \| 60.39 \| 99.79 \| 32.36 \| 94.07 \| 32.49 \| 88.15 \| \| ILD-Dense \| 88.18 \| 62.29 \| 62.72 \| 59.60 \| 99.76 \| 32.60 \| 80.92 \| 32.64 \| 83.59 \| \| ILD-Can \| 92.10 \| 85.74 \| 94.48 \| 72.95 \| 99.85 \| 70.84 \| 96.72 \| 64.99 \| 86.00 \| Figure 2: Domain counterfactuals with 3D Shapes and CausalIdent. Expanded figures can be found in Appendix D.5 (a) For 3D Shapes, only the object shape should change with domain counterfactuals – the other latent factors such as the hue of object, floor, background, should not change. (b) For CausalIdent, as the domain changes, the color of the background should change while holding all else unchanged. ILD-Can clearly performs better than the baseline ILD-Dense in terms of preserving non-domain features while changing domains for all datasets. Reversibility - whether the counterfactual model is cycle-consistent defined as $ P(h_{\text{class}}(\hat{x}_{d \rightarrow d'}) = y) $. For example, in the case of CRMNIST, a model might be able to rotate the image but cannot preserve the digit class during rotation, which would be high in effectiveness but low in preservation score. Details on the computation of these metrics and causal interpretations can be found in Appendix D.2 and Appendix D.3 respectively. Result Due to space constraints, we put all results with RMNIST and RFMNIST in Appendix D.5. In Figure 2 we can see examples of domain counterfactuals for both ILD-Dense and ILD-Can. We note that no latent information other than the domain label was seen during training, thus suggesting the intervention sparsity is what allowed the canonical models to preserve important non-domain-specific information such as class information when generating domain counterfactuals. In Table 2, we include quantitative results using our metrics, which shows ILD-Can having significantly better reversibility and preservation while maintaining similar levels of counterfactual effectiveness and composition than the non-sparse counterparts. In Appendix D.5, we further investigate our model’s sensitivity to the choice of sparsity by tracking how each metric change w.r.t. $ \|\mathcal{I}\| $. We observe that reversibility and preservation tends to decrease while effectiveness tends to increase as we increase $ \|\mathcal{I}\| $, which aligns with our findings here as ILD-Dense is equivalent to making $ \mathcal{I} $ contain all latent nodes. In summary, our results here indicate our theory-inspired model design leads to better domain counterfactual generation in the practical pseudo-invertible setting. 6 CONCLUSION In this paper, we show that estimating domain counterfactuals given only i.i.d. data from each domain is feasible without recovering the latent causal structure. We theoretically analyzed the DCF problem for a particular invertible causal model class and proved a bound on estimation error that depends on both a data fit term and an intervention sparsity term. 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zz61V8bIab	The aim of equation (3) is to ensure that the shared Feature Extractor F_s exactly extract the domain-invariant features. Thus the author maximum this loss to let the discriminator D be confused about the features coming from F_s. Here is the question: discriminator D may lack of capabilities to recognize the difference among domains as this loss function does not involve any domain knowledge.	Stochastic Adversarial Network for Multi-Domain Text Classification Anonymous authors Paper under double-blind review Abstract Adversarial training has played a pivotal role in the significant advancements of multi-domain text classification (MDTC). Recent MDTC methods often adopt the shared-private paradigm, wherein a shared feature extractor captures domain-invariant knowledge, while private feature extractors per domain extract domain-dependent knowledge. These approaches have demonstrated state-of-the-art performance. However, a major challenge remains: the exponential increase in model parameters as new domains emerge. To address this challenge, we propose the Stochastic Adversarial Network (SAN), which models multiple domain-specific feature extractors as a multivariate Gaussian distribution rather than weight vectors. With SAN, we can sample as many domain-specific feature extractors as necessary without drastically increasing the number of model parameters. Consequently, the model size of SAN remains comparable to having a single domain-specific feature extractor when data from multiple domains. Additionally, we incorporate domain label smoothing and robust pseudo-label regularization techniques to enhance the stability of the adversarial training and improve feature discriminability, respectively. The evaluations conducted on two prominent MDTC benchmarks validate the competitiveness of our proposed SAN method against state-of-the-art approaches. 1 Introduction Text classification has garnered considerable attention within Natural Language Processing (NLP) [Khurana et al., 2023]. Over the past decade, deep learning has propelled text classification forward, albeit at the expense of requiring extensive labeled data [Kowsari et al., 2019]. However, it is widely acknowledged that text classification is highly dependent on the specific domain. In other words, the same word can convey different sentiments across different domains [Wu et al., 2022b]. This can easily result in a model trained in one domain easily performing poorly when applied to another domain. Unfortunately, collecting a substantial amount of labeled data for each desired domain is often prohibitively expensive and unrealistic. Thus, it becomes crucial to investigate approaches for leveraging knowledge from related domains to enhance the classification accuracy in the target domain. Multi-domain text classification (MDTC) is proposed to address the problem stated above [Li & Zong, 2008]. Earlier MDTC methods employed a per-domain training approach and utilized ensemble learning strategies to generate final results [Li et al., 2012; Wu & Huang, 2015]. The most recent MDTC approaches can yield state-of-the-art performance by adopting adversarial training [Creswell et al., 2018; Ganin et al., 2016] and the shared-private scheme [Bousmalis et al., 2016b]. Adversarial training aligns different domains to extract domain-invariant features, while the shared-private scheme partitions the latent space into a shared component that captures common features across domains, and multiple domain-specific feature spaces that capture domain-unique features. The domain-invariant features are expected to be both discriminative and transferable, whereas the domain-specific features enhance the discriminability of the domain-invariant features [Bousmalis et al., 2016a]. However, these approaches face a challenge: the shared-private paradigm requires training domain-specific feature extractors for each domain, which often involves complex neural network architectures. As new domains emerge, incorporating numerous domain-specific feature extractors not only increases the number of model parameters (as depicted in Table 1), but also hampers training convergence. Table 1: The number of parameters of the shared feature extractor $F_s$, the domain-specific feature extractors $\{F_{di}\}_{i=1}^M$, the classifier $C$, and the domain discriminator $D$ in MAN (Chen & Cardie, 2018) on different tasks. Obviously, the domain-specific feature extractor parameters take the majority part in both tasks, demonstrating that tackling data from more domains in MDTC will drastically increase the model size. \| Task \| Amazon \| FDU-MTL \| \|-----------------------\|----------\|---------\| \| # Para. of $F_s$ \| 5.57M \| 20.20M \| \| # Para. of $\{F_{di}\}_{i=1}^M$ \| 22.13M=45.57M \| 322.65M=1620.20M \| \| # Para. of $C$ \| 0.04M \| 0.04M \| \| # Para. of $D$ \| 0.02M \| 0.02M \| \| # Total Para. \| 27.76M \| 342.91M \| To mitigate the aforementioned issue, we propose a novel approach called Stochastic Adversarial Network (SAN) that introduces a stochastic feature extractor to replace multiple domain-specific feature extractors. The stochastic feature extractor seamlessly integrates an infinite number of domain-specific feature extractors into existing MDTC methods, while keeping the model parameters unchanged. In SAN, instead of specific weight points used in previous MDTC approaches, the domain-specific feature extractors are represented by a weight distribution. Specifically, we model the domain-specific feature extractors using a Gaussian distribution, with the mean representing the final domain-specific feature extractor weight and the variance capturing the discrepancy among different domains. During training, the domain-specific feature extractor is sampled from the current distribution estimation, and the Gaussian distribution is optimized throughout the training. Consequently, the SAN model can extract domain-specific features across multiple domains using only one domain-specific feature extractor. Notably, this is achieved without the need to consider the number of required domain-specific feature extractors, while avoiding the negative impact of increasing the model size. To further enhance model performance, we incorporate domain label smoothing and robust pseudo-label regularization into the SAN method, ensuring stability in the adversarial training and improving feature discriminability. Through experiments conducted on two MDTC benchmarks, we demonstrate the effectiveness of our SAN approach, achieving competitive performance compared to state-of-the-art methods. Our contributions are summarized as follows: - We propose the Stochastic Adversarial Network (SAN) for MDTC, introducing a stochastic feature extractor mechanism. This enables MDTC models to learn domain-specific features from multiple domains using a single domain-specific feature extractor, thereby substantially reducing the number of model parameters. To the best of our knowledge, this study represents the first exploration of this matter in MDTC. - We incorporate domain label smoothing and robust pseudo-label regularization techniques to stabilize the adversarial training and enhance the discriminability of the acquired features, respectively. - The experimental results on two benchmarks illustrate the efficacy of the SAN method in comparison to state-of-the-art approaches. Additionally, we perform extensive experiments on multi-source unsupervised domain adaptation to highlight the generalization ability of our proposed SAN approach. 2 RELATED WORK Adversarial Training (AT). AT, initially introduced by the Generative Adversarial Network (GAN) (Creswell et al., 2018) for image generation, involves a generator synthesizing images and a discriminator distinguishing between generated and real images. Domain-Adversarial Neural Networks (DANN) (Ganin et al., 2016) apply AT to domain adaptation by training a feature extractor against a domain discriminator. The domain discriminator aims to distinguish source and target features, while the feature extractor aims to deceive the domain discriminator, generating domain-invariant features when the discriminator cannot discern the feature source. Conditional Adversarial Neural Networks (CDANs) (Long et al., 2018) employ multilinear conditioning to align conditional distributions and incorporate entropy conditioning to facilitate transfer learning. However, AT often exhibits oscillatory gradients during training, resulting in instability, slow convergence, and mode collapse (Arjovsky & Bottou, 2017; Mescheder et al., 2018). To overcome these limitations, Wasserstein GAN (Arjovsky et al., 2017) employs the earth mover distance to measure domain divergence. Additionally, Environment Label Smoothing (ELS) (Zhang et al., 2023) encourages the domain discriminator to output soft probabilities, enhancing the stability of AT. Stochastic Neural Network (SNN). The weight parameters of a neural network are typically treated as point estimates, limiting their ability to capture uncertainty and often resulting in overconfident predictions (Blundell et al., 2015). To address this limitation, SNNs are proposed, which consider weight parameters as random variables sampled from specific distributions. For example, Bayesian Neural Networks (BNNs) (Hernández-Lobato & Adams, 2015; Wang & Yeung, 2020) are widely used to represent intermediate outputs and final predictions as stochastic variables, providing richer representations. The Auto-Encoding Variational Bayes (AEVB) (Kingma & Welling, 2013) employs a Gaussian distribution to model latent variables in image inputs, serving as a form of data augmentation. Uncertainty-aware multi-modal BNNs (Subedar et al., 2019) combine deterministic and variational layers for activity recognition, while DistributionNet (Yu et al., 2019) models feature uncertainty in person re-identification using distributions. In unsupervised domain adaptation, the Stochastic Classifier (Lu et al., 2020) leverages a Gaussian distribution to model classifier parameters. Multi-domain text classifications (MDTC). MDTC aims to enhance overall classification accuracy by harnessing available resources from multiple domains (Li & Zong, 2008). Early MDTC methods employ transfer learning techniques to drive progress. The structural correspondence learning (SCL) (Blitzer et al., 2006) method computes relationships between different pivot features to learn correspondences among them. The collaborative multi-domain sentiment classification (CMSC) (Wu & Huang, 2015) method trains two types of classifiers: a shared classifier for all domains and a set of domain-specific classifiers for each domain, combining their outputs for final results. Recent MDTC approaches commonly adopt the adversarial training and shared-private paradigm, leading to significant advancements. The domain separation network (DSN) (Bousmalis et al., 2016a) first introduces the shared-private paradigm for adversarial domain adaptation and empirically demonstrates that domain-unique features can enhance the discriminability of domain-invariant features. The adversarial multi-task learning (ASP-MTL) method (Liu et al., 2017) applies adversarial training and the shared-private paradigm to MDTC. The multinomial adversarial networks (MANs) (Chen & Cardie, 2018) utilize the least square loss and negative log-likelihood loss to train the domain discriminator. The mixup regularized adversarial networks (MRANs) (Wu et al., 2021b) propose domain and category mixup regularizers for MDTC. The maximum batch Frobenius norm (MBF) (Wu et al., 2022b) method improves feature discriminability by maximizing the Frobenius norm of the intermediate feature matrix. In contrast to previous MDTC approaches that utilize separate domain-specific feature extractors for each domain, our proposed SAN method employs parameter sampling from a Gaussian distribution to model the domain-specific feature extractor. This approach enables the SAN method to acquire domain-specific knowledge through a single feature extractor, resulting in a significant reduction in the number of model parameters needed. ### 3 Method The MDTC task can be formulated as follows: given $M$ domains $\{D_i\}_{i=1}^M$, each domain contains a small amount of labeled data $L_i = \{x_j, y_j\}_{j=1}^{l_i}$ and a large amount of unlabeled data $U_i = \{x_j\}_{j=1}^{u_i}$. The primary objective of MDTC is to leverage these resources to enhance the average classification accuracy across all domains. #### 3.1 Adversarial Multi-Domain Text Classification Adversarial training has proven to be effective in mitigating domain discrepancies and has found widespread application in MDTC (Chen & Cardie, 2018; Wu & Guo, 2020; Wu et al., 2022b). Traditional adversarial MDTC models typically comprise four components: (1) a shared feature extractor $F_s$, (2) a collection of domain-specific feature extractors $\{F_{d_i}\}_{i=1}^M$, (3) a classifier $C$, and (4) a domain discriminator $D$. The objective of $F_s$ is to learn domain-invariant features capable of generalizing across diverse domains, while $\{F_d^i\}_{i=1}^M$ are designed to capture domain-unique features advantageous within their respective domains. $C$ serves as a binary classifier for sentiment prediction, while $D$ acts as an M-way classifier for domain identification. The feature extractors can adopt various neural network architectures, such as convolutional neural networks (CNNs) (Zhang et al., 2015), multi-layer perceptrons (MLPs) (Chen & Cardie, 2018), and transformers (Vaswani et al., 2017), to generate fixed-length feature representations. $D$ takes the shared feature vector as input, while $C$ takes the concatenation of the shared feature vector and the domain-specific feature vector. In conventional MDTC approaches, two primary objectives must be achieved: (1) minimizing the classification loss on labeled data, and (2) optimizing the adversarial loss on both labeled and unlabeled data. These objectives can be formulated as follows: $$\min_{F_s,\{F_d^i\}_{i=1}^M,C} \max_D J_C(F_s,\{F_d^i\}_{i=1}^M,C) + \lambda J_D(F_s,D)$$ $$J_C(F_s,\{F_d^i\}_{i=1}^M,C) = \sum_{i=1}^M \mathbb{E}_{(x,y)\sim L_i}[\mathcal{L}(C[F_s(x), F_d^i(x)], y)]$$ $$J_D(F_s,D) = \sum_{i=1}^M \mathbb{E}_{x\sim L_i\cup U_i}[\mathcal{L}(D(F_s(x)), d)]$$ Where $\mathcal{L}(\cdot,\cdot)$ is the canonical classification loss, $[\cdot,\cdot]$ represents the concatenation of two vectors, and $d$ is the ground-truth domain label of the corresponding instance $x$. 3.2 Stochastic Adversarial Network Given that feature extractors typically employ intricate neural network architectures to capture valuable information from input data, and MDTC models necessitate training domain-specific feature extractors for each domain, this approach leads to a significant increase in model parameter count and a slowdown in convergence speed. To overcome this problem, we propose the stochastic adversarial network (SAN) for MDTC, which introduces a stochastic feature extractor to replace multiple domain-specific feature extractors without compromising model performance. The architecture of our proposed SAN method is depicted in Figure 1. The fundamental concept behind our approach is to model a distribution of domain-specific feature extractors, where the domain-specific feature extractors utilized to learn domain-unique features are simply random samples drawn from this distribution. This design permits access to an infinite number of domain-specific feature extractors, as we can sample any desired quantity of them. Furthermore, it decouples the number of domain-specific feature extractors from the model parameter count, ensuring that the model size remains unchanged as new domains emerge. More specifically, we employ a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ represents the mean vector and $\Sigma$ corresponds to the diagonal covariance matrix. The parameters of the domain-specific feature extractors for each domain can be randomly drawn from $\mathcal{N}(\mu, \Sigma)$. The resulting loss is then back-propagated to update the learnable parameters $\mu$ and $\Sigma$. It is important to note that inferring and training a neural network that models the domain-specific feature extractor as a weight distribution pose significant challenges. The random sampling process impedes conventional end-to-end training. To overcome this, we utilize the reparameterization technique (Kingma & Welling, 2013) to train the multivariate Gaussian representing the domain-specific feature extractor distribution. This enables the SAN model to be trained effectively using backpropagation. Specifically, we express the last fully connected layer of the domain-specific feature extractor as $\phi_d$, which is obtained as $\phi_d = \mu + \sigma \odot \epsilon$, where $\epsilon$ is an independent sample drawn from a standard Gaussian, $\odot$ denotes element-wise multiplication, and $\sigma$ represents the diagonal elements of $\Sigma$. By adopting the stochastic domain-specific feature extractor, we can update Eq. 2 as: $$J_C(F_s, F_d, C) = \sum_{i=1}^{M} \mathbb{E}_{(x,y) \sim L_i} [L(C[F_s(x), F_d(x)], y)]$$ Utilizing the stochastic feature extractor in our SAN method enables us to achieve competitive outcomes when compared to multinomial adversarial networks (Chen & Cardie, 2018) (As shown in Sec. D.2), while also significantly reducing the number of model parameters (As shown in Sec. D.5). In our study, we represent the stochastic domain-specific feature extractor as $F_d$. To enhance the performance of our model further, we integrate domain label smoothing (Zhang et al., 2023) and robust pseudo-label regularization (Gu et al., 2020). These additions serve to stabilize the adversarial training and enhance the discriminability of features, respectively. ### 3.3 Enhancement via Domain Label Smoothing Despite AT has been empirically proven effective in minimizing domain divergence and capturing domain-invariant features (Ganin et al., 2016; Chen & Cardie, 2018), it is widely acknowledged that AT is challenging to train and converge (Roth et al., 2017; Jenni & Favaro, 2019; Arjovsky & Bottou, 2017). This difficulty arises from the use of one-hot domain labels in AT, which leads to highly over-confident output probabilities. Consequently, the over-confidence of the domain discriminator can result in significant oscillatory gradients (Arjovsky & Bottou, 2017; Mescheder et al., 2018), negatively impacting training stability. To address this issue, we incorporate the domain label smoothing (DLS) technique, which encourages the domain discriminator to estimate soft probabilities instead of relying on confident classifications (Zhang et al., 2023). DLS achieves this by employing a weighted soft-encoding approach to represent domain labels (as depicted in Figure 2). The DLS formulation is as follows: $$J_D^{dls}(F_s, D) = \sum_{i=1}^{M} \mathbb{E}_{x \sim L_i \cup U_i} [\gamma \log(D_i(F_s(x))) + \frac{1 - \gamma}{M - 1} \sum_{j=1, j \neq i}^{M} \log(D_j(F_s(x)))]$$ Where $D_i$ gives the $i$-th dimension of the domain discriminator’s output vector and $\gamma$ ($\gamma \in (0, 1)$) is a hyperparameter. The DLS is theoretically and empirically demonstrated to be capable of improving robustness to noisy domain labels, converging faster, attaining more stable training, and better generalization performance without extra parameters and optimization steps. With Eq. [4] and Eq. [5], the overall training objective can be updated as: $$\min_{F_s,F_d,C} \max_{D} J_C(F_s,F_d,C) + \lambda J_{DLs}(F_s,D)$$ (6) ### 3.4 Enhancement via Robust Pseudo-Label Regularization In MDTC, a considerable portion of each domain consists of unlabeled data, making it intuitive to leverage pseudo-labels, i.e., estimated labels of unlabeled data, to enhance feature discriminability. Nevertheless, since unlabeled data lack supervision, their pseudo-labels inevitably contain noise. To effectively select unlabeled data capable of generating reliable pseudo-labels and thereby improving feature discriminability, we integrate the robust pseudo-label regularization (RPLR) technique (Gu et al., 2020) into our proposed SAN method. The RPLR approach assesses the correctness of pseudo-labels for unlabeled data based on the feature distance to the corresponding class center in a spherical feature space. It treats incorrectly labeled data as outliers and models the conditional probability of outliers/inliers using a Gaussian-uniform mixture model. Specifically, $\hat{y}_j^u$ represents the generated pseudo-label for the input instance $x_j^u$: $\hat{y}_j^u = \arg\max_k[C[F_s(x_j^u), F_d(x_j^u)]_k]$, where $[\cdot]_k$ denotes the $k$-th element. To model the fidelity of the generated pseudo-label, a random variable $z_j \in \{0, 1\}$ is introduced, indicating whether the data is correctly or incorrectly labeled with values of 1 and 0, respectively. Consequently, RPLR is formulated as follows: $$J_{rplr}^{C}(F_s,F_d,C,\phi) = \sum_{i=1}^{M} E_{x_j^u \sim U_i}[w(x_j^u)L(C[F_s(x_j^u), F_d(x_j^u)], \hat{y}_j^u)]$$ (7) $$w(x_j^u) = \begin{cases} \beta_j & \text{if } \beta_j > 0.5 \\ 0 & \text{otherwise} \end{cases}$$ (8) Where $\beta_j$ represents the probability of correctly labeled data, i.e., $\beta_j = Pr(z_j = 1\|x_j^u, \hat{y}_j^u)$. In this manner, unlabeled data with a probability of correct labeling below 0.5 are discarded. The posterior probability of correct labeling, i.e., $Pr(z_j = 1\|X_j^u, \hat{y}_j^u)$, is modeled by the feature distance between the data and the class center to which it belongs, using a Gaussian-uniform mixture model based on pseudo-labels. Given a feature vector $f_j^u = [F_s(x_j^u), F_d(x_j^u)]$ of an unlabeled instance $x_j^u$, its distance to the corresponding class center $C_{\hat{y}_j^u}$ for category $\hat{y}_j^u$ is calculated as: $$d_j^u = \frac{f_j^u \cdot C_{\hat{y}_j^u}}{\\|f_j^u\\| \\|C_{\hat{y}_j^u}\\|}$$ (9) The class center $C_{\hat{y}_j^u}$ is defined in a spherical space as presented in (Gu et al., 2020), the details of computing $C_{\hat{y}_j^u}$ are available in the Appendix. The distribution of feature distance $d_j^u$ is modeled by the Gaussian-uniform mixture model, a statistical distribution considering outliers (Coretto & Henning, 2016; Lathuilière et al., 2018). $$p(d_j^u\|\hat{y}_j^u) = \pi_{\hat{y}_j^u} N^+(d_j^u\|0, \sigma_{\hat{y}_j^u}) + (1 - \pi_{\hat{y}_j^u})U(0, \delta_{\hat{y}_j^u})$$ (10) Where $N^+(d_j^u\|0, \sigma)$ denotes a density function that is proportional to Gaussian distribution when $d_j^u \geq 0$, otherwise the density is zero. $U(0, \delta_{\hat{y}_j^u})$ is uniform distribution defined on $[0, \delta_{\hat{y}_j^u}]$. Specifically, the Gaussian component captures the underlying probability distribution of correctly labeled data, while the uniform component provides a robust representation of the distribution for incorrectly labeled data. With equation [10], the posterior probability of correct labeling for unlabeled data $x_j^u$ is defined: \[ \beta_j = \frac{\pi_{\hat{y}_j^u} N^+(d_j^u \| 0, \sigma_{\hat{y}_j^u})}{p(d_j^u \| \hat{y}_j^u)} \] (11) The parameters of Gaussian-uniform mixture models are $ \phi = \{ \pi_k, \sigma_k, \delta_k \}_{k=1}^K $ where $ K $ is the number of classes. The details of approximating these parameters will be given in Sec. 3.5. In summary, the ultimate optimization objective is defined as: \[ \min_{F_s, F_d, C} \max_D J_C(F_s, F_d, C) + \lambda J_{Dls}(F_s, D) + \lambda_{rplr} J_{Cprlr}(F_s, F_d, C, \phi) \] (12) ### 3.5 Training Procedure In this section, we present how to optimize each component in the SAN model and estimate the parameters $ \phi $ of Gaussian-uniform mixture models. To optimize the ultimate object in Eq. [12], we alternatively optimize the networks and estimate parameters $ \phi $ by fixing other components following (Gu et al., 2020). We first initialize $ F_s, F_d, C, D $ with Eq. [6] via training strategies as in (Chen & Cardie, 2018), then we take the following two steps to make the optimization. #### (1) Estimating $ \phi $ with fixed $ F_s, F_d, C, D $. Fixing the parameters of $ F_s, F_d, C, D $, we generate the pseudo-label $ \hat{y}_j^u $ and calculate the distance $ d_j^u $ for all unlabeled data, then $ \phi $ is estimated using EM algorithm as below. Let $ \tilde{d}_j^u = (-1)^m_j d_j^u $, where $ m_j $ is sampled from Bernoulli distribution $ B(1, 0.5) $, and $ N_u $ denotes the number of unlabeled data, then $ \phi $ can be estimated as follows: \[ \beta_j^{l+1} = \frac{\pi_{\hat{y}_j^u} N(\tilde{d}_j^u \| 0, \sigma_{\hat{y}_j^u})}{\pi_{\hat{y}_j^u} N(d_j^u \| 0, \sigma_{\hat{y}_j^u}) + (1 - \pi_{\hat{y}_j^u}) U(-\delta_{\hat{y}_j^u}, \delta_{\hat{y}_j^u})} \] \[ \pi_k^{l+1} = \frac{1}{\sum_{j=1}^{N_u} I(\hat{y}_j^u = k)} \sum_{j=1}^{N_u} I(\hat{y}_j^u = k) \beta_j^{l+1} \] \[ \sigma_j^{l+1} = \frac{\sum_{j=1}^{N_u} I(\hat{y}_j^u = k) \beta_j^{l+1} (\tilde{d}_j^u)^2}{\sum_{j=1}^{N_u} I(\hat{y}_j^u = k) \beta_j^{l+1}}, \quad \delta_k^{l+1} = \sqrt{3(q_2 - q_1)} \] Where \[ q_1 = \frac{1}{\sum_{j=1}^{N_u} I(\hat{y}_j^u = k) \beta_j^{l+1}} \sum_{j=1}^{N_u} \frac{1 - \beta_j^{l+1}}{1 - \pi_k^{l+1}} I(\hat{y}_j^u = k) \tilde{d}_j^u \] \[ q_2 = \frac{1}{\sum_{j=1}^{N_u} I(\hat{y}_j^u = k) \beta_j^{l+1}} \sum_{j=1}^{N_u} \frac{1 - \beta_j^{l+1}}{1 - \pi_k^{l+1}} I(\hat{y}_j^u = k) (\tilde{d}_j^u)^2 \] We refer our readers to Gu et al. (2020) for the deduction details of the parameters $ \phi $. #### (2) Optimizing $ F_s, F_d, C, D $ with fixed $ \phi $. Given current pseudo-labels and estimated $ \phi $, we follow the standard MDTC training protocol (Chen & Cardie, 2018) to train $ F_s, F_d, C, D $ with Eq. [12]. ### 4 Experiment #### 4.1 Setup Datasets. We conducted experiments on two benchmark datasets for MDTC: the Amazon review dataset (Blitzer et al., 2007) and the FDU-MTL dataset (Liu et al., 2017). The Amazon review dataset comprises four domains: books, DVDs, electronics, and kitchen. Each domain consists of 2000 labeled data instances, with 1000 positive and 1000 negative examples. The data has been pre-processed into a bag-of-features representation, which includes unigrams and bigrams, without preserving word order information. The FDU-MTL dataset reflects real-world scenarios and contains raw text data. It encompasses 14 product review domains, including books, electronics, DVDs, kitchen, apparel, camera, health, music, toys, video, baby, magazine, software, sport, as well as two movie review domains: IMDB and MR. Each domain includes a validation set of 200 samples and a test set of 400 samples. The training and unlabeled sets vary in size across domains, but generally consist of approximately 1400 and 2000 instances, respectively. Implementation details. To ensure a fair comparison, we adopt identical network architectures as presented in MAN (Chen & Cardie [2018]). It is worth noting that we only replace the last fully connected layer of the domain-specific feature extractor with a stochastic layer. For the Amazon review dataset, we select the 5000 most frequent features and represent each review as a 5000-dimensional vector, where the feature values represent raw counts. Our feature extractors employ multi-layer perceptrons (MLPs) with an input size of 5000. Each feature extractor consists of two hidden layers with sizes of 1000 and 500, respectively. In the case of the FDU-MTL dataset, we employ a single-layer convolutional neural network (CNN) as the feature extractor. The CNN utilizes different kernel sizes (3, 4, 5) with a total of 200 kernels. The input to the CNN is a 100-dimensional embedding obtained by processing each word of the input sequence using word2vec (Mikolov et al. [2013]). For all experiments, we set the batch size to 8, the dropout rate for each component to 0.4, and the learning rate of the Adam optimizer (Kingma & Ba [2014]) to 0.0001. The size of the shared features is set to 128, and the size of the domain-specific features is set to 64. Both the classifier and discriminator are MLPs with hidden layer sizes matching their respective inputs (128+64 for the classifier and 128 for the domain discriminator). Furthermore, we set the hyperparameters $\lambda$ to 0.0001, $\gamma$ to 0.9, and $\lambda_{rpr}$ to 1. Comparison methods. In the MDTC tasks, we evaluate the SAN method against several state-of-the-art methods: The multi-task convolutional neural network (MT-CNN) (Collobert & Weston [2008]), the muti-task deep neural network (MT-DNN) (Liu et al. [2015]), the collaborative multi-domain sentiment classification method (CMSC) trained with the least square loss (CMSC-LS), the hinge loss (CMSC-SVM), and the log loss (CMSC-Log) (Wu & Huang [2015]), the pre-trained BERT-base model fine-tuned on each domain (BERT) (Devlin et al. [2018]), the adversarial multi-task learning for text classification method (ASP-MTL) (Liu et al. [2017]), the multinomial adversarial network (MAN) trained with the least square loss (MAN-L2) and the negative log-likelihood loss (MAN-NLL) (Chen & Cardie [2018]), the dynamic attentional sentence encoding method (DA-MTL) (Zheng et al. [2018]), the global and local shared representation-based dual-channel multi-task learning method (GLR-MTL) (Su et al. [2020]), the conditional adversarial network (CAN) (Wu et al. [2021a]), the co-regularized adversarial learning method (Wu et al. [2022a]). For MS-UDA experiments, the baselines involve the marginalized denoising autoencoder (mSDA) (Chen et al. [2012]), the domain adversarial neural network (Ganin et al. [2016]), the multi-source domain adaptation network (MDAN) (Wu et al. [2021b]), the MAN (MAN-L2 and MAN-NLL) (Chen & Cardie [2018]), the CAN (Wu et al. [2021a]) and CRAL (Wu et al. [2022a]). \| Domain \| CMSC-LS \| CMSC-SVM \| CMSC-Log \| MAN-L2 \| MAN-NLL \| CAN \| CRAL \| SAN(ours) \| \|--------\|---------\|----------\|----------\|--------\|---------\|-----\|------\|-----------\| \| Books \| 82.10 \| 82.26 \| 81.81 \| 82.46 \| 82.98 \| 83.76\| 85.26\| 86.29 ± 0.26 \| \| DVD \| 82.40 \| 83.48 \| 83.73 \| 83.98 \| 84.03 \| 84.68\| 85.83\| 86.43 ± 0.38 \| \| Electr.\| 86.12 \| 86.76 \| 86.67 \| 87.22 \| 87.06 \| 88.34\| 89.32\| 89.78 ± 0.12 \| \| Kit. \| 87.56 \| 88.20 \| 88.23 \| 88.53 \| 88.57 \| 90.03\| 91.60\| 91.31 ± 0.15 \| \| AVG \| 84.55 \| 85.18 \| 85.11 \| 85.55 \| 85.66 \| 86.70\| 88.00\| 88.45 ± 0.08 \| ### 4.2 Result Multi-Domain Text Classification. The experimental results on the Amazon review dataset and FDU-MTL dataset are reported in Table 2 and Table 3 respectively. We report the classification results of mean ± variance over five random runs. From Table 2, it can be noted that the SAN method obtains the best classification accuracy on 3 out of 4 domains, and yield state-of-the-art results for the average classification accuracy. For the experimental results on FDU-MTL, shown in Table 3, the Table 3: MDTC results on the FDU-MTL dataset \| Domain \| MT-CNN \| MT-DNN \| ASP-MTL \| BERT \| MAN-L2 \| MAN-NLL \| DA-MTL \| GLR-MTL \| SAN(Ours) \| \|------------\|--------\|--------\|---------\|------\|--------\|---------\|--------\|---------\|-----------\| \| books \| 84.5 \| 82.2 \| 84.0 \| 87.0 \| 87.6 \| 86.8 \| 88.5 \| 88.3 \| 90.5 ± 0.3 \| \| electronics\| 83.2 \| 88.3 \| 86.8 \| 88.3 \| 87.4 \| 88.8 \| 89.0 \| 90.3 \| 87.7±0.6 \| \| dvd \| 84.0 \| 84.2 \| 85.5 \| 85.6 \| 88.1 \| 88.6 \| 88.0 \| 87.3 \| 89.7 ± 0.5 \| \| kitchen \| 83.2 \| 80.7 \| 86.2 \| 91.0 \| 89.8 \| 89.9 \| 89.9 \| 89.8 \| 90.4±0.9 \| \| apparel \| 83.0 \| 85.0 \| 87.0 \| 90.0 \| 87.6 \| 88.5 \| 88.8 \| 88.2 \| 87.4±0.7 \| \| camera \| 86.0 \| 86.2 \| 89.2 \| 90.0 \| 91.4 \| 90.7 \| 91.8 \| 89.5 \| 91.1±0.6 \| \| health \| 87.2 \| 85.7 \| 88.2 \| 88.3 \| 89.8 \| 89.4 \| 90.3 \| 90.5 \| 90.3±0.3 \| \| music \| 83.7 \| 84.7 \| 82.5 \| 86.8 \| 85.9 \| 85.5 \| 85.0 \| 87.5 \| 85.9±0.8 \| \| toys \| 89.2 \| 87.7 \| 88.3 \| 91.0 \| 90.0 \| 90.1 \| 89.5 \| 89.5 \| 90.2±0.7 \| \| video \| 81.5 \| 85.0 \| 84.5 \| 88.0 \| 89.5 \| 89.6 \| 89.5 \| 90.8 \| 90.0±0.5 \| \| baby \| 87.7 \| 88.0 \| 88.2 \| 91.5 \| 90.0 \| 90.2 \| 90.5 \| 92.3 \| 90.7±0.8 \| \| magazine \| 87.7 \| 89.5 \| 92.2 \| 92.8 \| 92.5 \| 92.9 \| 92.0 \| 92.3 \| 92.3±0.1 \| \| software \| 85.4 \| 85.7 \| 87.2 \| 89.2 \| 90.4 \| 90.1 \| 90.8 \| 89.8 \| 89.5±0.4 \| \| sports \| 84.0 \| 83.2 \| 85.7 \| 90.8 \| 89.0 \| 89.0 \| 89.8 \| 87.8 \| 90.0 ± 0.2 \| \| IMDb \| 86.2 \| 83.2 \| 85.5 \| 85.8 \| 86.6 \| 87.0 \| 89.8 \| 87.5 \| 89.3±0.7 \| \| MR \| 74.5 \| 75.5 \| 76.7 \| 74.0 \| 76.1 \| 76.7 \| 75.5 \| 72.7 \| 76.5±0.9 \| \| AVG \| 84.5 \| 84.3 \| 86.1 \| 88.1 \| 88.2 \| 88.4 \| 88.2 \| 88.5 \| 88.8 ± 0.1 \| Table 4: Multi-source unsupervised domain adaptation results on the Amazon review dataset \| Domain \| mSDA \| DANN \| MDAN(H) \| MDAN(S) \| MAN-L2 \| MAN-NLL \| CAN \| CRAL \| SAN(Ours) \| \|------------\|------\|------\|---------\|---------\|--------\|---------\|-----\|------\|-----------\| \| Books \| 76.98\| 77.89\| 78.45 \| 78.63 \| 78.45 \| 77.78 \| 78.91\| 82.49 \| 81.48 \| \| DVD \| 78.08\| 78.86\| 77.97 \| 80.05 \| 81.57 \| 82.74 \| 83.37\| 85.53 \| 85.53 \| \| Electr. \| 81.98\| 81.98\| 84.83 \| 85.34 \| 83.37 \| 83.75 \| 84.76\| 87.12 \| 87.12 \| \| Kit \| 84.26\| 86.39\| 85.80 \| 86.26 \| 85.57 \| 86.41 \| 86.75\| 89.08 \| 89.00 \| \| AVG \| 80.46\| 82.01\| 81.76 \| 82.72 \| 82.24 \| 82.67 \| 83.45\| 85.67 \| 85.78 \| The proposed SAN method outperforms MT-CNN and MT-DNN consistently across all domains with notable large performance gains. When compared with the state-of-the-art MAN-L2, MAN-NLL, DA-MTL, and GLR-MTL, SAN achieves competitive results in terms of average classification accuracy. The experimental results on both benchmarks validate the efficacy of our proposed method. Multi-Source Unsupervised Domain Adaptation. In real application scenarios, it is not uncommon for the target domain to lack annotated data. Evaluating MDTC models under such circumstances is of utmost significance. In the multi-source unsupervised domain adaptation (MS-UDA) setting, we have multiple source domains, each containing both labeled and unlabeled data, and a target domain with only unlabeled data. Our MS-UDA experiments are conducted on the Amazon review dataset, following the same protocol as outlined in Chen & Cardie (2018). Specifically, in each experiment, three out of four domains were treated as source domains, while the remaining domain was treated as the target domain. As shown in Table 4, the proposed SAN method outperforms other baselines on two out of four domains as well as the average accuracy. It reveals that our SAN method has a good capacity for transferring knowledge to unseen domains. Further experimental results, including parameter sensitivity analysis, ablation study, convergence analysis, model runtime comparison and model parameter comparison, can be found in the Appendix. ## 5 Conclusion In this paper, we propose stochastic adversarial networks (SANs) for multi-domain text classification. In contrast to previous MDTC models that rely on multiple domain-specific feature extractors to capture domain-unique features, we introduce a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$ over the weights of the domain-specific feature extractor. This allows for the sampling of an arbitrary number of diverse domain-specific feature extractors, providing the ability to leverage an infinite number of such extractors without increasing the model size. Furthermore, we integrate domain label smoothing and robust pseudo-label regularization techniques to stabilize the adversarial training process and enhance feature discriminability. Experimental results on two MDTC benchmarks demonstrate the effectiveness of our SAN model in improving system performance on these benchmarks and its generalization ability to unseen domains. REFERENCES Martin Arjovsky and Léon Bottou. Towards principled methods for training generative adversarial networks. arXiv preprint arXiv:1701.04862, 2017. Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In International conference on machine learning, pp. 214–223. PMLR, 2017. John Blitzer, Ryan McDonald, and Fernando Pereira. 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NnYaYVODyV	The model uses patch size 4x4, which means the visual backbone only downsamples the image by 4. Intuitively, this model should be good at dense prediction tasks that require high feature resolution, e.g. semantic segmentation and object detection. The authors reported the results of semantic segmentation on ADE20K in Section 4.4, but it only outperforms the ViT-B by a smaller margin. I understand that the segmentation architecture is different. So it would be interesting to compare ViT-B vs PGT-B with the same segmentation architecture (linear classification layer).	Perceptual Group Tokenizer: Building Perception with Iterative Grouping Zhiwei Deng, Ting Chen, and Yang Li Google Research and Deepmind Abstract Human visual recognition system shows astonishing capability of compressing visual information into a set of tokens containing rich representations without label supervision. One critical driving principle behind it is perceptual grouping (Palmer, 2002; Wagemans et al., 2012; Herzog, 2018). Despite being widely used in computer vision in the early 2010s, it remains a mystery whether perceptual grouping can be leveraged to derive a neural visual recognition backbone that generates as powerful representations. In this paper, we propose the Perceptual Group Tokenizer, a model that entirely relies on grouping operations to extract visual features and perform self-supervised representation learning, where a series of grouping operations are used to iteratively hypothesize the context for pixels or superpixels to refine feature representations. We show that the proposed model can achieve competitive performance compared to state-of-the-art vision architectures, and inherits desirable properties including adaptive computation without re-training, and interpretability. Specifically, Perceptual Group Tokenizer achieves 80.3% on ImageNet-1K self-supervised learning benchmark with linear probe evaluation, establishing a new milestone for this paradigm. 1 Introduction Visual recognition mechanisms matter. The pursuit of advanced vision algorithms that encode an image to meaningful representations dates back to late 80s, with two paradigms marking the progress over the past 40 years: feature detection (LeCun et al., 1998; Lowe, 2004; He et al., 2016; Liu et al., 2022b) and perceptual grouping (Shi & Malik, 2000; Uijlings et al., 2013; Arbeláez et al., 2014), where feature detection focuses on specific distinctive patterns, while perceptual grouping considers similarities among all pixels to produce a compact set of tokens as proxies for image representation. Ever since the surge of deep learning, feature detection has predominated the vision field and become the main principle behind representation learning backbone designs and made impressive progress (Simonyan & Zisserman, 2014; Szegedy et al., 2015; He et al., 2016; Chen et al., 2017; Tan & Le, 2019; Qi et al., 2020; Liu et al., 2022b). The success of the former paradigm is, although striking, raising the question of whether perceptual grouping can also be used as the driving principle to construct a visual recognition model. Different from detecting and selecting distinctive features, perceptual grouping emphasizes on learning feature space where similarity of all pixels can be effectively measured (Uijlings et al., 2013; Arbeláez et al., 2014). With such a feature space, semantically meaningful objects and regions can be easily discovered with a simple grouping algorithm and used as a compact set to represent an image (Uijlings et al., 2013; Arbeláez et al., 2014; Locatello et al., 2020). This indicates that image understanding is essentially “pixel space tokenization”, and being able to produce generalizable feature representations is tightly connected to whether the correct contextual pixels are binded together (Hinton, 2022; Culp et al., 2022). The intriguing properties of perceptual grouping, including natural object discovery, deep connections with information theory and compression (Ma et al., 2007), and association with biological vision system (Herzog, 2018) or cognitive science explanations (Palmer, 2002), have led to a strong revival recently under deep learning frameworks (Locatello et al., 2020; Elsayed et al., 2022; Xu et al., 2022; Wu et al., 2022; Biza et al., 2023). However, these methods are either still focusing on small or toy datasets (Locatello et al., 2020; Chang et al., 2022; Biza et al., 2023), or used as an auxiliary component (Xu et al., 2022; Ke & Yu, 2022; Seitzer et al., 2022) to strengthen exist- Figure 1: Perceptual Group Tokenizer is entirely driven by grouping operations to perform representation learning. Group tokens (discovered objects) are shown above. See more in the appendix. In this paper, we propose Perceptual Group Tokenizer, a model trained under a self-supervised learning framework, which builds visual representation entirely based on perceptual grouping operations. Given an image, the core of our model is to understand each pixel or patch through hypothesizing its contexts with grouping operations. Starting from given input patches, the grouping operation performs an iterative binding process onto a set of randomly sampled group tokens to determine the affinity groups based on similarities. The group tokens are then used as hypothesized contexts to refine the feature representation for the image. We show that applying this simple principle can already produce expressive representations and works well with self-supervised pretraining on a large vision dataset. The grouping operation is also closely related to self-attention, a highly popular method commonly used in modern vision backbones. We build connection between the proposed grouping operation and self-attention and show that, if group tokens are treated as communication channels, self attention can potentially automatically emerge during learning processes as a special case, while the grouping operation can produce even richer interactions among tokens. Under this viewpoint, ViT (Dosovitskiy et al., 2020) can be considered as a grouping backbone, with a fixed number of grouping slots equal to the number of input tokens, and the binding is achieved through stacking more than one layer with non-shared weights. This provides one explanation on why grouping mechanism can be effective on visual representation learning and has the potential to be a promising competitive paradigm for vision architecture designs. The primary contribution of this work is proposing a new architecture derived purely by perceptual grouping that achieves competitive performance compared to other state-of-the-art architectures on self-supervised learning benchmarks, contributing to a new paradigm of developing vision architectures. The model has several key differences and advantages over ViT, including (1) explicit separating out the “group token” concept to allow for automatic image parsing and flexible customization on the number of groups without being binded to the number of patches; (2) much less peak memory usage during inference time given the same number of input tokens; (3) adaptive computation without re-training the model, leading to flexible usage according to domains and computes. 2 RELATED WORKS Vision architectures. There are two main frameworks for vision backbones. The first framework is Convolutional neural networks, which rely on local filters, sliding windows and translational equivariance to perform representation learning. Since the introduction of ConvNets in 1980s, ConvNet was repopularized by AlexNet (Krizhevsky et al., 2012). The line of ConvNet is a classical inheritance from traditional feature detection methods (Lowe, 2004; Dalal & Triggs, 2005; Rosten et al., 2008), where instead of hand crafting features, an overcomplete set of filters are automatically learned to obtain high-response regions. The object understanding is built along the depth axis (Simonyan & Zisserman, 2014; Szegedy et al., 2015; He et al., 2016), with early layers capturing low-level parts and higher-level layers producing object structure representations (Zeiler & Fergus, 2014; Zhou et al., 2014; Yosinski et al., 2015; Bau et al., 2017). In the feature detection framework, not every pixel is worth being used depending on particular tasks, leading to difficulty in obtaining representation for each pixel. Recently, Vision Transformer (ViT) (Dosovitskiy et al., 2020), a second vision backbone framework, shows impressive performance and has surpassed ConvNet on visual recognition. The core of ViT is the iterative applying of self-attention operations (Vaswani et al., 2017; Dosovitskiy et al., 2020). A direct usage of ViT on small patches (thus a high-resolution grid) is extremely computationally expensive due to its associated quadratic cost. Therefore, a common practice is often partitioning the image into large non-overlapping patches (Dosovitskiy et al., 2020; Touvron et al., 2021), or constrain the operation to local regions (Liu et al., 2021). Self-supervised learning. The field of representation learning has seen significant interest in self-supervised learning during the past few years. The main evaluation results using linear probe on ImageNet benchmarks is approaching the results obtained by supervised learning (Oquab et al., 2023). Contrastive representation learning is the early method that shows promising results (Oord et al., 2018; Chen et al., 2020a; Tian et al., 2020). BYOL (Grill et al., 2020) and DINO (Caron et al., 2021) propose to use a moving average target of an online network to perform self representation matching. Masked image modeling also shows to be effective on representation learning, where the masking is either at the pixel level (He et al., 2022) or the learned codebook level (Bao et al., 2021). Object discovery. The perceptual grouping is essentially performing “object and stuff” discovery in the pixel space. It has broad connections with the early works in computer vision (Shi & Malik, 2000; Uijlings et al., 2013; Levinstein et al., 2013; Arbeláez et al., 2014; Pont-Tuset et al., 2016), the recent progress on object-centric representation (Burgess et al., 2019; Locatello et al., 2020; Chang et al., 2022; Hinton, 2022; Hénaff et al., 2022; Culp et al., 2022; Elsayed et al., 2022), and biological or neural mechanisms on perceptual grouping (Palmer, 2002; Wagemans et al., 2012; Herzog, 2018; Kim et al., 2019). Despite the early popularity of perceptual grouping methods on various computer vision tasks (Shi & Malik, 2000; Uijlings et al., 2013; Levinstein et al., 2013; Krähenbühl & Koltun, 2011), it has not attracted significant attention until several recent works that apply it as a side component on top of another main backbone (Seitzer et al., 2022; Liu et al., 2022a; Xu et al., 2022; Ke & Yu, 2022). Some relevant works demonstrate alternative possibilities in architecture design, but only uses cross attention without refining the patch feature space (Jaegle et al., 2021), or apply it on diffusion tasks (Jabri et al., 2022). Other methods also attempt to use ad-hoc sparsification methods on top of ViT (Rao et al., 2021; Yin et al., 2022; Bolya et al., 2023) for efficiency and are orthogonal to our work. A most related work (Ma et al., 2023) focuses on supervised learning and relies on fixed-center pooling and less standard operations. In our proposed model, we adopt a design as ViT except for self attention, and highlight several key technical contributions, including multi-grouping with multi-seeding, adaptive computation without re-training, and other design choices for self-supervised representation learning. 3 MODELS In this section, we introduce Perceptual Group Tokenizer (PGT), a visual recognition architecture entirely driven by perceptual grouping principles. We discuss the core operations for grouping in section 3.1, the building blocks and network architectures in section 3.2, the loss function used for self-supervised learning in section 3.3, and the connections with other models in section 3.4. 3.1 PERCEPTUAL GROUPING We start with introducing notations for our method. Given an image $ x \in \mathbb{R}^{H \times W \times C} $, we first reshape it as a sequence of small patches$^1$. Each patch $ x_p \in \mathbb{R}^{h \times w \times c} $ has spatial shape $ h \times w $, where $ h \ll H $ and $ w \ll W $, leading to $ N = \frac{HW}{hw} $ number of patches per image. To represent a patch, we embed it into a high-dimensional vector $ h \in \mathbb{R}^d $. The set of embedded tokens $ \{ h_i \}_{i=1}^N $ is referred to as input tokens in later parts, and used as inputs for the following grouping blocks. Feature refinement through hypothesizing contexts. Individual pixels do not have meanings without putting it into contexts. At a high level, image understanding or feature learning is equivalent to binding the correct contextual pixels at all locations. The core idea of our model is to generate many (e.g. over-complete w.r.t number of objects in the image) hypothesized contexts and use the $^1$We use 4×4 patches as inputs in this work. Note that our method is generalizable to either pure pixels or other forms of superpixels given a proper patch-to-vector embedding layer. Figure 2: Perceptual Group Tokenizer takes in a sequence of patches (or pixels), generates high-dimensional embedding vectors for all patches, then passes through a series of grouping layers to refine the embedding vectors as feature representations. Each grouping layer performs $K$ rounds of binding from input tokens to group tokens. To consider various grouping possibilities, multiple grouping heads are adopted. Each group token provides a useful context for input tokens for feature refinement. The final output of the model contains refined input tokens, group tokens, and assignments between input tokens and groups tokens. hypothesized contexts as cues to refine the feature representation of each patch. This process is achieved through a grouping module. Given input tokens $\{h_i\}_{i=1}^N$, the grouping module starts from a set of random samples (referred as group tokens) from a random distribution, then performs binding process to aggregate information from input tokens to the group tokens, and ends up with a set of group tokens $c^* = \{c_j^\}_{j=1}^M$ representing hypothesized contexts among input tokens. The relation between $h_i$ and $c_j$ is soft assignment, indicating how likely an input token belongs to that context. Note that there are often various ways of generating groupings for an image, e.g., different semantics, colors, textures, etc., we propose the “multi-grouping operation” to hypothesize rich contexts for tokens. The overall model is shown in figure 2. Multi-grouping operation.* The building block of our model is the multi-grouping operation $G$, which contains multiple heads to perform the binding process in parallel. This design encourages the model to consider multiple ways of generating groups under different projection spaces. Each head owns a separate Gaussian distribution with learnable means and variance, similar to (Kingma & Welling, 2013; Locatello et al., 2020). Starting from a set of randomly sampled initial group tokens $c_{\text{HEAD}}^{(0)} \sim p_{\text{INIT}}(\cdot)$, the grouping operation uses doubly normalized attention weights to aggregate information from $h$, and the produced group tokens $c_{\text{HEAD}}^{(1)}$ are used for the next round binding. The attention normalization and feature projection are performed in all heads separately. $$c_{\text{HEAD}}^{(1)} = G(c_{\text{HEAD}}^{(0)}, h; \theta)$$ $$\vdots$$ $$c_{\text{HEAD}}^* = c_{\text{HEAD}}^{(K)} = G(c_{\text{HEAD}}^{(K-1)}, h; \theta)$$ where after $K$ steps the final group tokens $c^* = c^{(K)}$ is obtained, and $\theta$ is learnable parameters in $G$. The grouping operator is summarized in algorithm 1. The sampling distribution $p_{\text{INIT}}(\cdot)$ for initializing group tokens $c_{\text{HEAD}}^{(0)}$ needs to be lightweight. We explore two variations: (1) Gaussian distribution $p(\mu_{\text{HEAD}}, \sigma_{\text{HEAD}})$ with learnable means and variance, and a one-step normalizing flow module that transforms a unit Gaussian noise to a sample that follows more complex distributions. More details can be found in the appendix in section A.1. Implicit differentiation. The iterative grouping process unrolls $K$ steps per operation and leads to heavy burden in the training computation graph. Instead of explicitly backpropagating through the unrolled graph, we follow (Chang et al., 2022) and treat the multi-grouping process as a fixed point iteration per head. The gradient in the backpropagation is approximated using first-order Neumann series, which can be simply achieved by detaching the output before the final iteration. ### 3.2 Network architecture Similar to standard ViT, our model refines the hidden representation $h$ using $L$ model layers. We use $h^l$ to denote the representation after each layer, and explain the design in this section. Algorithm 1 Multi-grouping operation using $G$. ```python def multi_grouping(h_key, h_value, steps, num_tokens, num_heads): """ Input tensors: h_key and h_value are projected multi-head tensors with shape [num_heads x N x d]. # Initial M group tokens. group_tokens = sampling_distribution(nsamples=num_tokens, choice='Gaussian') # or 'Flow' group_tokens = group_tokens.reshape(num_heads, num_tokens, d) #[num_heads x M x d] # Binding process. for step in range(steps): # Implicit differentiation. if step == steps - 1: group_tokens = stop_gradient(group_tokens) # The following is a one-step grouping operation. # Attention operation for group assignment. attn_matrix = attention(group_tokens, h_key) #[num_heads x N x M] attn_matrix /= attn_matrix.sum(-2, keep_dim=True) h_updates = einsum('hij,hid->jhd', attn_matrix, h_value) #[num_heads x M x d] group_tokens = gru_cell(h_updates, group_tokens) # Grouped mlp/layernorm performs independent mlp/layernorm for each head. group_tokens = grouped_mlp(grouped_layer_norm(group_tokens)) + group_tokens return group_tokens ``` Grouping layer. Each grouping layer takes in $h_{l-1}$ as input, and uses the grouping operation in equation 1 to generate group tokens $c_{\text{HEAD}}^* = \{c_{j,\text{HEAD}}^\}_{j=1}^{M}$. To use the group tokens to provide context for each $h_i^{l-1}$, we perform another attention operation to obtain the attention matrix (only normalized over group token axis) $A \in \mathbb{R}^{N \times M}$ representing the assignment from input tokens to group tokens, and aggregate the feature back to the input token space: $$ h_{\text{HEAD}}^l = A[c_{1,\text{HEAD}}^; c_{2,\text{HEAD}}^; \ldots; c_{M,\text{HEAD}}^] \\ h^l = \text{Linear}([h_{\text{HEAD}}^l_1; \ldots; h_{\text{HEAD}}^l_M]) \\ h^l = h^{l-1} + \text{MLP}(\text{LN}(h^l)) $$ (3) (4) (5) This layer definition follows the standard ViT layer as close as possible, where features from each head are aggregated through concatenation and a linear layer transformation. Each token $h$ is further refined using a follow up multi-layer perceptron. Grouping blocks. Similar to previous architecture designs (He et al., 2016; Liu et al., 2021), we define blocks for the model. One block contains multiple grouping layers that share the same hyperparameters setups, i.e. the number of group tokens, and group token dimensions. The full model contains three grouping blocks. This increases the flexibility when exploring model design spaces. ### 3.3 Self-supervision loss We strictly follow the student-teacher self-supervision loss (Caron et al., 2021; Oquab et al., 2023), and use a moving average of online network (student model) as the teacher model to perform representation learning. To summarize group tokens outputted from the final layer, we use one multi-head attention layer with a learnable token to attend to all group tokens. The produced single vector is treated as the feature representation for the image and is input to the loss function. ### 3.4 Discussion Our proposed model, perceptual group tokenizer, does not contain self-attention operations and purely relies on grouping operations. In this section, we link the grouping process to several techniques and discuss the rationale on why this model can be effective on representation learning. Group tokens as “communication channels”. The core of feature representation learning is how information is exchanged among pixels. In perceptual grouping backbones, we can consider the set of group tokens as communication channels, where information from different input tokens are aggregated in various ways. Each group token represents a high-order channel that links input tokens with high affinity under certain projected space to exchange information among them. As a thought experiment, if each input token is... solely assigned to a different group token (given enough group tokens), then the perceptual grouping layer is equivalent to one self attention layer (up to some engineering design difference). While self attention layers mainly rely on pairwise communications, grouping operation, hypothetically, can automatically learn and emerge both pairwise and higher-order information exchange through the group token communication channels. This can also be linked to traditional factor graphs in probabilistic graphical models. Through the lens of that, grouping is forming factor nodes automatically through the learning processes. With a properly designed loss and grouping operation, it has the potential to be more effective if adopting a per-layer comparison with self-attention operations. Efficiency. Due to the flexibility in customizing number of group tokens (controlled by initial number of samples), grouping operation does not require a strict $O(N^2)$ operation and is $O(NM)$ on complexity. Furthermore, we show that in inference time, number of group tokens can even be adaptively customized, given an already trained model. ## 4 EXPERIMENTS We evaluate the representation learned by our model on standard benchmarks based on the ImageNet-1K dataset. We also explore and analyze the design space of perceptual group tokenizer in section 4.2, investigate its adaptive computation ability in section 4.3, demonstrate its generalization ability on semantic segmentation in section 4.4, and visualize learned attentions in section 4.5. ### 4.1 MAIN RESULTS Setup. The widely-adopted standard benchmark for evaluating self-supervised learning methods is ImageNet ILSVRC-2012 (ImageNet-1K) (Russakovsky et al., 2015). Performance of models are measured by top-1 classification accuracy. The pre-trained backbones are frozen, with a linear classifier trained on top. For fair comparison, we follow the standard data augmentation used in (Caron et al., 2021), with the same number of global views and local views. The model is optimized using AdamW (Loshchilov & Hutter, 2018) with learning rate 0.0005 and 1024 batch size for 600 epochs, trained with TPJuv5 for 21k core hrs (512 cores for 41 hrs). We use $4 \times 4$ patches as image tokens, which keeps as much details as possible while maintaining reasonable computation costs. \| Method \| Arch \| Param. \| Linear probe (top-1 acc) \| \|-----------------\|------------\|--------\|--------------------------\| \| (Other backbones with different losses within the same batch of DINO for reference) \| \| SCLR (Chen et al., 2020a) \| RN50W4 \| 375 \| 76.8 \| \| SwAV (Caron et al., 2020) \| RN50W2 \| 93 \| 77.3 \| \| BYOL (Caron et al., 2020) \| RN50W2 \| 93 \| 77.4 \| \| SwAV (Caron et al., 2020) \| RN50W5 \| 586 \| 78.5 \| \| BYOL (Caron et al., 2020) \| RN50W4 \| 375 \| 78.6 \| \| iBOT (Zhou et al., 2021) \| ViT-B/16 \| 85 \| 79.5 \| \| BYOL (Caron et al., 2020) \| RN200W2 \| 250 \| 79.6 \| \| SCLRV2 (Chen et al., 2020b) \| RN152w3+SK \| 794 \| 79.8 \| \| BEiTv2 (Peng et al., 2022) \| ViT-B/16 \| 85 \| 80.1 \| \| (Fair comparison under the DINO loss and framework) \| \| DINO (Caron et al., 2021) \| ViT-S/8 \| 21 \| 79.7 \| \| Ours (PGTG-S-1024) \| PGT-S \| 34 \| 79.8 \| \| DINO (Caron et al., 2021) \| ViT-B/16 \| 85 \| 78.2 \| \| DINO (Caron et al., 2021) \| ViT-B/8 \| 85 \| 80.1 \| \| Ours (PGTG-B-256) \| PGT-B \| 70 \| 79.7 \| \| Ours (PGTG-B-512) \| PGT-B \| 70 \| 79.9 \| \| Ours (PGTG-B-1024) \| PGT-B \| 70 \| 80.1 \| \| Ours (PGTF-B-256) \| PGT-B \| 115 \| 80.0 \| \| Ours (PGTF-B-512) \| PGT-B \| 115 \| 80.1 \| \| Ours (PGTF-B-1024) \| PGT-B \| 115 \| 80.3 \| Table 1: Comparison with strong baselines on ImageNet-1K under linear probe evaluation protocol. PGT$_{\text{DIST}}$-B-$X$ represents $X$ number of group tokens per grouping layer in inference (same trained model with 256 tokens is used). $\text{DIST}$: the distribution choice for group token initialization. G and F represent Gaussian and Flow, respectively. Our model achieves 80.3%, competitive with state-of-the-art vision backbones. \| \| Descend \| Flat \| Ascend \| \|----------------\|------------------\|-----------------\|-----------------\| \| Token size \| [576, 384, 192] \| [384, 384, 384] \| [192, 384, 576] \| \| Accuracy \| 62.0 \| 63.1 \| 63.4 \| \| Token shape \| [192, 128, 64] \| [128, 128, 128] \| [64, 128, 192] \| \| Accuracy \| 63.6 \| 63.7 \| 63.1 \| Table 2: Exploring the design choices for PGT. Token size: dimensions for group tokens in three grouping blocks. Token shape: number of tokens for group tokens in three grouping blocks. Accuracy measured on ImageNet-1K under linear probe protocol. Results indicate progressively large group token dimensions with flat or descend number of tokens arrangements work the best. Architecture details. In the experiments, we mainly evaluate two variants of PGT: the main model and a tiny version for exploring design choices. On the ImageNet-1K benchmark, we report the performance metrics of our main model. Three grouping blocks are used, with 10 grouping layers in each block. The dimension for input token is 384, with 256 group tokens per layer. The dimensions for group tokens are 98, 192, and 288 for the three blocks, respectively. There are 6 grouping heads used. For number of grouping iterations, we observe three rounds are sufficient to achieve good performance. The MLP hidden size for each layer is 384 as well, i.e., the MLP multiplication factor is 1. The final multihead attention layer uses a learnable token with 2048 dimensions to summarize all group tokens outputs from the model. The main results are summarized in table 1. We mainly compare with ResNet and ViT backbones, the two main stream vision architectures to show that perceptual grouping architecture can also achieve competitive results on the challenging ImageNet-1K benchmark. Although our model is trained with 256 group tokens, the model can use different numbers of group tokens in inference (more experiments in section 4.2). We evaluate PGT with 256, 512, and 1024 number of group tokens and observe that the model can achieve 80.3% top-1 accuracy, showing the self-supervised learned feature of PGT is as good as the ones learned by ViT architectures. 4.2 Ablations To explore design choices of PGT, we use a tiny version of PGT with 3 blocks, 2 layer in each block (6 layers in total), 256 hidden size for input tokens, and 3 number of grouping iterations. The learnable token in MAP head has 512 dimensions. There are ~10M parameters in this PGT-tiny. Group token layouts. Given a fixed number of budget on group tokens, we explore three choices on how they should be arranged across grouping blocks and layers: descend, flat and ascend. Intuitively, more group tokens will have higher capacity of capturing smaller parts and detailed visual features, while less group tokens are more prone to carry global information. As shown in table 2 bottom row, flat or descend number of group tokens performs the best. In practice, we find that using flat (same number of group tokens in three grouping blocks) version achieves better training stability. Group token dimension shapes. Similar to token number arrangements, we explore how group token dimensions should be set. Under three choices, progressively increasing the dimension size in the later layers performs the best, shown in first row of table 2. This also aligns with the intuition that later layers contain more information and requires higher capacity to represent groups. Multi-grouping vs single grouping. We further test whether multi-head grouping helps improve performance. As a fair comparison, we use 6 heads and 128 group tokens per head for a multi-grouping model, and 1 head with 6×128 group tokens for a single grouping model. We find that adopting multi-head design can improve the performance from 62.2% to 66.3%, a 4.1% accuracy boosts, showing that having multiple heads indeed helps with representation learning. Grouping distribution entropy. Will grouping process collapse to some specific group token during training? We visualize the entropy of marginal distribution over tokens $p(c)$ and conditional distribution $p(c\|x)$ in figure 4. Interestingly, we observe that conditional probability, i.e. the assignment to group tokens, tends to become more certain during training, while the marginal distribution remains having descend entropy, indicating collapses not happening in training. Peak memory usage. As discussed in section 3.4, given the same number of tokens, the grouping operation uses less memory than the self-attention operation. We show the percentage of peak memory usage in PGT$_G$-B compared to ViT-B with the same patch size (4×4) in table 3. usage is obtained from the forward inference graph, as in practice the underlying complex hardware optimizer is a less accurate measurement and varies across infrastructures. \| #group tokens \| 16 \| 32 \| 64 \| 128 \| 256 \| 384 \| 512 \| 768 \| 1024 \| ViT-B \| \|---------------\|----\|----\|----\|-----\|-----\|-----\|-----\|-----\|------\|-------\| \| Peak memory(%)\| 4.6\| 4.6\| 4.6\| 4.6 \| 4.6 \| 6.1 \| 8.2 \| 12.2\| 16.3 \| 100 \| Table 3: Peak memory usage of PGT-B compared to the baseline model ViT-B with $4 \times 4$ patch size. ### 4.3 Out-of-distribution adaptive computation One surprising and powerful ability of PGT is adaptive computation. For example, given a model trained using $M_1$ group tokens per layer, one can choose to use $M_2$ group tokens in inference, where $M_2 \neq M_1$. This is because the initial seeding group tokens are drawn from a probabilistic distribution, and the number of samples can be customized. This property leads to a highly customizable inference without re-training the model. When $M_1 \neq M_2$, the model copes with an out-of-distribution (OOD) problem where test time setting is different from training. We observe surprisingly strong generalization with our model. Specifically, with more tokens $M_2 > M_1$ in inference, the performance can actually outperform the setting ($M_2 = M_1$) used in training, even if it is OOD for the model. The results for OOD adaptive computation are summarized in table 4. We mainly test PGT-G-Tiny with a grid evaluation that varies the number of group tokens in training $M$ and the number of group tokens in inference $N$, and also show the main model’s results in the last row. When using the main model PGT-G-B to perform adaptive inference, with only 12.5% of the number of group tokens compared to training, the performance can still be maintained at 72.1% with only a $\sim$8% drop on top-1 accuracy. The adaptive computation ability is important for both general image understanding where images have varying number of objects and need different numbers of groups, and scenarios where test-time computational resource is constrained. This flexibility is an important advantage that grouping backbones hold. \| tr/inf \| 16 \| 32 \| 64 \| 128 \| 256 \| 384 \| \|--------\|----\|----\|----\|-----\|-----\|-----\| \| PGT-G-Ti-16 \| 57.4 ($\times 1$) \| 58.3 ($\times 2$) \| 58.5 ($\times 4$) \| 58.5 ($\times 8$) \| 58.5 ($\times 16$) \| 58.4 ($\times 24$) \| \| PGT-G-Ti-32 \| 57.3 ($\times \frac{1}{2}$) \| 59.9 ($\times 1$) \| 60.8 ($\times 2$) \| 61.0 ($\times 4$) \| 61.0 ($\times 8$) \| 60.9 ($\times 12$) \| \| PGT-G-Ti-64 \| 53.0 ($\times \frac{1}{4}$) \| 59.2 ($\times \frac{1}{2}$) \| 61.7 ($\times 1$) \| 62.6 ($\times 2$) \| 62.9 ($\times 4$) \| 62.9 ($\times 6$) \| \| PGT-G-Ti-128 \| 44.9 ($\times \frac{1}{8}$) \| 56.6 ($\times \frac{1}{4}$) \| 61.8 ($\times \frac{1}{2}$) \| 63.9 ($\times 1$) \| 64.7 ($\times 2$) \| 64.8 ($\times 3$) \| \| PGT-G-Ti-256 \| 27.2 ($\times \frac{1}{16}$) \| 47.4 ($\times \frac{1}{8}$) \| 58.8 ($\times \frac{1}{4}$) \| 63.3 ($\times \frac{1}{2}$) \| 65.1 ($\times 1$) \| 65.5 ($\times \frac{3}{2}$) \| \| PGT-G-Ti-384 \| 26.1 ($\times \frac{1}{32}$) \| 43.0 ($\times \frac{1}{16}$) \| 55.4 ($\times \frac{1}{8}$) \| 61.7 ($\times \frac{1}{4}$) \| 64.6 ($\times \frac{1}{2}$) \| 65.5 ($\times 1$) \| \| PGT-G-B-256 \| 60.4 ($\times \frac{1}{16}$) \| 72.1 ($\times \frac{1}{8}$) \| 77.1 ($\times \frac{1}{4}$) \| 78.9 ($\times \frac{1}{2}$) \| 79.7 ($\times 1$) \| 79.9 ($\times \frac{3}{2}$) \| Table 4: Out-of-distribution adaptive computation by selecting different numbers of initially sampled tokens. Row: number of tokens used for training. Column: number of tokens used for inference. Top-1 accuracy is reported under linear evaluation protocol using ImageNet-1K. The reported performance of first six rows is obtained using a tiny version of PGT, and last row is the main model. Number of group tokens is the same for underlined numbers in training and inference. Bold numbers are the best results. ### 4.4 Downstream task transfer: semantic segmentation on ADE20k To evaluate the generalizability of pretrained feature produced by PGT, we test the transfer performance of semantic segmentation with ADE20k. Following the standard setup, we finetune our model with the same data augmentation for 128 epoch. The baseline method uses DINO + ViT-B/16 (Zheng et al., 2021). For our model, we add one linear classification layer after the pre-trained PGTG-B for fine-tuning. To adapt to more objects and complex scenes in the segmentation datasets, we use 1024 group tokens for inference, benefiting from the adaptive computation ability of our model. We find that our model can obtain 45.1% on mean IoU while the baseline achieves 44.1% (Bao et al., 2021), leading to a 1.0% improvements. 4.5 GROUPING VISUALIZATION We visualize the attention maps calculated between group tokens and input tokens in figure 4.5. We find that (1) using multiple grouping heads can capture different information within each head. For example, in layer 0, the first head captures light and color, second head focuses on only spatial locations, and the third head potentially relies on textures; (2) group tokens can capture different semantic parts, for example, in the first image, group tokens separate apple, jar, handle, and background. In the second image, camel, legs, camel hump, and human are separately grouped. Compared to standard ViT in DINO (Caron et al., 2021) where only a single foreground can be extracted using [CLS] token, our model can flexibly group different parts given an image, leading to a set of tokens that are potentially more meaningful and customizable. Note that the grouping results are still different from human’s vision, and sometimes generates parts that seem to be “fragmented”. This is possibly due to the “parts-to-whole with data augmentation” training loss. Human vision, in contrast, is sensitive to moving objects and trained within a 4D space. Nevertheless, we believe with a similar dataset, environment and loss design, our grouping model can potentially produce groupings more coherent and sensitive to boundaries and moving objects. 5 CONCLUSION In this paper, we propose Perceptual Group Tokenizer (PGT), a new visual recognition architecture entirely built through perceptual grouping principles. 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ybavRGEmpw	when comparing the results shown in Tables 1 and 2, we can see that the best OT-regularized divergences methods in terms of accuracy are very different for image recognition and malware detection problem. What is the root cause behind it? No any written hypothesis or discussion.	Adversarially Robust Learning with Optimal Transport Regularized Divergences Anonymous authors Paper under double-blind review Abstract We introduce the ARMOR_D methods as novel approaches to enhancing the adversarial robustness of deep learning models. These methods are based on a new class of optimal-transport-regularized divergences, constructed via an infimal convolution between an information divergence and an optimal-transport (OT) cost. We use these as tools to enhance adversarial robustness by maximizing the expected loss over a neighborhood of distributions, a technique known as distributionally robust optimization (DRO). Viewed as a tool for constructing adversarial samples, our method allows samples to be both transported, according to the OT cost, and re-weighted, according to the information divergence; the addition of a principled and dynamical adversarial re-weighting on top of adversarial sample transport is the key innovation of ARMOR_D. We demonstrate the effectiveness of our method on malware detection and image recognition applications and find that it provides significant performance benefits. In malware detection, a discrete (binary) data domain, ARMOR_D improves the robustified accuracy under rFGSM^50 attack compared to the previous best-performing adversarial training methods by 22 percentage points while simultaneously lowering false negative rate from 4.99% to 2.44%. 1 Introduction Machine learning and specifically deep learning models are known to be vulnerable to adversarial samples: inputs intentionally and meticulously modified by an adversary to evade/mislead the classification model (Papernot et al., 2016; Goodfellow et al., 2014). One common and effective way to enhance a model’s robustness against this vulnerability is to include adversarial samples during the training process, known as adversarial training. However, adversarial training is often challenging, as it is hard to maintain the model’s performance generalizability while also enhancing its adversarial robustness (Carlini et al., 2019; Zhang et al., 2019). To date, the large body of prominent defense mechanisms for enhancing adversarial robustness includes certifiable approaches (Baharlouei et al., 2023; Raghunathan et al., 2018), which can guarantee the absence of adversarial examples misclassified by the model for a specific input, and adversarial training methods (Papernot et al., 2017; Madry et al., 2018; Hu et al., 2018; Wang et al., 2020; Zhang et al., 2019, 2020; Dong et al., 2020; Regniez et al., 2021; Bui et al., 2022; Dong et al., 2023) which somehow construct adversarial samples that are employed during training, with Sinha et al. (2018) having aspects of both categories. Despite attractive guarantees, the certifiable approaches often operate on a convex relaxation of the original model rather than the original model and tend to have inferior performance compared to approaches in the latter category (Wang et al., 2020; Athalye et al., 2018). In the pioneering robust optimization approach (Madry et al., 2018) to adversarial training, the loss function $ L_\theta $, depending on parameters $ \theta \in \Theta $, is maximized over a metric-space ball centered at the training samples $ x_i $, leading to the empirical risk minimization problem \[ \inf_\theta E_{P_n} \left[ \sup_{y: d(x,y) \leq \epsilon} L_\theta(y) \right], \] (1) where $ P_n = \frac{1}{n} \sum_{i=1}^{n} \delta_{x_i} $ is the empirical distribution. In Regniez et al. (2021) and Bui et al. (2022) it was recognized that (1) can be expressed as a distributionally robust optimization (DRO) problem over an optimal-transport (OT) neighborhood $ U(P_n) = \{ Q : C(Q,P_n) \leq \epsilon \} $ for an appropriate OT cost $C$; i.e., they noted that $\inf_\theta \sup_{Q: C(Q, P_\theta) \leq \epsilon} E_Q[L_\theta]$. DRO is a general framework for taking a stochastic optimization problem $\inf_\theta E_P[L_\theta]$ and regularizing (or robustifying) it by maximizing over a neighborhood of distributions, $U(P)$, around the baseline distribution $P$, leading to the general DRO problem $$\inf_\theta \sup_{Q \in U(P)} E_Q[L_\theta].$$ This formalizes an uncertainty in the underlying distribution $P$ and can protect against overfitting, leading to better out-of-sample performance; see Rahimian & Mehrotra (2022) for an overview of DRO. For general distribution neighborhoods (2) is an intractable infinite dimensional problem but if $U$ has the appropriate structure then one can derive tractable finite dimensional reformulations of (2). Prior approaches to the general theory of DRO employ various types of distribution neighborhoods, such as moment constraints Goh & Sim (2010); Delage & Ye (2010); Wiesemann et al. (2014), conditional moment constraints Blanchet et al. (2023), Kullback-Leibler (KL) and $f$-divergence neighborhoods Ben-Tal et al. (2010); Ahmadi-Javid (2012); Hu & Hong (2013); Ben-Tal et al. (2013); Lam (2019), MMD Staib & Jegelka (2019), Wasserstein neighborhoods Mohajerin Esfahani & Kuhn (2018); Shafieezadeh-Abadeh et al. (2019); Wu et al. (2022); Yu-Meng Li & Mao (2022); Gao & Kleywegt (2023), and more general optimal-transport (OT) neighborhoods Blanchet & Murthy (2019). In the present work we propose a novel class of divergences for comparing probability distributions, which we call the optimal-transport-regularized divergences, that combines features of both OT costs and information-theoretic divergences (such as KL) and use these to define distribution neighborhoods for use in DRO (2). This leads us to propose a novel class of adversarial training methods that simultaneously transport adversarial samples (with general OT cost) and re-weight the adversarial samples according to the information-theoretic divergence. The former feature is shared with the OT-DRO method Bui et al. (2022) (see also the related earlier work Sinha et al. (2018), which used a soft Wasserstein constraint), but the ability of our method to use information from the loss together with the OT cost in order to adversarially re-weight samples in a principled and dynamical manner during training is a qualitatively new feature of our method; this feature follows naturally from our more general DRO framework, which “mixes” information-theoretic and OT divergences via an infimal convolution (see Eq. (3) below). In practice, the adversarial re-weighting causes the optimization algorithm to focus on the samples in each minibatch that are more vulnerable to adversarial perturbation. The DRO-based methods are in contrast to methods which directly modify the loss $L_\theta$, such as TRADES Zhang et al. (2019), and MART Wang et al. (2020). In fact, the two types of techniques can be combined; in Bui et al. (2022) the combination of generalized OT costs with TRADES/MART was shown to lead to further performance gains, beyond either method individually. In this work we focus on evaluating the benefits of the adversarial re-weighting that is inherent to our method; we leave for future work the analysis of our DRO framework in combination with TRADES/MART-style loss modifications. Optimal-Transport-Regularized Divergences: The new divergences that we introduce in this work are defined as an infimal convolution between an optimal transport cost, $C$, and an information divergence, $D$, e.g., an $f$-divergence, $D = D_f$ Liese & Vajda (2006), of which the KL-divergence is one example. More precisely, given an OT cost function $c(x, y)$ and an information divergence, $D$, we define the OT-regularized divergence, $D^c$, of a distribution $Q$ with respect to a distribution $P$ by $$D^c(Q\|\|P) := \inf_{\eta \in \mathcal{P}(X)} \{D(\eta\|\|P) + C(\eta, Q)\},$$ where $\mathcal{P}(X)$ denotes the set of probability distributions on the space $X$ and the optimal transport cost associated with the cost function $c$ is given by $$C(\mu, \nu) := \inf_{\pi: \pi_1 = \mu, \pi_2 = \nu} \int c(x, y) \pi(dxdy)$$ ($\pi_i$ denote the marginals of $\pi \in \mathcal{P}(X \times X)$); the only assumptions we make regarding $c$ are non-negativity, lower semicontinuity, and that $c(x, x) = 0$ for all $x$. Intuitively, one can view (3) as specifying a cost via a two-step procedure for transforming $P$ into $Q$. First, one redistributes the probability-mass in $P$ to form an intermediate distribution $\eta$, paying the cost $D(\eta\|\|P)$ (we say redistribute because we focus on $D$ that are information divergences, meaning they are computable in terms of the likelihood ratio $d\eta/dP$, though most of our theorems in Appendix A apply more Generally). Second, one performs optimal transport to transform $ \eta $ into $ Q $, paying the cost $ C(\eta, Q) $. The optimal intermediate measure $ \eta_c $ determines the final cost $ D^c(Q \\| P) $. The infimal convolution structure, including the bound $ D^c(Q \\| P) \leq \min\{D(Q \\| P), C(P, Q)\} $, causes $ D^c $ to inherit properties from both $ D $ and $ C $ and allows it to interpolate between these two extremes; see Section 2.2. The OT-regularized divergences are related to the $ \Gamma $-divergences defined in Dupuis, Paul & Mao, Yixiang (2022), $(f, \Gamma)$-divergences defined in Birrell et al. (2022), and the IC-$ \Gamma $-Rényi divergences from Birrell et al. (2023), but here we utilize optimal transport costs as opposed to integral-probability-metric (IPM) regularization of information divergences. We found that OT-regularization is more naturally suited to adversarial robustness methods than IPM regularization from a mathematical perspective. Also, those prior works focused on the equality of the primal and dual formulas for the divergence, which facilitates applications to GANs; here we focus on adversarial robustness, which requires different techniques. We use the OT-regularized divergences to define distribution neighborhoods of size $ \epsilon > 0 $, leading to following DRO problem, which we will employ as a tool for enhancing adversarial robustness \[ \inf_{\theta} \sup_{Q: D^c(Q \\| P_n) \leq \epsilon} E_Q[\mathcal{L}_\theta]. \] The OT-regularized-divergence neighborhoods are qualitatively different from both $ f $-divergence and Wasserstein neighborhoods, as they allow for a combination of probability-mass transport and redistribution when forming the perturbed distributions, $ Q $. This allows for the support of $ Q $ to differ from that of $ P_n $ (as in Bui et al. (2022)), and also for the probability of widely separated modes to be re-weighted, something that is not possible with Wasserstein neighborhoods. When viewed as an adversarial training method, we call (5) the ARMOR_D methods, standing for Adversarially Robust Models with Optimal-Transport-Regularized Divergences. In Section 2 we show how (5) can be converted into a computationally tractable optimization problem and in Section 2.1 we provide a formal solution, thereby clarifying the manner in which our method combines optimal transport and adversarial re-weighting. In Section 2.2 we list a number of properties of the OT-regularized divergences, thus demonstrating that they are well-behaved mathematical objects; precise statements and proofs are found in Appendix A. In Section 3 we test the ARMOR_D methods on MNIST image classification as well as malware classification, where we find it offers significant performance gains. ## 2 OT-Regularized Divergences: DRO Identity and Properties In general, the DRO problem (5) is an intractable infinite dimensional optimization problem. However, for appropriate choices of $ D $ one can derive a finite dimensional reformulation that leads to computationally efficient implementations. In this section we provide a formal derivation of the key identity. For a rigorous proof, and statement of the required assumptions, see Appendix A.2. Noting that $ C $ is jointly convex and assuming that $ D $ is convex in its first argument (as is the case when $ D $ is an $ f $-divergence) one can see from (3) that $ D^c $ is convex in its first argument. Therefore the DRO problem is a convex optimization problem and one can compute \[ \sup_{Q: D^c(Q \\| P_n) \leq \epsilon} E_Q[\mathcal{L}_\theta] \] \[ = \inf_{\lambda > 0} \left\{ \lambda \epsilon + \sup_{Q \in \mathcal{P}(X)} \{E_Q[\mathcal{L}_\theta] - \lambda D^c(Q \\| P_n)\} \right\} \] \[ = \inf_{\lambda > 0} \left\{ \lambda \epsilon + \sup_{Q, \eta \in \mathcal{P}(X)} \{E_Q[\mathcal{L}_\theta] - \lambda D(\eta \\| P_n) - \lambda C(\eta, Q)\} \right\} \] \[ = \inf_{\lambda > 0} \left\{ \lambda \epsilon + \sup_{\eta \in \mathcal{P}(X)} \{-\lambda D(\eta \\| P_n) + \sup_{Q \in \mathcal{P}(X)} \sup_{\pi: \pi_1 = \eta, \pi_2 = Q} \{E_Q[\mathcal{L}_\theta] - \lambda \int c d\pi\}\} \right\} \] \[ = \inf_{\lambda > 0} \left\{ \lambda \epsilon + \lambda \sup_{\eta \in \mathcal{P}(X)} \{-D(\eta \\| P_n) + \sup_{\pi_x(dy)} \int \mathcal{L}_\theta(y)/\lambda - c(x, y)\pi_x(dy)\eta(dx)\} \right\} \] \[ = \inf_{\lambda > 0} \left\{ \epsilon \lambda + \lambda \sup_{\eta \in \mathcal{P}(X)} \int \sup_{y \in X} \{\lambda^{-1} \mathcal{L}_\theta(y) - c(x, y)\}\eta(dx) - D(\eta \\| P_n)\right\}. \] The equality (7) is obtained using strong duality, lines (8) and (9) are obtained using the definitions (3) and (4) of $ D^c $ and $ C $ along with properties of suprema and infima, (10) recognizes that the suprema over $Q$ and $\pi$ can be rewritten as a supremum over probability kernels $\pi_x(dy)$, and finally (11) uses the fact that the supremum over probability kernels achieves the pointwise supremum of the integrand. To this point, the derivation closely follows that of Mohajerin Esfahani & Kuhn (2018) for Wasserstein DRO, as well as the adversarial robustness approach by Bui et al. (2022). Note that effect of the OT cost is to replace the loss $L_\theta$ with what we call the OT-regularized loss $$L_{\theta,\lambda}(x) := \sup_{y \in X} \left\{ \lambda^{-1} L_\theta(y) - c(x, y) \right\},$$ which is known as the $c$-transform in the optimal transport literature; see Definition 5.2 in Villani (2008). The importance of the $c$-transformed loss for Wasserstein DRO is well known; see the references to prior work on Wasserstein and OT-DRO in the introduction. The supremum over $y \in X$ in (12) can be thought of as selecting an adversarial sample that is paired with each real sample, $x$. We note that our mathematical framework can be used to robustify any empirical risk minimization problem, not only classification, and so our notation does not yet explicitly decompose the variables into sample and label components, though we will do so when applying the method to classification problems in Section 3. The new ingredient in our OT-regularized-divergence DRO framework is the optimization over $\eta$ in (11). This can be recognized as the convex-conjugate of $\eta \mapsto D(\eta \|\| P_n)$ and for certain choices of $D$, in particular for the $f$-divergences which we now focus on, this term can be reformulated as a finite dimensional convex optimization problem. Using the generalization of the Gibbs variational principle to $f$-divergences, see Theorem 4.2 in Ben-Tal & Teboulle (2007), one has $$\sup_{\eta \in P(X)} \left\{ E_\eta[g] - D_f(\eta \|\| P) \right\} = \inf_{\rho \in \mathbb{R}} \left\{ \rho + E_P[f^(g - \rho)] \right\},$$ where $f^$ is the Legendre transform of $f$. Using this we obtain the following finite-dimensional reformulation of the DRO problem $$\inf_{\theta \in \Theta} \sup_{Q : D_f(Q \|\| P_n) \leq \epsilon} E_Q[L_\theta] = \inf_{\lambda > 0, \rho \in \mathbb{R}, \theta \in \Theta} \left\{ \epsilon \lambda + \rho + \frac{\lambda}{n} \sum_{i=1}^{n} f^(L_{\theta,\lambda}(x_i) - \rho/\lambda) \right\}. $$ Here we made the change of variables $\rho \rightarrow \rho/\lambda$ so that the objective function is jointly convex in $\lambda, \rho$ (see Corollary A.23). Note that the new variables $\lambda, \rho$ simply augment the minimization over model parameters $\theta$ by adding two real variables, which adds very small additional computational cost. The $\lambda$ parameter has the same interpretation as in the OT-DRO based method Bui et al. (2022); it can be viewed as a dynamical OT-cost weight, selected according to the optimization (11), which is tied to the neighborhood size $\epsilon$. This perspective is most apparent in (7). The significance of $\rho$ will be discussed in Section 2.1 below. In Section 3 we will experiment with the KL divergence and the family of $\alpha$-divergences (i.e., $f = f_\alpha$ as in Eq. (29), which we call the ARMOR$_{KL}$ and ARMOR$_\alpha$ methods respectively. An explicit formula for $f^$ in the case of $\alpha$-divergences is given in (30). In the KL-divergence case the minimization over $\rho$ can be evaluated analytically, yielding $$\inf_{\theta \in \Theta} \sup_{Q : KL_c(Q \|\| P_n) \leq \epsilon} E_Q[L_\theta] = \inf_{\lambda > 0, \theta \in \Theta} \left\{ \epsilon \lambda + \lambda \log \left( \frac{1}{n} \sum_{i=1}^{n} \exp(L_{\theta,\lambda}(x_i)) \right) \right\}. $$ We will refer to either of (14) or (15) as the outer minimization problem and will call (12) the inner maximization problem. While preparing this work a new DRO framework was proposed in Blanchet et al. (2023), employing conditional moment constraints, which was also motivated in part by the desire to combine transport and redistribution costs. Their approach reduces to the $D = D_f$ case of our DRO framework under appropriate assumptions; see their Theorems 4.1, 5.1, and Proposition 5.1 and compare with our Theorem A.22 and Eq. (14), (15). Our work is distinguished both mathematically, through the proofs of a number of properties of the OT-regularized divergences that do not have analogues in Blanchet et al. (2023) (see Section 2.2), and through our novel use of (14) and (15) as tools for enhancing adversarial robustness, where we find it leads to substantial performance gains. ### 2.1 Interpreting the Outer Minimizer: Adversarial Sample Weights In this section we give an intuitive interpretation of the minimization over the auxiliary parameters $\lambda, \rho$ in (14); they can be viewed as the computation of optimal adversarial weights for the adversarial samples, where optimality is defined in part by the chosen $f$-divergence. This is a complement to the inner maximizer (12) which constructs the optimally transported adversarial samples, according to the chosen OT cost function. This interpretation gives insight into the qualitatively novel nature of our method. Letting $ y_i(\lambda) $ be the solution to the inner maximizer (12) with $ x = x_i $ and $ \lambda_* $ and $ \rho_* $ be the optimal scaling and shift parameters for the outer minimizer at a fixed $ \theta $ (we suppress the $ \theta $-dependence of $ y_i $, $ \lambda_* $, and $ \rho_* $ in the notation) we derive the following reformulation of (14) in Appendix B, \[ \inf_{\lambda > 0, \rho \in \mathbb{R}} \left\{ \epsilon \lambda + \rho + \frac{\lambda}{n} \sum_{i=1}^{n} f^(L_{\theta, \lambda}(x_i) - \rho/\lambda) \right\} = E_{Q_{, \theta}}[L_{\theta}], \] where the optimal adversarial distribution is $ Q_{, \theta} := \sum_{i=1}^{n} p_{, i} \delta_{y_i(\lambda_)} $, having optimal adversarial weights \[ p_{, i} := \frac{1}{n} (f^)(L_{\theta, \lambda_}(x_i) - \rho_/\lambda_). \] This shows that the minimization over $ \theta $ in (14) solves the risk minimization problem for the ($ \theta $-dependent) optimal adversarial distribution $ Q_{, \theta} $. The optimal adversarial distribution is supported on the optimal adversarial samples $ y_i(\lambda_) $ and the weight of the $ i $'th sample is changed from $ 1/n $ to $ p_{, i} $ (17). To understand the significance of the re-weighting $ p_{, i} $, first recall that $ f^* $ is non-decreasing (see Definition A.2 and Corollary A.23), hence $ p_{, i} \geq 0 $. In addition, the $ p_{, i} $'s sum to 1 as shown in (90) below. Convexity of $ f^* $ implies that $ (f^)' $ is also non-decreasing, hence the $ p_{, i} $'s shift more weight towards the samples where the OT-regularized loss is larger, as would be expected for an adversarial re-weighting. In some cases, such as for the $ \alpha $-divergences, $ f^* $ is constant on $(-\infty, M)$ for some $ M $ ($ M = 0 $ when $ f = f_\alpha $, as seen in Eq. (30)). In such cases, samples with $ L_{\theta, \lambda_}(x_i) < \rho_/\lambda_* + M $ have their weighting changed to 0. Intuitively, one can consider those samples as having sufficiently small OT-regularized loss and hence the method moves its attention away from them to focus on more troublesome samples. These samples are only temporarily ignored; attention may return to them later in the training if their loss moves above the (dynamic) threshold. Part of the task of the outer minimizer is to dynamically determine the optimal threshold for "sufficient smallness", as set by $ \rho_/\lambda_ $. We emphasize that that this threshold changes with $ \theta $, as $ \lambda_* $ and $ \rho_* $ are both $ \theta $-dependent. The ability of the ARMOR_D methods to re-weight adversarial samples in addition to transporting them is the primary innovation of our approach, as compared to the prior OT-DRO based robustness method [Bui et al. (2022)] or the earlier soft constraint based method [Sinha et al. (2018)]. As we demonstrate in the examples in Section 3 and Appendix C.8, this is a powerful new ingredient and is made possible because our DRO neighborhoods incorporate both information-theoretic and OT components via the infimal convolution (5). Our approach is distinct from the re-weighting method proposed in [Guo et al. (2022)] for addressing the problem of class imbalance in the training data, which is not an adversarial re-weighting. Our method is also distinct from the approach in [Zhang et al. (2020)] where modified weights were introduced manually, based on an informal notion of distance to the decision boundary. In contrast, re-weighting in ARMOR_D is determined in a principled manner by the DRO framework, via the choice of $ f $ and $ c $; it uses information from the OT-regularized loss of each sample during training, along with a dynamic threshold, as seen in (17). In particular the adaptive threshold, which determines which samples the optimizer currently considers "troublesome", is a qualitatively novel feature of our method. ### 2.2 Properties of the OT-Regularized Divergences The OT-regularized divergences have many attractive mathematical properties, making them well suited to DRO as well as other statistical learning tasks. We summarize a number of these properties here; see Appendix A for precise statements of the required assumptions along with proofs. Given appropriate assumptions on $ D $ and $ c $ one has the following: 1. $ D^c(\nu \|\| \mu) \geq 0 $ and $ D^c(\nu \|\| \mu) = 0 $ if and only if $ \nu = \mu $; see Theorem A.7. This divergence property implies that $ D^c(\nu \|\| \mu) $ can be interpreted as measuring the discrepancy between $ \nu $ and $ \mu $. 2. There exists an optimal intermediate distribution that solves the minimization problem in the definition (3), i.e., there exists $ \eta_* $ such that \[ D^c(\nu \|\| \mu) = D(\eta_* \|\| \mu) + C(\eta_, \nu) \] and this $ \eta_n $ is unique under appropriate assumptions. See Theorem A.9. 3. $ D^c(\nu \\| \mu) $ is convex in $ \nu $ (see Lemma A.4). This implies that the DRO neighborhoods $ \{Q : D^c(Q \\| P_n) \leq \epsilon\} $ are convex sets and is also key in the derivation of the DRO identity (1). 4. $ D^c(\nu \\| \mu) $ is lower semicontinuous in $ \nu $ (see Theorem A.11). This property is useful for theoretical purposes and it implies that the DRO neighborhoods $ \{Q : D^c(Q \\| P_n) \leq \epsilon\} $ are closed sets. 5. $ D^c $ interpolates between $ D $ and $ C $ in the following sense: For $ r > 0 $ define the scaled cost function $ c_r = rc $. Then \[ \lim_{r \to 0^+} r^{-1} D^{c_r}(\nu \\| \mu) = C(\mu, \nu) \quad \text{(see Theorem A.12)}, \tag{19} \] \[ \lim_{r \to \infty} D^{c_r}(\nu \\| \mu) = D(\nu \\| \mu) \quad \text{(see Theorem A.13)}. \tag{20} \] Informally, this property implies that DRO over both $ D $ and $ C $ neighborhoods can be viewed as special cases of DRO over $ D^c $ neighborhoods. More specifically, (19) indicates that when $ r $ is sufficiently small, DRO over the neighborhood $ \{Q : D^{c_r}(Q \\| P_n) \leq r\epsilon\} $ is approximately the same as DRO over the neighborhood $ \{Q : C(P_n, Q) \leq \epsilon\} $. Similarly, (20) indicates that when $ r $ is sufficiently large, DRO over the neighborhood $ \{Q : D^{c_r}(Q \\| P_n) \leq \epsilon\} $ is approximately the same as DRO over the neighborhood $ \{Q : D(Q \\| P_n) \leq \epsilon\} $ (see Theorems A.24 and A.25 for precise statements). Therefore if one includes the scale factor $ r $ and neighborhood size $ \epsilon $ as tunable hyperparameters (as we do in the experiments in Section 3) then the special cases of $ C $ and $ D $ neighborhoods will be (approximately) explored in the process of tuning an ARMOR_D method. We note that these properties do not require the distributions to have compact support, except for the DRO interpolation results in Theorems A.24 and A.25. ### 3 EXPERIMENTS In this section we evaluate the ARMOR_D adversarial robustness methods on two classification problems: MNIST digit classification and malware detection, two common tasks featuring continuous and discrete data, respectively. Experimental Setup:* To evaluate the performance of our proposed method, we consider the application of adversarial robustness in two fundamental deep learning tasks: image recognition and malware detection. For the image recognition task we use the MNIST dataset with 50,000 digits in the training and 10,000 in the test set. For the malware detection task, we use a high-dimensional dataset with 22,761 features provided by Al-Dujaili et al. (2018), which includes a total of 54,690 binary encoded malware and benign Windows Portable Executables (PEs) partitioned into training (60%), validation (20%), and test set (20%). Each data point is represented as a 22,761-dimensional binary feature vector denoting the existence of a particular feature in the executable. The target detector models for image and malware data sets were a 4-layer convolutional neural network (CNN) and a 3-layer feed-forward network, respectively, for which the architecture details are given in Appendix C.3. In the binary encoded malware application there is an extra requirement: For the adversarial sample to be functional and preserve malicious malware functionality only bit flips from 0 to 1 are acceptable and not vice versa Al-Dujaili et al. (2018); this gives the problem an inherent asymmetry. Following the guidelines in Carlini et al. (2019), we consider a threat model characterizing the adversary’s goal, knowledge, and capabilities detailed in Appendix C.7. #### 3.1 Experiment 1: Illustrating the Importance of Adversarial Re-Weighting via Robust Image Detection In this experiment we focus on evaluating the benefit provided by the adversarial sample re-weighting component of the ARMOR_D method alone. Therefore we choose the OT-transport component so that the inner maximizer (12) agrees with the inner maximizer of the Madry et al. (2018) approach, (1), (which we call PGD-AT). Specifically, in this example we choose \[ c((x, y), (\tilde{x}, \tilde{y})) = \infty 1_{d(x, \tilde{x}) > \epsilon} + \infty 1_{y \neq \tilde{y}}, \tag{21} \] where $ x, \tilde{x} $ are samples and $ y, \tilde{y} $ are the corresponding labels. This cost allows for the adversarial sample to freely move within the $ \epsilon $-ball centered at the original sample, but not outside it, and does not allow for label modification; we let $ d $ be the metric induced by the $ \infty $-norm. Benchmark Methods and Evaluation Metrics: Following (Bui et al., 2022), we evaluate the methods against the Projected Gradient Method ($ PGD^{200} $) attack and the much stronger and more recent AutoAttack (Croce & Hein, 2020). In this experiment we are evaluating the $ ARMOR_D $'s adversarial re-weighting mechanism alone. Our primary comparison will be the recent OT-DRO based method Bui et al. (2022), called UDR, which like our method can also be used to enhance (1) or any other empirical risk minimization problem. The resulting $ UDR-PGD $ and $ ARMOR_D-PGD $ methods are compared on MNIST in Table 1. Following the experiment settings in Bui et al. (2022), all attacks were conducted by a neighborhood size of 0.3, $ \ell_\infty $ neighborhood, and 40 iterations of adversarial training. Implementation details are provided in Appendix C. Results: The best hyperparameters for $ ARMOR_D $ were attained via a small grid-search on a parameter space identified in Appendix C.4. To ensure a fair comparison, we closely followed the same settings as in (Bui et al., 2022); see Appendix C.3. We compare the performance of the methods under the attacks $ PGD^{200} $ and AutoAttack (Croce & Hein, 2020) as well as their performance when not under attack (Nat) and report the performance in Table 1. Our proposed method attains higher accuracy than the baseline $ PGD-AT $ method under both AutoAttack and $ PGD^{200} $. The $ ARMOR_\alpha $ augmentation of $ PGD $ also outperforms $ UDR-PGD $ on the stronger AutoAttack test. This indicates that the $ ARMOR_\alpha $ re-weighting mechanism is an effective tool for enhancing the adversarial robustness of an empirical risk minimization problem. The effects of $ ARMOR_\alpha $ can be combined with $ UDR $ by modifying the OT-cost function (21) while retaining the sample re-weighing provided by the $ f $-divergence component of our method; we intend to explore this for a wider variety of data sets in the future. In the present work, our second example in Section 3.2 explores the use of modified OT-costs within our method. Table 1: Enhancing adversarial robustness on MNIST: Here $ ARMOR_\alpha $ uses natural samples alongside the adversarial samples, as described in Appendix C.6. Best metrics are shown in bold font. \| Defense \| AutoAttack \| $ PGD^{200} $ \| Nat \| \|-------------\|------------\|-----------------\|-------\| \| $ PGD-AT $\| 88.9% \| 94.0% \| 99.4% \| \| $ UDR-PGD $\| 90.0% \| 94.3% \| 99.5% \| \| $ ARMOR_\alpha-PGD $\| 91.70% \| 94.24% \| 99.26% \| ### 3.2 Experiment 2: Enhancing the Adversarial Robustness of Malware Detectors using a Soft OT Constraint and Adversarial Labels Next we present our results on malware detection, a much higher dimensional and more realistic problem; we closely followed the settings from Al-Dujaili et al. (2018). Appendix C provides the implementation details. In this example we experiment OT cost modifications, which in our approach can be combined with the adversarial re-weighting. We consider two types of OT costs. Robust Classification Using Adversarial Samples: First consider optimal transport cost functions of the form \[ c((x, y), (\tilde{x}, \tilde{y})) = L \\| x - \tilde{x} \\|_q + \infty 1_{y \neq \tilde{y}} \] on the space $ X = D \times \{0, ..., N_c - 1\} $ (i.e., samples in $ D \subset \mathbb{R}^d $ with label from $ N_c $ classes). This applies a $ q $-Wasserstein cost on the first component (sample) but infinite cost on changing the second component (label); this is a form of soft-constraint, as opposed to the hard-constraint cost (21). The hyperparameter $ L > 0 $ allows one to choose how much weight is placed on the OT cost, as compared to the information divergence cost in $ D^c $. The OT-regularized loss is then \[ L_{\theta, \lambda}(x, y) = \sup_{\tilde{x} \in D} \left\{ \lambda^{-1} L_\theta(\tilde{x}, y) - L \\| x - \tilde{x} \\|_q \right\}, \] and corresponds to the construction of a new sample, $ \tilde{x} $, adversarial to the original sample $ x $ but keeping the original label $ y $. We consider the choice of vector norm to be a hyperparameter, selected from $ \ell^p, p \in [1, \infty] $, and use the cross-entropy loss, $ L_\theta(\tilde{x}, y) = CE(\phi_\theta(\tilde{x}), y) $ where $ \phi_\theta $ is the neural network (NN) classifier with NN-parameters $\theta$. The adversarial loss (23) can then be used in either of the outer minimizers (14) or (15) (or, more generally, Eq. [11] for some other $D$, provided one can compute its convex conjugate) to obtain an ARMOR$_D$ method. We use the notation $adv_s$ to denote methods that employ adversarial samples constructed via (23). Robust Classification Using Adversarial Class Labels and Adversarial Samples: We will also utilize OT cost functions that allow the class labels to be perturbed in the inner maximizer. To do this we consider the sample space to be $X = D \times P(\{0,...,N_c - 1\})$ where $P(\{0,...,N_c - 1\})$ is the space of probability vectors on the set of labels, with the original class labels mapped to the corresponding one-hot vectors. We relax the term $\infty 1_{y \neq \tilde{y}}$ in (22) to allow for the perturbation of class labels. Allowing for too much label uncertainty will destroy any predictive ability of the classifier, as is also the case with adversarial perturbation of samples, but we find that a small amount improves robustness. To this end, we consider OT cost functions of the form $$c((x,p), (\tilde{x},\tilde{p})) = L\\|x - \tilde{x}\\|^q + K g_\delta(OT(p,\tilde{p})),$$ where $OT$ is the optimal transport cost (with cost function $1_{i \neq j}$) between the probability vectors $p$ and $\tilde{p}$, i.e., $OT(p,\tilde{p}) = 1 - \sum_{i=1}^{N} \min\{p_i, \tilde{p}_i\}$, and $g_\delta : [0,\delta) \to [0,\infty)$ is increasing, continuous, and satisfies $g_\delta(0) = 0$, $\lim_{z \to \delta^-} g_\delta(z) = \infty$; we then extend the definition via $g_\delta([\delta,\infty)) := \infty$. $K > 0$ is a new cost coefficient hyperparameter and $\delta$ is a new hyperparameter that determines the maximum amount by which the class probabilities can change. More specifically, if the original sample has $p = 1_k$ (i.e., a one-hot vector with a 1 in the $k$'th position, corresponding to the label being $k$) then $OT(p,\tilde{p}) = 1 - \tilde{p}_k$ and so the cost (24) will force the adversarial label to have $\tilde{p}_k > 1 - \delta$. In particular we only use $\delta \in (0,1/2]$ so that the predicted class is never changed; the class probabilities are only relaxed from being either 0 or 1 to being in $[0,1]$. Therefore, we do not consider labels to be noisy in the sense discussed in, e.g., Natarajan et al. (2013); Shafieezadeh-Abadeh et al. (2019). We consider this only as a tool to enhance robustness. In our experiments we take $g_\delta(z) = z/(1-z/\delta)$; note this has a vertical asymptote at $z = \delta$, as required. The inner maximizer with original sample and label being $(x,1_k)$ is then $$L^c_{\theta,\lambda}(x,1_k) = \sup_{(\tilde{x},\tilde{p}) \in X : \tilde{p}_k > 1 - \delta} \left\{ \lambda^{-1} L_\theta(\tilde{x},\tilde{p}) - L\\|x - \tilde{x}\\|^q - K \frac{1 - \tilde{p}_k}{1 - (1 - \tilde{p}_k)/\delta} \right\}.$$ We let the baseline loss, $L_\theta$, be the KL divergence between the adversarial probability vector, $\tilde{p}$, and the classifier output $\phi_\theta(\tilde{x})$ (in the form of a probability vector); note that this is the same as the cross entropy when the label is one-hot but when the labels are relaxed to general probability vectors in the inner maximizer then they differ by the entropy of $\tilde{p}$. We use the notation $adv_{s,l}$ to denote methods that employ both adversarial labels and adversarial samples, constructed via (25). Benchmark Methods and Evaluation Metrics: We consider adversarial training with $r_FGSM^k$ (Al-Dujaili et al., 2018) and the method proposed by Grosse et al. (2017). We note that Al-Dujaili et al. (2018) proposes several variants for their adversarial training method among which $r_FGSM^k$ produces the best results. Consistent with Al-Dujaili et al. (2018), we consider three evaluation metrics: accuracy, false negative rate (FNR), and false positive rate (FPR) as well-established evaluation metrics. Results: Table 2 shows the malware experiment results for the non-robust model, benchmark models (the method proposed by Grosse et al. (2017) and $r_FGSM^k$), as well as variations of our ARMOR$_D$ method. The best hyperparameters for ARMOR$_D$ were attained via a small grid-search on the parameter space in Appendix C.5. As observed in Table 2, our proposed ARMOR$_\alpha(adv_{s,l})$ achieves the accuracy of 83.31%, FNR of 2.44%, and FPR of 42.0% against $r_FGSM^{50}$ attack outperforming, the benchmark methods and the non-robust model across all three evaluation metrics. ARMOR$_\alpha(adv_{s,l})$ also attains the lowest FNR against $r_FGSM^{50}$ and the lowest FNR against Grosse et al.’s attack. We note that, as shown in Table 2, the best performance under attack for Grosse et al. occurs when the adversarial training adopts the same method for inner maximizer. This is aligned with the findings in Al-Dujaili et al. (2018) (see their Table 3). In addition to these results, we also provide experiments to enhance the test generalizability of the malware detector in Appendix D. Table 2: Malware adversarial training to enhance performance under attack: Comparison of the performance of our proposed method in enhancing the robustness on the malware dataset. Hyperparameters were tuned to enhance performance under attack. $adv_s$ denotes the use of adversarial samples constructed via (23) and $adv_{s,l}$ denotes the use of both adversarial samples and labels, as in (25). $nat$ refers to the use of natural samples alongside the adversarial samples, as described in Appendix C.6. $adv^a$ refers to asymmetric methods, as described in Appendix C.7, with only the malicious samples robustified. See Table 5 for results tuned to maximize performance generalizability. \| Defense \| $rFGSM^{50}$ Attack \| Grosse et al. Attack \| No Attack \| \|--------------------------\|------------------------\|----------------------\|-----------\| \| Non-robust \| Acc FNR FPR \| Acc FNR FPR \| Acc FNR FPR \| \| \| 14.71% 77.85% 98.48% \| 33.03% 99.86% 8.53% \| 92.96% 5.30% 10.13% \| \| Grosse et al. \| 57.36% 10.96% 98.91% \| 91.08% 8.04% 10.48% \| 92.38% 5.47% 11.45% \| \| $rFGSM^{50}$ (Al-Dujaili et al.) \| 60.79% 4.99% 100.00% \| 74.74% 32.39% 12.59% \| 92.83% 5.20% 10.66% \| \| ARMOR$_KL$ ($adv_s$) \| 73.86% 2.80% 67.61% \| 87.59% 12.50% 12.26% \| 92.58% 5.44% 10.94% \| \| ARMOR$_KL$ ($adv^a_s$) \| 69.33% 5.39% 95.38% \| 85.53% 16.07% 11.62% \| 92.71% 5.14% 11.09% \| \| ARMOR$_KL$ ($adv_s + nat$) \| 84.25% 2.33% 2.28% \| 85.53% 20.50% 3.73% \| 92.90% 5.02% 10.81% \| \| ARMOR$_KL$ ($adv^a_s + nat$) \| 77.34% 3.14% 57.34% \| 72.45% 35.12% 14.11% \| 93.02% 5.39% 9.82% \| \| ARMOR$_c$ ($adv_s$) \| 83.23% 5.60% 36.62% \| 89.26% 9.67% 12.64% \| 92.92% 5.09% 0.63% \| \| ARMOR$_c$ ($adv^a_s$) \| 68.12% 5.23% 79.21% \| 88.57% 9.99% 13.98% \| 92.65% 4.99% 11.55% \| \| ARMOR$_c$ ($adv_s + nat$) \| 66.21% 6.72% 81.88% \| 86.79% 18.30% 4.16% \| 92.93% 5.13% 10.51% \| \| ARMOR$_c$ ($adv^a_s + nat$) \| 76.20% 2.86% 60.99% \| 78.68% 17.66% 27.82% \| 92.38% 2.71% 16.35% \| \| ARMOR$_c$ ($adv_{s,l}$) \| 83.31% 2.44% 42.00% \| 71.70% 1.83% 75.33% \| 91.08% 9.00% 8.78% \| Note: Best metrics are shown in bold font. The numbers for methods that outperform the non-robust model and prior adversarial robustness methods across all three metrics are underlined. 4 CONCLUSION In this work we proposed the ARMOR$_D$ methods for enhancing adversarial robustness of deep learning models. These methods are based on a new class of divergences for comparing probability distributions, the optimal-transport-regularized divergences $D^c$, which are defined as an infimal convolution between an information divergence $D$ (such as KL) and an optimal-transport cost $C$. The key innovation is the principled and dynamical manner in which the method combines transported adversarial samples, along with adversarial re-weighting of the samples via the information divergence. In practice, the adversarial re-weighting focuses the optimization towards improving the performance on the most troublesome adversarial samples. We demonstrated that these new tools have many attractive mathematical properties, making them well suited to applications in statistical learning. The ARMOR$_D$ methods were tested on classification problems representing both continuous (MNIST) and discrete data (malware), where we find that it provides significant performance benefits and outperforms existing methods at enhancing the robustness against adversarial attacks in most tests. For MNIST, we designed the test to isolate the effect of the adversarial sample re-weighting mechanism that is inherent to the ARMOR$_D$ framework. We find that, when used to augment PGD-AT, it increases the performance under AutoAttack by 2.8 percentage points, which is 1.7 points higher than achieved by the recent state-of-the-art OT-based augmentation method [Bui et al., 2022]. In malware detection, a discrete (binary) data domain, ARMOR$_D$ improves the robustified accuracy under $rFGSM^{50}$ attack compared to the previous best-performing adversarial training methods by 22 percentage points while simultaneously lowering false negative rate from 4.99% to 2.44%. These experiments were all done using ARMOR$_D$ where $D$ was an $f$-divergence, however the majority of the rigorous theoretical development we provide in Appendix A applies to a much more general class of $D$'s. Exploring cases beyond $D = D_f$ in the search for new variants of $D^c$ that can be efficiently and effectively applied to adversarial robustness, or to other statistical learning tasks, is an interesting direction for future work. In particular, the Rényi divergences are a natural candidate as their convex conjugate can be computed. Secondly, our method is based on a new general DRO framework of $D^c$ neighborhoods and hence can be used to augment any empirical risk minimization problem. Therefore our work can be used in a manner similar to [Bui et al., 2022], which used OT-DRO neighborhoods to obtain enhanced versions of TRADES, [Zhang et al., 2019], and MART, [Wang et al., 2020]; exploring such enhancements using the $D^c$-DRO framework is another promising direction for future work. REPRODUCIBILITY STATEMENT To facilitate reproducibility of the results presented in this paper we include implementation details in Appendix C. Specifically, Appendix C.2 contains pseudocode for the method, Appendix C.3 provides the target networks’ structure used for the malware and image applications, Appendix C.5 provides the hyperparameters that yielded the results reported in Tables 3, 2, 4 and 5. Appendix C.6 discusses the implementation of the $adv + nat$ methods, and Appendix C.7 discusses the implementation of the $adv^a$ methods. AUTHOR CONTRIBUTIONS All authors have made equal contribution to this work. REFERENCES A. Ahmadi-Javid. 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ZiHI6raor0	Intuitively, the idea of using conformal predictions to model the actions of other agents is to augment the agent’s state with the historical memory of other agents' behavior, which is similar to fictitious play in game theory. However, fictitious play only converges in some specific game settings, and MARL is known to be hard to converge in general settings. Moreover, the conformal predictions are based on previous observations but the other agent’s policy is also evolving. Is there any observation where the algorithm does not converge well or even has cycling behavior?	CAMMARL: Conformal Action Modeling in Multi Agent Reinforcement Learning Anonymous authors Paper under double-blind review Abstract Before taking actions in an environment with more than one intelligent agent, an autonomous agent may benefit from reasoning about the other agents and utilizing a notion of a guarantee or confidence about the behavior of the system. In this article, we propose a novel multi-agent reinforcement learning (MARL) algorithm CAMMARL, which involves modeling the actions of other agents in different situations in the form of confident sets, i.e., sets containing their true actions with a high probability. We then use these estimates to inform an agent’s decision-making. For estimating such sets, we use the concept of conformal predictions, by means of which, we not only obtain an estimate of the most probable outcome but get to quantify the operable uncertainty as well. For instance, we can predict a set that provably covers the true predictions with high probabilities (e.g., 95%). Through several experiments in two fully cooperative multi-agent tasks, we show that CAMMARL elevates the capabilities of an autonomous agent in MARL by modeling conformal prediction sets over the behavior of other agents in the environment and utilizing such estimates to enhance its policy learning. 1 Introduction Developing systems of autonomous agents capable of effective multi-agent interactions can be very useful in modern cooperative artificial intelligence (AI). For instance, service robots, surveillance agents, and many more similar applications require profound collaboration among agents (and with humans), without prior coordination. Now, to enable complex, constructive behaviors to emerge from unsupervised interactions among agents, an essential skill for an agent to have is the ability to reason about other agents in the environment. There has been considerable research addressing this problem of agent or opponent modeling (Albrecht & Stone [2018]). Generally, it involves constructing models of other agents that learn useful attributes to inform its own decision-making (such as the future actions of the other agents, or their current goals and plans) from current or past interaction history (such as the previous actions taken by other agents in different situations). We are interested in the particular aspect of an interactive, autonomous agent that involves learning an additional, independent model to make predictions about the actions of the other agents in the environment, supplemental to its reinforcement learning-based policy to make decisions related to its downstream task. An autonomous agent can then incorporate those estimates to inform its decision-making and optimize its interaction with the other agents. While there exist several methods for developing such models for other agents (Albrecht & Stone [2018]), there is currently no method or theory to the best of our knowledge that would allow an agent to consider the correctness or confidence of the predictions of the learned model. Conformal Predictions. Conformal predictions or inference is a fitting method for generating statistically accurate uncertainty sets for the predictions from machine learning classifiers. It is steadily gaining popularity owing to its explicit and non-asymptotic guarantees over the produced sets (Angelopoulos & Bates [2021]). In other words, we can obtain conformal sets that provably contain the true predictions with high probabilities, such as 95%, chosen in advance. This can be very useful and successful in high-risk learning settings, especially in decision-making in medical applications from diagnostic information, for instance, which demand quantifying uncertainties to avoid insufferable model failures. What if we only prefer to use the predictions when the model is Figure 1: Our proposed methodology of informing an autonomous agent’s decision-making by means of conformal predictions of action sets of other agents in the environment illustrated with two agents for simplicity. Two agents $(N_{self}, N_{other})$ receive their own partial observations from the environment $(o_{self}, o_{other})$ and take their actions $(a_{self}, a_{other})$. An independent conformal action prediction model $C$ learns to output a conformal action set, $\{a'_{other}\}$, corresponding to $N_{other}$ which are then used as additional inputs for training by $N_{self}$ to inform its policy and perform its action $a_{self}$. Confident? For example, doctors may only consider a predicted medical diagnosis when the model is at least 95% accurate, or may want to use the predicted set with high credence to consider ruling out relevant possibilities. So, in this article, we aim to enhance the capabilities of an agent in a multi-agent reinforcement learning (MARL) setting by modeling and using conformal prediction sets (or the latent representations learned in the process) over the behavior of an autonomous system. In particular, we model other agents’ actions in the form of confident sets, i.e., sets that contain other agents’ true actions with a high probability. We hypothesize that these estimated conformal sets would inform our learning agent’s decision-making and elevate its performance in MARL. Figure 1 shows the high-level idea of our proposed model for learning agents in any given environment. In this work, we aim to introduce a novel framework to train an autonomous agent that enhances its decision-making by modeling and predicting confident conformal actions of other agents in the environment — the CAMMARL algorithm (Section 3), and then empirically demonstrate that conformal action modeling used in CAMMARL indeed can help make significant improvements in cooperative policies learned by reinforcement learning agents in two multi-agent domains (Section 4). 2 RELATED WORKS Decision-making without reasoning about other agents in the environment can be very challenging, for instance, due to weak or no theoretical guarantees, non-stationarity (single agent’s perspective), and inefficient coordination for a considerable coherent joint behavior (Matignon et al., 2012). Modeling other agents in an environment is not new and has been studied in the past (Albrecht & Stone, 2018; Albrecht et al., 2020). However, our proposal of predicting conformal sets of actions of the other agents in the environment (with high probability) is novel and has not been attempted to the best of our knowledge. Learning world models. Model-based reinforcement learning (MBRL) has certainly shown its advantages in data efficiency, generalization, exploration, counterfactual reasoning, and performance in many tasks and domains (Hafer et al., 2020; 2021; Jain et al., 2022; Moerland et al., 2023; Pal & Leon, 2020; Polydoros & Nalpantidis, 2017) in single-agent RL, and now, it has also started to attract attention in MARL (Wang et al., 2022). However, most of the current works in model-based MARL do not yet focus on teammate or opponent modeling. Some recent works (Park et al., 2019b; Zhang et al., 2021) incorporated dynamics modeling and a prediction module to estimate the actions of other agents within the construction of the environment model. However, these --- 1 More details in Appendix B prediction models were trained without accessing the true trajectories from the other agents which can be problematic in several use cases. Learning agent models. A widely popular technique to reason about other agents in the environment is to learn representations of different properties of other agents. For instance, learning to reconstruct the actions of other agents from their partial observations (He et al., 2016; Mealing & Shapiro, 2015; Panella & Gmytrasiewicz, 2017; Albrecht & Ramamoorthy, 2015), modeling an agent or its policy using encoder-decoder-based architectures (Grover et al., 2018; Zintgraf et al., 2021), learning latent representations from local information with or without utilizing the modeled agent’s trajectories (Papoudakis et al., 2021; Xie et al., 2021) or modeling the forward dynamics of the system through relational reasoning using graph neural networks (Tacchetti et al., 2018). Theory-of-Mind Network or TomNet learned embeddings corresponding to other agents in the environment for meta-learning (Rabinowitz et al., 2018). Some works also constructed I-POMDPs to utilize recursive reasoning (Albrecht & Stone, 2018) assuming unrestricted knowledge of the observation models of other agents. Nevertheless, CAMMARL involves no form of reconstruction of other agent’s policy or rewards, or state models. Any of these techniques can be used with CAMMARL which, however, is not the objective of this work. Also, unlike CAMMARL, many of these aforementioned techniques evaluate in fully-observable environments or rely upon direct access to other agents’ experience trajectories even during execution. This can be infeasible in various settings. Multi-agent reinforcement learning (MARL). Numerous deep MARL research works that focus on partial observability in fully cooperative settings indirectly involve reasoning about the intentions of teammates or opponents in an environment (Gronauer & Diepold, 2022). For instance, many works allow agents to communicate, enabling them to indirectly reason about the others’ intentions (Lazaridou et al., 2016; Foerster et al., 2016; Sukhbaatar et al., 2016; Das et al., 2017; Mordatch & Abbeel, 2018; Gupta et al., 2021; Zhu et al., 2022). On the other hand, some studied the emergence of cooperative and competitive behaviors among agents in varying environmental factors, for instance, task types or reward structures (Leibo et al., 2017). Recent work in hierarchical reinforcement learning also attempts to develop a hierarchical model to enable agents to strategically decide whether to cooperate or compete with others in the environment and then execute respective planning programs (Kleiman-Weiner et al., 2016). However, none of these works study the improvement in an autonomous agent’s decision-making via directly modeling the other agents in the environment or predicting their actions or current or future intentions. Inverse reinforcement learning (IRL). Research in the field of IRL also relates to our work because we share the key motive of inferring other agents’ intentions and then use it to learn a policy that maximizes the utility of our learning agent (Arora & Doshi, 2021). However, IRL addresses this by deducing the reward functions of other agents based on their behavior, assuming it to be nearly optimal. On the other hand, in CAMMARL we directly model the other agent’s actions based on their observations and use these estimates to indirectly infer their goal in an online manner. Conformal prediction. Estimating well-grounded uncertainty in predictions is a difficult and unsolved problem and there have been numerous approaches for approximating it in research in supervised learning (Gawlikowski et al., 2021). Recent works in conformal predictions (Angelopoulos et al., 2020; Lei et al., 2018; Hechtinger et al., 2018; Park et al., 2019a; Cauchois et al., 2020; Messoudi et al., 2020) have now significantly improved upon some of the early research (Vovk et al., 2005; Platt et al., 1999; Papadopoulos et al., 2002), for instance in terms of predicted set sizes, improved efficiency, and providing formal guarantees. For this article, we adapt the core ideas from Regularized Adaptive Prediction Sets (RAPS) (Angelopoulos et al., 2020) to our setting owing to its demonstrated improved performance evaluation on classification benchmarks in supervised learning (Angelopoulos et al., 2020). ### 3 THE CAMMARL ALGORITHM #### 3.1 MATHEMATICAL MODEL Formally, we consider two agents in the environment — learning agent denoted by self and the other agent denoted by other. The partially observable Markov game (Littman, 1994) for our setting can then be defined using the following tuple: \[ \langle N_i, S, A_i, O_i, T, C, \pi_{\theta_i}, r_i \rangle_{i \in \{self, other_1, ..., other_{K-1}\}} \] With the set $ S $ describing the possible true states (or full observations) of the environment, $ K $ agents, $ N_{self} $ and $ (K-1)N_{other_j} $ ($ j \in [1, K] $), observe the environment locally using their sets of observations $ O_{self} $ and $ (K-1)O_{other_j} $, respectively, and act using their set of actions, $ A_{self} $ and $ (K-1)A_{other_j} $. Each agent $ i $ can select an action $ a_i \in A_i $ using their policy $ \pi_{\theta_i} $, and their joint action $ a \in A_{self} \times A_{other_1} \times \ldots \times A_{other_K} $ then imposes a transition to the next state in the environment according to the state transition function $ T $, defined as a probability distribution on the subsequent state based on current state and actions, $ T : S \times A_{self} \times A_{other_1} \times \ldots \times A_{other_K} \times S \rightarrow [0, 1] $. The agents use their individual reward function $ r_i(s, a) : O_i \times A_i \rightarrow \mathbb{R} $. Both agents aim to maximize their own total expected rewards $ R_i = \sum_{t=0}^{T} \gamma^t r_i^t $ where $ \gamma \in [0, 1) $ as the discount factor and $ T $ is the time horizon. In CAMMARL, at each time step $ t $, we also use a conformal prediction model for the $ j $-th agent, defined as a set-valued function, $ C : \mathbb{R}^d \rightarrow 2^{A_{other_j}} $ \[ C(o_{other_j}) \rightarrow \{A_{other_j}^t\} \] which outputs a conformal action predictive set $ \{A_{other_j}^t\} $ for each input of $ N_{other_j} $'s local observation $ o_{other_j}^t \in O_{other_j} $ at the time. ### 3.2 Conformal Action Modeling Algorithm 1 Conformal action modeling in MARL ``` 1: \( N_{self}, N_{other_j} \leftarrow \text{Initialize Actor-Critic networks for } N_{self} \text{ and } N_{other_j}, \text{ where } j \in [1, K] 2: (K-1) \text{conformalModels} \leftarrow \text{Initialize the conformal model to predict conformal action sets} 3: \textbf{for episode } = 1, 2, \ldots \textbf{ do} 4: \quad \text{Fetch observations } o_{self}, o_{other_1}, \ldots, o_{other_K} \text{ from environment} 5: \quad \textbf{for timesteps } = 1, 2, \ldots, T \textbf{ do} 6: \quad \quad \text{conformalActions } \leftarrow \text{conformalModels}(o_{other_j}); \text{ for } j \in [1, K] \quad \triangleright \text{Predict conformal action set} 7: \quad \quad o_{self} \leftarrow o_{self} + \text{conformalActions} \quad \triangleright \text{Concatenate conformal actions to } o_{self} 8: \quad \quad \text{Run agent policies in the environment} 9: \quad \quad \text{Collect trajectories of } N_{self} \text{ and } N_{other_1} \ldots N_{other_K} 10: \quad \textbf{if update interval reached} \textbf{ then} 11: \quad \quad \text{Train } \text{conformalModels} \text{ using } N_{other_j}'s \text{ state-action mappings; for } j \in [1, K] 12: \quad \quad \text{Train } N_{self} \text{ using PPO} 13: \quad \quad \text{Train } N_{other_j} \text{ using PPO; for } j \in [1, K] 14: \quad \textbf{end if} 15: \quad \textbf{end for} 16: \textbf{end for} ``` Now we formally describe our proposed algorithm — Conformal Action Modeling-based Multi-Agent Reinforcement Learning or CAMMARL. Our objective is to inform an $ N_{self} $'s decision-making by modeling the other agent’s actions in the environment as conformal prediction sets that contain the true actions with a high probability (for example, 95%). More specifically, $ N_{self} $ uses a separate conformal action prediction model to obtain sets of $ N_{other_j} $'s actions at each timestep that contains latter’s true action in the environment at a given time step with high prespecified probabilities. Algorithm 1 describes the complete workflow of training of agents in CAMMARL. We begin by initializing the actor-critic networks for both the agents in the environment, the conformal model, and the memory buffers for each of these. Now, at the beginning of each episode in the environment, both agents receive their own partial observations (line 4). Next, the conformal model predicts the actions of all the $ N_{other_j} $'s in the form of a set, which is then provided as an additional input to $ N_{self} $ (lines 6–7), whereas $ N_{other_j} $ has access to only its own partial observation, $ o_{other_j} $. Both agents now take actions in the environment and continue collecting their experiences (lines 8–9). The agents and the conformal model periodically train using their respective experience memory (lines 10–14). --- 2 A tabular version can be found in Section E Figure 2 shows a detailed illustration of our conformal action modeling that materializes internally at each time step. We use only one other agent $ N_{\text{other}} $ for simplicity. The conformal predictor $ C $ collects $ N_{\text{other}} $'s state-action pairs and periodically learns and updates a neural network classifier, $ f(\cdot) : \mathbb{R}^d \rightarrow \mathbb{R}^{\|\mathcal{A}_{\text{other}}\|} $ (where $ d $ is the number of dimensions in $ N_{\text{other}} $'s local observation and $ \|\mathcal{A}_{\text{other}}\| $ is the number of possible discrete actions available for $ N_{\text{other}} $), to predict action from a given state. Then, we adapt RAPS conformal calibration (Angelopoulos et al., 2020) to our setting. Considering $ o \in O_{\text{other}} $ as feature vectors, we use the updated $ f $ to compute action probabilities $ \hat{\pi}_o(a') \in \mathbb{R}^{\|\mathcal{A}_{\text{other}}\|} $. The probabilities are then ordered from most probable to least probable followed by the estimation of the predictive action set for the given feature inputs. To promote small predictive sets we also add a regularization term as also proposed in RAPS. Formally, for a feature vector $ o $ and the corresponding possible prediction $ a $, to estimate a set which includes all the actions that will be included before $ a $, let us define the total probability mass of the set of actions that are more probable than $ a $: \[ \rho_o(a) = \sum_{a' \neq a} \hat{\pi}_o(a') \mathbbm{1}_{\{\hat{\pi}_o(a') \geq \hat{\pi}_o(a)\}} \] Also, if we define a function to rank the possible action outcomes based on their probabilities $ \hat{\pi} $ as \[ z_o(a) = \|\{a' \in \mathcal{A}_{\text{other}} : \hat{\pi}_o(a') \geq \hat{\pi}_o(a)\}\| \] we can then estimate a predictive action set as follows: \[ C^(o) := \{a : \rho_o(a) + \hat{\pi}_o(a) \cdot u + \lambda \cdot (z_o(a) - k_{\text{reg}})^+ \leq \tau\} \] where $(x)^+$ denotes the positive portion of $ x $, and $ \lambda, k_{\text{reg}} \geq 0 $ are regularization hyperparameters to incentivize small set sizes. Here, $ u \sim \text{uniform}[0, 1] $ (used to allow for randomized procedures) and the tuning parameter $ \tau $ (the cumulative sum of the classifier scores after sorting and penalization) to control the size of the sets are identical to as used in RAPS for supervised tasks (Angelopoulos et al., 2020). To summarize, in CAMMARL, $ N_{\text{self}} $ gets to use estimates of $ N_{\text{other}} $'s actions at each time step to make informed decisions in the environment. Instead of modeling exact actions with no uncertainty... estimation, we prefer to produce an action set carrying desirable guarantees of containing $ N_{\text{other}} $'s true action with high probability, integrate it into an agent’s downstream task, and enable improved decision-making and collaboration with $ N_{\text{other}} $. 4 EXPERIMENTS In this section, we discuss the cooperative tasks with two agents used in this study (Figure 3). We note here that though we work in fully cooperative settings in this article, CAMMARL as an idea can be generalized to competitive or mixed settings too. 4.1 ENVIRONMENTS We focus on two cooperative multi-agent environments illustrated in Figure 3a & 3b. Further details on these environments are in Section A. ![Figure 3](image) Figure 3: Multi-agent cooperative environments used in this study: (a) OpenAI MPE’s Cooperative Navigation: Agents (blue) learn to cover the two landmarks (black) avoiding collisions. The figure shows cooperative navigation with 2 agents (N=2) and 2 landmarks (L=2). and (b) Level-based foraging: Agents must collect food and learn to cooperate using sparse rewards. This is a 12 × 12 level-based foraging grid-world with 2 cooperative players and 4 food locations. 4.2 RESULTS To show the benefits of conformal action set prediction, we compare CAMMARL with the performances of agents in different settings with varying pieces of information made available to $ N_{\text{self}} $ during training. No-Other-Agent-Modeling (NOAM).* At first, we train $ N_{\text{self}} $ without allowing it to model $ N_{\text{other}} $. This baseline, as expected, underperforms when compared to any other settings (where any kind of agent modeling is allowed). It is indicative of a lower bound to the learning performance of our model where no kind of benefit from agent modeling is utilized by $ N_{\text{self}} $. Results are shown in Figure 4. We call this baseline — No-Other-Agent-Modeling or NOAM. True-Action-Agent-Modeling (TAAM). Advancing from the inputs available in NOAM, we implement TAAM by allowing $ N_{\text{self}} $ to additionally utilize $ N_{\text{other}} $'s true actions to train. This baseline helps us evaluate CAMMARL against works that estimate other agents’ actions in the environment and use those predictions to enhance the decision-making of their controlled autonomous agents. By giving the true actions as inputs, this baseline can act as an upper bound to such works [He et al., 2016; Grover et al., 2018; Zintgraf et al., 2021; Mealing & Shapiro, 2015; Panella & Gmytrasiewicz, 2017; Albrecht & Ramamoorthy, 2015]. Figure 4 shows TAAM’s performance curve. Using additional information, it does reasonably better than NOAM in both tasks. True-Observation-Agent-Modeling (TOAM). As discussed in Section 2, learning world models often involves reconstructing observation as an additional task while learning task-related policies [Jain et al., 2022; Hafner et al., 2020, 2021]. Inspired by this research, we implement the TOAM baseline where we allow access to $ N_{\text{other}} $'s true observations to $ N_{\text{self}} $ during training and execution. In other words, we augment $ N_{\text{self}} $'s partial observations with the other agent’s local observations too. This baseline can act as an upper bound to the performances of research works that learn to reconstruct states for Figure 4: Comparison of agent performances (in terms of reward accumulation) in CN and LBF in different settings with varying pieces of information available to $N_{self}$ during training. CAMMARL’s performance is very close to the upper bound, GIAM, and is considerably better than the other extreme, NOAM. It also outperforms the other defined benchmarks (TAAM, TOAM, & EAP) in both tasks, along with the benefit of uncertainty quantification of its estimates. Interestingly, in CN, CAMMARL can be seen to learn arguably faster, but all methods converge to similar results, whereas in LBF, it actually seems to converge to a better policy. The curves are averaged over five independent trials and smoothed using a moving window average (100 points) for readability. agents (Hafner et al., 2020; 2021; Wang et al., 2022; Park et al., 2019b; Zhang et al., 2021). As expected, $N_{self}$ performs considerably better than NOAM in both environments (Figure 4). Here, the difference in returns in TOAM and TAAM can be attributed to the fact that the local observations of other agents include more information that can be useful to infer their behavior. For instance, in CN, knowing the relative positions of other agents with respect to the landmarks can be more useful to infer which landmark that agent might be approaching when compared to knowing its current (or history) actions. Global-Information-Agent-Modeling (GIAM). On the other extreme, we also implement GIAM, where $N_{self}$ trains with complete access to both (1) $N_{other}$’s true action trajectories ($a_{other}$), and (2) $N_{other}$’s true observations ($o_{other}$) as additional information. Figure 4 shows that GIAM achieves higher returns compared to all other settings in both environments. This is intuitive because it benefits from more information. GIAM is conditioned on $N_{other}$’s true experiences and consequently demands access to them even during execution. This can be infeasible in real-world scenarios, however, theoretically represents an upper bound on the performance of agents in CAMMARL and other settings. Exact-Action-Prediction (EAP). Building over the inputs of TOAM, we construct a stronger baseline, EAP, in which $N_{self}$ uses an additional neural network classifier to model a probability distribution over $N_{other}$’s actions. In other words, instead of predicting conformal sets of actions (like in CAMMARL), in this baseline, $N_{self}$ tries to model $N_{other}$’s actions from the latter’s observations without accounting for any uncertainty quantification. This baseline is inspired by works that explicitly model the other agent’s actions in the environments and utilize them to inform their controlled agent’s decision-making (for instance, He et al., 2016; Grover et al., 2018; Zintgraf et al., 2021). Hence, here, a cross-entropy loss is used to train the added sub-module that predicts the $N_{other}$’s actions along with a PPO loss to train $N_{self}$’s policy network. Figure 4 shows that CAMMARL agents are able to distinctly perform better than EAP in LBF, however, interestingly, the performance curve for this baseline nearly overlaps with CAMMARL in CN. Also, in LBF, the curves for TOAM and EAP seem to significantly overlap. We speculate that in a complicated task like LBF, estimating the exact action of $N_{other}$ can be difficult, and with unaccounted uncertainty in the predictions, $N_{self}$ suffers from a lower return. In CN, which is comparatively simpler, the closeness of returns in EAP and CAMMARL seem reasonable as even the conformal model predictions eventually start predicting the most probable action with higher probabilities and hence a set of size one (more on this in Section 6). CAMMARL. Now, we implement CAMMARL, where the conformal action prediction model periodically trains on collected observations of $N_{other}$ and predicts a corresponding conformal set of actions. $N_{self}$ uses these estimates of $N_{other}$’s actions along with its own observations and then decides upon its actions in the environment. Figure 4 shows that CAMMARL agents obtain returns that are much closer to the upper bound, GIAM, than the lower bound, NOAM. Furthermore, CAMMARL’s better performance compared to TOAM in both environments can be attributed to the fact that it can be difficult to predict the $N_{other}$’s intentions by only using $o_{other}$ without any information pertaining to its actions in those situations. And, in TAAM, $N_{self}$ is expected to implicitly encode information regarding $N_{other}$’s observations from its own local observations or in the latent space and map it to $N_{other}$’s true actions. We speculate that this could be a strong assumption and consequently very difficult, hence, CAMMARL agents outperform TAAM too. Note here that the sets output by the conformal action prediction model are of varying sizes in each iteration. Now, to be able to use these dynamically changing inputs for $N_{self}$ in CAMMARL, we convert the output sets to a corresponding binary encoding (by firing up the bits in a zero vector at indices corresponding to the actions predicted by the model). We discuss some more ways to be able to use conformal prediction sets with dynamic sizes and compare CAMMARL’s performances in all variations later in the supplementary. In summary, through experiments in two complex cooperative tasks, we show that (1) CAMMARL indeed works, (2) it outperforms common settings like NOAM, TOAM, and TAAM which assume the availability of other agents’ true trajectories during training and execution (generally infeasible in real-world scenarios), (3) Its performance is closest to our upper bound of performance (GIAM), (4) CAMMARL agents learn their policies faster than the other baselines, and (5) CAMMARL can be preferred over strong benchmarks such as EAP owing to its higher interpretability due to the theoretical guarantees of conformal predictions in terms of coverage (Angelopoulos et al., 2020) (discussed more in Section 6). 5 Motivating Conformal Predictions In Figure 4a and 4b, we observe the improved performance of CAMMARL by predicting conformal sets. One key ingredient to CAMMARL was the addition of uncertainty predictions to the actions of the other agents. Thus, we can attribute the increased performance of CAMMARL to this as well. To test this theory, we added one more baseline where similar to the action prediction baseline (EAP), we add the action prediction probabilities directly into the state space. We call this Action Prediction with Uncertainty (APU), and both APU and CAMMARL operate on the same information. From Figure 5, we can conclude that just adding uncertain predictions is not enough to achieve an uplift in performance that we see for CAMMARL and there is definite merit to using conformal predictions. Another reason for the poor performance of APU would be that the agent has to parse through the relevant information and learn from it, whereas they are readily provided in a concise manner for CAMMARL. 6 Discussion In this section, we dig deeper and try to analyze the inner components of CAMMARL. In particular, we plot some observable trends during the training of CAMMARL’s agents in both the tasks (Figure 6) and discuss each of them here. Set Sizes. We collected the set sizes produced in CAMMARL throughout the training and report them in Figure 6a and 6e. Smaller sets are preferred, as they carry specific information which can be more useful practically. The curves show a decreasing trend in the set sizes in CAMMARL in both CN and LBF respectively when tracked over the number of updates of the conformal prediction. model during training. This is a good sign for CAMMARL, as it shows that the conformal predictions are becoming more precise with continued training over time. ![Graphs showing trends in CN and LBF metrics](image) Figure 6: Analysing conformal prediction in CAMMARL over time during the training by looking at trends in conformal sets sizes, coverage of highly probable predictions, model loss and accuracy during training of CAMMARL agents. Coverage. As also discussed earlier, it is desirable for the predicted sets to provide $1 - \alpha$ coverage for a pre-defined user-specified $\alpha$ such as 10%. Formally, to map a feature vector, $o_{\text{other}} \in O_{\text{other}}$, to a subset of discrete responses, $a'_{\text{other}} \in A_{\text{other}}$, it is useful to define an uncertainty set function, $C(o_{\text{other}})$, such that $P(a'_{\text{other}} \in C(o_{\text{other}})) \geq 1 - \alpha$. Figure 6b and 6f shows the increasing trend of confidence coverage in CAMMARL. Model accuracy and loss. In Figure 6c and Figure 6d we show the conformal model’s accuracy and loss respectively for CN and LBF in Figure 6g and Figure 6h. The model accuracy, as expected, increases with more data coming in to train over time and the loss correspondingly decreases. 7 CONCLUSION In this article, we propose a novel MARL algorithm, CAMMARL, which calls for confident reasoning about other artificial agents in the environment and benefiting from inferences about their behavior. Through experiments in two cooperative multi-agent tasks, CN and LBF, we showed that guiding an agent’s decision-making by inferring other agents’ actions in the form of conformal sets, indeed helps in achieving better performances of the learning agents. By using conformal prediction, we were also able to ensure the estimation of predictive sets that covered the real predictions of the intentions of other agents with a very high pre-specified probability of 95%. Limitations and Future Works: In our paper, we analyzed CAMMARL with two agents, however, CAMMARL is certainly generalizable to bigger networks or more simple classifiers, and analyzing its changing performance on varying buffer sizes can help in better comprehending its efficiency. Second, it would be interesting to investigate the CAMMARL’s scalability to a system of many agents (say 100 or 1000) or on more complicated multi-agent environments such as tasks requiring a higher need for coordination. Thirdly, our mathematical model in Section 3.1 makes an assumption that the state space is accessible globally which may not be the case in some problems. Finally, in this work, we restricted the agents to infer the behavior of other agents only via conformal sets; it would be interesting to study the cases where more ways of sharing information or modeling agents’ behavior are additionally allowed. 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FOSBQuXgAq	If I understand the paper correctly, the estimates of the BNN posteriors come from the parameters of a 1000-member ensemble trained using maximum likelihood. Can we be sure this is a faithful representation of the posterior? For instance, do we know anything about the higher-order moments of those modes?	A Symmetry-Aware Exploration of Bayesian Neural Network Posteriors Olivier Laurent,1,2 Emanuel Aldea1 & Gianni Franchi2,† SATIE, Paris-Saclay University,1 U2IS, ENSTA Paris, Polytechnic Institute of Paris2 Abstract The distribution of modern deep neural networks (DNNs) weights – crucial for uncertainty quantification and robustness – is an eminently complex object due to its extremely high dimensionality. This paper presents one of the first large-scale explorations of the posterior distribution of deep Bayesian Neural Networks (BNNs), expanding its study to real-world vision tasks and architectures. Specifically, we investigate the optimal approach for approximating the posterior, analyze the connection between posterior quality and uncertainty quantification, delve into the impact of modes on the posterior, and explore methods for visualizing the posterior. Moreover, we uncover weight-space symmetries as a critical aspect for understanding the posterior. To this extent, we develop an in-depth assessment of the impact of both permutation and scaling symmetries that tend to obfuscate the Bayesian posterior. While the first type of transformation is known for duplicating modes, we explore the relationship between the latter and L2 regularization, challenging previous misconceptions. Finally, to help the community improve our understanding of the Bayesian posterior, we release the first large-scale checkpoint dataset, including thousands of real-world models, along with our code. Figure 1: Weight-space symmetries greatly impact the estimated Bayesian posterior. Permutation symmetries clearly increase the number of modes of the posterior distribution in the case of the last layer of a 2-hidden neuron perceptron, as detailed in Section 3.1. 1 Introduction Despite substantial advancements in deep learning, Deep Neural Networks (DNNs) remain black box models. Various studies have sought to explore DNN loss landscapes (Li et al., 2018; Fort & Jastrzebski, 2019; Fort & Scherlis, 2019; Liu et al., 2022) to achieve a deeper understanding of these models. Recent works have, for instance, unveiled the interconnection of the modes obtained with Stochastic Gradient Descent (SGD) via narrow pathways that link pairs of modes, or through tunnels that connect multiple modes simultaneously (Garipov et al., 2018; Draxler et al., †corresponding author – gianni.franchi@ensta-paris.fr This mode connectivity primarily arises from scaling and permutation invariances, which imply that numerous weights can represent the same exact function (e.g., Entezari et al. (2022)). Several studies have delved into the relationship between these symmetries and the characteristics of the loss landscape (Neyshabur et al., 2015; Brea et al., 2019; Entezari et al., 2022). Our work investigates the connections between these symmetries and the distribution of DNN weights, a crucial aspect for uncertainty quantification. These connections are highlighted in Figure 1. Uncertainty quantification plays a pivotal role in high-stakes industrial applications – such as autonomous driving (Levinson et al., 2011; McAllister et al., 2017; Sun et al., 2019) – where reliable predictions and informed decision-making are paramount. In such critical domains, understanding and effectively managing uncertainties, particularly the model-related epistemic uncertainties (Hora, 1996) arising from incomplete knowledge, is essential. Amongst the various methods introduced to address these challenges, Bayesian Neural Networks (BNNs) (Tishby et al., 1989) offer a principled and theoretically sound approach. BNNs quantify uncertainty by probabilistically modeling beliefs about parameters and outcomes (Tishby et al., 1989; Hinton & Van Camp, 1993). However, this perspective faces significant hurdles when applied to deep learning, primarily related to scalability (Izmailov et al., 2021) and the precision of approximations (MacKay, 1995). Due to their very high dimension, BNNs struggle to estimate the posterior distribution, i.e., the probability density that any set of model parameters/hypothesis $\omega$ generated the observed data $D$ with a given prior. Diverging from methods such as the Maximum Likelihood Estimate or Maximum A Posteriori (also Tishby et al. (1989)), which we typically derive through gradient descent optimization of cross-entropy (with L2 regularization for the latter), BNNs assign a probability to each possible model (or hypothesis) and offer predictions considering the full extent of possible models. In mathematical terms, denoting the target as $y$, the input vector as $x$, and the weight space as $\Omega$, we can express this approach through the following intractable formula, often referred to as the marginalization on the parameters of the model (Tishby et al., 1989; Rasmussen et al., 2006): $$p(y \mid x, D) = \int_{\omega \in \Omega} p(y \mid x, \omega)p(\omega \mid D)d\omega.$$ (1) The posterior distribution $p(\omega \mid D)$ assumes a central and arguably the most critical role in BNNs – and many successful methods for quantifying uncertainty can be viewed as attempts to approximate this posterior, each with its own trade-offs in terms of accuracy and computational efficiency, as illustrated in previous research (Blundell et al., 2015; Gal & Ghahramani, 2016; Lakshminarayanan et al., 2017). While prior work (Kuncheva & Whitaker, 2003; Fort et al., 2019; Ortega et al., 2022) has established the importance of achieving diversity in the sampled DNNs drawn from the posterior – particularly when dealing with uncertain input data – permutation and scaling symmetries amongst hidden units in neural networks may lead to an increased number of local minima (Zhao et al., 2023) with no diversity. In the context of BNNs, this phenomenon could result in a proliferation of functionally equivalent modes within the posterior distribution reducing the diversity within the inevitably limited number of samples, and degrading the quality of the uncertainty estimates. This paper delves into the impact of weight symmetries on the posterior distribution. While there have been numerous efforts to visualize the loss landscape, we explore the possibility of conducting similar investigations for the posterior distribution. Additionally, we introduce a protocol for assessing the quality of posterior estimation and examine the relationship between posterior estimation and the accuracy of uncertainty quantification. Specifically, our contributions are as follows: 1. We build a new mathematical formalism to highlight the different impacts of the permutation and scaling symmetries on the posterior and uncertainty estimation in DNNs. Notably, we explain the seeming equivalence of the marginals in Figure 1. We also perform the first in-depth exploration of the existence of scaling symmetries and their overlooked effect. 2. We evaluate the quality of various methods for estimating the posterior distribution on real-world applications using the Maximum Mean Discrepancy, offering a practical benchmark to assess their performance in capturing uncertainty. 3. We release Checkpoints, a new dataset including the weights of thousands of models across various computer vision tasks and architectures, ranging from MNIST to TinyImageNet. This dataset is intended to facilitate further exploration and collaboration in the field of uncertainty in deep learning. 4. Our investigation delves into the proliferation of modes in the context of posterior symmetries and exhibits the capacity of ensembles to converge toward non-functionally equivalent modes. Furthermore, we discuss the influence of symmetries in the training process. 2 RELATED WORK Epistemic uncertainty, Bayesian inference, and posterior. Epistemic uncertainty (Hora, 1996; Hüllermeier & Waegeman, 2021) plays a crucial role in accurately assessing predictive model reliability. However – and despite ongoing discussions – estimating this uncertainty remains a challenge. BNNs (Goan & Fookes, 2020) predominantly shape the landscape of methodologies that tackle epistemic uncertainties (Gawlikowski et al., 2023). Given the complexity of dealing with posterior distributions, these approaches have mostly been tailored for enhanced scalability. For instance, Hernández-Lobato & Adams (2015) introduced an efficient probabilistic backpropagation, and Blundell et al. (2015) developed BNNs by backpropagation to learn diagonal Gaussian distributions with the reparametrization trick. Similarly, Laplace methods (MacKay, 1992) estimate the posterior distribution, thanks to an approximation of the local curvature of the loss. They often focus on the final layer (Ober & Rasmussen, 2019; Watson et al., 2021), again for scalability. On a different approach, Monte Carlo Dropout, introduced by Gal & Ghahramani (2016) and Kingma et al. (2015), is a framework that, applied to fully-connected layers, models the posterior as a mixture of Dirac distributions. Broadening the spectrum, deep ensembles (Lakshminarayanan et al., 2017), arguably along with their more computationally efficient alternatives (Wen et al., 2019; Maddox et al., 2019; Franchi et al., 2020; 2023; Havasi et al., 2021; Laurent et al., 2023), have been interpreted by Wilson & Izmailov (2020) as Monte Carlo estimates of Equation 1. Markov-chain-based Bayesian posterior estimation. Neal et al. (2011) introduced Hamiltonian Monte Carlo (HMC) – based on Monte Carlo Markov Chains (MCMC) – as an accurate method for estimating distributions, but its application to large-scale problems, such as the posterior of modern DNNs, remains challenging due to its exceptionally high computational demands. In response to these challenges, stochastic approximations of MCMC have gained attention for their ability to provide computationally feasible solutions. A prominent example is the stochastic version of Langevin dynamics (Roberts & Tweedie, 1996) by Welling & Teh (2011). By adding noise into the dynamics, stochastic Langevin allows for more practical implementation on large datasets. In addition, other stochastic gradient-based methods have been introduced to improve the efficiency of MCMC sampling. Chen et al. (2014) presented Stochastic Gradient Hamiltonian Monte Carlo (SGHMC), and Zhang et al. (2020) designed C-SGLD and C-SGHMC (Cyclic Stochastic Gradient Langevin Dynamics), introducing controlled noise via cyclic preconditioning. While stochastic approximation methods offer computational convenience, they come with the trade-off of slowing down the convergence and potentially introducing bias into the resulting inference (Bardenet et al., 2017; Zou & Gu, 2021). As such, the suitability of these approaches depends on the specific application and the level of acceptable bias in the analysis. Izmailov et al. (2021) estimated the Bayesian posterior scaling full-batch HMC to CIFAR-10 thanks to 512 TPUv3. While we also estimate posteriors, we select another more scalable method supported and compared to HMC in Appendix B. Furthermore, we bring a novel focus that remained mostly uncharted: theoretically and empirically quantifying the impact of symmetries on the posterior. Symmetries in neural networks. The seminal work from Hecht-Nielsen (1990) established a foundational understanding by investigating permutation symmetries and setting a lower bound on symmetries in multi-layer perceptrons. Albertini et al. (1993) extended this work and studied flip-sign symmetries in neural networks with odd activation functions. These works were further generalized to a broader range of activation functions by Kůrková & Kainen (1994), who suggested symmetry removal to streamline evolutionary algorithms. Recent advancements have generalized symmetries to modern neural architectures. Neyshabur et al. (2015) explored the scaling symmetries that arise in architectures containing non-negative homogeneous activation functions, including Nair & Hinton (2010)’s ubiquitous Rectified Linear Unit (ReLU). This perspective extends our understanding of symmetries to ReLU-powered architectures, e.g., AlexNet (Krizhevsky et al., 2012), and ResNet architectures (He et al., 2016). This paper focuses on scaling and permutation symmetries, but other works, such as Rolnick & Kording (2020); Grigsby et al. (2023), unveil less apparent symmetries. Closer to our work, Wiese et al. (2023) demonstrated that taking weight-space symmetries into account could reduce the support of the Bayesian posterior and improve MCMC posterior estimation. 3 SYMMETRIES INCREASE THE COMPLEXITY OF BAYESIAN POSTERIORS We study scales and permutations, the most influential weight-space symmetries and their properties related to the Bayesian posterior. Since there is no posterior without prior, we advise the reader that we will work on maxima a posteriori and take the most common weight-prior amongst practitioners, the Gaussian prior on the weights, which is equivalent to L2 regularization. We detail the role of the priors in Appendix D.4. Now, let us start with a definition: weight-space symmetries transform the parameters of the neural networks while keeping the networks functionally invariant. Definition 3.1. Let $ f_\omega $ be a neural network of parameters $ \omega $ taking $ n $-dimensional vectors as inputs. We say that the transformation $ T $ modifying $ \omega $ is a weight-space symmetry operator, iff \[ f_T(\omega) = f_\omega, \text{ or } \forall x \in \mathbb{R}^n, \quad f_T(\omega)(x) = f_\omega(x). \] (2) With the notation $ f_T(\omega)(x) $, we apply the symmetry operator $ T $ on the weights $ \omega $, resulting in a set of modified weights. In the following, we show that scaling and permutation symmetries have different impacts on the posterior of neural networks. They can, for instance, complicate Bayesian posteriors, creating artificial functionally equivalent modes. 3.1 AN INTRODUCTORY EXAMPLE OF ARTIFICIAL SYMMETRY-DRIVEN POSTERIOR MODES To illustrate the considerable impact of symmetries on the Bayesian posterior, we showcase a small-scale classification example in Figure 1. We generate this example by training two-hidden-neuron perceptrons on linearly separable data. The figure presents the estimation of a bivariate marginal of the posterior of the output weights with 10,000 independently-trained samples (left) when successively removing scaling (center) and then permutation symmetries (right). This figure shows that the scaling symmetries seem to disperse the points from the modes and that the most important mode is duplicated due to the (here, unique) permutation symmetry, which symmetrizes the graph. We detail this toy experiment in Appendix A. In the following, we develop a new mathematical framework tailored to help understand the impact of these symmetries on the posterior, devise mathematical insights explaining these intuitions, and explore more empirical dimensions. 3.2 BACKGROUND AND DEFINITIONS The full extent of this formalism (including sketches of proofs, other definitions, properties, and propositions) is developed in Appendix E. Here, we summarize the minimal information to understand the impact on the Bayesian posterior of the two main symmetries – scales and permutations. This part summarizes the most important results for multi-layer perceptrons, but we provide leads for generalizing our results to modern DNNs such as convolutional residual networks in Appendix D.11. 3.3 SCALING SYMMETRIES For clarity, the following definitions and properties are provided for two-layer fully connected perceptrons, without loss of generality. We first denote the line-wise and column-wise as $ \triangledown $ and $ \triangleright $, respectively (see Definition E.2). Given that the rectified linear unit $ r $ is non-negative homogeneous – i.e., for all non-negative $ \lambda $, $ r(\lambda x) = \lambda r(x) $ – we have the following core property for scaling symmetries (Neyshabur et al., 2015), trivially extendable to additive biases: Property 3.1. For all $ \theta \in \mathbb{R}^{m \times m}, \omega \in \mathbb{R}^{m \times n}, \lambda \in (\mathbb{R}_{>0})^m $, \[ \forall x \in \mathbb{R}^n, \quad (\lambda^{-1} \triangledown \theta) \times r(\lambda \triangleright \omega x) = \theta \times r(\omega x). \] (3) Denoting the transformation of Equation 3 by $ T_s $ – in the case of a two-layer perceptron – the core property directly follows, with the set of parameters $ \Lambda = \{\lambda\} $: Property 3.2. For any usual neural network with non-negative homogenous activations $ f_\omega $, the scaling operation $ T_s $ with a set of non-negative parameters $ \Lambda $ is a symmetry, i.e., $ \forall x \in \mathbb{R}^n, \quad f_{T_s(\omega,\Lambda)}(x) = f_\omega(x) $. 3.4 Permutation symmetries We also present an intuitive formalism for permutation symmetries, multiplying the weights by permutation matrices. For two-layer perceptrons, with $ P_m $ the set of permutation matrices, we have that: Property 3.3. For all $ \theta \in \mathbb{R}^{m \times m}, \omega \in \mathbb{R}^{n \times n} $, and permutation matrices $ \pi \in P_m $, \[ \forall x \in \mathbb{R}^n, \quad \theta \pi^\top \times r(\pi \times \omega x) = \theta \times r(\omega x). \] (4) The left term of Equation 4 is the definition of the permutation symmetry operator of parameter $ \Pi = \{\pi\} $ for a network including two layers. In general, we have the following property: Property 3.4. For any usual neural network $ f_\omega $, the permutation operation $ T_p $ with a set of parameters $ \Pi $ is a symmetry, i.e., $ \forall x \in \mathbb{R}^n, \quad f_{T_p(\omega, \Pi)}(x) = f_\omega(x) $. 3.5 The Bayesian posterior as a mixture of distributions With this formalism, we can establish the following proposition, a formalization and extension of Kurle et al. (2021), clarifying the impact of weight-space symmetries on the Bayesian posterior. Proposition 1. Define $ f_\omega $ a neural network and $ f_{\tilde{\omega}} $ its corresponding identifiable model – a network transformed for having sorted unit-normed neurons. Let us also denote $ \Pi $ and $ \Lambda $, the sets of permutation sets and scaling sets, respectively, and $ \tilde{\Omega} $ the random variable of the sorted weights with unit norm. The Bayesian posterior of a neural network $ f_\omega $ trained with stochastic gradient descent can be expressed as a continuous mixture of a discrete mixture: \[ p(\Omega = \omega \mid D) = \int_{\Lambda \in \Lambda} \|\Pi\|^{-1} \sum_{\Pi \in \Pi} p(\tilde{\Omega} = T_p(T_s(\omega, \Lambda), \Pi), \Lambda \mid D) d\Lambda. \] (5) Proposition 1 provides an expression of the Bayesian posterior that highlights the redundancy of the resulting distribution, explaining the symmetry in Figure 1 (left). Interestingly, a direct corollary of this formula is that layer-wise, all marginal inbound posteriors are identical. This has practical consequences: in Appendix B, we show that HMC-based posterior estimation breaks this corollary. In Equation 5, the permutations play a transparent role, being independent of $ \omega $ (except for strongly quantized spaces for $ \omega $ that we leave for future works). On the other hand, the part played by scaling symmetries is more complex, and we discuss their impact in the following section. 3.6 On the effective impact of scaling symmetries While the equiprobability of permutations in Equation 5 leads to a simple balanced mixture of $ \|\Pi\| $ permuted terms, we have no such result on scaling symmetries since the standard L2-regularized loss is not invariant to scaling symmetries (and the initialization is not “uniform”). This absence of invariance obscures the impacts of scaling symmetries, which mostly remains to be addressed, although the “reduction” of their effect due to regularization was mentioned in, e.g., Godfrey et al. (2022). To the best of our knowledge, we provide the first analysis of the tangibility of scaling symmetries and their impact on the Bayesian posterior. With this objective in mind, we define the following problem. Definition 3.2. Let $ f_\omega $ be a neural network and $ \tilde{\omega} $ its weights without the biases. We define the scaled network representation cost problem (or the scaled-representation problem) as the minimization of the L2-regularization term of $ f_\omega $ (the “mass”) under scaling transformations. In other words, \[ m^* = \min_{\Lambda \in \Lambda} \|T_s(\tilde{\omega}, \Lambda)\|^2. \] (6) This problem – a restriction of the representation cost minimization, e.g. (Jacot, 2022) – has interesting properties. Notably, we show in Appendix E.5 that the scaled-representation problem is log-log strictly convex (Boyd & Vandenberghe, 2004; Agrawal et al., 2019): Proposition 2. The scaled-representation problem is log-log strictly convex: it is equivalent to a strictly convex problem on $ \mathbb{R}^{\|\Lambda\|} $ and admits a single global minimum attained at $ \Lambda^* $. It follows from Proposition 2 that, if not already optimal at convergence, there is an infinite number of equivalent networks with training loss lower than the original network. We put the proposition into practice in Figure 2 using trained OptuNets (see Appendix C.2.1): we measure their mass distribution and compare them to the masses of the optimal networks found with convex optimization. Figure 2: OptuNets trained with weight decay never converge to the minimum scaled representation. We also note that the maxima of the weights of scaled OptuNets at the minimum scaled representation – referred to as “at opt.” in (right) – tend to be greater for layers with fewer parameters than in the original networks. The effect of scaling symmetries remains even with weight decay: neural networks seem to persist in being subject to scaling symmetries as shown in Figure 2 (left). Figure 2 (center) depicts that the ratios of the mass at convergence on the minimum representation are non-negligible, the converged networks being consistently heavier. Figure 2 (right) displays the values of the largest elements of each layer: the minimization of the mass tends to increase the heaviest weights of the layers with fewer parameters (here, the convolutional layers) but does not seem to promote unfeasible values. Finally, we provide a loss landscape interpretation of this property in Appendix D.5 and explain the inability of DNNs to converge to the minimum mass by its corresponding gradient being lower than SGD noise. We generalize this result to ResNet-18 on CIFAR-100 in Appendix D.3 and Figure 11. In this case, the networks at minimal representation costs have extremely low mass weights due to the sequences of convolution and batch normalization layers (see Section D.11.3). Until now, we provided theoretical insights on the contrasted impacts of both scaling and permutation symmetries. In the following, we develop more empirical studies and explore the link between posterior quality and performance. 4 COMPARING POSTERIOR ESTIMATIONS BY APPROXIMATE BAYESIAN METHODS In this section, we leverage symmetries to compare popular single-mode methods, namely, Monte Carlo Dropout (Gal & Ghahramani, 2016), Stochastic Weight Averaging Gaussian (SWAG) by Maddox et al. (2019), variational inference BNNs (viBNNs) (Blundell et al., 2015), and Laplace methods (Ritter et al., 2018). We also include their multi-modal variations, corresponding to the application of these methods on ten different independently trained models, as well as SGHMC (Chen et al., 2014), preconditioned SGLD (Li et al., 2016) and deep ensembles (DE) highlighted by Hansen & Salamon (1990) and Lakshminarayanan et al. (2017). We compare these methods on three image classification tasks with different levels of difficulty, ranging from MNIST (LeCun et al., 1998) with OptuNet (392 parameters) to CIFAR-100 (Krizhevsky, 2009) and Tiny-ImageNet (Deng et al., 2009) with ResNet-18 (He et al., 2016). To this extent, we leverage maximum mean discrepancies (MMD) to estimate the dissimilarities between the high-dimensional posterior distributions. We estimate the target posterior using 1000 independently trained neural networks and compare it to 100 samples from all previously mentioned techniques. This choice – compared to the more theoretically-grounded HMC (Neal et al., 2011; Izmailov et al., 2021) – is supported by theoretical aspects on the estimation of high-dimensional distributions (Wild et al., 2023), by computational constraints that would shatter mini-batched HMC’s guarantees in practice, and by experiments showing that full-batch HMC’s theoretical performance may not be fully achieved in our real-world settings. However, we stress that sampling from independently trained checkpoints to estimate the posterior remains imperfect and can be debated. We discuss these limitations in-depth in Appendix B. 4.1 EVALUATING THE QUALITY OF THE ESTIMATION OF THE BAYESIAN POSTERIOR One approach to assess the similarities between distributions involves estimating the distributions and subsequently quantifying the distance between these estimated distributions (Smola et al., Table 1: Comparison of popular methods approximating the Bayesian posterior. All scores are expressed in %, except the ACEs and the MMDs for ResNet-18 networks, expressed in %c. Acc stands for accuracy, and IDMI and OODMI are in-distribution and out-of-distribution mutual information. NS is the MMD computed after removing the symmetries, and DE stands for Deep Ensembles. Multi-mode methods are based on ten independently trained models. \| Method \| MMD ↓ \| NS ↓ \| Acc ↑ \| ECE ↓ \| ACE ↓ \| Brier ↓ \| AUPR ↑ \| FPR95 ↓ \| IDMI ↓ \| OODMI ↑ \| \|--------\|-------\|------\|-------\|-------\|-------\|---------\|--------\|---------\|--------\|--------\| \| Dropout \| 15.0 \| 14.3 \| 83.3 \| 26.1 \| 60.0 \| 33.4 \| 96.4 \| 98.6 \| 26.1 \| 22.2 \| \| viBNN \| 18.8 \| 17.1 \| 78.1 \| 7.4 \| 17.6 \| 30.9 \| 67.9 \| 93.7 \| 0.1 \| 0.1 \| \| SWAG \| 16.0 \| 14.6 \| 88.3 \| 4.9 \| 11.9 \| 17.7 \| 73.4 \| 68.6 \| 4.0 \| 8.7 \| \| Laplace \| 10.6 \| 9.5 \| 87.9 \| 4.8 \| 15.1 \| 18.1 \| 48.2 \| 74.6 \| 6.2 \| 5.9 \| \| SGHMC \| 16.7 \| 17.7 \| 95.1 \| 2.8 \| 3.2 \| 7.6 \| 73.7 \| 98.4 \| 4.3 \| 14.5 \| \| p5GLD \| 15.1 \| 17.3 \| 88.1 \| 3.8 \| 9.1 \| 17.7 \| 49.2 \| 75.5 \| 1.0 \| 0.9 \| \| Dropout \| 2.1 \| 2.1 \| 92.1 \| 36.8 \| 67.5 \| 29.2 \| 97.2 \| 78.2 \| 36.6 \| 52.5 \| \| viBNN \| 2.8 \| 2.5 \| 86.5 \| 17.5 \| 31.3 \| 24.4 \| 96.9 \| 27.2 \| 21.1 \| 52.3 \| \| SWAG \| 1.8 \| 1.3 \| 95.0 \| 17.5 \| 27.6 \| 13.1 \| 88.7 \| 24.6 \| 27.6 \| 62.2 \| \| Laplace \| 1.8 \| 0.8 \| 94.8 \| 15.8 \| 24.5 \| 12.8 \| 95.4 \| 32.1 \| 21.1 \| 52.2 \| \| DE \| 0.0 \| 0.0 \| 95.3 \| 10.9 \| 21.0 \| 13.5 \| 95.7 \| 12.8 \| 19.3 \| 62.6 \| \| Dropout \| 4.5 \| 7.5 \| 74.2 \| 14.7 \| 3.2 \| 38.8 \| 76.4 \| 47.7 \| 5.7 \| 9.1 \| \| viBNN \| 9.0 \| 10.2 \| 57.9 \| 24.6 \| 3.0 \| 63.7 \| 60.9 \| 79.1 \| 2.7 \| 4.2 \| \| SWAG \| 6.7 \| 7.2 \| 70.9 \| 2.3 \| 1.2 \| 38.9 \| 86.2 \| 48.0 \| 2.4 \| 6.3 \| \| Laplace \| 5.7 \| 7.0 \| 75.1 \| 0.9 \| 0.9 \| 34.6 \| 81.3 \| 42.4 \| 27.6 \| 63.3 \| \| SGHMC \| 7.5 \| 7.9 \| 73.7 \| 4.9 \| 1.0 \| 36.2 \| 79.4 \| 62.3 \| 0.2 \| 0.5 \| \| Dropout \| 0.7 \| 4.5 \| 79.5 \| 4.3 \| 1.0 \| 29.2 \| 78.2 \| 48.1 \| 20.5 \| 46.3 \| \| viBNN \| 6.1 \| 5.6 \| 66.5 \| 2.8 \| 2.0 \| 45.3 \| 71.9 \| 71.7 \| 45.5 \| 81.1 \| \| SWAG \| 5.0 \| 5.4 \| 72.8 \| 1.5 \| 1.1 \| 36.9 \| 89.1 \| 50.6 \| 6.5 \| 19.7 \| \| Laplace \| 0.6 \| 4.3 \| 78.9 \| 6.9 \| 0.8 \| 30.3 \| 82.9 \| 41.3 \| 44.1 \| 98.5 \| \| DE \| 0.0 \| 0.0 \| 79.5 \| 1.6 \| 0.6 \| 28.7 \| 81.1 \| 45.6 \| 22.5 \| 58.0 \| \| Dropout \| 9.5 \| 4.9 \| 63.2 \| 16.4 \| 2.4 \| 53.9 \| 48.8 \| 81.1 \| 8.3 \| 8.4 \| \| viBNN \| / \| / \| / \| / \| / \| / \| / \| / \| / \| / \| \| SWAG \| 9.1 \| 3.9 \| 66.4 \| 10.5 \| 0.7 \| 46.2 \| 61.9 \| 57.7 \| 3.0 \| 4.5 \| \| Laplace \| 5.5 \| 6.1 \| 33.1 \| 6.0 \| 3.6 \| 77.1 \| 48.8 \| 77.7 \| 200.7 \| 228.0 \| \| SGHMC \| 9.8 \| 5.3 \| 58.3 \| 2.6 \| 1.0 \| 54.1 \| 56.3 \| 72.7 \| 0.24 \| 0.30 \| \| Dropout \| 4.3 \| 1.8 \| 70.2 \| 9.9 \| 1.2 \| 42.1 \| 74.8 \| 58.2 \| 34.1 \| 60.0 \| \| viBNN \| / \| / \| / \| / \| / \| / \| / \| / \| / \| / \| \| SWAG \| 6.7 \| 5.4 \| 69.3 \| 3.6 \| 0.6 \| 41.3 \| 96.5 \| 55.9 \| 17.6 \| 32.1 \| \| Laplace \| 0.5 \| 3.1 \| 37.0 \| 10.9 \| 3.3 \| 75.1 \| 48.4 \| 72.5 \| 219.5 \| 254.7 \| \| DE \| 0.0 \| 0.0 \| 70.3 \| 6.5 \| 0.7 \| 40.9 \| 86.3 \| 50.2 \| 38.4 \| 83.4 \| However, these methods can become impractical when dealing with distributions in extremely high-dimensional spaces, such as the posterior of modern DNNs. An alternative solution is to embed the probability measures into reproducing kernel Hilbert spaces (RKHS) (Bergmann, 1922; Schwartz, 1964). Within this framework, a distance metric, the maximum mean discrepancy (MMD) (Song, 2008) - defined as the distance between the respective mean elements within the RKHS - is used to quantify the dissimilarity between the distributions. Appendix C.4 formalizes and explains our implementation of the MMD reported in Table 1: we follow Schrab et al. (2023) and use multiple Gaussian and Laplace kernels. For tractability, we report the mean – weighted by the number of parameters of each layer – of the median over twenty MMD kernels between the layer-wise DE-based posterior estimation and the approximation provided by each method. The NS metric corresponds to the MMD computed after a posteriori-symmetry removal using the algorithms detailed in Appendix D.2. Appendix C gathers all details concerning these experiments (including the means and maxima over the kernels of the MMDs). 4.2 Performance metrics and OOD datasets On top of the MMD quantifying the difference between the posterior estimations, we measure several empirical performance metrics. We evaluate the overall performance of the models using the accuracy and the Brier score (Brier, 1950; Gneiting et al., 2007). Furthermore, we choose the binned expected calibration error (ECE) (Naeini et al., 2015) and adaptive calibration error (ACE) (Nixon et al., 2019) for top-label calibration and measure the quality of the out-of-distribution (OOD) detection using the area under the precision-recall curve (AUPR) and the false positive rate at 95% recall (FPR95), as recommended by Hendrycks & Gimpel (2017). We expect the OOD detection abilities... Figure 3: Experiments show no hint of a functional collapse between couples of independently trained ResNet-18 on CIFAR-100. Moreover, the in-distribution and out-of-distribution mutual information (IDMI, resp. OODMI) exhibit different variances but do not seem correlated. of the models to correlate with the quality of the estimated posterior. Finally, we report the mean diversity of the predictions in each ensemble through the mutual information (MI) (e.g., Ash (1965)), often used to measure epistemic uncertainty (Kendall & Gal, 2017; Michelmore et al., 2018). We use FashionMNIST (Xiao et al., 2017), SVHN (Netzer et al., 2011), and Textures (Cimpoi et al., 2014) as OOD datasets for MNIST, CIFAR-100, and TinyImageNet, respectively. 4.3 RESULTS Table 1 demonstrates that multi-mode techniques consistently exhibit superior performance in terms of MMD when compared to their single-mode counterparts. This trend holds true in accuracy, negative log-likelihood, and calibration (ECE and ACE) for ResNet architectures. We provide further details on the calibration performance for OptuNets and Laplace methods in Appendix C.3. Turning our attention to the assessment of epistemic uncertainty, as quantified by AUPR and FPR95, multi-mode techniques, notably multi-SWAG and DE, consistently outperform other methods. This underscores the strong connection between posterior estimation and the accuracy of epistemic uncertainty quantification. However, we note that the quality of aleatoric uncertainty quantification does not steadily correlate with that of the posterior distribution estimation. The final two columns of the table shed light on the diversity of the models sampled from the posterior. The objective is to minimize in-distribution mutual information (IDMI) while maximizing out-of-distribution mutual information (OODMI). An analysis shows that mono-mode methods yield lower values than multi-mode methods, suggesting inferior diversity for the former. 5 DISCUSSIONS We develop further insights on the posterior of Bayesian neural networks in relationship with symmetries. Notably, we evaluate the risk of functional collapse, i.e., of training very similar networks in Section 5.1, and discuss the frequency of weight permutations in 5.2. We showcase “Checkpoints” our dataset in Section D.1. In Appendix B, we discuss using independent checkpoints for the posterior estimation. Appendix D expands these discussions, discusses the impact of the chosen prior in D.4, links the posterior to recent works on the loss landscapes (Section D.5), adds observations on the number of modes of the posterior in D.8, and provides visualizations (Section D.10). 5.1 FUNCTIONAL COLLAPSE IN ENSEMBLES: A STUDY OF ID AND OOD DISAGREEMENTS Given the high number of equivalent modes due to permutation symmetries (see Section 3.5), we support broadening the concept of collapse in the parameter-space – e.g., in D’Angelo & Fortuin (2021) – to functional collapse to account for the impact of symmetries on the posterior. Parameter-space collapse is more restrictive and may not be formally involved when ensemble members lack diversity. It is also harder to characterize as it would require an analysis of the loss landscape, at least. We quantify functional collapse as a potential ground for the need for more complex repulsive ensembling methods (Masegosa, 2020; Rame & Cord, 2021). We take 1000 ResNet-18 trained to estimate the Bayesian posterior in Section 4 and compute the mean over the test set of their pairwise mutual information (see Section D.12), quantifying the divergence between the single models and their average. We measure these values on in-distribution and OOD data with CIFAR-100 and SVHN, respectively. In Figure 3 (left), we see that the in-distribution MI between any two networks has a very low variance. Given that the usual diversity between models is satisfactory (Lakshminarayanan et al., 2017), there is an extremely low probability of training two similar networks, despite the huge number of potential symmetric models. These results hint that the complexity of the posterior is orders of magnitude greater than what we grasp with symmetries: large DNNs seem to empirically never fall into these numerous symmetric local minima. This may be explained by the high complexity of the network (here, a ResNet-18), and we refer to Appendix D.9 for results on a smaller architecture. Interestingly, we note in Figure 3 (right) that in contrast to intuition, we have no significant correlation between the in-distribution and the OOD MI. This highlights that measuring the in-distribution diversity may be a very poor indicator of the OOD detection performance of a model. 5.2 Frequency of Weight Permutations During Training We devise a new protocol to evaluate if a network tends to permute during training. Given a DNN $f_\omega$, we compute, for each step $s$ of the training, the permutation set $\Pi_s$ sorting its weights (and removing the corresponding symmetries). If the DNN tends to permute during the training, this implies a variation in the $\Pi_s$. We measure the extent of the variations using Kendall’s $\tau$ correlation between successive permutations $\Pi_s$ and $\Pi_{s+1}$. We plot the variation of the mean over several training instances, and the Kendall’s $\tau$ of each element of the permutation sets in Figure 4. We see that on MNIST (left), the variations of the permutation sets are scarce and gathered around points of instability. These instabilities are due to the sorting mechanism, which is based on the maximum values of the neurons’ weights; we have tried other statistics on the values of the weights, but taking the maximum remains the most stable. Moreover, the weights nearly never permute in the last training phase. The analysis differs for ResNet-18 (right) since the number of degrees of freedom is much greater. We see a lot of variation during the phases with a high learning rate (reduced after 25 and 50 epochs). However, as for the first case, we do not see any particular sign of permutations in the last part of the training. This evidence is in favor of weight-averaging methods such as SWAG (Maddox et al., 2019), which, therefore, have only very limited risks of averaging symmetric networks. 6 Conclusion In this study, we have examined Bayesian neural network posteriors, which are pivotal for understanding uncertainty. Our findings suggest that part of the complexity in these posteriors can be attributed to the non-identifiability of modern neural networks viewing the posterior as a mixture of permuted distributions. To explore this further, we introduce the scaled representation problem and investigate the real impact of scaling symmetries. Using real-world applications, we design a method to assess the quality of the posterior distribution and study its correlation with model performance, particularly regarding uncertainty quantification. While considering symmetries has provided valuable insights, our discussions hint at a more profound complexity going beyond these weight-space symmetries. In future work, we plan to continue our exploration of this intriguing area. 7 ACKNOWLEDGEMENTS This work was performed using HPC resources from GENCI-IDRIS (Grant 2023-[AD011011970R3]). 8 REPRODUCIBILITY STATEMENT To ensure transparency and accessibility, we use publicly available datasets, including MNIST, FashionMNIST, CIFAR100, SVHN, ImageNet-200, and Textures. Please refer to Appendix C.2.2 for details on these datasets. Our detailed experimental methods are outlined in Appendices A and C, and the proofs for our theoretical results are provided in Appendix E. To help replicate our work, we share the source code of our experiments on GitHub, notably including code to remove symmetries from neural networks \textit{a posteriori}. For our experiments in Section 4, we rely exclusively on open-source libraries. Most of our experiments are performed with TorchUncertainty, including the training of the standard and dropout models, but also the evaluation of their Deep Ensembles versions. For the rest, we use the GitHub repository Bayesian-Neural-Networks for SGHMC, BLiTZ for variational Bayesian neural networks, and Laplace Redux (Daxberger et al., 2021) for Laplace evaluations. We also use the publicly available code from the original paper (Maddox et al., 2019) for the SWAG method. Finally, we estimate the maximum mean discrepancies with a homemade torch version of the code from Schrab et al. (2023) and solve our convex optimization problems (see Definition E.6) with cvxpy (Diamond & Boyd, 2016). The statistical experiments, such as Pearson’s $\rho$ and Kendall’s $\tau$, are performed with SciPy. 9 ETHICS Our primary goal in this paper is to improve our comprehension of the Bayesian posterior, which we argue is a fundamental element to understand to contribute to the reliability of machine-learning methods. We note that training a substantial number of checkpoints for estimating the posterior, especially in the case of the thousand models trained on TinyImageNet, was energy intensive (around 3 Nvidia V100 hours per training). We opted for the Jean-Zay supercomputer, a carbon-efficient cluster to mitigate the environmental impact of our research.
yxKZGQLzOP	I couldn't understand the argument for sampling a subset of the consistency matrix $M$ in 3.3. If $M$ is sparse, why does that allow us to sample a subset of the rows and columns? Wouldn't we mostly sample zeros? Or is there a strategy for sampling (e.g., sampling dense areas) which is not mentioned?	Generating Pragmatic Examples to Train Neural Program Synthesizers Saujas Vaduguru Carnegie Mellon University svadugur@cs.cmu.edu Daniel Fried Carnegie Mellon University dfried@cs.cmu.edu Yewen Pu Autodesk Research yewen.pu@autodesk.com Abstract Programming-by-example is the task of synthesizing a program that is consistent with a set of user-provided input-output examples. As examples are often an under-specification of one’s intent, a good synthesizer must choose the intended program from the many that are consistent with the given set of examples. Prior work frames program synthesis as a cooperative game between a listener (that synthesizes programs) and a speaker (a user choosing examples), and shows that models of computational pragmatic inference are effective in choosing the user intended programs. However, these models require counterfactual reasoning over a large set of programs and examples, which is infeasible in realistic program spaces. In this paper, we propose Prax, a novel way to amortize this search with neural networks. We sample pairs of programs and examples via self-play between listener and speaker models, and use pragmatic inference to choose informative training examples from this sample. We then use the informative dataset to train models to improve the synthesizer’s ability to disambiguate user-provided examples without human supervision. We validate Prax on the challenging task of synthesizing regular expressions from example strings, and find that our method (1) outperforms models trained without choosing pragmatic examples by 23% (a 51% relative increase) (2) matches the performance of supervised learning on a dataset of pragmatic examples provided by humans, despite using no human data in training. 1 Introduction In program synthesis – specifically programming-by-example-(PBE) – a user describes a target program using input-output examples (i.e. test cases) and the synthesizer finds a program that is consistent with these input-output examples. In PBE, the users directly express the semantics of the intended program (what it should do) without having to understand the syntax of the program (what it should look like). Such systems have found real-world use in a variety of scenarios such as spreadsheet formulas [Chen et al., 2021; Gulwani, 2011] and data wrangling [Feng et al., 2018]. An important aspect of inferring programs from examples is dealing with ambiguity. Given a set of examples, there can be many spurious programs consistent with the set, and picking out the right one the user has in mind is a long-standing challenge. For example, when describing the regular expression $a+b$, an informative user might provide the example $(ab,\checkmark)$ indicating that the string $ab$ matches the target regular expression. However, to the program synthesizer, both $a+b$ and $ab+c?$ would be among many acceptable answers based on this example. Pu et al. [2020] resolves this ambiguity by framing program synthesis as a cooperative communicative game: the user chooses an informative set of examples to convey the program to the synthesizer, and the synthesizer chooses a program under the assumption that these examples were chosen informatively. Models of pragmatic inference, specifically, the Rational Speech Acts (RSA) framework [Frank & Goodman, 2012] can then be used to build a program synthesizer that can resolve ambiguity via recursive Bayesian inference. The RSA framework allows the synthesizer to reason about 1 or more $a$s followed by 0 or more $b$s $c?$ means optionally having a $c$ at the end Figure 1: PRAX iteratively generates datasets containing increasingly informative program specifications (lists of examples consistent with the program), and updates models on the generated datasets. ① We use a Speaker model — that generates an example consistent with a target PROGRAM — to propose a set of candidate specifications. Using the Rational Speech Acts model of pragmatic reasoning (red box; described in Figure 2), we choose the example that is most informative to a Listener model that synthesizes programs consistent with a given specification. In this manner, we incrementally build the list of examples SPEC for the PROGRAM. We repeat this for different programs to create a dataset of informative PROGRAM-SPEC pairs. ② We use the dataset to update the Speaker and Listener models. We train the speaker to generate the selected pragmatic examples, and the listener to synthesize the target program given the generated examples. what program a user intended, given that they chose that particular set of examples rather than a different one. For example, the synthesizer could reason that a user that wanted to describe $a \times b + c$ would have chosen an example containing the character $c$. However, this approach requires the synthesizer to perform expensive counterfactual inference over the space of all programs and examples to resolve ambiguity, making it difficult to scale to realistic programming domains. To scale to realistic program spaces, modern approaches of PBE systems have relied on training neural networks to efficiently search through large program spaces for consistent programs given examples [Balog et al., 2017; Devlin et al., 2017]. In this paper, we explore whether we can use simulated reasoning in communication games using the RSA framework as a way to generate training data consisting of pragmatic examples. The generated data is then used to train neural networks to enable scaleable pragmatic program synthesis. We dub this approach PRAX. We hypothesize that since the RSA framework computationally models how a human chooses examples to communicate a program, end users would succeed more often when communicating with a neural synthesizer trained on pragmatic data (our work) as compared to a neural synthesizer trained on non-pragmatic data [Balog et al., 2017; Devlin et al., 2017]. An overview of PRAX is shown in Figure 1. We start with a neural literal listener — a synthesizer trained in the style of [Devlin et al., 2017] — and a neural literal speaker that generates examples consistent with a given program. We generate a sequence of pragmatic examples incrementally to obtain a training pair (program, examples). This pair is then added to an aggregate training set, which is used to finetune both the speaker model — making it more likely to generate pragmatic examples — and the listener model — making it more likely to recover the intended program given pragmatic examples. We validate the effectiveness of PRAX on the well-studied PBE task of inferring regular expressions from a set of examples. Each example is a pair $(\text{string}, \text{bool})$, indicating whether a particular string matches the regex. To compare our training algorithm to standard supervised learning from human-annotated data, we collect a novel dataset of human annotations, consisting of $(\text{program}, \text{examples})$ where the examples were given by a person for a total of 440 regular expressions. We find that with only a small number (40) of human annotations — used only for model selection — our method is able to outperform a system that is fine-tuned using 400 annotated regexes from this dataset. We conduct human evaluation of PRAX by giving a user a target regex to communicate interactively to Figure 2: An illustration of how the Rational Speech Acts framework is used to select an informative example for a given program. We start with the matrix corresponding to the consistency relation between the sample of programs and examples shown in Figure 1. We obtain a literal listener distribution $ L_0 $ over programs for each example by normalizing the rows of this matrix. Since the $ M $ matrix is binary, each row in $ L_0 $ is a uniform distribution over consistent programs in the sample — any of the consistent programs is equally likely to be the intended program. We then obtain a pragmatic speaker distribution $ S_1 $ by normalizing the columns of the $ L_0 $ matrix: modeling the probability an informative speaker might have for choosing each example when communicating a program to a literal listener. RSA outputs the highest-probability example in $ S_1 $ (e.g., $(aa, ✓)$) in the column corresponding to the target program (e.g., $ab$). the synthesizer using examples, and find that the informative examples generated by our procedure substantially improve the performance of a regular expression synthesizer, with improvements of 22.8% absolute (51.4% relative) in accuracy (11 participants, 340 regexes total). Prax, despite not using human-provided data during training, matches the performance of a model fine-tuned on a large dataset of human-written pragmatic examples. Our code and data are available at https://github.com/saujasv/generating-pragmatic-examples 2 BACKGROUND Programming-by-Example In this paper, we tackle the task of finding a program $ \in P $, where $ P $ is a space of possible programs. As a specification of intent, a user provides a sequence of input-output examples $ \in E^+ $, where $ E = X \times Y $ is the space of all possible input-output pairs that programs in $ P $ operate over. For example, $ P $ may be a space of regular expression programs, $ X $ the space of all strings in the alphabet that the regular expressions are defined over, and $ Y \in \{✓, ✗\} $ where output $ ✓ $ indicates whether the input string matches the regular expression. For simplicity of explanation, in this section we consider cases where the specification consists of a single example, deferring the cases of multiple examples to the next section. The semantics of programs are captured by the consistency relation $ M $ between $ P $ and $ E $: \[ M = \{(program, example) \| example = (x, y) \in X \times Y, program(x) = y\} \] A program is consistent with an example iff executing the program on the input produces the intended output. We can view $ M $ as a consistency matrix where each row corresponds to an example, each column corresponds to a program, and an element is 1 if the program is consistent with the example and 0 otherwise (Figures 1 and 2 left). Literal model of program synthesis A minimal requirement of a program synthesizer is that it finds any program that is consistent with the given specification. We refer to such a synthesizer as the literal listener $ L_0 $, which naively assigns equal probability to any consistent program. \[ L_0(program\|example) \propto M(example, program)P(program) \] This literal listener distribution is given by normalizing the rows of the consistency matrix to produce uniform probability distributions ($ L_0 $ in Figure 2). However, this literal listener cannot resolve $^3$Here the notation $ X^+ $ indicates a sequence of 1 or more elements belonging to $ X $ ambiguity when interacting with users, as it places equal probability on all consistent programs. A literal speaker – one that generates any consistent examples for a given program – is defined analogously as $ S_0(\text{example}\|\text{program}) \propto M(\text{example}, \text{program})P(\text{example}) $. Pragmatic model of program synthesis* When interacting with a synthesizer, users choose examples that are informative – those that distinguish the program they desire from others. For example, given the specification $(ab, \lor)$, a user is likely wants the regular expression $a+b+$ than $a+b+c?$, since they probably would have included the character $c$ if they wanted the latter expression. To leverage the informativity of examples, Pu et al. (2020) use the Rational Speech Acts (RSA; Frank & Goodman, 2012) framework to derive a pragmatic program synthesizer that resolves ambiguity by modeling how people choose examples informatively. First, they construct a pragmatic speaker $S_1$ that chooses an example in proportion to the likelihood that it would make the literal listener infer the intended program: \[ S_1(\text{example}\|\text{program}) \propto L_0(\text{program}\|\text{example})P(\text{example}) \] Using a uniform prior $P(\text{example})$, the $S_1$ distribution is given by normalizing the columns of the $L_0$ matrix ($S_1$ in Figure 2). As we can see, given a program $S_1$ selects an example in proportion to the likelihood of $L_0$ recovering the program given the example, choosing examples that are informative to the listener.$^4$ Finally, a pragmatic listener (program synthesizer) $L_1$ is built on top of $S_1$: \[ L_1(\text{program}\|\text{example}) \propto S_1(\text{example}\|\text{program})P(\text{program}) \] using a prior $P(\text{program})$ over programs. This listener resolves ambiguity by choosing that program that an informative speaker (modeled by $S_1$) would have described using the chosen example. Pu et al. (2020) demonstrated that building a pragmatic synthesizer $L_1$ in this way allows for users to communicate the target program to the synthesizer using fewer examples without training a model on human-produced examples or explicitly defining a prior over the space of programs. However, in realistic domains, enumerating large numbers of programs and examples is intractable, preventing the application of this framework to a broader range of tasks. ### 3 METHOD In this section, we describe the iterative process by which we bootstrap a pragmatic neural program synthesizer by generating informative specifications, without human supervision (Figure 1). The full algorithm is detailed in Appendix C. #### 3.1 SPEAKER AND LISTENER MODELS We build on past work that uses neural models as specification-conditioned proposal distributions over programs (Balog et al., 2017; Devlin et al., 2017). Our listener (synthesizer) models represent distributions over programs $L_\theta(\text{program}\|\text{examples})$. Our speaker (specification generation) models generate the sequence of examples in a specification autoregressively: \[ S_\phi(\text{example}_i\|\text{program}, \text{examples}_{1:i-1}) \] While all our listener and speaker models share the same architecture and initialization, we vary their training data, as described below. #### 3.2 TRAINING BASE MODELS As a foundation for our approach, we train base listener and speaker models to approximate literal speakers and listeners $S_0$ and $L_0$ (Equation 1). Since we cannot enumerate the consistency matrix completely and normalize rows, we obtain these approximate models by training on data obtained by randomly sampling an input from the space of inputs $\mathcal{X}$, and executing the program on the input. $^4$For a sequence of examples, Pu et al. (2020) propose factoring the pragmatic speaker distribution autoregressively as $S_1(\text{examples}\|\text{program}) = \prod_{i=1}^{N_{\text{examples}}} S_1(\text{example}_i\|\text{program}, \text{examples}_{<i})$ $^5$We train the speaker to predict the input-output pair to encourage the model to capture aspects of the domain semantics to obtain the output (e.g., by checking if an sampled example string is matched by a sampled regular expression). This lets us generate as many samples from $M$ as we can, which we can use to train a base listener to approximate $L_0$ (Equation 1), and a base listener to approximate an analogous $S_0$, using standard maximum likelihood training. This is essentially the method proposed by Devlin et al. (2017). We denote the resulting base listener model as $L_{\theta_0}$ and the base speaker model as $S_{\phi_0}$, and use these as the initial models in our iterative model bootstrapping procedure. 3.3 Generating Informative Examples The crux of our algorithm is using the existing $S_{\phi}$ and $L_{\theta}$ to approximate $S_1$, which can then be used to generate training data to improve $S_{\phi}$ and $L_{\theta}$ over rounds of training. At each round $r$ of our approach, we use the current speaker and listener models, together with the RSA procedure, to create a dataset of informative examples specifying programs (① of Figure 1). We incrementally generate examples to create a specification. Given a partial specification of $i$ examples $\text{examples}_{1:i}$, we sample a set of additional candidate examples: $S_{\phi_r}(\text{example}_{i+1}\|\text{program, examples}_{1:i})$. Similarly, we sample a set of alternative programs: $L_{\theta_r}(\text{program}\|\text{examples}_{1:i})$ from the partial specification. We can then compute the sampled consistency matrix over the generated examples and programs, and use RSA inference as shown in Figure 2 to choose the highest scoring example from the approximate $S_1$ distribution. This example is added to the partial specification, and we repeat until a maximum number of examples are reached. The completed program-specification pair is then added to a dataset $D_r$ of examples from that round of training. This process amounts to choosing an example proposed by the current speaker model that minimizes ambiguity among programs that the current listener infers to be likely. The full algorithm for incrementally generating a sequence of examples is presented in Algorithm 2 (Appendix C). 3.4 Model Updates We use the dataset $D_r$ to update both the speaker and listener models as sketched in part ② of Figure 1 using standard maximum likelihood training. In each round $r$ we further fine-tune the speaker and listener models on the generated data to obtain the updated parameters $\theta_{r+1}$ and $\phi_{r+1}$. The full algorithm to iteratively generate examples and update the model is presented in Algorithm 1 (Appendix C). To select the model that works best with human-provided examples, we choose the model that maximizes a model selection metric computed over a small set of programs paired with human-provided examples. Note that this validation set is never used to update the model parameters, and only is used to choose a model. 4 Experiments 4.1 Regular Expressions We validate the training algorithm we propose on the task of synthesizing regular expressions ('regexes') as formally defined in Section 2. We use the regular expression domain-specific language presented by Ye et al. (2020). In addition to defining a regular expression specification language, they also define a sampling distribution over the space of regular expressions that we use to sample programs for training and evaluating our model. This distribution uses templates that generalize types of regular expressions that people ask about on fora such as StackOverflow. Further details are provided in Appendix A. --- 6 We follow prior work (Pu et al., 2020) and impose a uniform, rather than learned, prior over this sample. 7 Ideally, one could perform exact inference to draw samples from $S_1$ directly. As stated earlier, this is intractable. Therefore, we first sample a subset of the rows and columns in the consistency matrix, then perform the RSA inference over this much smaller and denser sampled matrix. However, the consistency matrix $M$ is sparse (mostly 0s) – most programs are inconsistent with any non-trivial set of examples – allowing for reasoning about a sample of the matrix. 8 Since the models are used only to generate the programs and utterances that are used to create the lexicon, we can draw examples from other sources, including models other than $S_{\phi_r}(\text{example}_i\|\text{program, examples}_{1:i})$ and $L_{\theta_r}(\text{program}\|\text{examples}_{1:i})$. 9 This amounts to performing early stopping on the validation metric. 4.2 Models for comparison Base models We use ByT5-small models (Xue et al., 2022) as the backbone for all speaker and listener models. To obtain the base speaker $S_{\theta_0}$ and listener $L_{\phi_0}$ models that approximate the literal speaker and listener respectively, we use a set of 300,000 randomly-generated program–specification pairs (with varying numbers of examples in each specification) and finetune the pretrained ByT5 checkpoint. Full details of training are provided in Appendix B. The $L_{\theta_0}$ model acts as the LITERAL model in our experiments. PRAX We start with the base models $S_{\theta_0}$ and $L_{\phi_0}$, and use the iterative data generation and fine-tuning algorithm to obtain a sequence of synthesis models $L_{\phi_r}$ for rounds $r = 0, \ldots, R_{max}$. We use the TOP-1 metric evaluated on a small validation set to choose the best model which we refer to as the PRAX model. Finetuning on human-provided specifications We obtain an HFT model by fine-tuning $L_{\phi_0}$ on a curated high-quality human-provided specifications (Section 4.5). This model allows us to compare how well our approach of using model generated informative examples compares with sourcing more expensive human annotations. GPT-3.5 We evaluate GPT-3.5 by using the program-specification pairs we obtain as users interact with the other three models, and revealing each specification to the GPT-3.5 model in the order the user provided them, one example at a time. We stop when the model guesses the correct regular expression, or when all the examples are presented. We can think of this as a form of interaction where the human doesn’t observe the outputs of this model while giving examples. Further details of how the model is prompted are in Appendix F. Inference Crucial to our approach is the ability to generate programs from examples, and vice versa. To generate programs from a specification (sequence of examples from either self-play or human), we present it to the listener, and sample 500 programs using top-$p$ sampling (Holtzman et al., 2020) with $p = 0.9$. We then deduplicate the set of sampled programs, and filter out programs inconsistent with the given specification. We can then sort the remaining programs by their score under the model to obtain a ranked list of consistent programs. Similarly, to generate specifications from a program, we sample 500 examples using top-$p$ sampling with $p = 1$ and check that the examples are consistent with the program. 4.3 Procedure We evaluate the model on the basis of successful communications on 11 human participants. A sampled regex $p$ is given to a human participant, whom describes it using a sequence of examples, providing one example in each turn for up to a maximum of 10 turns. The synthesizer takes the examples provided and generates a ranked list of inferred programs, of which the top-1 regex $p'$ is shown to the participant. The communication is successful when $p = p'$, at which point the interaction ends.\footnote{We used the greenery Python library to identify regex matches in terms of semantic similarity, and not just surface form match.} A total three synthesizers were considered – the LITERAL model, the human fine-tuned HFT model, and the PRAX model. The identity of the models is referred to the users only as differently colored robots. The study yielded communication history over a total of 340 regexes (109 to the LITERAL model, 113 to the HFT model, and 118 to the PRAX model).\footnote{A small bug caused us to collect few extra interactions for some of the models, which does not change the measurements on the models’ relative performances.} Further details about how the user study is conducted are provided in Appendix E. 4.4 Measurement We consider the following metrics. TOP-1@t measures whether the model’s top-1 matches intended regular expression at any point at turn $t$ of the interaction. We can also consider the average value of TOP-1 by aggregating across the turns $t \in \{1, \ldots, 10\}$. Averaging across turns rewards models that pass success criteria given fewer turns — a model that can infer the target regex at turn 4 on average... \| Model \| Top-1@10 (SE) \| Top-10@10 (SE) \| Edit Distance ≤ 1@10 (SE) \| \|-----------\|---------------\|----------------\|--------------------------\| \| LITERAL \| 0.434 (0.047) \| 0.522 (0.047) \| 0.513 (0.047) \| \| GPT-3.5* \| 0.074 (0.014) \| 0.189 (0.021) \| 0.082 (0.015) \| \| HFT \| 0.587 (0.047) \| 0.623 (0.047) \| 0.614 (0.047) \| \| PRAX \| 0.661 (0.043) \| 0.703 (0.042) \| 0.694 (0.042) \| Table 1: Success metrics at the end of 10 turns of interaction with each model, with standard errors computed using bootstrap sampling. * indicates that the results are in replay. \| Model \| Top-1 (SE) \| Top-10 (SE) \| Edit Distance ≤ 1 (SE) \| \|-----------\|------------\|-------------\|------------------------\| \| LITERAL \| 0.233 (0.028) \| 0.333 (0.034) \| 0.296 (0.031) \| \| GPT-3.5* \| 0.048 (0.010) \| 0.122 (0.014) \| 0.056 (0.011) \| \| HFT \| 0.349 (0.032) \| 0.424 (0.035) \| 0.390 (0.034) \| \| PRAX \| 0.373 (0.028) \| 0.430 (0.030) \| 0.432 (0.031) \| Table 2: Average success metric over 10 turns. We see that the PRAX training method we propose is on par with HFT, and outperforms other baselines significantly for all success criteria. * indicates that the results are in replay. is better than a model that can only infer the target at turn 10. We use Top-1 over a validation set as the model selection criterion for our proposed method. Top-10@t and Top-10 are similarly defined. Edit Distance ≤ 1@t measures whether the model’s highest scoring prediction in any of the turns up to t is at most a 1 token edit from the intended program, and Edit Distance ≤ 1 is the average of this value over t ∈ {1, . . . , 10}. 4.5 Human-provided Specifications An alternative to generating informative examples using the method we propose is to have human annotators provide examples. We collect a new dataset of high-quality program-specification pairs. Procedure We present a participant with a sampled regular expression, and instruct them to provide examples that they might use to illustrate the regular expression to another person. Participants are asked to provide at least 5-7 examples. We verify whether the examples are informative by checking whether a different annotator is able to identify the program which the given set of examples describes. Further details about the data collection process are presented in Appendix D. Usage of data We collect a total of 440 program-specification pairs. We sample a small subset of 40 pairs that received 2 “correct” verifications as a validation set for model selection. We use the other 400 pairs as a training set to finetune the $L_{\theta_0}$ models on human-provided informative examples, obtaining HFT (see Appendix B). 4.6 Results Table 1 shows the rate of success for different models at the end of 10 turns of interaction. We see that training on informatively examples results in large gains in performance. Both the PRAX and HFT models significantly outperform the literal model for all three criteria of success. Looking at the aggregate success rate across turns in Table 2 reveals that it is not just that the PRAX synthesizer eventually catches up to the HFT model, but also performs on par with it over the course of the interaction. Figure 3 shows the progression of each metric over the course of interaction. In contrast, we see that GPT-3.5 performs worse than the literal model. One reason for this could be that the distribution of regular expressions that the GPT-3.5 encounters in its training data could be quite different, leading to worse performance. In conclusion, the experiments validate our hypothesis that humans communicate more effectively with the model trained on informative examples (the PRAX and HFT models) than with a model trained on randomly chosen examples (the LITERAL model). Figure 3: Performance of various models as a function of turns, measured in (a) Top-1@t, (b) Top-1@f, and (c) Edit Distance ≤ 1@t. Lines show averages, and bands are standard errors. Our model Prax, trained entirely from self-play and RSA inference without using human-provided data performs better than the non-pragmatic Literal model across all turns and metrics, and matches the performance of HFT tuned on a human-provided examples. \| Target program \| Examples \| Literal \| Prax \| \|----------------\|----------\|---------\|------\| \| 4A(2,) \| 4AAAAAAA \| ✓ \| 4A(1,) \| \| \| 4AA \| ✓ \| 4A(2,) \| \| ---------------\|----------\|---------\|------\| \| [A-Z](1,)i(2,4)\| Aii \| ✓ \| (A(1,)B(1,))i(2,4) \| \| \| Bi \| ✓ \| [A-Z](1,)i(2,4) \| \| \| Biii \| ✓ \| \| \| \| AAAAAAAAl \| ✓ \| \| Figure 4: Example specifications for two programs provided during the user study, along with the highest ranked guess from the Literal and the Prax models. Examples of synthesized programs Figure 4 shows examples of guesses by the Literal and Prax models given the same sequence of examples. In the first case, we see that the Prax model is able to infer that if a user wanted a regular expression that accepted 4A, they would have specified that, and instead correctly guesses that the user wanted at least two A’s in the string. The second example also shows how the Literal model synthesizes a regular expression that is correct, but is too specific, while the Prax model recovers the correct generalization. 5 ANALYSIS OF TRAINING Figure 5 shows the progression of the Top-1 metric over the course of different rounds of training. We see that as we train the model for more rounds, the performance of the model generally increases, and then tapers off. This shows that as the model is trained for more rounds, it gets increasingly pragmatic. Since the model we choose for the user study is trained for 5 rounds, on $5 \times 1024 = 5120$ programs, we also compare to training the model for only a single round on the same number of programs. In a replay study (similar to how we evaluated GPT-3.5; Figure 5), we find that iteratively generating data and updating the model performs better. We also see that finetuning the base model on 400 examples (to match the HFT setting) from a later round of training also results in a strong model, suggesting that as the speaker is trained more, it generates examples that are useful to finetune a listener model. 6 RELATED WORK Neural network models of pragmatic reasoning Prior work has applied the RSA pragmatic reasoning framework to improve neural models at inference time for tasks including image captioning (Andreas & Klein 2016; Cohn-Gordon et al. 2018), instruction generation (Fried et al. 2018a), vision-and-language navigation (Fried et al. 2018b), and machine translation (Cohn-Gordon & Goodman 2019). RSA is used at inference time to re-rank multiple outputs from the neural models. Figure 5: Top-1 metric over the course of rounds of training of the Prax model. We report the metric on the validation set as well evaluating on all interactions from the user study in the replay setting (similar to how we evaluated GPT-3.5). We compare the accuracy over rounds of training to generating specifications and updating the models only once, amounting to a single round of the procedure with more programs (Prax-single-round). We also compare to fine-tuning the base model on 400 pairs (same number as HFT) generated by the speaker in the 5th round of training (Prax-HFT-match) to assess the quality of our speaker-generated examples. Prax has two advantages over these works: (1) it requires no human-provided data during training; (2) it uses RSA at training time via data generation, amortizing expensive RSA computation. Other approaches have used pragmatically-motivated training procedures. The closest works to ours are White et al. (2020) and Lazaridou et al. (2020), who use reinforcement learning approaches to fine-tune a speaker model using reward from a fixed listener model. Monroe & Potts (2015) and McDowell & Goodman (2019) backpropagate through the RSA procedure at training time to reason counterfactually about pragmatically produced utterances from humans. Again, Prax is unique in that it does not require human-provided training data. Finally, Andreas & Klein (2016) find that amortizing pragmatic reasoning during training does not perform as well as explicit pragmatic reasoning during inference in the domain of image captioning. We demonstrate that amortization is in fact effective for the domain programming-by-example. Pragmatic reasoning for program synthesis Similar to our work, Vaithilingam et al. (2023) conduct a study of how users interact with an exact RSA pragmatic regular expression synthesizer over a toy domain of ~1000 regexes total over strings of only 0s and 1s. Pu et al. (2023) propose a way to make pragmatic PBE more efficient by inferring a global ranking function, but still relies on the expensive exact RSA during training. Our approach is different in that by using neural models for speakers and listeners at training time, we are able to scale to a realistic regex domain. Pertseva et al. also present an approach to version space algebra-based approach to regular expression synthesis from examples by explicitly modeling the probability of examples describing programs (as in our speaker models), but they work with only positive examples (a subset of our example space with examples that have the output ✓, excluding those with the output ✗). Ferreira et al. (2021) present an SMT-based method that reasons about distinguishing inputs to synthesize regular expressions. We discuss connections to iterated bootstrapped training for program synthesis in Appendix H. 7 Conclusion We present Prax, a novel algorithm that bootstraps pragmatic program synthesizers by (1) generating datasets using self-play between a speaker (program → examples) and a listener model (examples → programs), and (2) training on the generated data. Crucial to our approach is the use of pragmatic inference to make the generated data more informative. Prax produces pragmatic program synthesizers with minimal supervision: in a challenging regular expression domain, matching the performance of synthesizers fine-tuned on human-produced examples, despite not using any human-provided data during training. Future work might explore scaling pragmatic program synthesis to open-ended Python code generation, and application to multimodal specifications — e.g. with natural language and examples (Ye et al., 2020). ETHICS STATEMENT Our dataset collection process and user study constitute human subjects research. Our studies were deemed exempt from full IRB review by our institution. All participation was voluntary. Participants signed an online consent form, and were compensated fairly for their time. ACKNOWLEDGEMENTS The authors would like to thank Xi Ye for help with sampling regular expression programs, Eric Lu and Kevin Ellis for initial discussions, Priyan Vaithilingam for inputs on the interface and user study, Kira Jones for help with compensating participants, Catherine Copetas for help with advertising, Alex Xie and Simran Khanuja for help with testing the user study interface, and Vijay Viswanathan, Jared Fernandez, Zhiruo Wang, Harshita Diddee, and Lindia Tjuatja for feedback on drafts. SV was supported by a gift from Autodesk Research. REFERENCES Jacob Andreas and Dan Klein. Reasoning about pragmatics with neural listeners and speakers. 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In Doina Precup and Yee Whye Teh (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 990–998. PMLR, 06–11 Aug 2017. URL https://proceedings.mlr.press/v70/devlin17a.html Kevin Ellis, Catherine Wong, Maxwell Nye, Mathias Sablé-Meyer, Lucas Morales, Luke Hewitt, Luc Cary, Armando Solar-Lezama, and Joshua B. Tenenbaum. Dreamcoder: Bootstrapping inductive program synthesis with wake-sleep library learning. In Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation, PLDI 2021, pp. 835–850, New York, NY, USA, 2021. Association for Computing Machinery. ISBN 9781450383912. doi: 10.1145/3453483.3454080. URL https://doi.org/10.1145/3453483.3454080
JdWpIe70FL	In standard ensemble learning, variability is normally enforced through randomisation, e.g., by resampling the training data or randomly initialising weights in a neural network. Here, the risk is that different randomisations may lead to different variability, making (epistemic) uncertainty arbitrary to some extent. So how is the variability produced by the NF ensemble controlled, and in which sense is it “meaningful” or “natural”?	ESCAPING THE SAMPLE TRAP: FAST AND ACCURATE EPISTEMIC UNCERTAINTY ESTIMATION WITH PAIRWISE-DISTANCE ESTIMATORS Anonymous authors Paper under double-blind review ABSTRACT In machine learning, the ability to assess uncertainty in model predictions is crucial for decision-making, safety-critical applications, and model generalizability. This work introduces a novel approach for epistemic uncertainty estimation for ensemble models using pairwise-distance estimators (PaiDEs). These estimators utilize the pairwise-distance between model components to establish bounds on entropy, which are then used as estimates for information-based criterion. Unlike recent deep learning methods for epistemic uncertainty estimation, which rely on sample-based Monte Carlo estimators, PaiDEs are able to estimate epistemic uncertainty up to 100 times faster, over a larger input space (up to 100 times) and perform more accurately in higher dimensions. To validate our approach, we conducted a series of experiments commonly used to evaluate epistemic uncertainty estimation: 1D sinusoidal data, Pendulum-v0, Hopper-v2, Ant-v2 and Humanoid-v2. For each experimental setting, an Active Learning framework was applied to demonstrate the advantages of PaiDEs for epistemic uncertainty estimation. 1 INTRODUCTION In this paper, we propose Pairwise-Distance Estimators (PaiDEs) as a non-sample based alternative for estimating epistemic uncertainty in deep ensembles with probabilistic outputs. Epistemic uncertainty, often distinguished from aleatoric uncertainty, pertains to model ignorance and can be reduced by increasing the amount of data available [Hora 1996; Der Kiureghian & Ditlevsen 2009; Hüllermeier & Waegeman 2021]. Traditionally, in multi-dimensional regression tasks, epistemic uncertainty has been estimated using Monte Carlo (MC) methods because closed-form expressions are generally lacking in most modeling scenarios [Depeweg et al., 2018; Berry & Meger, 2023]. However, as the number of dimensions increases, these MC methods become increasingly reliant on a large number of samples. PaiDEs offer a non-sample based alternative for estimating information-based criterion in ensemble models with probabilistic outputs [Kolchinsky & Tracey, 2017; Kulak & Calmon, 2021; Kulak et al., 2021]. Ensembles can be conceptualized as committees, with each ensemble component serving as a committee member [Rokach, 2010]. PaiDEs can synthesize the consensus amongst committee members by calculating the distributional distance between each pair of committee members. Distributional distance is a measure of the distance between two probability distributions. These pairwise-distances are aggregated in a way that accurately estimates the differential entropy of the entire ensemble. Assuming that the pairwise distances can be efficiently calculated, PaiDEs provide an efficient way to estimate epistemic uncertainty that is not sample-dependent. In this study, we showcase the application of PaiDEs for epistemic uncertainty estimation for ensembles with probabilistic outputs, specifically Normalizing Flows (NFs). Prior research has demonstrated the effectiveness of NFs in capturing heteroscedastic and multi-modal aleatoric uncertainty [Kingma & Dhariwal, 2018; Rezende & Mohamed, 2015]. In the context of robotic systems, these characteristics are particularly relevant as robots frequently encounter nonlinear stochastic dynamics. We evaluate our method on an array of regression tasks on robotic datasets in the context of active learning. Our contributions are as follows: We establish the framework for the application of PaiDEs in the context of estimating epistemic uncertainty for deep ensembles with probabilistic outputs (Section 6). We extend previous epistemic uncertainty estimation methods from 11 to 257 dimensions, and demonstrate how PaiDEs outperform MC methods in the higher dimensional setting with rigorous statistical testing (Section 7). We provide an analysis of the time saving advantages offered by PaiDEs compared to MC estimators for epistemic uncertainty estimation (Section 7.4). 2 Problem Statement This section provides an overview of the problem at hand. Following a supervised learning framework, let $ D = \{x_i, y_i\}_{i=1}^N $ denote a dataset, where $ x_i \in \mathbb{R}^K $ and $ y_i \in \mathbb{R}^D $, and our objective is to approximate the conditional probability $ p(y\|x) $. Let $ f_\theta(y, x) $ denote our approximation to the conditional probability density, where $ \theta $ is a set of parameters to be learned. The ground-truth distribution, $ p(y\|x) $, is assumed to take any form including complex multi-modal distributions. To enable our methods to capture epistemic uncertainty, in addition to complex multi-modal aleatoric uncertainty, we employ ensembles. Ensembles leverage multiple models to obtain the estimated conditional probability by weighting the result output from each ensemble component, \[ f_\theta(y, x) = \sum_{j=1}^{M} \pi_j f_{\theta_j}(y, x) \quad \text{subject to} \quad \sum_{j=1}^{M} \pi_j = 1, \] where $ M $ and $ 0 \leq \pi_j \leq 1 $ are the number of model components and the component weights, respectively. In order to create an ensemble, one of two ways is typically chosen: randomization (Breiman, 2001) or boosting (Freund & Schapire, 1997). While boosting has led to widely used machine learning methods (Chen & Guestrin, 2016), randomization has been the preferred method in deep learning due to its tractability and ease of implementation (Lakshminarayanan et al., 2017). 3 EPISTEMIC UNCERTAINTY Uncertainty is grounded in probability theory and is often analyzed from this perspective (Cover & Thomas [2006]; Hüllermeier & Waegeman [2021]). When capturing uncertainty in supervised learning, one common measure is conditional differential entropy, $$H(y\|x) = - \int p(y\|x) \ln p(y\|x) dy.$$ Utilizing conditional differential entropy, we can establish an estimate for epistemic uncertainty as introduced by Houltsby et al. (2011), expressed as: $$I(y, \theta\|x) = H(y\|x) - E_{p(\theta)}[H(y\|x, \theta)],$$ (2) where $I(\cdot)$ refers to mutual information and $\theta \sim p(\theta)$. Equation (2) demonstrates that epistemic uncertainty, $I(y, \theta\|x)$, can be represented by the difference between total uncertainty, $H(y\|x)$, and aleatoric uncertainty, $E_{p(\theta)}[H(y\|x, \theta)]$. Mutual information measures the information gained about one variable by observing another. When all components produce the same $f_{\theta_i}(y, x)$, $I(y, \theta\|x)$ is zero, indicating no epistemic uncertainty. Conversely, when the components have non-overlapping supports, epistemic uncertainty is high. Epistemic uncertainty is valuable in decision-making, particularly in active learning (MacKay [1992]; Settles [2009]). We select data points that maximize Equation (2) as in Bayesian Active Learning by Disagreement (BALD) to improve the model’s performance (Houlsby et al. [2011]). It’s worth noting that in the realm of continuous outputs and ensemble models, Equation (2) often lacks a closed-form solution, primarily because total entropy cannot be expressed in closed form, $$H(y\|x) = \int_D \sum_{j=1}^{M} \pi_j f_{\theta_j}(y, x) \ln \sum_{j=1}^{M} \pi_j f_{\theta_j}(y, x) dy.$$ Hence, prior methods have resorted to Monte Carlo (MC) estimators for the estimation of epistemic uncertainty (Depeweg et al. [2018]; Postels et al. [2020]). The Monte Carlo method samples $K$ points from our model, $y_j \sim f_{\theta}(y, x)$, and then estimates the total uncertainty, $$\hat{H}_{MC}(y\|x) = -\frac{1}{K} \sum_{j=1}^{K} \ln f_{\theta}(y_j, x).$$ MC estimators are convenient for estimating quantities through random sampling and are more apt for high-dimensional integrals compared to other numerical methods. However, as the number dimensions increase, MC methods typically require a greater number of samples (Rubinstein & Glynn [2009]). 4 PAIRWISE-DISTANCE ESTIMATORS Unlike MC methods, PaiDEs completely remove this dependence on sampling by leveraging (generalized) distance functions between model component distributions. They can be applied when estimating entropy of mixture distributions as long as the pairwise-distances have a closed-form. Their derivation and properties follow from Kolchinsky & Tracey (2017); we are extending the use of PaiDEs to a supervised learning problem and epistemic uncertainty estimation. 4.1 PROPERTIES OF ENTROPY One can treat a mixture model as a two step process: first a component is drawn and, second, a sample is taken from the corresponding component. Let $p(y, \theta\|x)$ denote the joint of our output and model components given input $x$, $$p(y, \theta\|x) = p(\theta_j\|x)p(y\|\theta_j, x) = \pi_{\theta_j} p(y\|\theta_j, x).$$ Now that we have a representation of the joint, following principles of information theory (Cover & Thomas [2006]), we can write its entropy as, $$H(y, \theta\|x) = H(\theta\|y, x) + H(y\|x).$$ (3) Additionally, one can show the following bounds for $ H(y\|x) $, \[ H(y\|\theta, x) \leq H(y\|x) \leq H(y, \theta\|x). \] (4) Intuitively, the lower bound can be justified by the fact that conditioning on more variables can only decrease or keep entropy the same and the upper bound follows from Equation (3) and $ H(\theta\|y, x) \geq 0 $. ### 4.2 PaiDEs Definition Let $ D(p_i \\| p_j) $ denote a (generalized) distance function between the probability distributions $ p_i $ and $ p_j $, which for our case represent $ p_i = p(y\|x, \theta_i) $ and $ p_j = p(y\|x, \theta_j) $, respectively. More specifically, $ D $ is referred to as a premetric, $ D(p_i \\| p_j) \geq 0 $ and $ D(p_i \\| p_j) = 0 $ if $ p_i = p_j $. The distance function need not be symmetric nor obey the triangle inequality. As such, PaiDEs can be defined as, \[ \hat{H}_D(y\|x) = H(y\|\theta, x) - \sum_{i=1}^{M} \pi_i \ln \sum_{j=1}^{M} \pi_j \exp(-D(p_i \\| p_j)). \] (5) PaiDEs have many options for $ D(p_i \\| p_j) $ (Kullback-Leibler divergence, Wasserstein distance, Bhattacharyya distance, Chernoff $ \alpha $-divergence, Hellinger distance, etc.). Theorem 4.1. Using the extreme distance functions, \[ D_{\text{min}}(p_i \\| p_j) = 0 \quad \forall i, j \] \[ D_{\text{max}}(p_i \\| p_j) = \begin{cases} 0, & \text{if } p_i = p_j, \\ \infty, & \text{o/w}, \end{cases} \] one can show that PaiDEs lie within bounds for entropy established in Equation (4). Refer to Kolchinsky & Tracey (2017) for the proof. This provides a general class of estimators but a distance function still needs to be chosen. Certain distance functions improve the bounds in Equation (4) and we will use them to guide our choice. ### 4.3 Improved Bounds for PaiDEs Let the Chernoff $ \alpha $-divergence be defined as (Nielsen, 2011), \[ C_\alpha(p_i \\| p_j) = -\ln \int p^\alpha(y\|x, \theta_i)p^{1-\alpha}(y\|x, \theta_j)dx, \] where $ \alpha \in [0, 1] $. Corollary 4.2. When applying Chernoff $ \alpha $-divergence as our distance function in Equation (5), we achieve a tighter lower bound than $ H(y\|\theta, x) $, \[ \hat{H}_{C_\alpha}(y\|x) = H(y\|\theta, x) - \sum_{i=1}^{M} \pi_i \ln \sum_{j=1}^{M} \pi_j \exp(-C_\alpha(p_i \\| p_j)), \] (6) \[ H(y\|\theta, x) \leq \hat{H}_{C_\alpha}(y\|x) \leq H(y\|x). \] (7) Refer to Kolchinsky & Tracey (2017) for the proof. In addition, the tightest lower bound can be shown to be $ \alpha = 0.5 $ for certain situations (Kolchinsky & Tracey, 2017). Note that when $ \alpha = 0.5 $, the Chernoff $ \alpha $-divergence is known as the Bhattacharyya distance, \[ D_B(p_i \\| p_j) = -\ln \int \sqrt{p(y\|x, \theta_i)p(y\|x, \theta_j)}dx. \] (8) We utilized PaiDEs with the Bhattacharyya distance, $ \hat{H}_{Bhatt}(y\|x) = \hat{H}_{C_{0.5}}(y\|x) $, as one proposed improvement to MC estimators. In addition to the improved lower bound, there is an improved upper bound as well. Let Kullback-Liebler (KL) divergence be defined as follows, \[ D_{KL}(p_i \\| p_j) = \int p(y\|x, \theta_i) \ln \frac{p(y\|x, \theta_i)}{p(y\|x, \theta_j)}dx. \] Note that the KL divergence does not satisfy the triangle inequality nor is it symmetric, thus it is not metric but does suffice as a (generalized) distance function. Corollary 4.3. When applying Kullback-Liebler divergence as our distance function in Equation (5), we achieve a tighter upper bound than $H(y, \theta \| x)$, $$\hat{H}_{KL}(y \| x) = H(y \| \theta, x) - \sum_{i=1}^{M} \pi_i \ln \sum_{j=1}^{M} \pi_j \exp(-D_{KL}(p_i \\| p_j)), \quad (9)$$ $$H(y \| x) \leq \hat{H}_{KL}(y \| x) \leq H(y, \theta \| x). \quad (10)$$ Refer to Kolchinsky & Tracey (2017) for the proof. In addition to Bhattacharyaa distance, we applied PaiDEs with KL divergence as another proposed improvement to Monte Carlo estimation. 5 NORMALIZING FLOW ENSEMBLES In this study, we utilize an ensemble technique named Nflows Base, which has previously shown robust performance in estimating both aleatoric and epistemic uncertainty on robotic datasets by leveraging normalizing flows (NFs) to create ensembles (Berry & Meger, 2023). PaiDEs can be employed with any ensemble possessing probabilistic outputs and closed-form distributional distance between ensemble components. 5.1 Nflows Base NFs have been classically applied to unsupervised tasks (Tabak & Vanden-Eijnden, 2010; Tabak & Turner, 2013; Rezende & Mohamed, 2015), though NFs have been adapted to a supervised learning setting (Winkler et al., 2019; Ardizzone et al., 2019). Using the structure of Winkler et al. (2019) one can define a supervised NF as, $$p_{y\|x}(y\|x) = p_{b\|x}(g^{-1}_\theta(y, x)) \times \|\det(J(g^{-1}_\theta(y, x)))\|,$$ $$\log(p_{y\|x}(y\|x)) = \log(p_{b\|x}(g^{-1}_\theta(y, x))) + \log(\|\det(J(g^{-1}_\theta(y, x)))\|),$$ where $p_{y\|x}$ is the output distribution, $p_{b\|x}$ is the base distribution, $J$ refers to the Jacobian, and $g^{-1}_\theta : y \times x \mapsto b$ is the bijective mapping. For a more comprehensive review of NFs, refer to Papamakarios et al. (2021). Nflows Base creates an ensemble in the base distribution, $$p_{y\|x,\theta}(y\|x, \theta) = f_\theta(y, x) = p_{b\|x,\theta}(g^{-1}_\theta(y, x))\|\det(J(g^{-1}_\theta(y, x)))\|,$$ where $p_{b\|x,\theta}(b\|x, \theta) = N(\mu_{\theta,x}, \Sigma_{\theta,x})$, $\mu_{\theta,x}$ and $\Sigma_{\theta,x}$ denote the mean and covariance conditioned on both $x$ and $\theta$. These parameters are modeled using a neural network with fixed dropout masks to establish an ensemble and ensemble diversity is created by randomization and bootstrapping. By constructing the ensemble within the base distribution, we can leverage closed-form pairwise-distance formulae. Berry & Meger (2023) showed that Nflows Base outperforms previous methods when estimating epistemic uncertainty, as the aleatoric uncertainty from Equation (2) can be estimated in the base distribution space and therefore allow for aleatoric uncertainty to be computed analytically. This does not apply to the other quantity of Equation (2), total uncertainty, and thus samples still need to be drawn in order to estimate epistemic uncertainty. Figure 2: In the right graphs, the blue dots are sampled from Nflows Base and the 3 lines depict epistemic uncertainty corresponding to different estimators. The left graphs depicts the ground-truth data as the blue dots and its corresponding density as the orange histogram. Note the legend refers to the lines in the right graphs. 6 Epistemic Uncertainty Estimation with PaiDEs 6.1 Estimators As mentioned in Section 3, the quantity of interest is mutual information rather than entropy. By applying our definition of PaiDEs to Equation (2), we obtain the following expression: \[ \hat{I}_\rho(y, \theta) = \tilde{H}_\rho(y\|x) - E_{p(w)}[H(y\|x, \theta)] = -\sum_{i=1}^{M} \pi_i \ln \sum_{j=1}^{M} \pi_j \exp(-D(p_i \\| p_j)), \] (11) as $ E_{p(\theta)}[H(y\|x, \theta)] = H(y\|x, \theta) $. PaiDEs provide a succinct estimator that can estimate epistemic uncertainty with only the pairwise distances between components, thus eliminating reliance on sample-based techniques. We propose the following specific estimators: \[ \hat{I}_{Bhatt}(y, \theta) = -\sum_{i=1}^{M} \pi_i \ln \sum_{j=1}^{M} \pi_j \exp(-D_{Bhatt}(p_i \\| p_j)), \] \[ \hat{I}_{KL}(y, \theta) = -\sum_{i=1}^{M} \pi_i \ln \sum_{j=1}^{M} \pi_j \exp(-D_{KL}(p_i \\| p_j)), \] where $ D_{Bhatt}(p_i \\| p_j) $ and $ D_{KL}(p_i \\| p_j) $ are defined for Gaussians in Appendix A.1. Note that our proposed estimators can be applied to any ensemble model whose output distributions have closed-form pairwise-distances as such, we have included experiments using probabilistic network ensembles (PNEs) in Appendix A.4. 6.2 Combination of PaiDEs & Nflows Base Berry & Meger (2023) demonstrate that estimating Equation (2) in the base distribution is equivalent to estimating it in the output distribution. Consequently, by combining Nflows Base and PaiDEs, we construct an expressive non-parametric model capable of capturing intricate aleatoric uncertainty in the output distribution while efficiently estimating epistemic uncertainty in the base distribution. Unlike previously proposed methods, we are able to estimate epistemic uncertainty without taking a single sample. Figure 1 shows an example of the distributional pairs that need to be considered in order to estimate epistemic uncertainty for an Nflows Base model with 3 components. 7 Experimental Results To evaluate our method, we tested each PaiDE (KL $ \hat{I}_{KL}(y, \theta) $ and Bhatt $ \hat{I}_{Bhatt}(y, \theta) $) on two 1D environments, as has been previously proposed in the literature (Depeweg et al., 2018). Additionally, we present 4 multi-D environments. In contrast to previous papers (Berry & Meger, 2023), we increased the number of dimensions by more than an order of magnitude, from 11 to 257, to demonstrate the utility of PaiDEs in higher dimensions. The ensembles used in our experiments were constructed by randomly initializing the weights and creating bootstrapped samples of the training dataset. Also note that, for all experiments, the model components are assumed to be uniform, $ \pi_j = \frac{1}{M} $, independent of $ x $. In addition, all model hyper-parameters are contained in Appendix A.1 and the code can be found at (added upon publication). 7.1 Data We evaluated PaiDEs on two 1D benchmarks, hetero and bimodal. The ground-truth data for hetero and bimodal can be seen in Figure 2 on the left graphs with the blue dots with the orange bar chart corresponding to the density. For hetero, there are two regions with low density (2 and -2). In these regions, we would expect a model to have high epistemic uncertainty. For bimodal, the number of data points drops off as $ x $ increases, thus we would expect a model to have epistemic uncertainty grow as $ x $ does. All details for data generation are contained in Appendix A.2. In addition to the 1D environments, we tested our methods over four multi-dimensional environments (Pendulum-v0, Hopper-v2, Ant-v2, and Humanoid-v2) (Todorov et al., 2012). Replay buffers were Table 1: Mean RMSE on the test set for the last (100\textsuperscript{th}) Acquisition Batch for Nflows Base. Experiments were across ten different seeds and the results are expressed as mean plus minus one standard deviation with results that are statistically significant highlighted. \| Env. \| Output Dim. \| Random \| MC \| KL \| Bhatt \| \|---------------\|-------------\|------------\|------------\|------------\|------------\| \| hetero \| 1 \| 1.6 ± 0.19 \| 1.58 ± 0.32\| \textbf{1.42} ± 0.16 \| 1.43 ± 0.18 \| \| bimodal \| 1 \| 6.4 ± 0.62 \| 6.01 ± 0.04\| 6.01 ± 0.04 \| \textbf{6.0} ± 0.04 \| \| Pendulum-v0 \| 3 \| 0.55 ± 0.17\| \textbf{0.09} ± 0.02\| 0.11 ± 0.03 \| 0.12 ± 0.04 \| \| Hopper-v2 \| 11 \| 1.58 ± 0.3 \| 0.61 ± 0.05\| \textbf{0.53} ± 0.05 \| 0.56 ± 0.05 \| \| Ant-v2 \| 32 \| 2.16 ± 0.06\| 2.3 ± 0.09 \| \textbf{2.06} ± 0.08 \| 2.11 ± 0.1 \| \| Humanoid-v2 \| 257 \| 8.06 ± 1.63\| 7.78 ± 1.41\| \textbf{3.88} ± 1.47 \| \textbf{4.96} ± 2.76 \| gathered from an agent and the dynamics model for each environment was modeled, $ f_\theta(s_t, a_t) = \hat{s}_{t+1} $. We evaluated on multi-dimensional environments because they are routinely used as benchmarks and provide us a higher dimensional output space to validate our methods. Also note that, for Ant-v2 and Humanoid-v2, the dimensions representing their contact forces were eliminated as Mujoco-v2 had a bug, always returning zero for those dimensions.\footnote{More information can be found here: https://github.com/openai/gym/issues/1541.} ### 7.2 1D Experiments Our 1D environments provide empirical proof that PaiDEs can accurately measure epistemic uncertainty. Figure 2 depicts that both KL and Bhatt are proficient at estimating the epistemic uncertainty as each method shows an increase in epistemic uncertainty around 2 and -2 on the hetero setting. This can be seen from the orange and gray lines. KL and Bhatt perform indistinguishably from MC, as shown by the blue line. A similar pattern can be seen for the bimodal setting in Figure 2, which shows that both Bhatt and KL can accurately capture epistemic uncertainty. Each estimator shows the pattern of increasing epistemic uncertainty where the data is more scarce. Both examples show accurate epistemic uncertainty estimation with no loss in aleatoric uncertainty representation, as demonstrated in the right graphs in Figure 2; the blue dots closely match the blue dots on their corresponding left graphs. ### 7.3 Active Learning While the 1D experiments provide evidence of PaiDEs’ effectiveness for estimating epistemic uncertainty, the active learning experiments extend this evaluation to higher-dimensional data. Nflows Base started with 100 or 200 data points depending on the setting. At the end of each training epoch, the MC estimator sampled 1,000 unseen inputs and estimated their epistemic uncertainties, except for the Humanoid-v2 environment where only 100 new inputs were sampled due to computational Figure 3: RMSE on the test set at the 100\textsuperscript{th} acquisition batch of the MC estimator on the Hopper-v2 environment as the number of samples increases. Experiment run across 10 seeds and the mean is being reported. Figure 4: On the left, the amount of time taken for each estimator across the different settings (1, 3, 11, 27, 257 dimensions). On the right, the amount of time taken for PaiDEs as the number of ensemble components increases for the 257 dimensional setting. Results are averaged over 10 seeds and shown on a log scale. On the other hand, PaiDEs sampled 10,000 new inputs and estimated their epistemic uncertainties for each environment. This highlights one advantage PaiDEs over MC estimators, as PaiDEs are able to estimate epistemic uncertainty over larger regions at lower computational cost than their MC counterparts. Upon estimating epistemic uncertainty, the 10 data points with the highest epistemic uncertainty (50 data points for Humanoid-v2) were added to the training set. Additionally, the root mean squared error (RMSE) on the test set was calculated at each acquisition batch. Table 1 displays the performance of each estimator on the 100th acquisition batch. In each environment, we conducted a Welch’s t-test that compares both PaiDE estimators against the two baselines. Note that we included a Holm–Bonferroni correction to control the family-wise error rate (FWER), for more information refer to Appendix A.6. For each data setting, the PaiDEs reach lower or comparable RMSEs to MC estimators, thus demonstrating that PaiDEs can be used to estimate epistemic uncertainty. In addition, PaiDEs are more effective in higher dimensions as can be seen by the fact that PaiDEs outperform MC estimates in statically significant manner for Humanoid-v2 and Ant-v2. A random acquisition function was included as a baseline. To conduct a more in-depth analysis of our proposed method, we compared PaiDEs to MC estimators with a varying sample size in the Hopper-v2 environment. We expected that MC estimators would perform on par with PaiDEs with a sufficient number of samples. However, as illustrated in Figure 3, MC estimators fell short of achieving the same level of performance as PaiDEs in this particular scenario. This suggests that, taking into consideration hardware constraints, PaiDEs begin to outperform MC estimators when dealing with 11 dimensional outputs. 7.4 TIME ANALYSIS & LIMITATIONS In addition to benchmarking PaiDEs on active learning experiments, we provide an analysis of the time gains across our experiments. The left hand side of Figure 4 depicts the speed increase that can be gained using PaiDEs over an MC approach. A 1-2 order magnitude of improvement can be seen. The estimates are obtained from the active learning experiments, and the number of dimensions corresponds to each of the environments. A weakness of PaiDEs is that as the number of components increases, the computational cost rises. Therefore algorithms like MC dropout (Gal & Ghahramani 2016) may not be suitable. In the instance where the distance is not symmetric, KL-divergence, $M^2 - M$ pairwise-distances need to be computed. For symmetric distances, such as Bhattacharyaa distance, only $\frac{M^2 - M}{2}$ distances need to computed. The right hand side of Figure 4 shows an analysis of the time taken as the number of ensembles grow. Note that for Bhatt, the time costs could be improved upon using the symmetry logic described as the results shown calculated all pairwise-distances. Despite the growing complexity of PaiDEs with the number of components, this is normally not a problem for deep learning ensembles as they have a relatively low number (5-10) of components (Osband et al., 2016; Chua et al., 2018). An additional limitation is the bias introduced by PaiDEs, which MC estimators do not suffer from. It is essential to note that, in the context of active learning, epistemic uncertainty serves as a relative quantity for comparing potential acquisition points. The introduction of bias from PaiDEs does not impact the relative relationship of epistemic uncertainty between different data points. We demonstrate that the relative relationship of epistemic uncertainty remains intact in Appendix A.3. 8 RELATED WORK Researchers have employed Bayesian neural networks alongside information-based criteria for active learning in image classification problems (Gal et al., 2017; Kendall & Gal, 2017; Kirsch et al., 2019). These studies utilize epistemic uncertainty estimation with MC dropout to gauge uncertainty in image classification tasks. In contrast, our research focuses on estimating uncertainty within a continuous output space. Our experiments encompassed tasks where the output spans continuous distributions for 1 to 257 dimensions, as opposed to the aforementioned methods that primarily address classification problems with a 1D categorical output. In addition to Bayesian methods, ensembles have been harnessed for epistemic uncertainty estimation (Lakshminarayanan et al., 2017; Choi et al., 2018; Chua et al., 2018). Specifically related to our work, ensembles have been leveraged to quantify epistemic uncertainty in regression problems and active learning (Depeweg et al., 2018; Postels et al., 2020; Berry & Meger, 2023). Depeweg et al. (2018) employed Bayesian neural networks to model mixtures of Gaussians and demonstrated their ability to measure uncertainty in low-dimensional environments (1-2D). Building upon this foundation, Postels et al. (2020) and Berry & Meger (2023) extended the research by developing efficient Normalizing Flow (NF) ensemble models that effectively captured epistemic uncertainty. Our work advances this line of research by eliminating the need for sampling to estimate epistemic uncertainty, resulting in a faster and more effective method, especially in higher dimensions. Entropy estimators, which do not rely on sampling, is an active area of research (Jebara & Kondor, 2003; Jebara et al., 2004; Huber et al., 2008; Kolchinsky & Tracey, 2017). Kulak et al. (2021) and Kulak & Calinon (2021) demonstrated the utility of Pairwise-Distance Estimators (PaiDEs) within Bayesian contexts, employing PaiDEs to estimate conditional predictive posterior entropy. In contrast, our approach provides a more general estimate of epistemic uncertainty, as defined in Equation (2), which can be applied to both ensemble and Bayesian methods. Furthermore, our method is adaptable to flexible deep learning models, a capability that was previously unavailable in the approach presented by Kulak et al. (2021) and Kulak & Calinon (2021). Several methods have emerged in the literature for estimating epistemic uncertainty without relying on sampling techniques (Van Amersfoort et al., 2020; Charpentier et al., 2020). Both Van Amersfoort et al. (2020) and Charpentier et al. (2020) focus on classification tasks with 1D categorical outputs. Charpentier et al. (2021) extends the work of Charpentier et al. (2020) to regression tasks but is limited to modeling outputs as members of the exponential family. In contrast, our approach can handle more complex output distributions by directly considering the outputs from Normalizing Flows (NFs). This flexibility is particularly valuable in scenarios involving intricate non-linear robotic dynamics, as demonstrated in our experiments. 9 CONCLUSIONS In this study, we introduced two epistemic uncertainty estimators and applied them to expressive ensemble models. We depicted how our method can be used to more efficiently quantify uncertainty by leveraging closed-form pairwise-distance instead of sampling. This led to improvements in computational speed and accuracy, especially in larger dimensions. We addressed the problem of epistemic uncertainty estimation in high-dimensional problems by building effective epistemic uncertainty estimators without sampling. REFERENCES Lynton Ardizzone, Carsten Lüth, Jakob Kruse, Carsten Rother, and Ullrich Köthe. Guided image generation with conditional invertible neural networks. arXiv preprint arXiv:1907.02392, 2019. Lucas Berry and David Meger. Normalizing flow ensembles for rich aleatoric and epistemic uncertainty modeling. 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tm8s3696Ox	If it is truly OFL, what does the epoch correspond to? Is it the server model training? This question also relates to Figure 1 (a) and (d). If there are multiple epochs of both the client and server, how is it considered OFL?	Enhancing One-Shot Federated Learning Through Data and Ensemble Co-Boosting Rong Dai\textsuperscript{1,2}, Yonggang Zhang\textsuperscript{3}, Ang Li\textsuperscript{3}, Tongliang Liu\textsuperscript{4}, Xun Yang\textsuperscript{1,}, Bo Han\textsuperscript{2} \textsuperscript{1}University of Science and Technology of China, \textsuperscript{2}TMLR Group, Hong Kong Baptist University \textsuperscript{3}ECE Department, University of Maryland College Park, \textsuperscript{4}Sydney AI Centre, The University of Sydney rongdai@mail.ustc.edu.cn {csygzhang, bhanml}@comp.hkbu.edu.hk angliece@umd.edu tongliang.liu@sydney.edu.au xyang21@ustc.edu.cn Abstract One-shot Federated Learning (OFL) has become a promising learning paradigm, enabling the training of a global server model via a single communication round. In OFL, the server model is aggregated by distilling knowledge from all client models (the ensemble), which are also responsible for synthesizing samples for distillation. In this regard, advanced works show that the performance of the server model is intrinsically related to the quality of the synthesized data and the ensemble model. To promote OFL, we introduce a novel framework, Co-Boosting, in which synthesized data and the ensemble model mutually enhance each other progressively. Specifically, Co-Boosting leverages the current ensemble model to synthesize higher-quality samples in an adversarial manner. These hard samples are then employed to promote the quality of the ensemble model by adjusting the ensembling weights for each client model. Consequently, Co-Boosting periodically achieves high-quality data and ensemble models. Extensive experiments demonstrate that Co-Boosting can substantially outperform existing baselines under various settings. Moreover, Co-Boosting eliminates the need for adjustments to the client’s local training, requires no additional data or model transmission, and allows client models to have heterogeneous architectures. 1 Introduction Federated learning (FL) [Mcmahan et al., 2017] has emerged as a prominent distributed machine learning framework to train a global server model via collaboration among users without sharing their dataset. Though the multi-round parameter-server communication paradigm offers the benefit of effectively exchanging information among clients and the central server, it might not be feasible in the real world. This paradigm brings forth significant challenges: 1) heavy communication burden and the risk of connection drop errors between clients and the server [Li et al., 2020a; Kairouz et al., 2021; Dai et al., 2022], and 2) potential risk for man-in-the-middle attacks [Wang et al., 2021] and various other privacy or security concerns [Mothukuri et al., 2021; Yin et al., 2021]. One-shot FL (OFL) [Guha et al., 2019] has emerged as a solution to these issues by restricting communication rounds to a single iteration, thereby mitigating errors arising from multi-round communication and concurrently diminishing the vulnerability to malicious interception. Furthermore, OFL is more practical, particularly within contemporary model market scenarios [Vartak et al., 2016] where clients predominantly offer pre-trained models. In OFL, the server model is aggregated by distilling knowledge from all client models, commonly using the ensemble, while the ensemble is also responsible for synthesizing data samples for knowledge distillation. Consequently, as illustrated in [Guha et al., 2019] and [Zhang et al., 2022a], the server model’s performance is intricately linked to both the quality of synthesized data and the ensemble. Thus, the primary challenge in improving performance lies in the process of improving the data and the ensemble. Existing approaches tend to tackle this challenge by exclusively concentrating on either enhancing the quality of the ensemble or improving the quality of synthetic data. For instance, to bolster ensemble, prior works including [Dennis et al., 2021; Heimbaugh et al., 2023], and [Diao et al., 2023] Corresponding author. Work done during Rong’s visit to TMLR Group at HKBU. modify the local training phase and require additional transmissions. In terms of improving synthetic data, Li et al. (2021) utilizes auxiliary public datasets, Zhou et al. (2020) proposes transmitting distilled datasets to the server, Yang et al. (2023) proposes to use auxiliary diffusion model and Zhang et al. (2022a) employs data-free data generation methods to synthesize data directly from averaged ensemble models. While the distilled server may improve through the above methods, it is noteworthy that these approaches typically follow a sequential process, which means the enhancement of data or the ensemble is a prerequisite step before the server model can benefit, omitting the crucial relationship between them. What’s more, in contemporary model market scenarios where only well-pre-trained models with diverse architectural possibilities are accessible, any modifications to local training or additional data or model transmissions are discouraged and often disallowed. To address these challenges, we propose Co-Boosting, a novel one-shot federated learning algorithm as in Fig. 1(a), in which the synthesized data and the ensemble model mutually boost each other progressively. More specifically, in each training epoch, higher-quality hard samples are generated based on the previous epoch’s ensemble and server model. Based on these hard samples, the aggregation weight for each client model is adjusted, forming a better ensemble. Subsequently, the server model is updated by distilling knowledge from both the enriched data and the refined ensemble. As a result, with the continuous enhancement of both data and the ensemble, the final server model naturally improves. As depicted in Fig. 1(b), (c), and (d), with a better weighted ensemble model and better-quality hard samples, Co-boosting naturally achieves state-of-art performance. Thorough experiments on multiple benchmark datasets demonstrate the superiority of the proposed Co-Boosting. What’s more, due to its inherent nature, our proposed Co-Boosting is more practical to today’s model market scenarios. In summary, our main contributions can be summarized as follows: 1) We demonstrate that it is possible to simultaneously improve the quality of the synthesized data and the ensemble, which are two key elements in OFL. This discovery could spur progress in OFL methods, highlighting the need to optimize their interaction. 2) Within an adversarial paradigm, we introduce Co-Boosting, a novel one-shot federated learning method. Periodically, in Co-Boosting, hard samples are generated from the current ensemble, which, in turn, are used to reweight clients, forming an improved ensemble. This mutual enhancement of synthetic data quality and the ensemble collectively contributes to the natural emergence of a high-performing distilled server model. 3) Our proposed method Co-Boosting, is highly practical to the contemporary model market scenarios as it eliminates the necessity for client-side training adjustments, entails no extra data or model transmissions, and accommodates diverse client model architectures. 4) Extensive experiments confirm the effectiveness of Co-Boosting, consistently outperforming other baselines thanks to the improved quality of both the synthetic data and ensemble. 2 RELATED WORKS 2.1 ONE-SHOT FEDERATED LEARNING Guha et al. (2019) originally proposes OFL which collects local models as an ensemble for the final prediction and further proposes to use knowledge distillation (KD) on such ensemble with public data. This paradigm, which is followed by most works, inherently relates the performance of the server model to the data and ensemble used in the KD stage. Li et al. (2021) proposes to improve the ensemble on the public data. Instead of using public data, Zhou et al. (2020) proposes to transmit the distilled local dataset for the server. Yang et al. (2023) proposes to use auxiliary pre-trained diffusion model, while Zhang et al. (2022a) generates fake data scouring from the direct ensemble. Regarding the improvement of the ensemble, Dennis et al. (2021) utilizes a cluster-based method and requires uploading the cluster means. Diao et al. (2023) and Heinbaugh et al. (2023) modify the local training phase of each client by introducing placeholders, or conditional variation auto-encoders. However, none of the aforementioned methods simultaneously address improvements in both data and the ensemble. Moreover, few works can be practically applied, especially in contemporary model-market scenarios (Vartak et al., 2016) where only well-pretrained models are provided to the server. This situation implies constraints such as no alterations to the client’s local training, no additional transmissions, and the possibility of client model heterogeneity. 2.2 KNOWLEDGE DISTILLATION Knowledge distillation (KD) (Hinton et al., 2015) is proposed to transfer knowledge from one or more networks (teacher) to another (student). Taking the same spirit, KD in federated learning focuses on transferring knowledge from multiple local clients to the global server model. Lin et al. (2020) initially introduced to utilize KD at the server side based on an unlabeled auxiliary dataset. In an effort to reduce reliance on proxy datasets, generators that are locally updated and globally aggregated are used in Zhu et al. (2021) and Zhang et al. (2022b) to synthesize distillation samples. Wang et al. (2023) further enhances the basic ensemble distillation by using weighted averaging based on locally trained discriminators. However, in the context of OFL, conducting multiple rounds of training or transmitting generators and discriminators is not practical. Additionally, the need for an additional local client component violates the constraints in modern model-market OFL settings. More seriously, the generator trained locally has direct access to the training samples, potentially leading to privacy leakage through its ability to remember all the training data (Liu et al., 2019). On the other hand, the generator in OFL is trained without access to even one single raw data. 3 METHODOLOGY In this section, we first introduce the general process of one-shot federated learning (OFL). Then we detail the proposed method, Co-Boosting, in how we generate high-quality data, high-quality ensemble, and how to link and make them boost each other as illustrated in Fig. 1(a). 3.1 ONE-SHOT FEDERATED LEARNING Suppose we have a set of clients $ C $, with $ n = \|C\| $ clients in total. Each client $ c_k \in C $ has a local private dataset $ D^k = \{(x_i, y_i)\}_{i=1}^{n_k} $, where $ n_k = \|D^k\| $ is the number of local data samples $ x_i $ with the corresponding label $ y_i $. OFL’s goal is to train a good machine learning model with parameter $ \theta_S $ over $ D \triangleq \bigcup_{k=1}^{n} D^k $ with the help of a server in only one communication, as in \[ \min_{\theta_S} L(\theta_S) \triangleq \frac{1}{\|D\|} \sum_{(x_i, y_i) \in D} \ell_{CE}(f_S(x_i; \theta_S), y_i), \] where $ \ell_{CE}(\cdot, \cdot) $ is the cross-entropy function, $ f_S(x_i; \theta_S) $ is the prediction function of the server that outputs the logits (i.e., outputs of the last fully connected layer) of $ x_i $ given parameter $ \theta_S $. Noticeably, in one-shot federated learning, the original training set $ D^k $ cannot be accessed, and only well-pretrained models parameterized by $ \theta_k $, are provided. Here, we define the ensemble as: \[ A_w(x; \{\theta_k\}_{k=1}^{n}) \triangleq \sum_{k=1}^{n} w_k f_k(x; \theta_k), \] where $ f_k(x; \theta_k) $ denotes the prediction function that outputs the logits of $ x $ given $ \theta_k $, while $ w = [w_1, w_2, ..., w_n] $ adjusts the weights of each local client logits. When $ w_k = 1/n $, the ensemble is the same as the averaged ensemble, while when $ w_k = n_k / \sum_{k=1}^{n} n_k $, the ensemble becomes weighted according to the data amount. For simplicity, in the rest paper, we use $ A_w $ to denote the ensemble and $ A_w(x) $ to denote $ A_w(x; \{\theta_k\}_{k=1}^{n}) $, which means the output logits of the ensemble given $ x $. When aggregating pre-trained models $ \{\theta_k\}_{k=1}^{n} $ into one server model $ \theta_S $, existing works mostly follow a two-stage framework. The first is to synthesize data $ D_S $ based on the ensemble output. In particular, giving a random noise $ z $ sampled from a standard Gaussian distribution and a random uniformly sampled label $ y_s $, the generator $ G(\cdot) $ with $ \theta_G $ is responsible for generating the data $ x_S = G(z) $, forming the synthetic dataset $ D_S $. Typically, to make sure the synthetic data can be classified correctly with a high probability by the ensemble $ A_w $, the following loss is adopted: \[ L(\theta_G) \triangleq \frac{1}{\|D_S\|} \sum_{(x_s, y_s) \in D_S} \ell_{CE}(A_w(x_s), y_s). \] After getting the synthetic dataset $ D_S $ based on the generator in Eq.(3), OFL intends to distill the ensemble $ A_w $ into the final server model $ \theta_S $ with the help of these synthetic data, as in: \[ \min_{\theta_S} L(\theta_S) \triangleq \frac{1}{\|D_S\|} \sum_{(x_s, y_s) \in D_S} \ell_{KL}(A_w(x_s), f_S(x_s; \theta_S)), \] where $ \ell_{KL}(\cdot, \cdot) $ denotes the Kullback-Leibler (KL) divergence. Existing works illustrate that the performance of the server model is intrinsically related to the synthetic data $ D_S $ and the ensemble $ A_w $, which can also be concluded according to Eq.(4). ### 3.2 Boosting the Data Quality Synthesizing data $ D_S $ is used to distill the ensemble model into the final server model as in Eq.(4). The quality of these synthesized data has been demonstrated vital to the distillation stage [Lin et al., 2020]. Moreover, since these data are also generated sourcing from the ensemble as in Eq.(3), it is of great importance to make these data embed as much of the knowledge of the ensemble as possible and make them transferable to the final server model. However, as hinted in [Wang et al., 2020] and [Zhang et al., 2022a], by utilizing only the CE loss, the synthesized data can be easily fitted by the server model, resulting in poor performance in the knowledge distillation stage. Therefore, to improve the quality of the generated data and make them focus more on transferable components, taking inspiration from [Dong et al., 2020] and [Li et al., 2023a], we increase the importance of hard samples while suppressing the importance of easy-to-fit samples in the generation stage. More specifically, given a prediction function $ f $ which outputs logits, we employ the GHM introduced in [Li et al., 2019] to measure the sample difficulty $ d $ of $ x $: \[ d(x, f) = 1 - \sigma(f(x; \theta))_y, \] where $ \sigma(f(x; \theta))_y $ is the probability on label $ y $ predicted by the function $ f(\cdot) $ with $ \theta $. Built upon the sample difficulty, we propose a hard-sample-enhanced loss $ L_H $ to synthesize data: \[ L_H(x_s, y_s; \theta_G) \triangleq \frac{1}{\|D_S\|} \sum_{(x_s, y_s) \in D_S} d(x_s, A_w) \ell_{CE}(A_w(x_s), y_s), \] Moreover, to make synthesized samples hard for the server model to fit, an adversarial loss [Zhang et al., 2022c] is also introduced to generate hard samples. We try to maximize the differences in predictions between the ensemble model and the server model when generating data as follows: \[ L_A(x_s, \theta_S; \theta_G) \triangleq \frac{1}{\|D_S\|} \sum_{(x_s, y_s) \in D_S} -\ell_{KL}(A_w(x_s), f_S(x_s; \theta_S)). \] By combining the above losses, we can obtain the loss used to train the generator as follows: \[ L(\theta_G) \triangleq L_H(x_s, y_s; \theta_G) + \beta L_A(x_s, \theta_S; \theta_G), \] where $ \beta $ is the scaling factor for the losses, which is set as 1 in the implementations. Though samples synthesized using Eq.(8) are hard to fit for the current ensemble model, their difficulties for the server model are still lacking. This stems from the fact that the server model can easily fit these limited unchanged data during multiple distillation steps. To further promote the sample difficulty and diversity on the fly, we draw inspiration from adversarial learning (Goodfellow et al., 2014; Tashiro et al., 2020) to generate hard and diverse samples for the server model to learn. More specifically, we diversify and increase the sample difficulty $d(x_s, A_w)$ on the fly by introducing a perturbation $\delta_i$ for each $x_s$: \[ \delta_i = \arg \max_{\\|\delta'\\|_\infty \leq \epsilon} d(x_s + \delta', A_w), \] (9) where $\\|\cdot\\|_\infty$ represents the $L_\infty$-norm and $\epsilon$ controls the strength of perturbation. To simplify the computation and enhance the diversity, instead of the iterative adversarial attacks, we take only one step of loss backward to seek the direction in the input space that maximizes the similarity between the model output and a randomly sampled vector. The hard samples are constructed as follows: \[ \tilde{x}_s \triangleq x_s + \epsilon \frac{\nabla_x (u^\top A_w(x_s))}{\\|\nabla_x (u^\top A_w(x_s))\\|_2}, \] (10) where $u \sim \text{Unif}([-1, 1]^d)$ is a randomly sample vector with dimension $d$. Following these, the originally generated hard samples are further harder and more diverse due to the randomness in $u$. By replacing each sample $x_s$ in $D_S$ into $\tilde{x}_s$, we can achieve a more hard and diverse synthetic dataset $D_S$. Utilizing these hard samples, the knowledge of the ensemble model is transferred to the server model with parameter $\theta_S$ by knowledge distillation the same as in Eq.(4). Overall, with the hard sample technique embedding in the data synthesizing stage (replacing the generator loss in Eq.(3) with Eq.(8) and making them diverse in the distillation stage (reconstruct the synthetic dataset $D_S$ according to Eq.(10)), the quality of data generated and used for distillation becomes better, naturally boosting the performance of the server model. ### 3.3 Boosting the Ensemble Quality The ensemble model takes the role of aggregating knowledge from all pre-trained models $\{\theta_k\}_{k=1}^n$ and forms a virtually best-performance teacher. A straightforward method is to obtain the global model by averaging the parameters of all client models (e.g., FedAvg (McMahan et al., 2017)). However, FedAvg may fail to deliver a good performance when data among clients are non-IID (Karimireddy et al., 2020; Acar et al., 2021) and cannot handle the challenge of client model heterogeneity. Recent works (Heinbaugh et al., 2023; Diao et al., 2023) intend to construct a better ensemble model by altering the local training phase of each client, which may be unreliable, especially in today’s model market scenarios. To tackle the client model heterogeneity and make the ensemble more practical, Guha et al. (2019); Zhang et al. (2022a) utilizes the direct ensemble $A_w$ with $w_k = 1/n$ as the teacher, which means averaging the logits or weighted averaging them according to the number of the client data $(w_k = n_k / \sum_{k=1}^n n_k)$. However, as suggested by Zhang et al. (2023) and Wang et al. (2023), the simple averaging or weighted averaging based on the client data amount may not be effective, especially in non-IID settings. There exists a better weighted combination of each client’s contribution. Yet, their methods either need to alter the local training or transmit additional information, therefore their methods cannot be applied to one-shot FL. To this end, we propose to boost the ensemble quality by searching for a more effective weighted ensemble of logits. As demonstrated by our experimental results in Fig. 1(b), given high-quality data (validation data), we can achieve a better ensemble with weights different from simple averaging or data amount based averaging. Fortunately, instead of using auxiliary data, we actually can acquire high-quality generated data from the hard synthesized samples set $D_S$. Therefore, to get the best weights $w = [w_1, w_2, \cdots, w_N]$ on $D_S$, we need to solve the following optimization problem: \[ \min_w L_w(w) \triangleq \frac{1}{\|D_S\|} \sum_{(x_s, y_s) \in D_S} \sum_{k=1}^N w_k f_k(x_s; \theta_k), y_s), \] (11) where $y_s$ is the corresponding label to each synthesized hard samples $x_s$. Exploring the optimal $w$ requires multiple inner steps, leading to the training time to increase exponentially. Also inspired by methods of adversarial attacks (Goodfellow et al., 2014), we use the gradient’s direction and fixed step size $\mu$ to update $w$ every time after getting each batch of the synthesized data $D_S$: $$w^t = \text{Normalize}(w^{t-1} - \mu \text{sign}(\nabla_w L_w(w))),$$ where Normalize denotes bounding each $w_k$ into $[0, 1]$ and $\text{sign}(\cdot)$ means the sign function. The reweighting of each client’s logit results in a superior ensemble model, which will naturally benefit the server model. Moreover, since the operations are done on the logit layer, this reweighting technique can be easily applied to both heterogeneous and homogeneous client model settings. ### 3.4 Co-Boosting the Data and the Ensemble As aforementioned, we introduce how to boost the data quality by utilizing hard sample techniques with a fixed ensemble and how to boost the ensemble with a fixed synthetic dataset. Actually, these two stages are inherently entangled and can boost each other at the same time. To get high-quality ensemble $A_w$ and synthesized data $D_S$, we are in fact trying to solve the following problem: $$\min_w \frac{1}{\|D_S\|} \sum_{(x_s, y_s) \in D_S} \max_{\delta \in S} \ell_{CE} \left( \sum_{k=1}^{n} w_k f_k(x_s + \delta; \theta_k), y_s \right),$$ where $y_s$ is the label of sample $x_s$, $\delta$ is the perturbation constrained in $S$. This problem can be addressed adversarially, which means the improvement of the data and the ensemble can be done simultaneously. With better quality data, the weighted ensemble can reach higher performance, while with this better ensemble, the data synthesized sourcing from this ensemble can further embed more knowledge. Therefore, by mutually boosting the quality of the synthesized data and the ensemble, we can naturally get a better-performance server model through the distillation in Eq. (4). The overall algorithm is summarized in Algorithm 1. In each epoch, we first generate hard samples based on the current ensemble model and the last epoch server model. With these generated data, an enhanced ensemble model is cultivated by searching for the optimal ensembling weights of each client’s logits. Utilizing the generated data and the upgraded ensemble, the final server model is trained by distilling the ensemble on these data. As illustrated in Sec. 3.2 and Sec. 3.3, with either one fixed, one can benefit from the other, thus by mutually boosting each other in the proposed Co-Boosting, we can achieve both better quality data and the ensemble periodically. Therefore, the global server model trained on them will inherently become better than other methods. Algorithm 1 Co-Boosting 1: Input: Clients’ local models $\{\theta_1, \ldots, \theta_n\}$, server model $\theta_S$, synthetic dataset $D_S = \emptyset$, ensemble $A_w$, generator $\theta_G$, perturbation strength $\epsilon$, step size $\mu$, learning rate of generator and server $\eta_G$ and $\eta_S$, generation iterations $T_G$, global model training epochs $T$, and batch size $b$ 2: Output: Global server model $\theta_S$ 3: for epoch = 0 to $T - 1$ do 4: // Generate hard synthetic samples 5: Sample a batch of noises and labels $\{z_i, y_i\}_{i=1}^b$ 6: for $t_g = 0$ to $T_G - 1$ do 7: Generate $\{x_s\}_{s=1}^b$ with $\{z_i\}_{i=1}^b$ and $\theta_G$ 8: Update the generator: $\theta_G \leftarrow \theta_G - \eta_G \nabla_{\theta_G} L(\theta_G)$, where $L(\theta_G)$ is defined in Eq.(8) 9: end for 10: $D_S \leftarrow D_S \cup \{x_s\}_{s=1}^b$ 11: Diverse each sample $\{x_s\}$ in $D_S$ to $\{\tilde{x}_s\}$ according to Eq.(10) 12: // Obtain a better ensemble 13: Update the mixing weights with $D_S$ according to Eq.(12) 14: Construct an updated ensemble $A_w$ with updated $w$ according to Eq.(2) 15: // Obtain the final server model 16: for sampling batch $\{x_s\}$ in $D_S$ do 17: Update the server model: $\theta_S \leftarrow \theta_S - \eta_S \nabla_{\theta_S} L(\theta_S)$, where $L(\theta_S)$ is defined in Eq.(4) 18: end for 19: end for Table 1: Test accuracy of the server model of different methods over five datasets and across three levels of statistical heterogeneity (lower $\alpha$ is more heterogeneous). \| Method \| $\alpha$ \| FedAvg \| FedDF \| F-ADI \| F-DAFL \| DENSE \| Co-Boosting \| \|----------\|----------\|------------\|------------\|------------\|------------\|------------\|-------------\| \| MNIST \| 0.05 \| 46.35±0.98 \| 80.73±1.08 \| 80.12±1.76 \| 78.49±1.36 \| 81.06±1.12 \| 93.93±0.69 \| \| \| 0.1 \| 75.68±0.82 \| 87.91±0.92 \| 85.92±0.82 \| 87.44±0.61 \| 87.83±1.38 \| 94.44±1.02 \| \| \| 0.3 \| 78.97±0.82 \| 97.66±0.10 \| 96.34±0.83 \| 96.36±0.98 \| 96.96±0.53 \| 97.25±0.44 \| \| FMNIST \| 0.05 \| 20.07±1.98 \| 44.73±0.40 \| 42.25±2.01 \| 41.66±0.83 \| 44.77±1.87 \| 50.62±1.13 \| \| \| 0.1 \| 46.61±1.94 \| 68.40±1.80 \| 63.19±1.99 \| 67.81±0.93 \| 69.43±1.94 \| 74.86±1.70 \| \| \| 0.3 \| 60.13±2.62 \| 83.14±0.47 \| 74.80±1.72 \| 78.68±0.49 \| 81.31±0.83 \| 83.11±1.28 \| \| SVHN \| 0.05 \| 39.41±2.19 \| 60.79±0.52 \| 56.58±1.05 \| 59.38±1.19 \| 60.24±1.31 \| 65.40±0.86 \| \| \| 0.1 \| 46.22±1.92 \| 68.98±0.63 \| 66.33±1.69 \| 67.77±0.34 \| 68.30±1.01 \| 72.88±1.19 \| \| \| 0.3 \| 72.61±2.06 \| 79.78±0.55 \| 76.75±1.47 \| 78.01±1.02 \| 78.73±0.84 \| 81.31±1.09 \| \| CIFAR-10 \| 0.05 \| 17.49±2.51 \| 37.53±0.67 \| 36.94±1.70 \| 37.82±1.30 \| 38.37±1.08 \| 47.20±0.81 \| \| \| 0.1 \| 27.54±1.80 \| 49.63±0.80 \| 47.19±0.97 \| 46.32±0.97 \| 47.80±1.21 \| 57.09±0.94 \| \| \| 0.3 \| 46.39±2.37 \| 67.18±0.60 \| 60.60±1.32 \| 65.89±1.69 \| 66.77±1.55 \| 70.24±1.56 \| \| CIFAR-100\| 0.05 \| 6.45±0.92 \| 16.07±0.54 \| 13.75±1.01 \| 15.79±0.21 \| 16.17±1.33 \| 19.24±1.42 \| \| \| 0.1 \| 10.28±1.70 \| 22.07±0.43 \| 19.44±1.66 \| 20.99±1.17 \| 22.21±1.41 \| 23.59±1.27 \| \| \| 0.3 \| 15.22±2.08 \| 30.71±0.53 \| 26.14±1.37 \| 28.79±1.25 \| 30.33±1.24 \| 31.30±1.30 \| 4 EXPERIMENTS 4.1 Experimental Details Datasets and partitions. We conduct experiments on five real-world image datasets that are standard in the FL literature: MNIST (LeCun et al., [1998]), FMNIST (Xiao et al., [2017]), SVHN (Netzer et al., [2011]), CIFAR10, and CIFAR100 (Krizhevsky et al., [2009]). To simulate statistical heterogeneity, we use Dirichlet distribution to generate disjoint non-IID client training datasets as in Zhang et al. ([2022a]) and Heinbaugh et al. ([2023]). In particular, we sample $p_k \sim \text{Dir}(\alpha)$ and allocate a $p_k^i$ proportion of the data of class $i$ to client $k$. The parameter $\alpha$ controls the level of statistical imbalance, with a smaller $\alpha$ inducing more skewed label distributions among clients. Baselines. Within the contemporary model-market scenarios, we compare the performance of Co-Boosting against two existing methods: FedAvg (McMahan et al., [2017]) and DENSE (Zhang et al., [2022a]). Similar in Zhang et al. ([2022a]), we also introduce two prevailing data-free KD methods DAFL (Chen et al., [2019]) and ADI (Yin et al., [2020]) and apply these to one-shot FL, giving F-DAFL and F-ADI. We also include FedDF (Lin et al., [2020]) using the real validation dataset as the baseline. Configurations. Following McMahan et al. ([2017]), we use CNN with 5 layers for SVHN, CIFAR10, and CIFAR100. LeNet-5 (LeCun et al., [1998]) for MNIST and FMNIST. All available test data is used to evaluate the final server model (or ensemble). Unless otherwise stated, experiments are done with 10 clients and $\text{Dir}(0.1)$-parted. Results are reported averaged across at least 3 random seeds. 4.2 General Results Overall Comparison. To evaluate the effectiveness of our method, we conduct experiments under various non-IID settings by varying $\alpha = \{0.05, 0.1, 0.3\}$ and report the performance across different datasets and methods in Table 1. Notice, that we use the validation set in FedDF, which is not practical in the real world. From the table, we can conclude that Co-Boosting consistently outperforms all other baselines in all settings. Notably, in many settings, Co-Boosting achieves over a 5% accuracy improvement compared to the best baseline, DENSE. In cases of extreme statistical heterogeneity, such as when $\alpha = 0.05$, Co-Boosting surpasses the best baseline by substantial margins with 12.87%, 5.85%, 5.16%, 8.83%, and 3.07% on MNIST, FMNIST, SVHN, CIFAR-10, and CIFAR-100, respectively. We also compare the performance of the ensemble used in different methods on SVHN and CIFAR-10 in Table 2; others please refer to the Appendix. FedENS denotes averaged ensemble. The superior performance of the server model can be attributed to the enhanced 1 Code is available at https://github.com/rong-dai/Co-Boosting Table 2: Test accuracy of the ensemble in SVHN, and CIFAR-10 in Dir-parted setting. \| Dataset \| SVHN \| CIFAR-10 \| \|---------\|------\|----------\| \| Method \| α=0.05 \| α=0.1 \| α=0.3 \| α=0.05 \| α=0.1 \| α=0.3 \| \| FedENS \| 61.62±1.61 \| 69.71±0.68 \| 80.54±0.57 \| 42.34±0.67 \| 49.99±0.85 \| 69.61±0.50 \| \| Co-Boosting \| 65.69±1.48 \| 73.52±1.71 \| 82.90±1.35 \| 48.75±1.25 \| 59.86±1.76 \| 72.67±1.27 \| quality of synthesized data and the ensemble. While all compared methods, except FedAvg, utilize the direct logits averaging ensemble as the ensemble and aim to distill knowledge from it in a data-free manner. Co-Boosting, with its co-enhancing technique, results in a superior ensemble teacher, surpassing FedENS significantly. In a word, the superiority of our proposed method can be owed to the enhanced data and ensemble quality, which naturally translates into a better server model. Adaptation to Client Model Heterogeneity. To evaluate our proposed method in a potential client heterogeneity setting, we apply five different models under a CIFAR-10, Dir(0.1)-parted setting. The heterogeneous models include CNN1 in McMahan et al. (2017), CNN2 in the pytorch tutorial (Paszke et al., 2019), ResNet (He et al., 2016), MobileNet (Howard et al., 2019), and ShuffleNet (Ma et al., 2018). Table 3 demonstrates the results of comparable methods, where Local denotes directly taking the pre-trained model for testing. We take ResNet as the server architecture and omit FedAvg as it does not support this setting. We use the same optimization hyperparameter for all methods and across all model architectures. We remark that as suggested in Zhang et al. (2022a) and Diao et al. (2021), FL under both non-IID data and different model architecture is a quite challenging task. Even under this setting, our proposed Co-Boosting still consistently outperforms other baselines by a large margin thanks to the benefits of making the ensemble and data improve together. Table 3: Test accuracy of the server model architected in RestNet in CIFAR-10 across three levels of statistical heterogeneity under a heterogeneous client model setting. \| Dir(·) \| Local \| FedDF \| F-ADI \| F-DAFL \| DENSE \| Co-Boosting \| \|--------\|-------\|-------\|-------\|--------\|-------\|-------------\| \| α=0.05 \| 46.49±1.97 \| 56.51±1.87 \| 55.64±1.96 \| 55.14±1.51 \| 55.79±1.75 \| 58.64±1.83 \| \| α=0.1 \| 51.70±1.14 \| 59.49±1.06 \| 60.30±1.70 \| 58.58±1.49 \| 58.67±0.88 \| 62.30±0.90 \| \| α=0.3 \| 67.61±1.54 \| 70.53±1.19 \| 71.61±1.24 \| 71.92±1.95 \| 72.98±1.61 \| 75.02±1.37 \| 4.3 IN-DEPTH STUDY Different Local Data Amount. To further assess the effectiveness of Co-Boosting, which involves a better-weighted ensemble, we conduct experiments in an unbalanced local data setting. Similar to Acar et al. (2021), we sample data amounts for each client from a lognormal distribution. Higher values of σ result in more unequal data distribution. Table 4 and Fig. 2 display the performance of the ensemble and the server, where the prefix ‘DW-’ signifies weighted averaging based on the local data amount. As observed, reweighting clients based on their data amount yields some benefits, but it falls short of achieving the best ensemble. Furthermore, when using the averaged ensemble FedENS as the teacher, all baseline methods perform poorly due to the suboptimal teacher. In contrast, benefiting from simultaneously boosting data and ensemble, the ensemble we get consistently outperforms FedENS. This leads to a substantial performance gain of the server model achieved in Co-Boosting over all baselines, with a margin of at least 10%. Different Data Distribution Shift. Following Diao et al. (2023), we also conduct experiments on a C_cls partition setting, which means each client only possesses data of C out of all classes. Results in Table 5 further demonstrate the superiority of our proposed method. With better data quality and better ensemble, our method consistently achieves the best server model. Table 4: Test accuracy of ensemble. \| Method \| σ=0.4 \| σ=0.8 \| σ=1.2 \| \|----------\|-------\|-------\|-------\| \| FedENS \| 46.87±1.02 \| 41.86±1.20 \| 37.88±1.38 \| \| DW-FedENS\| 47.80±1.21 \| 53.25±0.52 \| 47.52±0.40 \| \| Co-Boosting \| 58.94±0.50 \| 57.41±1.12 \| 55.27±1.72 \| Figure 2: Test accuracy of server Table 5: Test accuracy of the server of different methods in CIFAR-10 under $C_{cls}$-parted setting. \| $C_{cls}$ \| FedAvg \| FedDF \| F-ADI \| F-DAFL \| DENSE \| Co-Boosting \| \|----------\|----------\|----------\|----------\|----------\|----------\|-------------\| \| 2 \| 16.15±2.61 \| 23.07±0.83 \| 23.16±1.12 \| 24.53±0.57 \| 23.85±0.93 \| 36.37±1.85 \| \| 3 \| 26.47±1.46 \| 38.39±0.64 \| 36.06±1.93 \| 38.13±1.17 \| 38.14±1.38 \| 53.91±1.80 \| \| 4 \| 33.78±2.42 \| 54.51±1.12 \| 51.04±1.40 \| 52.53±0.97 \| 51.53±1.79 \| 58.00±1.59 \| \| 5 \| 35.95±1.96 \| 58.34±1.58 \| 55.27±2.06 \| 54.67±1.26 \| 56.79±1.03 \| 62.52±1.75 \| Different Number of Clients. We evaluate the performance of these methods by varying the number of clients participating in OFL in Table 6. From the table, the final sever model still achieves the best accuracy when increasing the number of clients. This again validates that the increment quality of the ensemble model and data naturally brings a better server model. Table 6: Test accuracy of the server model in CIFAR-10 across different numbers of clients. \| n \| FedAvg \| FedDF \| F-ADI \| F-DAFL \| DENSE \| Co-Boosting \| \|-------\|----------\|----------\|----------\|----------\|----------\|-------------\| \| 5 \| 36.94±1.74 \| 50.62±0.98 \| 48.76±0.78 \| 49.76±1.42 \| 50.53±1.02 \| 54.29±1.38 \| \| 10 \| 27.54±1.80 \| 49.63±0.80 \| 47.19±0.97 \| 46.32±0.97 \| 47.80±1.21 \| 57.09±0.94 \| \| 20 \| 26.34±1.97 \| 38.98±0.99 \| 38.93±0.64 \| 36.28±1.39 \| 38.86±0.42 \| 49.56±0.98 \| \| 50 \| 23.01±0.94 \| 29.52±0.62 \| 27.45±1.13 \| 29.41±0.90 \| 28.51±0.54 \| 42.29±1.43 \| Effects of the proposed components. We further study the effectiveness of our proposed hard sample generation loss in Sec 3.2, on-the-fly sample difficulty promotion in Sec 3.2, and ensemble enhancing in Sec 3.3. Table 7 shows the experimental results on SVHN and CIFAR-10 in a 10-client Dir(0.05) parted setting. The results in the table illustrate that individually improving either data or ensemble leads to noticeable enhancements in the final server model performance. However, the most remarkable results are achieved when both data quality and ensemble capability are improved simultaneously. This finding strongly aligns with the underlying motivation of our study. Table 7: Ablations on different components of our method. “GHS” for hard sample generation in the generator loss, “DHS” for on-the-fly diverse hard sample creation, and “EE” for ensemble enhancement through reweighting. \| GHS \| DHS \| EE \| SVHN \| CIFAR-10 \| \|-----\|-----\|----\|------\|---------\| \| ✔ \| \| \| 58.46 \| 39.72 \| \| \| ✔ \| \| 61.18 \| 42.85 \| \| \| \| ✔ \| 61.38 \| 43.75 \| \| \| ✔ \| ✔ \| 62.67 \| 41.45 \| \| ✔ \| \| ✔ \| 63.42 \| 45.81 \| \| \| ✔ \| ✔ \| 62.46 \| 44.36 \| \| ✔ \| ✔ \| ✔ \| 64.40 \| 46.74 \| \| ✔ \| ✔ \| ✔ \| 65.40 \| 47.20 \| More facets. For a thorough and comprehensive understanding, we operate sensitivity analyses of hyperparameters, compare with multi-round federated learning, and conduct experiments with heavier local models. Please refer to the Appendix for the results. Moreover, since our Co-Boosting needs no alternation of the local training, it can be combined with advanced local training. The results attached in the Appendix further demonstrate the superiority of our proposed Co-Boosting. Limitation. The mixing weights are determined using synthetic samples. Though promising, there is still some disparity when compared to a mixing-weighted ensemble trained on real training data, as in Fig. 1(b). One possible way is to introduce virtual data (Yang et al., 2022). The exploration of methods to generate data capable of bridging this gap remains an avenue for further research. 5 CONCLUSION In this paper, we seek to tackle the inherent bottleneck of one-shot federated learning, where the performance of the server model is inextricably linked with the quality of the generated data and the ensemble. We propose Co-Boosting, a novel method that facilitates a mutually beneficial relationship between data generation and ensemble improvement. By iteratively generating hard samples from the ensemble and enhancing the ensemble based on these data, Co-Boosting adversarially improves the quality of both the data and the ensemble, leading to the natural refinement of the server model. Extensive experiments across various settings validate the efficacy of our method and demonstrate that our method can be practically applied to contemporary model-market scenarios. ACKNOWLEDGEMENTS RD, YGZ and BH were supported by the NSFC General Program No. 62376235, Guangdong Basic and Applied Basic Research Foundation No. 2022A1515011652, CCF-Baidu Open Fund, HKBU Faculty Niche Research Areas No. RC-FNRA-IG/22-23/SCI/04, and HKBU CSD Departmental Incentive Scheme. RD and XY were supported by National Natural Science Foundation of China (NSFC) under Grant U22A2094. TL is partially supported by the following Australian Research Council projects: FT220100318, DP220102121, LP220100527, LP220200949, IC190100031. 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Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. Domain-adversarial training of neural networks. The journal of machine learning research, 17(1):2096–2030, 2016.
ueTdErd5Ib	In addition, I suspect that Lemma A.1 is not actually true. In particular, the Hypothesis Stability provides a convergence rate $\beta_N$ on the change in predicted probability by omitting a point for a single binary classification problem. However, it seems to me for this lemma to work, the authors would need a uniform convergence rate $\beta_N$ that applies to all $N$ binary classification problems (note here that the number of binary classification problems appearing in the authors' method increases with sample size). This seems to me to be a much stronger property.	A Discretization Framework for Robust Contextual Stochastic Optimization Rares Cristian, Georgia Perakis Operations Research Center Massachusetts Institute of Technology, Cambridge, MA, USA {raresc,georgiap}@mit.edu Abstract We study contextual stochastic optimization problems. Optimization problems have uncertain parameters stemming from unknown, context-dependent distributions. Due to the inherent uncertainty in these problems, one is often interested not only in minimizing expected cost, but also to be robust and protect against worst case scenarios. We propose a novel method that combines the learning stage with knowledge of the downstream optimization task. The method prescribes decisions which aim to maximize the likelihood that the cost is below a (user-controlled) threshold. The key idea is (1) to discretize the feasible region into subsets so that the uncertain objective function can be well approximated deterministically within each subset, and (2) devise a secondary optimization problem to prescribe decisions by integrating the individual approximations determined in step (1). We provide theoretical guarantees bounding the underlying regret of decisions proposed by our method. In addition, experimental results demonstrate that our approach is competitive in terms of average regret and yields more robust solutions than other methods proposed in the literature, including up to 20 times lower worst-case cost on a real-world electricity generation problem. 1 Introduction In recent years, the field of machine learning (ML) has made remarkable strides in developing powerful algorithms that can automatically extract patterns and insights from data. While prediction is often the primary focus in many ML applications, the ultimate goal is to make optimal decisions based on these predictions. For example, one may predict the hourly electricity demand of a power plant for the next day. But more importantly, based off of this forecast, the operator must decide how much electricity to generate in order to minimize cost while staying within the operational constraints of the plant. Another key factor is the possible distribution of the uncertain parameters and being able to protect against worst-case scenarios. In the previous example, the decision-maker may have the goal to maximize the probability that their operational cost is below a certain threshold. Robustness is a crucial property in real-world decision-making since a single significantly poor decision may have separate damaging effects. For instance, it could damage a company’s reputation and trust with its customers. Traditionally, a predict-then-optimize approach has been used in practice: the learning stage is performed separately from the optimization task. First one trains a model to predict the uncertain parameters (such as the electricity demand), then independently solve the corresponding optimization problem. Recent work in end-to-end learning has focused on how to train a model with a loss function that is meant to explicitly approximate the true decision cost a prediction would produce. However these approaches only target minimizing the average cost, and do not, in general, take robustness into account. In this work, we propose a different paradigm for combining the learning and optimization tasks. In particular our paper makes the following contributions. 1) Novel approach to contextual stochastic optimization problems that is robust and data driven: We propose a novel data-driven method to tackle contextual stochastic optimization problems. The proposed method is directly applicable to any class of optimization problems, including linear, nonlinear and discrete optimization problems. Furthermore, it gives rise to solutions that are robust against uncertainty in the objective function using a single user-defined parameter to control the degree of robustness. In contrast to more traditional robust optimization methods, our proposed approach does not rely on constructing uncertainty sets. It is data driven and does not need to make any assumptions about the structure of the data itself and its distribution. 2) Analytical guarantees on the regret and the stability of the proposed approach: We prove analytical guarantees on the regret of the proposed method. For instance, we show that the difference between the in-sample cost and the out-of-sample cost decreases on the order of $1/\sqrt{n}$, for $n$ datapoints. Furthermore, we prove the proposed method is stable against noise, showing that the decisions prescribed do not change significantly if the dataset is perturbed by noise. 3) Computational experiments on a variety of applications: We show with computational experiments that the proposed method is competitive in terms of the average error relative to existing approaches in the literature. In addition to testing our approach for linear optimization applications such as portfolio optimization using historical stock data, we also consider nonlinear optimization problem applications such as inventory allocation and electricity generation using real-world data. Finally, through these experiments, we show significant improvement in terms of robustness. We obtain as much as 20 times lower cost in the worst case when compared to other end-to-end learning methods and 5 times lower than other robust approaches. 2 RELATED WORK Due to space limitations we will keep this section short. End-to-End Learning: Traditionally, the simplest way to learn the uncertain parameters is to do so independently of the optimization problem by minimizing a loss function such as mean-squared error between predictions and observed realizations. However, it has been shown that solving the predictive and decision-making problems independently, can produce significantly suboptimal decisions Cameron et al. (2021). As such, a large stream of the literature includes end-to-end methods where the goal is to propose predictions whose corresponding optimal decisions minimize the downstream task loss (e.g., the objective function). One of the earlier works related to end-to-end learning is Kao et al. (2009) which trains a model to minimize task loss of an unconstrained quadratic optimization problem. In general, the primary difficulty in end-to-end learning approaches is the differentiability of the constrained optimization task. Amos & Kolter (2017) extends the setting in Kao et al. (2009) to constrained quadratic optimization, computing the gradient by differentiating through the KKT system of equations at the optimal solution. Unfortunately, for linear optimization, the problem becomes more complex since the gradient of the output of a linear problem with respect to its objective coefficients is either zero everywhere or undefined. Wilder et al. (2019) addresses the issue by taking a similar approach to Amos & Kolter (2017) for linear optimization problems but also adding a quadratic regularization term to the objective function. Other approaches have focused on other methods of altering the loss function or the objective function to compute more useful gradients end-to-end. For instance, for linear optimization problems, Elmachtoub & Grigas (2022) constructs a surrogate loss function that is a convex and differentiable approximation of the objective function. Elmachtoub et al. (2020) takes this approach and proposes a method to train decision trees with this surrogate loss. Mandi & Guns (2020), Vlastelica et al. (2020), Berthet et al. (2020) take different approaches to address this issue. Kotary et al. (2021) and the references therein provide a general survey for end-to-end combinatorial learning problems. The approach we propose is applicable directly to any class of optimization problems, while individual end-to-end methods are usually restricted to certain sub-classes of problems. Moreover, a major difference in the approach proposed in this paper, is that it is non-parametric and proposes decisions directly from data without requiring an intermediate forecast. Prescriptive Analytics and Robust Optimization: To solve a stochastic optimization problem one can apply well-known methods such as contextual Sample Average Approximation (SAA) (see for example, Kleywegt et al. (2002)). The work of Bertsimas & Kallus (2020) extends SAA to take advantage of the contextual nature of the problem by using covariates and weighing samples in a non-uniform way (unlike SAA) using ML methods such as $k$-nearest neighbors or decision trees. For instance, for an out-of-sample set of features $\mathbf{x}$, use the $k$-nearest observations in the training data to make a decision. Alternatively, Bertsimas & Koduri (2021) generates weights using global methods, not only by using data around a neighborhood of out-of-sample $\mathbf{x}$. Bertsimas et al. (2019a) extends the general methodology by introducing an optimal prescriptive tree framework to produce weights that are directly dependent on the optimization problem and minimize task loss. Furthermore, Kallus & Mao (2022) considers a similar framework using a random forest. Finally, Bertsimas & McCord (2019) applies these prescriptive ideas to the multi-period setting. There has been significant work within the robust optimization literature over the years (see for example, the books by Ben-Tal et al. (2009), Bertsimas & den Hertog (2021) as well as the survey paper by Bertsimas et al. (2010) and the references within). In robust optimization, careful construction of the underlying uncertainty sets is required to ensure the models are not overly conservative. Various formulations have been proposed, starting for example, with Soyster (1973), Ben-Tal & Nemirovski (2000), and Bertsimas & Sim (2004). Nevertheless, uncertainty sets can be learnt as we gain information from data. Earlier papers uses estimates of the mean and standard deviation from the available data such as for example, Bertsimas et al. (2013) who takes a data-driven robust optimization view. Uncertainty sets could vary as a function of the features, as for example in Bertsimas & Van Parys (2021), Kannan et al. (2020) and Bertsimas et al. (2019b). ### 3 THE FRAMEWORK In this section, we first formally describe the problem and the data-driven setting we study in this paper. Given a feasible region $\mathcal{P}$ and decision variables $\mathbf{w} \in \mathcal{P}$, the goal is to minimize an objective function $g_{\nu}(\mathbf{w})$ parameterized by uncertain parameters $\nu$. If we have exact knowledge of the realized uncertainty $\nu$ values, the optimal decision could be determined through the following problem $$w^(\nu) = \arg \min_{\mathbf{w} \in \mathcal{P}} g_{\nu}(\mathbf{w}).$$ If the problem above does not a unique solution, we can instead assume that $w^(\nu)$ is an oracle providing any optimal solution. For example, $g_{\nu}(\mathbf{w})$ can correspond to a linear optimization problem objective, $g_{\nu}(\mathbf{w}) = \nu^T \mathbf{w}$ or a quadratic optimization problem objective, $g_{\nu}(\mathbf{w}) = \mathbf{q}^T \mathbf{w} + \mathbf{w}^T \mathbf{Q} \mathbf{w}$, where $\nu = (\mathbf{q}, \mathbf{Q})$ corresponds to the linear and quadratic objective coefficients respectively. In a shortest path example, the uncertainty parameters $\nu$ would correspond to the unknown travel times along each edge while the decision $\mathbf{w}$ would be a vector determining which path to take. Naturally set $\mathcal{P}$ constrains $\mathbf{w}$ to properly satisfy the path related constraints. Finally, we formally define the notion of regret of a decision: Definition 1 The regret $R_{\nu}(\mathbf{w})$ of a decision $\mathbf{w}$ with respect to a parameterization $\nu$ is given by the difference between its objective value and the optimal one corresponding to $\nu$: $$R_{\nu}(\mathbf{w}) = g_{\nu}(\mathbf{w}) - g_{\nu}(w^(\nu))$$ We make the following assumptions regarding the optimization problem: Assumption 3.1* We assume that the maximum regret $R_{\nu}(x) = g_{\nu}(\mathbf{w}) - g_{\nu}(w^(\nu))$ is bounded and at most $M_1 > 0$ for any $\nu$, $\mathbf{w}$. In the data-driven setting, we assume that the objective’s uncertain parameters are distributed according to an unknown distribution $D_x$ which depends on features $\mathbf{x}$. Given some vector of features $\mathbf{x}$, we need to compute decision $\hat{\mathbf{w}}(\mathbf{x})$. Only afterwards can we observe the realization of $\nu_x \sim D_x$ and incur a cost of $g_{\nu_x}(\hat{\mathbf{w}}(\mathbf{x}))$. We take a data driven approach and do not assume we know the distribution $D_x$ of the cost vector for any given feature vector $\mathbf{x}$. Rather we assume we are given $N$ data points $(\mathbf{x}_1, \nu_1), \ldots, (\mathbf{x}_N, \nu_N)$ consisting of observed covariates $\mathbf{x}_i$ and observed realizations $\nu_i \sim D_x$. One objective is to minimize the expected regret of the decision: $$\min_{\mathbf{w} \in \mathcal{P}} \mathbb{E}_{\nu_x \sim D_x}[R_{\nu_x}(\mathbf{w})]$$ while another would be to provide a solution that is more robust to uncertainty. One may wish to minimize the probability that the regret is above a certain threshold: $$\min_{\mathbf{w} \in \mathcal{P}} \mathbb{P}_{\nu_x \sim D_x}(R_{\nu_x}(\mathbf{w}) \geq \phi).$$ However, this formulation, even given perfect knowledge of the distribution of $ w^(\nu_x) $ is not tractable. For instance, for discrete $ D_x $, it may be solved only by a mixed integer optimization problem (see Appendix E). We propose an approximation to this objective: to minimize the expected regret that is larger than $ \phi $, which we denote as the minimum violation solution. If the regret of $ w $ is below $ \phi $ for a given $ \nu_x $, we treat it as having no regret. Otherwise, we assign it a regret of $ R_{\nu_x}(w) - \phi $, the amount by which its regret surpasses $ \phi $. This is the same as in formulation (4) except the cost of having regret greater than $ \phi $ is simply a constant of 1 in (4). Moreover, this is now a convex problem whenever $ g_{\nu}(x) $ is convex. Definition 2 (minimum-violation) The minimum violation optimization problem is given by \[ \min_{w \in P} \mathbb{E}_{\nu_x \sim D_x} \left[ \max \{ R_{\nu_x}(w) - \phi, 0 \} \right] \] (5) To gain additional intuition into the choice of this formulation, we can also rewrite this objective as \[ \min_{w \in P} \mathbb{P}_{\nu_x \sim D_x} (R_{\nu_x}(w) \geq \phi) \cdot \mathbb{E}_{\nu_x \sim D_x} [R_{\nu_x}(w) - \phi \| R_{\nu_x}(w) \geq \phi] \] (6) Notice that for $ \phi = 0 $, the above formulation reduces to simply minimizing the expected regret. In addition, for $ \phi $ large enough (so that regret is always bounded by $ \phi $), the problem becomes fully robust and produces the same solution as (4). In between, this is a combination of two objectives. The left term is the original one in (4) to minimize the probability that regret is larger than $ \phi $. The second term is similar to conditional value at risk (CVaR) [Rockafellar & Uryasev (2002)]. The CVaR objective minimizes $ \mathbb{E}_{\nu_x} [R_{\nu_x}(w) \| R_{\nu_x}(w) \geq q_\alpha(w)] $ where $ q_\alpha(w) $ is the $ \alpha $-th quantile of the regret distribution of taking decision $ w $. To contrast this approach with ours, note that $ q_\alpha(w) $ can change for each $ w $ while the $ \phi $ term remains constant throughout. We see in the computational experiments (section 5.1 and Figure 2) that the CVaR approach produces decisions that change discretely as the robustness parameter changes (the quantile being targeted), whereas the minimum-violation objective produces more continuously changing decisions. Overview The key idea is to coarsen the problem and discretize the feasible region $ P $ into $ K $ subsets $ H_1, \ldots, H_K $ and determine the probabilities $ \mathbb{P}(w^(\nu_x) \in H_k) $ that the optimal solution $ w^(\nu_x) $ belongs to each $ H_k, k = 1, \ldots, K $. We can use these discrete $ H_k $ as building blocks to approximate the expectations in (3) and (5). Intuitively, we would like to construct $ H_k $ so that, if $ w^(\nu_x) $ belongs to $ H_k $, then $ R_{\nu_x}(w) $ can be well approximated by some deterministic function of $ w $. Then, we minimize the expected regret based off of these individual approximations. Discretization* Consider constructing $ H_k^\epsilon $ for each datapoint $(x^k, \nu^k)$ corresponding to the set of points whose regret is at most $ \epsilon $: \[ H_k^\epsilon = \{ w \in P : g_{\nu^k}(w) - g_{\nu^k}(w^(\nu^k)) \leq \epsilon \} = \{ w \in P : R_{\nu^k}(w) \leq \epsilon \} \] (7) Now it remains to approximate the probability that $ w^(\nu_x) $ belongs to each $ H_k^\epsilon $. We approximate $ \mathbb{P}(w^(\nu_x) \in H_k^\epsilon) $ by leveraging the data we already have for training. For the training data, we compute point estimates of this probability since we have access to the realized cost parameters. That is, for every feature point $ x^n $, we can determine whether the optimal decision $ w^(\nu^n) $ either belongs to set $ H_k^\epsilon $ or not. For each pair $(x^n, \nu^n)$ and $ H_k^\epsilon $, we generate the following labels: \[ p_k^n = \begin{cases} 1, & \text{if } w^(\nu^n) \in H_k^\epsilon \\ 0, & \text{otherwise} \end{cases} \] (8) This creates a new multi-label data set $(x^n, (p_k^n)_{k=1,\ldots,N})$. We can then learn a mapping $ \hat{p}_k(x) $ which approximates $ \mathbb{P}(w^(\nu_x) \in H_k^\epsilon) $. We accomplish this using any classification method such as for example, logistic regression, decision trees, k-nearest neighbors, and neural networks among others. Figure 1 provides an illustration and the corresponding labels we create. Algorithm We summarize the algorithm as the following steps: (i) Define subsets $ H_k^\epsilon = \{ w \in P : R_{\nu^k}(w) \leq \epsilon \} $ for each datapoint. (ii) Construct labels $ p_k^n $ to indicate whether $ w^(\nu^n) \in H_k^\epsilon $. Figure 1: $ w^(\nu^1) $ and $ w^(\nu^2) $ belong to $ H_1^\epsilon $, but $ w^(\nu^3) $ does not. Thus, points $ x^1, x^2 $ are labeled with 1, and $ x^3 $ is labeled as 0. (iii) Train ML model $ \hat{p}_k(x) $ on multi-label dataset $(x^n, (p_k^n)_{k=1,\ldots,N})$. (iv) For out-of-sample $ x $, take decision \[ \hat{w}_{\epsilon,\phi}(x) = \arg\min_{w \in P} \sum_{k=1}^{N} \hat{p}_k(x) \cdot \max\{R_{\nu^k}(w) - \phi, 0\} \] (9) This optimization problem in (9) combines the individual predictions of which sets $ H_k^\epsilon $ the solution should belong to. To gain some intuition on this last step, consider the case that the sets $ H_k^\epsilon $ form a partition of the feasible region $ P $. Note that we will relax this to the general case in Theorem 4.1. Then, conditioning on the events $ w^(\nu_x) \in H_k^\epsilon $, the minimum-violation problem in (5) can be rewritten as \[ E_{\nu_x \sim D_x} [\max\{R_{\nu_x}(w) - \phi, 0\}] = \sum_k P(w^(\nu_x) \in H_k^\epsilon) E[\max\{R_{\nu_x}(w) - \phi, 0\} \| w^(\nu_x) \in H_k^\epsilon]. \] (10) In Theorem 4.1, we will show that the term $ \max\{R_{\nu^k}(w) - \phi, 0\} $ approximates the value of $ \max\{R_{\nu_x}(w) - \phi, 0\} $, whenever $ w^(\nu_x) \in H_k^\epsilon $. Moreover, these terms are weighted by $ \hat{p}_k(x) $, the approximations of $ P(w^(\nu_x) \in H_k^\epsilon) $. Alternative interpretation for $ \phi = \epsilon $: We would also like to present an alternative viewpoint of our proposed method which connects the choice of objective $ E[\max\{R_{\nu_x}(w) - \phi, 0\}] $ to the rest of the method and how the weights $ \hat{p}_k(x) $ are generated. This interpretation is unique to our proposed method, and differs from that of Bertsimas & Kallus (2020) and related literature. We can alternatively view the problem as follows. For out-of-sample features $ x $, the goal is to find a feasible solution $ w $ that best matches the predictions $ \hat{p}_k(x) $ in terms of which sets $ H_k^\epsilon $ the optimal solution $ w^(\nu_x) $ should belong to. For example, if we predict that $ w^(\nu_x) $ belongs to $ H_1^\epsilon $ and to $ H_2^\epsilon $ with high probability (meaning weights $ \hat{p}_1(x), \hat{p}_2(x) $ are high), then the solution we propose should belong to the intersection of sets $ H_1^\epsilon $ and $ H_2^\epsilon $. The formulation in (9) performs this by implicitly scoring each feasible solution: if a feasible solution $ w $ does not belong to $ H_k^\epsilon $, we penalize it by our approximation $ \hat{p}_k(x) $ that it should have belonged to it multiplied by the distance from $ H_k^\epsilon $. However, if $ w $ does belong to $ H_k^\epsilon $, then there is no penalty. This score exactly corresponds to each term $ \hat{p}_k(x) \cdot \max\{R_{\nu^k}(w) - \epsilon, 0\} $. We choose the feasible solution that minimizes overall penalty. 4 THEORETICAL REGRET BOUND AND PRACTICAL APPLICATION Theorem 4.1 Under Assumption 3.1, the expected regret of a decision $ w \in P $ can be bounded above by the approximate problem in (9) with probability $ 1 - \delta $ as follows: \[ E[\max\{R_{\nu_x}(w) - \theta, 0\}] \leq c_\epsilon \cdot \alpha \left( OJB(w) + M_1 \mathcal{E} + \sqrt{\frac{\log 1/\delta}{2N}} \right) \] (11) for $ \theta = \alpha(\beta_\epsilon + \phi) $ and where $ OJB(w) = \frac{1}{N} \sum_{k=1}^{N} \hat{p}_k(x) \max\{R_{\nu^k}(w) - \phi, 0\} $ is the approximation we optimize over in (9). $ \mathcal{E} $ is the mean prediction error $ \mathcal{E} = \frac{1}{N} \sum_{k=1}^{N} \|\hat{p}_k(x) - P(w^(\nu_x) \in H_k^\epsilon)\| $. and constants $ \alpha, \beta_\epsilon $ depending on the optimization problem. In particular, for the following classes of objectives we have 1. bi-lipschitz objectives: for any bi-lipschitz objective, with constants $ L, \mu $ such that \[ \mu \\| w^1 - w^2 \\| \leq \| g_w(w^1) - g_w(w^2) \| \leq L \\| w^1 - w^2 \\|, \] we have $ \alpha = L/\mu $ and $ \beta_\epsilon = \epsilon $. As an example, this holds for any quadratic optimization problem with bounded feasible region having a positive definite quadratic term. 2. quantile loss function: given a prediction $ w $ and outcome $ v_x $, the regression loss for quantile $ q $ is given by \[ g_{v_x}(w) = \max \{ q(v_x - w), (1 - q)(w - v_x) \}. \] Then $ \alpha = 1 $ and \[ \beta_\epsilon = \max \left\{ q/(1-q), (1-q)/q \right\} \epsilon. \] For example, this is also applicable to the inventory stock problem in section 5.1. Furthermore, $ c_\epsilon $ is a constant factor describing for any $ x $, how often $ w^(v_x) $ will have regret at most $ \epsilon $ with respect to a random other cost vector. In particular, $ c_\epsilon = 1/\min_x P_y(R_{v_y}(w^(v_x)) \leq \epsilon) $. The full proof can be found in Appendix A. Moreover, we prove stability of the output of our proposed model under perturbations in the data. This can be found in the appendix (see Appendix B). In short, the stability result describes the change in the decision $ \hat{w}_{\epsilon,0}(x) $ when the dataset is perturbed by noise. If each of the learning algorithms used to train $ \hat{p}_k(x) $ have hypothesis stability (defined in the appendix, see definition 3), then the output $ \hat{w}_{\epsilon,0}(x) $ also changes by a small amount when the dataset is perturbed. Before the computational section, we discuss some of the main practical issues and take-aways from Theorem 4.1. In particular, we discuss how to practically choose $ \epsilon $ and how this affects each term in the bound in theorem 4.1. Discussion on choosing $ \epsilon $. By definition, $ \epsilon $ determines the size of sets $ H_k^\epsilon $. This in turn affects the resulting multi-labelling $ p_n^k, k = 1, \ldots, N $. If for example, the sets $ H_k^\epsilon $ do not intersect, then each vector $ p^n = (p_n^k)_{k=1,\ldots,N} $ has a single non-zero entry. This would make $ \hat{p}_k(x) $ impossible to learn. As such, we propose the following method of choosing $ \epsilon $: choose $ \epsilon $ large enough so that for each vector $ p^n $, at least some fraction, which we denote by $ \gamma_\epsilon $, of its entries are non-zero. Then, from Theorem 4.1 we argue that given $ N $ datapoints, one should choose $ \epsilon $ large enough so that $ \gamma_\epsilon \geq 1/\sqrt{N} $. We can see this as follows by viewing the impact of $ \epsilon $ on each term of the bound: Effect on $ c_\epsilon $. Recall that if $ p_n^k = 1 $, this is equivalent to $ R_{v_y}(w^(v_x)) \leq \epsilon $. Therefore, a $ \gamma_\epsilon $ chosen in this way implies that \[ P_y(R_{v_y}(w^(v_x)) \leq \epsilon) \approx \gamma_\epsilon. \] Moreover, this results in $ c_\epsilon \approx 1/\gamma_\epsilon $. Effect on $ E $. $ \gamma_\epsilon $ also affects the error of the ML models, namely $ E $. For instance, labelling everything with a zero will have an error of $ E = \gamma_\epsilon $. In general, any ML model that improves beyond this baseline will have $ E \leq \gamma_\epsilon $. Effect on $ OBJ(w) $. We can bound $ OBJ(w) $ by \[ OBJ(w) = \frac{1}{N} \sum_{k=1}^{N} \hat{p}_k(x) \max \{ R_{v_y}(w) - \phi, 0 \} \] \[ \leq M_1 \frac{1}{N} \sum_{k=1}^{N} \hat{p}_k(x) \leq M_1 \left( E + \frac{1}{N} \sum_{k=1}^{N} P(w^(v_x) \in H_k^\epsilon) \right). \] Finally, the term $ \frac{1}{N} \sum_{k=1}^{N} P(w^(v_x) \in H_k^\epsilon) $ concentrates around its expectation which is $ \gamma_\epsilon \approx P_y(R_{v_y}(w^(v_x)) \leq \epsilon) $. It follows that $ OBJ(w) \lesssim 2M_1 \gamma_\epsilon $, given that $ E \leq \gamma_\epsilon $ as argued previously. Overall, this implies that the right hand side of the bound in theorem 4.1 has $ c_\epsilon $ on the order of $ 1/\gamma_\epsilon $, while $ OBJ(w) $ and $ E $ are both on the order of $ \gamma_\epsilon $. Putting these together, we see that these cancel out to be overall on the order of $ O(M_1) $. The only remaining term is $ c_\epsilon \cdot \sqrt{\log(1/\delta)/2N} $. Hence, we need enough data so that this term also becomes on the order of $ O(1) $. Therefore, we need on the order of $ 1/\gamma_\epsilon^2 $ datapoints or that $ \gamma_\epsilon \geq 1/\sqrt{N} $. We present results in the computational experiments section about the effect of $ \epsilon $ on the quality of decisions produced. The remaining question is now, how small can $ \epsilon $ be so that $ \gamma_\epsilon \geq 1/\sqrt{N} $? In practice, this can be determined on a case-by-case basis for each dataset, for instance by using binary search. Of course, $ \gamma_\epsilon \geq 1/\sqrt{N} $ is only a rough guide to understand the magnitude needed. A process like cross-validation can ultimately be used to choose $ \epsilon $. Theoretically, in the worst case either (1) such an $ \epsilon $ needs to be a constant fraction of the diameter of the feasible region or (2) $ \epsilon $ is kept small but we need an amount of data that is exponential in the dimension of the decisions. However, this is often not the case in practice when presented with real-world data. Ultimately this largely depends on the distribution of the decisions $ w^(\nu) $ and $ P $ itself. In real-world data, the distribution of $ w^(\nu) $ would not pathologically uniformly cover the feasible region, but rather be more clustered around smaller regions. Moreover, the feasible region itself plays a crucial role. Consider the case that decisions $ w \in \mathbb{R}^d $ are constrained to a feasible region $ P $ that is an $ s $-dimensional subspace of the ambient space $ \mathbb{R}^d $. For instance, if $ P = \{ w : Aw = b, w \geq 0 \} $ where the null space of $ A $ has dimension $ s $. 5 COMPUTATIONAL RESULTS We consider four applications of the proposed approach. Two applications come from [Donti et al. (2017)]: a synthetic inventory stock problem and a real-world energy scheduling task. We show that this discretization approach is competitive with other methods in terms of expected cost, and has significant improvement in terms of robustness with up to 20 times lower cost in worst-case scenarios. Moreover, we also compare against other non-contextual robust optimization methods. Due to space limitations, we present this last experiment in appendix C.1. We also put into practice the earlier discussion regarding the choice of $ \epsilon $ and how its value affects the quality of the decisions ranging from performing better at minimizing the expected cost or being robust. 5.1 INVENTORY STOCK PROBLEM Consider the classical newsvendor problem in which a given product has uncertain demand $ d $ as well as observed covariates $ x $. Each day we observe the new features $ x $ and must make a decision $ w $ for the amount of product to supply. Afterwards, the true demand is realized. For each unit of supply above the demand, there is a unit cost of $ c_h $ (for holding an item overnight in the store) and for each unit of supply below the demand, we incur a backorder or lost sales unit cost $ c_b $ (for example, cost of expedited shipping to ensure the product arrives the next day). It has been shown that the optimal order quantity is the $ c_b/(c_b + c_h) $ quantile of the demand distribution (see Arrow et al. (1951)). In principle, one could apply the quantile loss function to predict this quantity and solve the problem in a two-stage manner. However, to consider the identical problem also presented in [Donti et al. (2017)], we consider a version with additional quadratic costs to over or under-stocking as well as an ordering cost. The objective is given as \[ g_d(w) = c_0w + c_qu^2 + c_b \max\{d - w, 0\} + \frac{1}{2} q_b \max\{d - w, 0\}^2 + \\ c_h \max\{w - d, 0\} + \frac{1}{2} q_h \max\{w - d, 0\}^2 \] Experimental setup: We use the same unit cost parameters as well as data, and compare against the same models as in [Donti et al. (2017)]. However, we present not only the average cost incurred on the testing data but also on various quantiles of the cost distribution. We plot on the $ x $-axis the mean cost incurred by each method, and on the $ y $-axis the cost at the $ q^{th} $ quantile. The problem of $ \min_{w \in P} \mathbb{E}[\max\{R_{\nu_x}(w) - \phi, 0\}] $ can also be solved by existing methods such as [Bertsimas & Kallus (2020)] by thinking of the objective function not as $ g_{\nu}(x) $ but rather as $ \max\{R_{\nu_x}(w) - \phi, 0\} $ directly. In this approach, weights are generated by K-nearest neighbors (or other ML models like linear or tree models). However, these methods do not generate these weights based on the optimization problem itself. In contrast our proposed method generates weights explicitly depending on the optimization task. We denote this approach as KNN + minimum-violation in the experiments. We also compare against the method of [Bertsimas & Kallus (2020)] where we use CVaR as the objective, namely $ \mathbb{E}[R_{\nu_x}(w)\|R_{\nu_x}(w) \geq q_{\alpha}(R_{\nu_x}(w))] $, where $ q_{\alpha}(Z) $ is the $ \alpha $-quantile of a random variable $ Z $. We consider a range of values for the quantile $ \alpha $ in the experiments. In addition, we also compare against another data-driven contextual robust optimization methods where one solves the following problem. For an out-of-sample $ x $, find the $ K $ nearest neighbors in the data, Figure 2: Comparison of average cost of each method vs. 85th, 90th, 95th, 99th and 100th quantile cost. We vary the degree of robustness for each method. As this robustness parameter increases, the mean cost increases, but the quantile cost (generally) decreases for all methods. namely, $\mathcal{N}(x, K)$. Unlike Bertsimas & Kallus (2020), which minimizes the average cost, we weight the remaining $K$ datapoints adversarially according to a distribution $\pi$ that is, within a Kullback-Leibler divergence distance of at most $r$ from a uniform distribution, assigning weights $1/K$ to each datapoint (so that $D_{KL}(\pi \|\| 1/K) \leq r$). Due to space limitations, details and formulations for each of these approaches can be found in appendix C.1. Moreover, we compare against the method in Doni et al. (2017). This is an end-to-end method which trains a neural network to predict a discrete probability distribution for the possible values of demand. We refer to this as the task-based method in Figure 2. This work makes use of the OptNet framework in Amos & Kolter (2017) to compute gradients of the loss function with respect to the predicted demand $d$. Furthermore, we also compare against a policy optimizer approach. Here, one does not make a forecast for demand, but rather the neural network model directly outputs the policy/decision to take. Each of these methods use a linear model to make predictions (or decisions, as in the case of the policy optimizer). For the other weight-based methods (KNN + KL divergence and those based on Bertsimas & Kallus (2020)) we use a KNN method to generate weights (with $K = 10$). For consistency, our approach also uses a KNN model to predict the $\hat{p}_k(x)$, also using $K = 10$ neighbors. Each $\hat{p}_k(x)$ predicts a value between 0 and 1 based on the average label assigned to the $K$-nearest neighbors of $x$ in the training data. This is done independently for each $k$. In contrast, for the following experiment in section 5.2, the models $\hat{p}_k(\cdot)$ are trained simultaneously by a neural network. Results: In Figure 2, we report the mean and the 85th, 90th, 95th, 99th and 100th quantile (out-of-sample) cost of each decision made by the approaches as we vary the level of robustness for each approach ($\phi$ for the minimum-violation objective and $\alpha$ in for CVaR). The KNN + minimum-violation method and our proposed method have the same objective, but use different methods of solving it. We see that both approaches produce similarly shaped mean vs. quantile cost curves but the discretization method consistently has lower $q^{th}$ quantile cost for the same average cost for all $q = 85$ to $q = 100$. As this quantile approaches 100, the gap between the two decreases but of note, at $q = 100$, all approaches, other than the discretization method performs poorly, worse than even the policy method in terms of robustness. The CVaR and traditional robust methods also deteriorate in performance as the quantile increases. 5.2 LOAD FORECASTING AND GENERATOR SCHEDULING Next, we consider a real-world problem for generator scheduling using 8 years of real electrical grid data from PJM, an electricity routing company coordinating the movement of electricity throughout Figure 3: Reporting for each hour of the day the mean, 99th quantile, and maximum cost of each method. A fixed $\phi = 0.5$ value for the discretization method was used for all results. 13 states. We use the same range of data as also used in [Donti et al., 2017]. Here, we must make decisions $w \in \mathbb{R}^{24}$ for the amount of electricity generation for each hour of the following day. Similar to the inventory problem, the operator incurs a cost $\gamma_e$ for excess generation and a cost $\gamma_s$ for a shortage in generation. In addition, power plants have physical limitations prohibiting large changes in generation from one hour to the next. The objective and constraints are given as: $$g_d(w) = \sum_{i=1}^{24} \gamma_s \max\{d_i - w_i, 0\} + \gamma_e \max\{w_i - d_i, 0\}, \quad \|w_{i+1} - w_i\| \leq r, \ i = 1, \ldots, 23$$ Experimental setup: We use the same setup used in [Donti et al., 2017], using the same parameters for the problem as well as data. We make use of the same data preprocessing and feature engineering. They use a two-layer (each layer of width 200) network with an additional residual connection from the inputs to the outputs. We use the same architecture to learn the labeling $\hat{p}_k(x)$. In addition, we compare against a cost-weighted model minimizing mean-squared error which periodically reweights training samples based on their task-based cost. Finally, we also compare against the method described in the previous section with objective to minimize CVaR. Results: We compare the average cost as well as the 98 – 100th quantiles of the cost distribution for each method. In particular, we also present results for different choices of $\epsilon$. Following the discussion of theorem 4.1, a starting point would be to choose $\epsilon$ so that the average number of positive labels, $\gamma_e$, is around $1/\sqrt{N}$. We choose different epsilon so that $\gamma_e = 0.015, 0.025, 0.04, 0.05$ where $1/\sqrt{N} \approx 0.02$ (here we have $N = 2,553$ training points). As $\epsilon$ and $\gamma_e$ decrease, we find that the solutions better target minimizing the expected cost, while increasing $\epsilon$ will improve performance on higher quantiles of the cost distribution. In particular, when $\gamma_e = 0.025$, the method outperforms even the task-based method on average cost at peak demand hours of the day (hours 15-22). At the other extreme at $\epsilon = 0.05$, we find that the worst case cost across the entire day is nearly constant and up to more than 20 times lower than the CVaR and other methods. However, setting $\epsilon = 0.01$ to be too small does not introduce enough robustness. While it performs best in terms of average cost, its worst-case cost spikes suddenly. 6 CONCLUSIONS We proposed a novel method for contextual stochastic optimization based off of discretizing the feasible region into subsets and learning how the optimal solution maps to each subset. We proved analytical guarantees on the bound between the expected out-of-sample cost compared to the approximate objective proposed in [9]. Finally, we present computational experiments on three datasets, including a real-world electricity generation problem, and show our proposed method is competitive against other end-to-end approaches and provides significantly more robust solutions, even when compared to other robust optimization methods. Future directions of research may include devising different constructions of the subsets $H_k^\epsilon$ and to consider uncertainty in constraints as well. REFERENCES Brandon Amos and J Zico Kolter. Optnet: Differentiable optimization as a layer in neural networks. In International Conference on Machine Learning, pp. 136–145. PMLR, 2017. Kenneth J Arrow, Theodore Harris, and Jacob Marschak. Optimal inventory policy. Econometrica: Journal of the Econometric Society, pp. 250–272, 1951. A. Ben-Tal, L.E. Ghaoui, and A. Nemirovski. Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, 2009. ISBN 9781400831050. Aharon Ben-Tal and Arkadi Nemirovski. 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rUH2EDpToF	- In section 4.3, point 3), I hold a different perspective regarding the efficiency of MAMs compared to ARMs, especially in high-dimensional scenarios. While it's true that in ARM-Full, you do require D feed-forward runs for gradient computation, in MAMs, you also necessitate Gibbs sampling to generate samples from the model. Even if you employ block-wise Gibbs sampling, it still demands multiple steps to guarantee MCMC chain convergence. Hence, I doubt that MAMs also face challenges in high-dimensional problems.	GENERATIVE MARGINALIZATION MODELS Anonymous authors Paper under double-blind review ABSTRACT We introduce marginalization models (MAMs), a new family of generative models for high-dimensional discrete data. They offer scalable and flexible generative modeling with tractable likelihoods by explicitly modeling all induced marginal distributions. Marginalization models enable fast evaluation of arbitrary marginal probabilities with a single forward pass of the neural network, which overcomes a major limitation of methods with exact marginal inference, such as autoregressive models (ARMs). We propose scalable methods for learning the marginals, grounded in the concept of “marginalization self-consistency”. Unlike previous methods, MAMs also support scalable training of any-order generative models for high-dimensional problems under the setting of energy-based training, where the goal is to match the learned distribution to a given desired probability (specified by an unnormalized (log) probability function such as energy or reward function). We demonstrate the effectiveness of the proposed model on a variety of discrete data distributions, including binary images, language, physical systems, and molecules, for maximum likelihood and energy-based training settings. MAMs achieve orders of magnitude speedup in evaluating the marginal probabilities on both settings. For energy-based training tasks, MAMs enable any-order generative modeling of high-dimensional problems beyond the capability of previous methods. 1 INTRODUCTION Deep generative models have enabled remarkable progress across diverse fields, including image generation, audio synthesis, natural language modeling, and scientific discovery. However, there remains a pressing need to better support efficient probabilistic inference for key questions involving marginal probabilities $p(x_s)$ and conditional probabilities $p(x_u \| x_v)$, for appropriate subsets $s, u, v$ of the variables. The ability to directly address such quantities is critical in applications such as outlier detection [50, 40], masked language modeling [11, 72], image inpainting [73], and constrained protein/molecule design [69, 55]. Furthermore, the capacity to conduct such inferences for arbitrary subsets of variables empowers users to leverage the model according to their specific needs and preferences. For instance, in protein design, scientists may want to manually guide the generation of a protein from a user-defined substructure under a particular path over the relevant variables. This requires the generative model to perform arbitrary marginal inferences. Towards this end, neural autoregressive models (ARMs) [3, 30] have been developed to facilitate conditional/marginal inference based on the idea of modeling a high-dimensional joint distribution as a factorization of univariate conditionals using the chain rule of probability. Many efforts have been made to scale up ARMs and enable any-order generative modeling under the setting of maximum likelihood estimation (MLE) [30, 66, 20], and great progress has been made in applications such as masked language modeling [72] and image inpainting [20]. However, marginal likelihood evaluation in the most widely-used modern neural network architectures (e.g., Transformers [68] and U-Nets [53]) is limited by $\mathcal{O}(D)$ neural network passes, where $D$ is the length of the sequence. This scaling makes it difficult to evaluate likelihoods on long sequences arising in data such as natural language and proteins. In contrast to MLE, in the setting of energy-based training (EB), instead of empirical data samples, we only have access to an unnormalized (log) probability function (specified by a reward or energy function) that can be evaluated pointwise for the generative model to match. In such settings, ARMs are limited to fixed-order generative modeling and lack scalability in training. The subsampling techniques developed to scale the training of conditionals for MLE are no longer applicable when matching log probabilities in energy-based training (see Section 4.3 for details). Figure 1: Marginalization models (MAMs) enable estimation of any marginal probability with a neural network $\theta$ that learns to “marginalize out” variables. The figure illustrates marginalization of a single variable on bit strings (representing molecules) with two alternatives (versus $K$ in general) for clarity. The bars represent probability masses. To enhance scalability and flexibility in the generative modeling of discrete data, we propose a new family of generative models, marginalization models (MAMs), that directly model the marginal distribution $p(x_s)$ for any subset of variables $x_s$ in $x$. Direct access to marginals has two important advantages: 1) significantly speeding up inference for any marginal, and 2) enabling scalable training of any-order generative models under both MLE and EB settings. The unique structure of the model allows it to simultaneously represent the coupled collection of all marginal distributions of a given discrete joint probability mass function. For the model to be valid, it must be consistent with the sum rule of probability, a condition we refer to as “marginalization self-consistency” (see Figure 1); learning to enforce this with scalable training objectives is one of the key contributions of this work. We show that MAMs can be trained under both maximum likelihood and energy-based training settings with scalable learning objectives. We demonstrate the effectiveness of MAMs in both settings on a variety of discrete data distributions, including binary images, text, physical systems, and molecules. We empirically show that MAMs achieve orders of magnitude speedup in marginal likelihood evaluation. For energy-based training, any-order generative models to high-dimensional problems that previous methods fail to achieve. 2 BACKGROUND We first review two prevalent generative modeling settings. Then we introduce autoregressive models under two training settings. Maximum likelihood (MLE) Given a dataset $D = \{x^{(i)}\}_{i=1}^N$ drawn from a data distribution $p = p_{\text{data}}$, we aim to learn the distribution $p_\theta(x)$ that maximizes the probability of the data under our model. Mathematically, we aim to learn the parameters $\theta^\star$ that maximize the log-likelihood: $$\theta^\star = \arg\max_\theta \mathbb{E}_{x \sim p_{\text{data}}} [\log p_\theta(x)] \approx \arg\max_\theta \frac{1}{N} \sum_{i=1}^N \log p_\theta(x^{(i)})$$ which is also equivalent to minimizing the Kullback-Leibler divergence under the empirical distribution, i.e., minimizing $D_{\text{KL}}(p_{\text{data}}(x)\|\|p_\theta(x))$. This is the setting that is most commonly used in generation of images (e.g., diffusion models [59, 18, 60]) and language (e.g. GPT [49]) where we can empirically draw observed data from the distribution. Energy-based training (EB) In this setting, we do not have data from the distribution of interest. Instead, we have access to the unnormalized (log) probability mass function $f$, usually in the form of reward function or energy function, that are defined by humans or by physical systems to specify how likely a sample is. Mathematically, we can define the target probability mass function to be $f(x) = \exp(r(x)/\tau)$, where $r(x)$ is the reward function and $\tau > 0$ is a temperature parameter. This expresses the intuitive idea that we would like the model to assign higher probability to data with larger reward. For example, the reward function can represent human preferences in alignment of large language models [43, 42]. In molecular/material design applications, scientists can specify the reward according to how close a particular sample’s measured or calculated properties are to some functional desiderata. When modeling the thermodynamic ensemble of physical systems, $ r(x) $ is defined to be the (negative) energy function of a given state [41]. Mathematically, we aim to learn the parameters $ \theta $ such that $ p_\theta(x) \approx f(x)/Z $, where $ Z $ is the normalization constant of $ f $. A common training criteria is to minimize the KL divergence [41, 71, 9]: \[ \min_\theta D_{KL} \left( p_\theta(x) \parallel f(x)/Z \right) = \mathbb{E}_{x \sim p_\theta(x)} \left[ \log p_\theta(x) - \log f(x)/Z \right]. \] Autoregressive models Autoregressive models (ARMs) [3, 30] model a complex high-dimensional distribution $ p(x) $ by factorizing it into univariate conditionals using the chain rule: \[ \log p(x) = \sum_{d=1}^{D} \log p(x_d \| x_{<d}), \] where $ x_{<d} = \{x_1, \ldots, x_{d-1}\} $. Recently there has been great success in applying autoregressive models to discrete data, such as natural language, proteins [58, 32, 36], and molecules [56, 15]. Due to their sequential nature via modeling the conditionals, evaluation of (joint/marginal) likelihood requires up to $ D $ neural network evaluations. This is costly for long sequences, leading to limitations that prevent ARMs to be scalable for marginal inference and energy-based training. Any-order ARMs (AO-ARMS) Under the MLE setting, Uria et al. [66] propose to learn the conditionals of ARMs for arbitrary orderings that include all permutations of $ \{1, \ldots, D\} $. The model $ \phi $ can be trained by maximizing a lower-bound objective [66, 20] that takes an expectation under a uniform distribution on orderings. This objective allows scalable training of AO-ARMs, leveraging efficient parallel evaluation of multiple one-step conditionals for each token in one forward pass with architectures such as the U-Net [53] and Transformers [68]. However, under the EB setting, training AO-ARMs presents challenges, which we will discuss in details in Section 4.3. ### 3 MARGINALIZATION MODELS We propose marginalization models (MAMs), a new type of generative model that enables scalable any-order generative modeling as well as efficient marginal evaluation, for both maximum likelihood and energy-based training. The flexibility and scalability of marginalization models are enabled by the explicit modeling of the marginal distribution and enforcing marginalization self-consistency. In this paper, we focus on generative modeling of discrete structures using vectors of discrete variables. The vector representation encompasses various real-world problems with discrete structures, including language sequence modeling, protein design, and molecules with string-based representations (e.g., SMILES [70] and SELFIES [29]). Moreover, vector representations are inherently applicable to any discrete problem, since it is feasible to encode any discrete object into a vector of discrete variables. Definition We are interested in modeling the discrete probability distribution $ p(x) $, where $ x = [x_1, \ldots, x_D] $ is a $ D $-dimensional vector and each $ x_d $ takes $ K $ possible values, i.e. $ x_d \in \{1, \ldots, K\} $. Marginalization Let $ x_s $ be a subset of variables of $ x $ and $ x_{s^c} $ be the complement set, i.e. $ x_s \subseteq \{x_1, \ldots, x_D\} $ and $ x_{s^c} = \{x_1, \ldots, x_D\} \setminus x_s $. The marginal of $ x_s $ is obtained by summing over all values of $ x_{s^c} $: \[ p(x_s) = \sum_{x_{s^c}} p(x_s, x_{s^c}) \] We refer to (4) as the “marginalization self-consistency” that any valid distribution should follow. The goal of a marginalization model $ \theta $ is to estimate the marginals $ p(x_s) $ for any subset of variables $ x_s $ as closely as possible. To achieve this, we train a deep neural network $ p_\theta $ that minimizes the distance of $ p_\theta(x) $ and $ p(x) $ on the full joint distribution while enforcing the marginalization self-consistency. Parameterization To approximate arbitrary marginals over $ x_s $ with a single neural network forward pass, we additionally include the “marginalized out” variables $ x_{s^c} $ in the input by introducing a special symbol “?” to denote the missing values. By doing this, we create an augmented $ D $-dimensional vector representation $ x_s^{\text{aug}} \in X^{\text{aug}} \triangleq \{1, \ldots, K, ?\}^D $ and feed it to the NN. For example, for a binary vector $ x $ of length 4, for $ x_s = \{x_1, x_3\} $ with $ x_1 = 0 $ and $ x_3 = 1 $, $ x_s^{\text{aug}} = [0, ?, 1, ?] $ where “?” denotes $ x_2 $ and $ x_4 $ being marginalized out. From here onwards we will use $ x_s^{\text{aug}} $ and $ x_s $ interchangeably. --- 1 An alternative is to consider minimizing distance over some marginal distribution of interest if we only cares about a specific marginal. Note this is impractical under the energy-based training setting, when the true marginal $ p(x_s) $ is intractable to evaluate in general. A marginalization model parameterized by a neural network $ \theta $ takes in the augmented vector representation $ x^{\text{aug}} \in \{1, \ldots, K, ?\}^D $, and outputs the marginal log probability $ f_\theta(x_s) = \log p_\theta(x_s) $ that satisfy the marginalization self-consistency constraints: \[ \sum_{x_{s'}} p_\theta([x_s, x_{s'}]) = p_\theta(x_s) \quad \forall x_s \in \{1, \ldots, K, ?\}^D \] where $[x_s, x_{s'}]$ denotes the concatenation of $x_s$ and $x_{s'}$. Given a random ordering of the variables $ \sigma \in S_D $ where $ S_D $ defines the set of all permutations of $1, 2, \cdots, D$, let $ \sigma(d) $ denote the $d$-th element in $ \sigma $ and $ \sigma(< d) $ be the first $d - 1$ elements in $ \sigma $. The marginalization can be imposed over one variable at a time, which leads to the following one-step marginalization constraints: \[ p_\theta(x_{\sigma(<d)}) = \sum_{x_{\sigma(d)}} p_\theta(x_{\sigma(\leq d)}), \quad \forall \sigma \in S_D, x \in \{1, \cdots, K\}^D, d \in [1 : D]. \] (5) Sampling Given the learned marginalization model, one can sample from the learned distribution by picking an arbitrary order $ \sigma $ and sampling one variable at a time. To evaluate the conditionals at each step of the generation, we can use the product rule of probability: \[ p_\theta(x_{\sigma(d)} \| x_{\sigma(<d)}) = \frac{p_\theta(x_{\sigma(\leq d)})}{p_\theta(x_{\sigma(<d)})}. \] However, the above is not a valid conditional distribution if the marginalization in (5) is not strictly enforced, since it might not sum up exactly to one. Hence we use following normalized conditional: \[ p_\theta(x_{\sigma(d)} \| x_{\sigma(<d)}) = \frac{p_\theta([x_{\sigma(<d)}, x_{\sigma(d)}])}{\sum_{x_{\sigma(d)}} p_\theta([x_{\sigma(<d)}, x_{\sigma(d)}])}. \] (6) In this paper, we focus on the sampling procedure that generates one variable at a time, but marginalization models can also facilitate sampling multiple variables at a time (See Appendix B.2). Scalable learning of marginals with conditionals In training, we impose the marginalization self-consistency by minimizing the squared error of the constraints in (5) in log-space. Evaluation of each marginalization constraint in (5) requires $K$ NN forward passes, where $K$ is the number of discrete values $x_d$ can take. This makes training challenging to scale when $K$ is large. To address this issue, we augment the marginalization models with learnable conditionals parameterized by $ \phi $. The marginalization constraints in (5) can be decomposed into $K$ parallel marginalization constraints, which makes it highly scalable to subsample from for training: \[ p_\theta(x_{\sigma(<d)})p_\phi(x_{\sigma(d)} \| x_{\sigma(<d)}) = p_\theta(x_{\sigma(\leq d)}), \quad \forall \sigma \in S_D, x \in \{1, \cdots, K\}^D, d \in [1 : D]. \] (7) During training, we need to specify a distribution $q(x)$ for subsampling the marginalization constraints to optimize on. In practice, it can be set to the distribution we are interested to perform marginal inference on, such as $p_{\text{data}}$ or the distribution of the generative model $p_{\theta,\phi}$. ### 4 Training the Marginalization Models #### 4.1 Maximum Likelihood Estimation Training In this setting, we train MAMs with the maximum likelihood objective while additionally enforcing the marginalization constraints in Equation (5): \[ \max_{\theta, \phi} \mathbb{E}_{x \sim p_{\text{data}}} \log p_\theta(x) \] subject to \[ p_\theta(x_{\sigma(<d)})p_\phi(x_{\sigma(d)} \| x_{\sigma(<d)}) = p_\theta(x_{\sigma(\leq d)}), \quad \forall \sigma \in S_D, x \in \{1, \cdots, K\}^D, d \in [1 : D]. \] (8) Two-stage training A typical way to solve the above optimization problem is to convert the constraints into a penalty term and optimize the penalized objective, but we empirically found the learning to be slow and unstable. Instead, we identify an alternative two-stage optimization formulation that is theoretically equivalent to Equation (8), but leads to more efficient training: Claim 1. Solving the optimization problem in (8) is equivalent to the following two-stage optimization procedure, under mild assumption about the neural networks used being universal approximators: Stage 1: $ \max_{\theta, \phi} \mathbb{E}_{x \sim p_{\text{data}}} \mathbb{E}_{\sigma \sim U(S_D)} \sum_{d=1}^{D-1} \log p_\phi(x_{\sigma(d)} \| x_{\sigma(<d)}) $ Stage 2: $ \min_{\theta} \mathbb{E}_{x \sim q(x)} \mathbb{E}_{\sigma \sim U(S_D)} \mathbb{E}_{d \sim U(1, \cdots, D)} \left( \log[p_\theta(x_{\sigma(<d)})p_\phi(x_{\sigma(d)} \| x_{\sigma(<d)})] - \log p_\theta(x_{\sigma(\leq d)}) \right)^2 $. To make sure $p_\theta$ is normalized, we can either additionally enforce $p_\theta([? ? \cdots ?]) = 1$ or let $Z_\theta = p_\theta([? ? \cdots ?])$ be the normalization constant. The first stage can be interpreted as fitting the conditionals in the same way as AO-ARMs [66, 20] and the second stage acts as distilling the marginals from conditionals. The intuition comes from the chain rule of probability: there is a one-to-one correspondence between optimal conditionals $\phi$ and marginals $\theta$, i.e. $\log p_\theta(x) = \sum_{d=1}^{D} \log p_\phi(x_{\sigma(d)} \| x_{\sigma(<d)})$ for any $\sigma$ and $x$. By assuming neural networks are universal approximators, we can first optimize for the optimal conditionals, and then optimize for the corresponding optimal marginals. We provide more details in Appendix A.1. ### 4.2 ENERGY-BASED TRAINING In this setting, we train MAMs using the energy-based training objective in Equation (2) with a penalty term to enforce the marginalization constraints in Equation (5): $$\min_{\theta,\phi} D_{KL}(p_\theta(x) \\| p(x)) + \lambda \mathbb{E}_{x \sim q(x)} \mathbb{E}_\sigma \mathbb{E}_d (\log[p_\theta(x_{\sigma(d)} \| x_{\sigma(<d)})] - \log p_\theta(x_{\sigma(d)}))^2,$$ where $\sigma \sim U(S_D)$, $d \sim U(1, \cdots, D)$ and $q(x)$ is the distribution of interest for evaluating marginals. Scalable training We use REINFORCE to estimate the gradient of the KL divergence term: $$\nabla_\theta D_{KL}(p_\theta(x) \|\| p(x)) = \mathbb{E}_{x \sim p_\theta(x)} [\nabla_\theta \log p_\theta(x) (\log p_\theta(x) - \log f(x))]$$ $$\approx \frac{1}{N} \sum_{i=1}^{N} \nabla_\theta \log p_\theta(x^{(i)}) (\log p_\theta(x^{(i)}) - \log f(x^{(i)}))$$ For the penalty term, we subsample the ordering $\sigma$ and step $d$ for each data $x$. Efficient sampling with persistent MCMC We need cheap and effective samples from $p_\theta$ in order to perform REINFORCE, so a persistent set of Markov chains are maintained by randomly picking an ordering and taking block Gibbs sampling steps using the conditional distribution $p_\phi(x_{\sigma(d)} \| x_{\sigma(<d)})$ (full algorithm in Appendix A.5), in similar fashion to persistent contrastive divergence [64]. The samples from the conditional distribution $p_\phi$ serve as approximate samples from $p_\theta$ when they are close to each other. Otherwise, we can additionally use importance sampling for adjustment. ### 4.3 ADDRESSING LIMITATIONS OF ARMs We discuss in more detail about how MAMs address some limitations of ARMs. The first one is general to both training settings, while the latter two are specific to energy-based training. 1) Slow marginal inference of likelihoods Due to sequential conditional modeling, evaluation of a marginal $p_\phi(x_\sigma)$ with ARMs (or an arbitrary marginal with AO-ARMs) requires applying the NN $\phi$ up to $D$ times, which is inefficient in time and memory for high-dimensional data. In comparison, MAMs are able to estimate any arbitrary marginal with one NN forward pass. 2) Lack of support for any-order training In energy-based training, the objective in Equation (2) aims to minimize the distance between $\log p_\phi(x)$ and $\log p(x)$, where $\phi$ is the NN parameters of an ARM. However, unless the ARM is perfectly self-consistent over all orderings, it will not be the case that $\log p_\phi(x) = \mathbb{E}_\sigma \log p_\phi(x \| \sigma)$. Therefore, the expected $D_{KL}$ objective over the orderings $\sigma$ would not be equivalent to the original $D_{KL}$ objective, i.e., $$\mathbb{E}_{p_\phi} [\mathbb{E}_\sigma \log p_\phi(x \| \sigma) - \log p(x)] \neq \mathbb{E}_{p_\phi} [\log p_\phi(x) - \log p(x)].$$ As a result, ARMs cannot be trained with the expected $D_{KL}$ objective over all orderings simultaneously, but instead need to resort to a preset order and minimize the KL divergence between $\log p_\phi(x \| \sigma)$ and the target density $\log p(x)$. The self-consistency constraints imposed by MAMs address this issue. MAMs are not limited to fixed ordering because marginals are order-agnostic and we can optimize over expectation of orderings for the marginalization self-consistency constraints. 3) Training not scalable on high-dimensional problems When minimizing the difference between $\log p_\phi(x \| \sigma)$ and the target $\log p(x)$, ARMs need to sum conditionals to evaluate $\log p_\phi(x \| \sigma)$. One might consider subsampling one-step conditionals $p_\phi(x_{\sigma(d)} \| x_{\sigma(<d)})$ to estimate $p_\phi(x)$, but this leads... to high variance of the REINFORCE gradient in Equation (9) due to the product of the score function and distance terms, which are both high variance (We validate this in experiments, see Figure 3). Consequently, training ARMs for energy-based training necessitates a sequence of $D$ conditional evaluations to compute the gradient of the objective function. This constraint leads to an effective batch size of $B \times D$ for batch of $B$ samples, significantly limiting the scalability of ARMs to high-dimensional problems. Furthermore, obtaining Monte Carlo samples from ARMs for the REINFORCE gradient estimator is slow when the dimension is high. Due to the fixed input ordering, this process requires $D$ sequential sampling steps, making more cost-effective sampling approaches like persistent MCMC infeasible. Marginalization models circumvent this challenge by directly estimating the log-likelihood with the marginal neural network. Additionally, the support for any-order training enables efficient sampling through the utilization of persistent MCMC methods. 5 RELATED WORK Autoregressive models Developments in deep learning have greatly advanced the performance of ARMs across different modalities, including images, audio, and text. Any-order (Order-agnostic) ARMs were first introduced in [66] by training with the any-order lower-bound objective for the maximum likelihood setting and recently seen in ARDM [20] with state-of-the-art performance for any-order discrete modeling of image/audio. Germain et al. [16] train an auto-encoder with masking that outputs the sequence of all one-step conditionals for a given ordering, but does not generate as well as methods [67, 72, 20] that predict one-step conditionals under the given masking. Douglas et al. [14] trains an AO-ARM and use importance sampling to estimate arbitrary conditional posteriors, but with limited experiment validation on a synthetic dataset. Shih et al. [57] utilizes a modified training objective of ARMs for better marginal inference performance but loses any-order generation capability. Comparisons of MAMs and ARMs are discussed in detail in Section 4.3. Arbitrary conditional/marginal models For continuous data, VAEAC [25] and ACFlow [31] extends the idea of conditional variational encoder and normalizing flow to model arbitrary conditionals. ACE [62] improves the expressiveness of arbitrary conditional models through directly modeling the energy function, which puts less constraints on parameterization but comes at the cost of approximating the normalizing constant. Instead of using neural networks as function approximators, probabilistic circuits (PCs) [6, 45] offer tractable probabilistic models for both conditionals and marginals by building a computation graph with sum and product operations following specific structural constraints. Examples of PCs include Chow-Liu trees [7], arithmetic circuits [10], sum-product networks [47], etc. Peharz et al. [45] have improved the scalability of PCs through combining arithmetic operations into a single monolithic einsum-operation and automatic differentiation. More recently, [33, 34] demonstrated the potential of PCs with distilling latent variables from trained deep generative models on continuous image data. However, their expressiveness are limited by the structural constraints. All methods mentioned above focus on MLE settings, except ARMs are explored in energy-based training of science problems [9, 71], but suffer in scaling when $D$ is large. GFlowNets GFlowNets [2, 4] formulate the problem of generation as matching the probability flow at terminal states to the target normalized density. Compared to ARMs, GFlowNets allow flexible modeling of the generation process by assuming learnable generation paths through a directed acyclic graph (DAG). The advantages of learnable generation paths come with the trade-off of sacrificing the flexibility of any-order generation and exact likelihood evaluation. Under fixed generation path, GFlowNets are reduced to fixed-order ARMs [74]. In Appendix A.3, we further identify the connections and differences between GFlowNets and AO-ARMS/MAMs. For discrete problems, Zhang et al. [75] train GFlowNets on the squared distance loss with the trajectory balance objective [38], which is less scalable for large $D$ (due to the same reason as ARMs in Section 4.3) and renders direct access to marginals unavailable. For the MLE setting, an energy function is additionally learned from data such that training is reduced to energy-based training. 6 EXPERIMENTS We conduct experiments with marginalization models (MAM) on both MLE and EB settings for discrete problems including binary images, text, molecules and physical systems. We consider the following baselines for comparison: Any-order ARM (AO-ARM) [20], ARM [30], GFlowNet [39, 75], Discrete Flow[65] and Probabilistic Circuit (PC) [45]. MAM, PC and Figure 4: An example of the data generated (with 100/400/700 pixels masked) for comparing the quality of likelihood estimate. Numbers below the images are LL estimates from MAM’s marginal network (left) and AO-ARM-E’s ensemble estimate (right). \| Model \| NLL (bpd) ↓ \| Spearman’s ↑ \| Pearson ↑ \| Marg. inf. time (s) ↓ \| \|------------------------\|-------------\|--------------\|-----------\|----------------------\| \| AO-ARM-E-U-Net \| 0.148 \| 1.0 \| 1.0 \| 661.98 ± 0.49 \| \| AO-ARM-S-U-Net \| 0.149 \| 0.996 \| 0.993 \| 132.40 ± 0.03 \| \| GflowNet-MLP \| 0.189 \| – \| – \| – \| \| PC-Image (EiNets)$^4$ \| 0.187 \| 0.716 \| 0.752 \| 0.015 ± 0.00 \| \| MAM-U-Net \| 0.149 \| 0.992 \| 0.993 \| 0.018 ± 0.00 \| (AO-)ARM support arbitrary marginal inference. Discrete flow$^3$ allows exact likelihood evaluation while GFlowNet needs to approximate the likelihood with sum using importance samples. For evaluating AO-ARM’s marginal inference, we can either use an ensemble model by averaging over several random orderings (AO-ARM-E) or use a single random ordering (AO-ARM-S). In general, AO-ARM-E should always be better than AO-ARM-S but at a much higher cost. Neural network architecture and training hyperparameter details can be found in Appendix C. Ablation studies on measuring marginal self-consistency and sampling with marginals are in Appendices B.1 and B.2. Guidance on picking $q$ is in Appendix B.3. Appendix C.3 contains more results on CIFAR-10. 6.1 Maximum Likelihood Estimation Training Binary MNIST We report the negative test likelihood (bits/digit), marginal estimate quality and marginal inference time per minibatch (of size 16) in Table (1). To keep GPU memory usage the same, we sequentially evaluate the likelihood for ARMs. Both MAM and AO-ARM use a U-Net architecture with 4 ResNet Blocks interleaved with attention layers (see Appendix C). GFlowNets fail to scale to large architectures as U-Net, hence we report GFlowNet results using an MLP from Zhang et al. [75]. For MAM, we use the conditional network to evaluate test likelihood (since this is also how MAM generates data). The marginal network is used for evaluating marginal inference. The quality of the marginal estimates will be compared to the best performing model. In order to evaluate the quality of marginal likelihood estimates, we employ a controlled experiment where we randomly mask out portions of a test image and generate multiple samples with varying levels of masking (refer to Figure 4). This process allows us to obtain a set of distinct yet comparable samples, each associated with a different likelihood value. For each model, we evaluate the likelihood of the generated samples and compare that with AO-ARM-E’s estimate since it achieves the best likelihood on test data. We repeat this controlled experiment on a random set of test images. The mean Spearman’s and Pearson correlation are reported to measure the strength of correlation in marginal inference likelihoods between the given model and AO-ARM-E. MAM achieves close to 4 order of magnitude speed-up in marginal inference while at comparable quality to that from AO-ARM-S. PCs are also very fast in marginal inference but there remains a gap in terms of quality. Generated samples and additional marginal inference on partial images are in Appendix C. Molecular sets (MOSES) We test generative modeling of MAM on a benchmarking molecular dataset [46] refined from the ZINC database [61]. Same metrics are reported as Binary-MNIST. Likelihood quality is measured similarly but on random groups of test molecules instead of generated ones. The generated molecules from MAM and AO-ARM are comparable to standard state-of-the-art molecular generative models, such as CharRNN [56], JT-VAE [26], and LatentGAN [48] (see Appendix C), with additional controllability and flexibility in any-order generation. MAM supports $^3$Results are only reported on text8 for discrete flow since there is no public code implementation. $^4$We adopt the SOTA implementation of PCs from EiNets [45]. Results are reported on Binary MNIST using the image-tailored PC structure [47]. For text and molecular data, designing tailored PC structures that deliver competitive performance remains an open challenge. Table 2: Performance Comparison on Molecular Sets \| Model \| NLL (bpd) ↓ \| Spearman’s ↑ \| Pearson ↑ \| Marg. inf. time (s) ↓ \| \|------------------------\|-------------\|--------------\|-----------\|----------------------\| \| AO-ARM-E-Transformer \| 0.652 \| 1.0 \| 1.0 \| 96.87 ± 0.04 \| \| AO-ARM-S-Transformer \| 0.655 \| 0.996 \| 0.994 \| 19.32 ± 0.01 \| \| MAM-Transformer \| 0.655 \| 0.998 \| 0.995 \| 0.006 ± 0.00 \| Table 3: Performance Comparison on text8 \| Model \| NLL (bpc) ↓ \| Spearman’s ↑ \| Pearson ↑ \| Marg. inf. time (s) ↓ \| \|------------------------\|-------------\|--------------\|-----------\|----------------------\| \| Discrete Flow (8 flows)\| 1.23 \| – \| – \| – \| \| AO-ARM-E-Transformer \| 1.494 \| 1.0 \| 1.0 \| 207.60 ± 0.33 \| \| AO-ARM-S-Transformer \| 1.529 \| 0.982 \| 0.987 \| 41.40 ± 0.01 \| \| MAM-Transformer \| 1.529 \| 0.937 \| 0.945 \| 0.005 ± 0.000 \| Table 4: Performance Comparison on Ising model (10 × 10) \| Model \| NLL (bpd) ↓ \| KL divergence ↓ \| Marg. inf. time (s) ↓ \| \|------------------------\|-------------\|-----------------\|----------------------\| \| ARM-Forward-Order-MLP \| 0.79 \| -78.63 \| 5.29 ± 0.07e-01 \| \| ARM-MC-Forward-Order-MLP\| 24.84 \| -18.01 \| 5.30 ± 0.07e-01 \| \| GFlowNet-Learned-Order-MLP\| 0.78 \| -78.17 \| – \| \| MAM-Any-Order-MLP \| 0.80 \| -77.77 \| 3.75 ± 0.08e-04 \| Table 5: Performance Comparison on Target Lipophilicity \| Distribution \| KL divergence ↓ \| \|--------------\|-----------------\| \| logP = 4, τ = 1.0 \| logP = −4, τ = 1.0 \| logP = 4, τ = 0.1 \| logP = 4, τ = 0.1 \| \| ARM-FO-MLP \| -174.25 \| -168.62 \| -167.83 \| -160.2 \| \| MAM-AO-MLP \| -173.07 \| -166.43 \| -165.75 \| -157.59 \| much faster marginal inference, which is useful for domain scientists to reason about likelihood of (sub)structures. Generated molecules and property histogram plots of are available in Appendix C. Text8 Text8 [37] is a widely used character level natural language modeling dataset. The dataset comprises of 100M characters from Wikipedia, split into chunks of 250 character. We follow the same testing procedure as Binary-MNIST and report the same metrics. The test NLL of discrete flow is from [65], for which there are no open-source implementations to evaluate additional metrics. 6.2 ENERGY-BASED TRAINING We compare with ARM that uses sum of conditionals to evaluate $\log p_\phi$ with fixed forward ordering and ARM-MC that uses a one-step conditional to estimate $\log p_\theta$. ARM can be regarded as the golden standard of learning autoregressive conditionals, since its gradient needs to be evaluated on the full generation trajectory, which is the most informative and costly. MAM uses marginal network to evaluate $\log p_\theta$ and subsamples a one-step marginalization constraint for each data point in the batch. The effective batch size for ARM and GFlowNet is $B \times O(D)$ for batch of size $B$, and $B \times O(1)$ for ARM-MC and MAM. MAM and ARM optimizes KL divergence using REINFORCE gradient estimator with baseline. GFlowNet is trained on per-sample gradient of squared distance [75]. Ising model Ising models [24] model interacting spins and are widely studied in mathematics and physics (see MacKay [35]). We study Ising model on a square lattice. The spins of the $D$ sites are represented a $D$-dimensional binary vector and its distribution is $p^(x) \propto f^(x) = \exp(-E_J(x))$ where $E_J(x) \triangleq -x^\top J x - \theta^\top x$, with $J$ the binary adjacency matrix. These models, although simplistic, bear analogies to the complex behavior of high-entropy alloys [9]. We compare MAM with ARM, ARM-MC, and GFlowNet on a $10 \times 10$ ($D=100$) and a larger $30 \times 30$ ($D=900$) Ising model where ARMs and GFlowNets fail to scale. 2000 ground truth samples are generated following Grathwohl et al. [17] and we measure test negative log-likelihood on those samples. We also measure $D_{KL}(p_\theta(x)\|\|p^)$ by sampling from the learned model and evaluating $\sum_{i=1}^{M} (\log p_\theta(x_i) - \log f^(x_i))$. Figure 5 contains KDE plots of $-E_J(x)$ for the generated samples. As described in Section 4.3, the ARM-MC gradient suffers from high variance and fails to converge. It also tends to collapse and converge to a single sample. MAM has significant speedup in marginal inference and is the only model that supports any-order generative modeling. The performance in terms of KL divergence and likelihood are only slightly worse than models with fixed/learned order, which is expected since any-order modeling is harder than fixed-order modeling, and MAM is solving a more complicated task. of jointly learning conditionals and marginals. On a $30 \times 30$ ($D = 900$) Ising model, MAM achieves a bpd of 0.835 on ground-truth samples while ARM and GFlowNet fails to scale. Distribution of generated samples is shown in Figure 5. Molecular generation with target property In this task, we are interested in training generative models towards a specific target property of interest $g(x)$, such as lipophilicity (logP), synthetic accessibility (SA) etc. We define the distribution of molecules to follow $p^(x) \propto \exp\left(-\frac{(g(x) - g^)^2}{\tau}\right)$, where $g^$ is the target value of the property and $\tau$ is a temperature parameter. We train ARM and MAM for lipophilicity of target values 4.0 and −4.0, both with $\tau = 1.0$ and $\tau = 0.1$. Both models are trained for 4000 iterations with batch size 512. Results are shown in Figure 6 and Table 5 (additional figures in Appendix C). Findings are consistent with the Ising model experiments. Again, MAM performs just marginally below ARM. However, only MAM supports any-order modeling and scales to high-dimensional problems. Figure 6 (right) shows molecular generation with MAM for $D = 500$. 7 CONCLUSION* In conclusion, marginalization models are a novel family of generative models for high-dimensional discrete data that offer scalable and flexible generative modeling with tractable likelihoods. These models explicitly model all induced marginal distributions, allowing for fast evaluation of arbitrary marginal probabilities with a single forward pass of the neural network. 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z8Wva86JLB	In the paper, the authors touches on the crucial aspect of balancing model fairness and performance, but it does not delve into a comprehensive discussion or provide methodologies to measure and visualize this trade-off. I appreciate if the authors could expand on this topic, providing detailed explanations, methodologies, or visual aids to help readers understand the implications of this trade-off on the model's outputs?	FAIRNESS MITIGATION VIA A GEOMETRIC FRAMEWORK FOR FAIRNESS (GEOFFAIR) Anonymous authors Paper under double-blind review ABSTRACT Fairness is a critical concern in Machine Learning, impacting its applications across domains. Existing fairness analyses often rely on complex mathematics, lacking of intuitive understanding. In this study, we introduce GEOFFair, a Geometric Framework for Fairness. It represents Machine Learning elements as vectors and sets, offering a more intuitive understanding of fairness related concepts. GEOFFair visualizes fairness mitigation techniques as vector projections, it provides a solid base to investigate the bias injection, aiding in constructing proofs, and it enables the study of fairness properties by means of geometric considerations. The main contribution of the work is to highlight GEOFFair’s effectiveness in fairness studies, demonstrating that solely maximizing accuracy based on observed labels may not always be optimal for fairness. 1 INTRODUCTION Fairness concerns within the realm of machine learning (ML) have recently emerged as a prominent and critical challenge, pitching a shadow over the widespread adoption of data-driven AI in critical domains such as healthcare, economics, welfare, and policy-making [Mehrabi et al., 2021]. Usually, these concerns are approached from a mathematical standpoint presenting challenging complexities, as it necessitates scuffling with statistical distributions and, in some cases, non-linear models [Srivastava et al., 2019]. Many of the existing analyses rely on sophisticated methodologies, which may prove less than intuitive for an exhaustive comprehension of the components involved in addressing fairness. We argue that the field could benefit from a streamlined framework offering an intuitive and sound grasp of fundamental fairness concepts and mechanisms in the realm of AI. Accordingly, in this work we present GEOFFair, a Geometric Framework for Fairness, that casts distributions, functions (such as ML models), fairness constraints, and hypothesis spaces into vectors and sets. The main advantage of geometric frameworks lies in their capacity for visualization, facilitating insights into both data and model behaviour. Our motivation for adopting this approach stems from successful applications in other facets of ML [Kansizoglou et al., 2021; Bronstein et al., 2017], wherein the mapping of models into vector spaces, often achieved by concatenating their parameters, has simplified their representation and analysis. Through this lens, concepts like distance metrics, projections, similarities, and algorithms can be visualized to gain valuable insights [Shahmirzadi et al., 2019]. In this paper we showcase the practical utility of our framework by revisiting fairness mitigation techniques within the context of the geometric framework. By visualizing mitigation as vector movements within the space, we highlight properties and unveil the effects of actions underpinned by the mitigation process. We delve into bias injection and debiasing, two facets of the same coin, to underscore the analytical prowess afforded by our framework. 2 GEOFFAIR: A GEOMETRIC FRAMEWORK FOR FAIRNESS The following section aims to introduce the GEOFFair formal framework. To achieve this goal, we will concentrate on two primary aspects. First, we introduce the key components of a typical setting for studying fairness of Machine Learning models; then, in Subsection 2.2 we introduce a vector representation for some key statistical concepts, which serves as the basis for the framework proper in Subsection 2.3. The vector representation enables us to formalize fairness concepts and metrics in a clear and precise mathematical language. Finally, we will discuss how these vector representations exist within the same space, providing a common basis for comparing and contrasting different fairness metrics (Subsection 2.4). 2.1 Formalization of a Generic Fairness Problem Traditionally, the task of learning fair representations has been always formulated using probability theory and statistical analysis. Within this context, both the input data and the ground truth are commonly represented as aleatory variables with their own probability distribution; machine learning models are then seen as parameterized functions over those distributions, whose aim is to minimize a measure of likelihood – i.e., the training loss; and, finally, fairness metrics are seen as functions operating on the conditional expectations of the variables. Before delving into the specifics and the key differences of our framework, let us recall the main concepts starting from the definition of our data. Let $X = (X_1, \ldots, X_n)$ be a multivariate random variable with support $\mathcal{X}$ and distribution $P(X)$. This variable represents our input distribution and, in the context of learning fair representations, it must have one (or more) feature $X_i$ which is considered protected, namely it represents a sensitive attribute of the input against which we want to ensure a non-discriminative behaviour. Similarly, let $Y$ with support $\mathcal{Y}$ and distribution $P(Y)$ be another random variable, typically but not necessarily univariate. This represents our target distribution, i.e., the value on which we aim to forecast given the input distribution $X$. In our analysis we focus on a supervised learning setting. This means that, at training time, we have information, typically in the form of a sample, on the joint probability $P(X,Y)$, which we can use to learn the machine learning model $M$ that better approximates the conditional probability $P(Y \| X)$. From this perspective, $M$ is a function $f : \mathcal{X} \rightarrow \mathcal{Y}$. While this is enough to formally represent the task of unconstrained machine learning, when taking into account fairness requirements we also need to introduce one or more predicates $\Pi$ defined over the conditional expectations of $X$ and $Y$; more specifically, we aim at maximizing the likelihood subject to the constraint $\Pi(X,Y)$. This predicates are often based on a divergence metric $K(\cdot)$ which measures the difference between the conditional distributions of the target variable $Y$ with respect to all those values that the protected feature $X_i$ could assume. For example, if $X_i \sim \{0, 1\}$, then a valid predicate could be: $$\Pi(X,Y) = K(Y \| X_i = 0, Y \| X_i = 1) < t$$ In the context of learning fair representations, $K$ is referred to as a fairness metric and $t$ is a threshold applied to the reported unfairness level. To avoid excess of notation, in the rest of the paper we will use $K(Y)$ to indicate that the fairness metric is applied to a random variable $Y$ according to the protected attributes of the input distribution $X$ to which it is paired to. 2.2 From Distribution to Vectors in the Space Due to the complexity of such a statistical viewpoint on the fairness-aware learning task, the main purpose of our framework is to switch to an alternative representation designed to enhance clarity and to enable visualization of the underlying processes. As a first step is that direction, we introduce a vector-based representation for some key probabilistic concepts that can be used in our context without any significant loss of generality. Probability Distributions and Functions The main idea we rely on is to represent probability distributions to arbitrary precision via an infinite sample. Formally: Notation 1 (Probability Distributions). Given the two data distributions $X$ and $Y$, we encode them as a vector $(x,y) = \{x_i, y_i\}_{i=1}^n$, with $x_i, y_i \sim P(X,Y)$ and $n \rightarrow \infty$. Intuitively, $X$ represents an observable that may serve as the input for an ML model, while $Y$ represents the quantity (or class) to be estimated. The same representation can be applied for the individual distributions of $X$ and $Y$, which are therefore denoted as $x$ and $y$. Our approach makes it particularly easy to represent functions over random variables (e.g., Machine Learning models evaluated over their input). Formally: Notation 2 (Functions). A deterministic function $f$ over $X$ and $Y$ can then be naturally viewed as a vector $f(x, y) = \{f(x_i, y_i)\}_{i=1}^n$ with $n \to \infty$, i.e., just the vector with the function evaluation over all the samples. Functions that depend only on $X$ or only on $Y$ are sub-cases of the above definitions and are respectively denoted as $f(x)$ and $f(y)$. There are a few observations worth making. First, while we use the term “vector” for simplicity, our definitions are closer to functions that map an index $i$ to an object such as $x_i$ or $y_i$. In other words, $x$, $y$, $f(x)$, etc. can be thought of as points in a Hilbert space. Second, our representations are not exact, but they will be sufficient to approximate key statistical properties with arbitrarily high probability. Exact representations for distributions exist and are well known, e.g., the Probability Mass Function or Probability Density Function; however, they do not enable constructing a simple 1-1 mapping between components in the vector (e.g., $x_i$) and function evaluations (e.g., $f(x_i)$), which is instead trivial with our approach. Equivalence of Expectation Predicates Many of the existing fairness metrics are expressed in terms of (conditional) expectations, i.e., averages, or can be reduced in such a form. For example, assuming $X$ is a binary protected attribute, the DIDI metric from Aghaei et al. (2019) is defined in terms of the discrepancy between the global average outcome and the average outcome for each protected group, i.e., $\|\mathbb{E}[Y \| X = 0] - \mathbb{E}[Y]\| + \|\mathbb{E}[Y \| X = 1] - \mathbb{E}[Y]\|$. Statistical parity in classification, which advocates for similar probabilities of a positive outcome across all groups, can be defined as $\|\mathbb{E}[Y \| X = 0] - \mathbb{E}[Y \| X = 1]\|$, and so on. Intuitively, this means that many fairness constraints can be viewed as predicates over (conditional) expectations. The sample expectation function, represented by $\mu(\cdot)$, tends to converge towards the true expectation $\mathbb{E}[\cdot]$ as the sample size grows: we use this result to establish a form of equivalence between predicates expressed over a distribution and those expressed over a sample. Theorem 1. Let $\Pi(X, Y)$ be a predicate over (conditional) expectations for $X$ and $Y$ and let $\pi(\{x_i\}_{i=1}^n, \{y_i\}_{i=1}^n)$ be its sample counterpart. Then we have that: $$P\left(\Pi(X, Y) \iff \lim_{n \to \infty} \pi(\{x_i\}_{i=1}^n, \{y_i\}_{i=1}^n)\right) = 1$$ (1) i.e., the two predicates are equivalent almost surely as the sample size grows if the involved expectations are finite. Proof. The two predicates are identical except for the use of the true and sample expectations. For the sake of simplicity and without loss of generality, let us assume the involved expectations are respectively $\mathbb{E}[Y]$ and $\mu(\{y_i\}_{i=1}^n)$. Since the samples are drawn independently from the same distribution, due to the strong law of large numbers we have that: $$P\left(\lim_{n \to \infty} \mu(\{y_i\}_{i=1}^n) = \mathbb{E}[Y]\right) = 1$$ (2) Equivalence of the sample and true expectations then implies equivalence of $\Pi$ and $\pi$. □ Notation [1] and Notation [2] give us the ability to transition from the conventional distribution paradigm of ML to the realm of vector spaces. Theorem 1 enables reasoning over the vector representation and translates almost certainly any result to the original distribution, at least as far as fairness metrics are concerned. Together, these tools allow us to leverage the power and interpretability of vector space representations in the context of fairness metrics, expanding the scope of analysis and decision-making. ### 2.3 The Formal Model As mentioned in Section 2.2, we focus on a supervised learning setting where the goal is to learn a model that maps inputs (always observable) to outputs (observable at training time and to be estimated at inference time). In this context, we introduce four key mathematical objects that play a major role in the analysis of fairness issues in AI. We represent the infinite input distribution by means of an infinite input vector $ x_n \in X^n $, with $ n \to \infty $, according to Notation 1. For consistency reason with the following notation, we identify the infinite input vector as $ x_\infty \in X^\infty $. Concerning the output, we make a distinction between the distribution that can actually be observed and the one that we ideally wish to estimate. We start by introducing the following concept: Definition 1 (Ground Vector). A ground vector $ y^+_\infty \in Y^\infty $ represents data that can be observed and used as ground truth to learn machine learning models. It is paired with the input vector $ x $. As inspired by Dutta et al. (2020), we model the fact that the ground truth might be subject to systemic social biases, but with a key difference. That is, we directly define an “unbiased” output vector rather than an unbiased input matrix, as our framework allows us to reason in terms of vector components within the output space. Definition 2 (Gold Vector & Biased Mapping). A gold vector $ y^_\infty \in Y^\infty $ represents the “unbiased” data obtained by sampling the fair output distribution before it is corrupted by social biases; accordingly, we can derive a ground vector $ y^+_\infty $ by considering the application of a biased mapping over the gold vector, i.e.: \[ y^+_\infty = b_\infty(y^_\infty), \] where $ b_\infty : Y^\infty \to Y^\infty $ is called “biased mapping” and takes the input vector as parameter. Since we only work in the output space, $ x $ is required to compute the fairness measure but can be considered as a constant for the purpose of our framework. For this reason, we use the compact notation $ b_\infty $. Note that in practical applications, the gold vector is typically unobservable and therefore not accessible at training time. Still, explicitly modeling the unbiased distributions allows us to study in deeper detail the interplay between bias and fairness constraints. In our framework, an ML model can be viewed as a function that maps input to output data. In supervised learning, the training process is typically viewed as that of selecting one model out of a pool of candidates, so as to minimize a loss metric. Formally, training amounts to solving in an exact or approximate fashion: \[ \arg\min_{f \in F} L(f(x), y^+_\infty) \] where $ f $ is the ML model, $ L $ is the chosen loss metric and $ F $ represents the set of possible models, usually defined by specifying an architecture (e.g. a number and size of layers in a feed-forward neural network, number of estimators and maximum depth in a random forest). In our framework, however, the input vector $ x_\infty $ is by construction fixed, thus making the model output the only relevant factor. In other words, two models are equivalent as long as they have the same output. This observation allows us to introduce a simplified representation of the classical notion of hypothesis space. Definition 3 (Hypothesis Space). The hypothesis space $ \hat{Y}_\infty $ is the set of possible infinite-dimensional outputs for the chosen class of ML models, i.e. \[ \hat{Y}_\infty = \{ y \in Y^\infty \mid \exists f \in F_\infty : f(x) = y \} \] Intuitively, the hypothesis space can be viewed as the set of possible model outputs for the considered sample. A linear regression model will have a limited hypothesis space due to its ability to represent linear relationships only, while more complex models such as random forests and neural networks will have a much larger hypothesis space. Finally, as we are considering a fairness scenario, we need to model a final mathematical object in order to guarantee a proper analysis of the phenomenon, namely the region in the output space that is considered fair. Definition 4 (Fair Space). Let $ \bar{Y}_\infty \subseteq Y^\infty $ be the set containing all the infinite-dimensional output vectors that are aligned with the fairness requirements. We make no assumption on the mathematical definition of the fair space. Nonetheless, it is worth noting that in many practical cases, this set is defined by means of a threshold $t$ on a fairness metric $K$, i.e., $\mathbb{Y}_\infty = \{y \in \mathcal{Y}^\infty \| K(y) \leq t\}$. Once all the elements are defined, we can examine how they interact with each other. In the most general setup, we cannot make any assumption about the relationships between $y^_\infty$, $y^+_\infty$, $\hat{\mathbb{Y}}_\infty$, and $\mathbb{Y}_\infty$. Without specific contextual information on data, models, and constraints, the relationships between these entities can vary significantly. It is worth mentioning that, in the defined framework, all vectors and sets we introduced exist in the same space, which facilitates easy visualization (see Figures in Section 3). This visual representation can assist with proof-by-witness, allowing us to analyze and demonstrate relationships between these vectors more effectively. ### 2.4 Relationships Between Elements Table 1: Possible one-to-one relationships. When comparing the two sets, we use $\cap$ and $\cap$ as aliases for $\hat{\mathbb{Y}}_\infty \cap \mathbb{Y}_\infty = \emptyset$ and $\hat{\mathbb{Y}}_\infty \cap \mathbb{Y}_\infty \neq \emptyset$, $\hat{\mathbb{Y}}_\infty$, $\mathbb{Y}_\infty$. \| \| $\hat{\mathbb{Y}}_\infty$ \| $\mathbb{Y}_\infty$ \| $y^+_\infty$ \| $y^_\infty$ \| \|-------\|--------------------------\|---------------------\|--------------\|--------------\| \| $\hat{\mathbb{Y}}_\infty$ \| $\cap$, $\subseteq$, $\supseteq$, $\cap$ \| $\exists$, $\exists$ \| $\exists$, $\exists$ \| \| $\mathbb{Y}_\infty$ \| $\exists$, $\exists$ \| $\exists$, $\exists$ \| \| $y^+_\infty$ \| \| $\equiv$, $\neq$ \| \| $y^_\infty$ \| \| \| It is worth noting that the relationships among the objects in the framework allow us to establish interesting properties and interactions among the objects involved in fairness mitigation techniques. A full discussion is beyond the scope of this contribution; however, a summary of every potential one-to-one relationship is presented in Table 1. The complete list is provided in Appendix A. While the number of possible combinations is not small, it is nevertheless finite, which can be helpful for proving universally quantified statements (i.e., $\forall$ and $\exists$). While we do not examine each possible scenario, it is worth to highlight some very common or interesting cases. - If $\hat{\mathbb{Y}}_\infty \subseteq \mathbb{Y}_\infty$, the machine learning model is said to be fair-by-design [Nurock et al., 2021]. While achieving this is challenging in many practical cases, it can be attained by incorporating explicit rules into the model, ensuring that certain deontological fair principles are always upheld. - If $\hat{\mathbb{Y}}_\infty \supseteq \mathbb{Y}_\infty$, the machine learning models can cover all existing fair outputs. This can be the case when employing powerful models like large neural networks. - If $y^+_\infty \in \hat{\mathbb{Y}}_\infty$, it can be perfectly represented by the machine learning model, although this representation is not guaranteed to be fair unless $y^+_\infty$ is already in the Fair Space. Conversely, when the model lacks the capacity to represent $y^+_\infty$ adequately, it will be trained to minimize the loss $L$ between the labels and the model outputs. The same considerations applies also to the relationship between $y^_\infty$ and $\hat{\mathbb{Y}}_\infty$. The only difference is that, in this case, the analysis is purely theoretical since no model can be trained on $y^_\infty$, which is not observable in real-world scenarios. - If $y^_\infty \not\in \mathbb{Y}_\infty$, the fairness metric is not aligned with the true distribution and/or the threshold is too small, then searching for a fairer vector is an ill posed problem. ### 2.5 Finite Samples All the possible applications and advantages (i.e., visualization) introduced by adopting the geometric framework require that the previous considerations hold also in a real-world scenario. To understand how we can apply GEOFFair to a finite dataset, we need to add a specific notation to differentiate a real case from the ideal one. Thus, we represent the finite input distribution by means of a finite input vector $ x_n \in X^n $, with $ n \in \mathbb{N} $. Concerning the output, we make a distinction between the distribution that can actually be observed and the one that we ideally wish to estimate. Moreover, we highlight that the notations refers to finite vectors. Thus, we start by introducing a new concept: Definition 5 (Finite Vectors). Given a ground vector $ y^_\infty \in Y^\infty $ and the respective gold vector $ y^_n \in Y^n $ both for $ n \to \infty $, we analogously define the finite ground vector $ y^+_n \in Y^n $ and the respective finite gold vector $ y^_n \in Y^n $ for $ n \in \mathbb{N} $. It still holds the same relationship of the infinite case: \[ y^+_n = b(y^_n), \quad \text{where } b_n : Y^n \to Y^n \text{ is called finite biased mapping} \] We can demonstrate that when $ n $ increases, the statistical and analytical properties of the data tend to converge to the ones we obtained in the infinite case. In other words, for a finite value of $ n $, we are performing a sampling of a continuous distribution which implies the validity of the transition from the conventional distribution paradigm of ML to the realm of vector spaces is no more guaranteed. In the following section we will show for as many statements as possible that the considerations assuming $ n \to \infty $ can be generalized also to the finite samples scenario. Definition 6 (Finite Spaces). As done for the ground vector and the gold vector, we can define the finite Hypothesis Space $ \hat{\Psi}_n \subseteq Y^n $ for $ n \in \mathbb{N} $ as the set of possible finite-dimensional outputs for the chosen class of ML models, i.e. \[ \hat{\Psi}_n = \{ y \in Y^n \mid \exists f \in F_n : f(x) = y \} \] Finally, let $ \bar{\Psi}_n \subseteq Y^n $ for $ n \in \mathbb{N} $ be the set containing all the finite-dimensional output vectors that are aligned with the fairness requirements called finite Fair Space. ### 3 Fairness Mitigation Through the Lens of GEOFFair In this section, we will utilize the GEOFFair framework to analyze fairness mitigation techniques. In a previous work by Dutta et al. [Dutta et al., 2020], it was demonstrated that maximizing accuracy solely based on the observed labels vector may not always be the optimal choice. They employed statistical distributions and mathematical tools from probability theory to establish this result. Rather than extending their findings, our objective is to employ our proposed geometric framework to support and validate them. By leveraging the GEOFFair framework, we aim to present similar conclusions in a more accessible and interpretable way and can bridge the gap between complex mathematical concepts and practical implications. This allows for a clearer comprehension of the challenges associated with fairness and the potential solutions that can be pursued. First, we propose the use of projections applied to the finite-dimensional case where standard numerical techniques can be adopted to algorithmically find the solution, then we extend the properties in the ideal case of an infinite-dimensional space. Finally, we introduce the problem of data polarization both as bias injection in a synthetic case and bias removal in a real-world case. #### 3.1 Mitigation as Projection for Finite Samples Mitigation, in the AI fairness context, refers to the process of reducing unfairness by either transforming the biased distribution or by ensuring that the ML model behaviour is compatible with the fairness constraints. From a geometric point of view, such techniques can be viewed as projecting either the ground vector or the ML output onto the Fair Space. Analogously, training an ML model can be viewed as the problem of finding a vector in the Hypothesis Space that is closest to the ground vector in terms of the loss function, i.e. as projecting the ground vector onto the Hypothesis Space. Therefore, in the context of GEOFFair, projections provide a convenient lens through which we can study mitigation at pre-processing, training, and post-processing time in a uniform fashion. We focus our analysis on the more widespread case where learning a fair ML model is possible (i.e. $ \hat{\Psi}_n \cap \bar{\Psi}_n \neq \emptyset $). We start by introducing two additional vectors, i.e. the projections of the ground truths and the gold standard vector, respectively. These projections will be onto the intersection space between the Hypothesis and the Fair Space. Definition 7 (Ground and Gold Fair Projections). The optimal fair predictions $ p_n^+ $ and $ p_n^* $ obtained from the ground ($ y_n^+ $) and gold ($ y_n^* $) vectors for $ n \in \mathbb{N} $, i.e.: \[ p_n^+ = \arg\min_v \{ L(v, y_n^+) \mid v \in \hat{Y}_n \cap Y_n \} \tag{5} \] \[ p_n^* = \arg\min_v \{ L(v, y_n^) \mid v \in \hat{Y}_n \cap Y_n \} \tag{6} \] Intuitively, $ p_n^+ $ represents the outcome of training an ML model under fairness constraints, or equivalently of training an ML model over a ground distribution transformed so as to enforce the fairness restrictions. The $ p_n^ $ vector represents the best fair model that we could learn for the (typically unobservable) “unbiased” distribution. It is worth noting that $ p_n^+ $ and $ p_n^* $ might not be unique, as equally accurate outputs that are both fair and representable by the model can exist. Moreover, the finite gold vector $ y_n^* $ is inherently not unique since all the vectors obtained by sampling the fair distribution are, by definition, gold vectors; this trivially leads to multiple projections $ p_n^* $. Furthermore, for the purpose of our theoretical analysis, we will assume that $ p_n^+ $ and $ p_n^* $ are obtained from exact and globally optimal algorithms. However, it is important to acknowledge that many machine learning models, especially larger ones, do not guarantee this optimality property in practice. Additionally, to avoid trivial cases, we assume that the biased mapping function $ b_n : Y^n \rightarrow Y^n $ applies a modification to the input vector, i.e. that $ y_n^* \neq y_n^+ $. This assumption narrows down our analysis to even fewer cases than those defined in Subsection 2.4, and let us draw the following conclusion: \[ L(y_n^+, y_n^) > 0 \tag{7} \] where $ L $ is any non-negative loss function such that $ L(y_n^+, y_n^) = 0 $ iff $ y_n^+ \equiv y_n^* $. Basic Properties of Fair Projections Let us consider the optimization problems defined in Equations (5), (6) and examine the behaviour of $ p_n^+ $ and $ p_n^* $ in terms of fairness based on the position of $ y_n^+ $ and $ y_n^* $, respectively. We will rely on the formulation of the Fair Space based on a fairness metric $ K(\cdot) $ that we introduced in Section 2, i.e.: \[ \overline{Y}_n = \{ y \in Y^n \mid K(y) \leq t \} \tag{8} \] Property 1 (Fair Projections). Given a vector $ y $ and its projection $ y' $ onto the Fair Space as defined in Equation (8), we know that: \[ y \in \overline{Y}_n \implies y' \equiv y \implies K(y') = K(y) \tag{9} \] \[ y \not\in \overline{Y}_n \implies K(y') = t \tag{10} \] meaning that any vector lying within the Fair Space will be projected onto itself (thus exhibiting the same fairness level); conversely, if the vector is outside the Fair Space, its projection will be on the boundary of the Fair Space, resulting in threshold-level fairness. This is a well-known property in both convex and non-convex optimization, whose proof can be found in Jain & Kar (2017). Now, if we take into account the capabilities of the ML model, we can extend Property 1 as follows: Property 2 (Representable Fair Projections). Given a vector $ y $ and its projection $ y' $ onto the intersection between the Fair and Hypothesis Space, we know that: \[ y \in \overline{Y}_n \lor \hat{Y}_n \subseteq \overline{Y}_n \implies K(y') \leq t \tag{11} \] \[ y \not\in \overline{Y}_n \land \hat{Y}_n \supseteq \overline{Y}_n \implies K(y') = t \tag{12} \] It is important to note that when the Fair Space and the Hypothesis Space have a non-trivial intersection – i.e. neither space is a subset of the other –, we cannot draw conclusions about $ K(y') $ since points in the boundary of the intersection can exhibit different fairness levels. 3.2 Mitigation as Projection for Infinite Samples All the considerations made in the previous paragraph can be generalized for the infinite-dimensional case with $ n \rightarrow \infty $. In particular, we define $ p_n^+ $ and $ p_n^* $ in the same way as Equations (5) and (6). respectively. One of the main differences lies between $ y_n^* $ and $ y_\infty^* $: in the finite-dimensional case it is easy to conclude that there can be multiple gold vectors due to the distribution sampling operation, each generated vector is probably close to the others in the output space (for example considering the loss function as distance measure) but we can assert that is almost impossible for all the existing gold vectors to coincide in the very same vector. Property 3 (Multiple finite gold vectors). Given a set of $ m $ finite-dimensional gold vectors sampled from the same fair distribution, the probability that at least two of them do not coincide is 1, i.e.: \[ \lim_{m \to \infty} P\left( \exists i, j \mid 0 \leq i < j < m \land L(\{y_n^\}_i, \{y_n^\}_j) > 0 \right) = 1 \] In the realm of infinite-dimensional samples, a different phenomenon emerges: the distance between any two gold vectors tends to zero, as they nearly perfectly represent the same distribution, as demonstrated in Theorem 1. Property 4 (Multiple infinite gold vectors). Given a set of $ m $ infinite-dimensional gold vectors sampled from the same fair distribution, the probability that at least two of them do not coincide is 0, i.e.: \[ P\left( \exists i, j \mid 0 \leq i < j < m \land L(\{y_\infty^\}_i, \{y_\infty^\}_j) > 0 \right) = 0 \quad \forall m \in [2, \infty) \] Using the notation $ y_\infty^* $, this definition implies $ \lim n \to \infty $. Finally, as a consequence of the fact that the mutual distance between any pair $ (\{y_\infty^\}_i, \{y_\infty^\}_j) $ goes to zero, we can conclude that it exists a vector $ u_\infty^* $ towards all the gold vectors converge. Studying the properties of $ u_\infty^* $ allows us to make considerations for a single vector and then generalise them to any gold vector introducing an arbitrary small error. Properties 1 and 2 are still valid in a Hilbert Space, keeping substantially unchanged the considerations made about the value of $ K(\cdot) $ for the projections with respect to the threshold $ t $. We can assess that the mitigation process remains substantially unchanged in the infinite-dimensional cases despite the different properties of the gold vector. ### 3.3 Injecting Bias – Polarization Throughout the whole paper, we have consistently emphasized the distinction between two crucial elements: the gold vector, a $ n $-dimensional sample of the target fair distribution, and the ground vector, the real-world data derived from the gold vector by means of a biased mapping $ b $ (Definition 5). This biasing effect is often linked to distortions resulting from the data collection process, such as imbalances in the population, or even deliberate unfair practices driven by human bias or a combination of both factors. Consequently, it is quite natural to think about how to solve the challenge of reversing the biased mapping to retrieve the gold vector. However, it is worth mentioning that we can also explore a complementary perspective on this issue. Instead of unbiasing the model, we can investigate methods to intentionally introduce controlled bias into a given gold vector, thereby generating a desired ground vector. This operation of “controlled bias injection” can be referred to as polarization. The geometric framework we illustrated in the previous sections not only offers an intuitive visual comprehension of why a biased mapping significantly impacts an ML model’s ability to attain an optimal and fair solution but also may provide insights how is possible to polarize a dataset (more specifically the target feature) to obtain a certain configuration between the ground vector and the Fair o the Hypothesis Spaces. This type of investigation can be conducted using synthetic datasets, which offer the capability to deliberately introduce controlled, arbitrary biases. This approach allows for the precise manipulation of both the distance and the orientation of the newly perturbed vector in relation to the original one. In the highly simplified example shown in Figure 1, we can intuitively understand the relevance of controlling the process of bias injection and its pivotal role in examining the interplay between the ground vector and the gold vector in a geometric sense. It’s worth noting that in this example, we solely assess which segment of the output space contains a particular entity. We do not take into account the effectiveness or the quality of the mitigation; instead, our focus is solely on evaluating the ML model’s ability to predict the projected vector while adhering to the fairness constraint. Figure 1: The picture illustrates three possible results of a polarization applied to the same gold vector (for simplicity reasons, we assume that $y_n^*$ is unique). In this example the gold vector belongs to the Fair Space but not to the Hypothesis Space. In (a) we notice that the biased mapping produces a different vector, but the same relationships with the other entities of the original one still hold. In (b) the biased mapping eases the problem since the ground vector belongs not only to the Fair Space but also to the Hypothesis Space (becoming a valid output for an ML model). Finally, case (c) leads to the worst scenario where the ground vector still cannot be returned by an ML model, but now the fairness constraint is violated as well. It’s important to emphasize that the three depicted polarizations share an equal magnitude (in terms of the Euclidean distance), yet merely altering the direction of the biased mapping leads to significant disparities. Possible applications Creating a synthetic dataset with well-defined properties is a common challenge that developers and researchers have to address in many ML applications. Among all the possible examples which demonstrate the large employment of generated data, we can mention the what-if analyses where emulating specific conditions is required to perform experiments, or we can even think about the benchmarking and evaluation phase of a traditional unbiasing technique; both these cases can rarely rely on extensive and representative historical datasets. Thanks to the polarization process, it is possible to tune the alteration of the biased target feature in order to generate a problem which can result particularly demanding for a certain mitigation technique, showing the weakness (or the strengths) of the approach that needs to be validated on a wide number of scenarios. Therefore, even when historical data are available it might be convenient to tackle the problem using properly generated synthetic datasets. Highly replicability on different domains allows to better generalize the quality of a novel methodology, in fact the polarization technique can quite intuitively show when a injected bias is actually moving a ground vector further from its projection or it is just producing equivalent vectors according to the entities we defined, and if the belonging of a vector to its original output segment is changing. This interesting double point of view on polarization (studying how to retrieve the gold vector, and enforcing the validation procedure of already existent techniques) highlight not only the agnostic nature of this framework but also one of the most salient and direct applications to real-world use cases. 4 CONCLUSION This study has introduced GEOFFair, a novel GEometric Framework for Fairness, which harnesses geometric principles to offer a robust and intuitive comprehension of fairness in the realm of AI. We have highlighted the benefits of adopting such a geometric framework for addressing fairness concerns. Furthermore, our analysis in this study has employed GEOFFair to examine fairness mitigation strategies and bias injection, ultimately yielding polarized datasets that serve as valuable tools for assessing and testing fairness in AI systems. REFERENCES Sina Aghaei, Mohammad Javad Azizi, and Phebe Vayanos. Learning optimal and fair decision trees for non-discriminative decision-making. In The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, January 27 - February 1, 2019, pp. 1418–1426. AAAI Press, 2019. doi: 10.1609/aaai.v33i01.33011418. URL https://doi.org/10.1609/aaai.v33i01.33011418. 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TW0MVSflg5	Could the authors clarify how they determine that some viewpoints are reliable and others are not? How generalizable is this approach to diverse real-world cases with more pronounced viewpoint variety?	SELF-EVOLVING NEURAL RADIANCE FIELDS Anonymous authors Paper under double-blind review ABSTRACT Recently, neural radiance field (NeRF) has shown remarkable performance in novel view synthesis and 3D reconstruction. However, it still requires abundant high-quality images, limiting its applicability in real-world scenarios. To overcome this limitation, recent works have focused on training NeRF only with sparse viewpoints by giving additional regularizations, often called few-shot NeRF. We observe that due to the under-constrained nature of the task, solely using additional regularization is not enough to prevent the model from overfitting to sparse viewpoints. In this paper, we propose a novel framework, dubbed Self-Evolving Neural Radiance Fields (SE-NeRF), that applies a self-training framework to NeRF to address these problems. We formulate few-shot NeRF into a teacher-student framework to guide the network to learn a more robust representation of the scene by training the student with additional pseudo labels generated from the teacher. By distilling ray-level pseudo labels using distinct distillation schemes for reliable and unreliable rays obtained with our novel reliability estimation method, we enable NeRF to learn a more accurate and robust geometry of the 3D scene. We show and evaluate that applying our self-training framework to existing models improves the quality of the rendered images and achieves state-of-the-art performance in multiple settings. 1 INTRODUCTION Novel view synthesis that aims to generate novel views of a 3D scene from given images is one of the essential tasks in computer vision fields. Recently, neural radiance field (NeRF) (Mildenhall et al., 2021) has shown remarkable performance for this task, modeling highly detailed 3D geometry and specular effects solely from given image information. However, the requirement of abundant high-quality images with accurate poses restricts its application to real-world scenarios, as reducing the input views causes NeRF to produce broken geometry and undergo severe performance degradation. Numerous works (Kim et al., 2022; Jain et al., 2021; Wang et al., 2023; Niemeyer et al., 2022; Yu et al., 2021) tried to address this problem, known as few-shot NeRF, whose aim is to robustly optimize NeRF in scenarios where only a few and sparse input images are given. To compensate for the few-shot NeRF’s under-constrained nature, they either utilize the prior knowledge of a pre-trained model (Jain et al., 2021; Yu et al., 2021) such as CLIP (Radford et al., 2021) or 2D CNN (Yu et al., 2021) or introduce an additional regularization (Niemeyer et al., 2022; Kim et al., 2022; Kwak et al., 2023), showing compelling results. However, these works show limited success in addressing the fundamental issue of overfitting as NeRF tends to memorize the input known viewpoints instead of understanding the geometry of the scene. In our toy experiment, this behavior is clearly shown in Figure 1, where existing methods (even with regularization (Fridovich-Keil et al., 2023; Niemeyer et al., 2022; Kim et al., 2022)) trained with 3-views show a noticeable drop in PSNR even with slight changes of viewpoints. Utilizing additional ground truth data for viewpoints that were unknown to the few-shot setting, we compare the rendered images from few-shot NeRF with the ground truth images and verify that there are accurately modeled regions even in unknown viewpoints that are far from known ones. This indicates that if we can accurately identify reliable regions, the rendered regions can be utilized as additional data achieved with no extra cost. Based on these facts, we formulate the few-shot NeRF task into the self-training framework by considering the rendered images as pseudo labels and training a new NeRF network with confident pseudo labels as additional data. Figure 1: Toy experiment to verify the robustness of models trained with sparse views. (Left) The red camera (a) indicates the camera position used for training and cameras from (b-e) are used to verify the robustness of models when the novel viewpoint gets further from the known viewpoint. (Middle) For each viewpoint (a-e), we visualize the rendered images by RegNeRF (Niemeyer et al., 2022), baseline ($K$-Planes (Fridovich-Keil et al., 2023)), and SE-NeRF from top to bottom rows. (Right) Starting from viewpoint (a), we show the PSNR graph of the rendered images as the viewpoint moves gradually from (a-e). Existing models show extreme PSNR drops, even with slight movements. Expanding upon this idea, we introduce a novel framework, dubbed Self-Evolving Neural Radiance Fields (SE-NeRF), which enables a more robust training of few-shot NeRF in a self-supervised manner. We train the few-shot NeRF under an iterative teacher-student framework, in which pseudo labels for geometry and appearance generated by the teacher NeRF are distilled to the student NeRF, and the trained student serves as the teacher network in the next iteration for progressive improvement. To estimate the reliability of the pseudo labels, we utilize the semantic features of a pre-trained 2D CNN to measure the consistency of the pseudo labels within multiple viewpoints. We also apply distinct distillation schemes for reliable and unreliable rays, in which reliable ray labels are directly distilled to the student, while unreliable rays undergo a regularization process to distill more robust geometry. Our experimental results show that our framework successfully guides existing NeRF models towards a more robust geometry of the 3D scene in the few-shot NeRF setting without using any external 3D priors or generative models (Xu et al., 2022). Also, we show the versatility of our framework, which can be applied to any existing models without changing their structure. We evaluate our approach on synthetic and real-life datasets, achieving state-of-the-art results in multiple settings. 2 RELATED WORK Neural radiance fields (NeRF). Synthesizing images from novel views of a 3D scene given multi-view images is a long-standing goal of computer vision. Recently, neural radiance fields (NeRF) (Mildenhall et al., 2021) has achieved great success by optimizing a single MLP that learns to estimate the radiance of the queried coordinates. The MLP learns the density $\sigma \in \mathbb{R}$ and color $c \in \mathbb{R}^3$ of continuous coordinates $x \in \mathbb{R}^3$, and is further utilized to explicitly render the volume of the scene using ray marching (Kajiya & Von Herzen, 1984). Due to its impressive performance in modeling the 3D scene, various follow-ups (Deng et al., 2022; Jain et al., 2021; Kim et al., 2022; Fridovich-Keil et al., 2023; Niemeyer et al., 2022; Wang et al., 2023; Roessle et al., 2022; Yang et al., 2023) adopted NeRF as their baseline model to solve various 3D tasks. Few-shot NeRF. Although capable of successfully modeling 3D scenes, NeRF requires abundant high-quality images with accurate poses, making it hard to apply in real-world scenarios. Several methods have paved the way to circumvent these issues by showing that the network can be successfully trained even when the input images are limited. One approach addresses the problem using prior knowledge from pre-trained local CNNs (Yu et al., 2021; Chibane et al., 2021; Kwak et al., 2023). PixelNeRF (Yu et al., 2021), for instance, employs a NeRF conditioned with features extracted by a pre-trained encoder. Another line of research introduces a geometric or depth-based regularization to the network (Jain et al., 2021; Kim et al., 2022; Niemeyer et al., 2022; Deng et al., 2022). DietNeRF (Jain et al., 2021) proposes an auxiliary semantic consistency loss to encourage realistic renderings at novel poses. RegNeRF (Niemeyer et al., 2022) regularizes the geometry and appearance of patches rendered from unobserved viewpoints. DS-NeRF (Deng et al., 2022) introduces additional depth supervision from sparse point clouds obtained in the COLMAP (Schonberger & Frahm, 2016) process. Self-training. Self-training is one of the earliest semi-supervised learning methods (Fralick, 1967; Scudder, 1965) mainly used in settings where obtaining sufficient labels is expensive (e.g., Instance segmentation). Self-training exploits the unlabeled data by pseudo labeling with a teacher model, which is then combined with the labeled data and used in the student training process. Noisy student (Xie et al., 2020) succeeds in continually training a better student by initializing a larger model as the student, and injecting noise into the data and network. Meta pseudo labels (Pham et al., 2021), on the other hand, optimizes the teacher model by evaluating the student’s performance on labeled data, guiding the teacher to generate better pseudo labels. We bring self-training to NeRFs by formulating the few-shot NeRF task as a semi-supervised learning task. Our approach can be seen as an analogous method of noisy student (Xie et al., 2020) that exploits NeRF as the teacher and student model, with teacher-generated unknown views as the unlabeled data. 3 PRELIMINARIES AND MOTIVATION 3.1 Preliminaries Given a set of training images $ S = \{ I_i \| i \in \{1, \ldots, N\} \} $, NeRF (Mildenhall et al., 2021) represents the scene as a continuous function $ f(\cdot; \theta) $, a neural network with parameters $ \theta $. The network renders images by querying the 3D points $ x \in \mathbb{R}^3 $ and view direction $ d \in \mathbb{R}^2 $ transformed by a positional encoding $ \gamma(\cdot) $ to output a color value $ c \in \mathbb{R}^3 $ and a density value $ \sigma \in \mathbb{R} $ such that $ \{c, \sigma\} = f(\gamma(x), \gamma(d); \theta) $. The positional encoding transforms the inputs into Fourier features (Tancik et al., 2020) that facilitate learning high-frequency details. Given a ray parameterized as $ r(t) = o + td $, starting from camera center $ o $ along the direction $ d $, the expected color value $ C(r; \theta) $ along the ray $ r(t) $ from $ t_n $ to $ t_f $ is rendered as follows: \[ C(r; \theta) = \int_{t_n}^{t_f} T(t)\sigma(r(t); \theta)c(r(t), d; \theta)dt, \quad T(t) = \exp \left( -\int_{t_n}^{t} \sigma(r(s); \theta)ds \right), \] where $ T(t) $ denotes the accumulated transmittance along the ray from $ t_n $ to $ t $. To optimize the network $ f(\cdot; \theta) $, the photometric loss $ L_{\text{photo}}(\theta) $ enforces the rendered pixel color value $ C(r; \theta) $ to be consistent with the ground-truth pixel color value $ C_{gt}(r) $: \[ L_{\text{photo}}(\theta) = \sum_{r \in R} \\|C_{gt}(r) - C(r; \theta)\\|_2^2, \] where $ R $ is the set of rays corresponding to each pixel in the image set $ S $. 3.2 Motivation Despite its impressive performance, NeRF has the critical drawback of requiring large amounts of posed input images $ S $ for robust scene reconstruction. Naively optimizing NeRF in a few-shot setting (e.g., $ \|S\| < 10 $) results in NeRF producing erroneous artifacts and undergoing major breakdowns in the geometry due to the task’s under-constrained nature (Niemeyer et al., 2022; Kim et al., 2022). A closer look reveals important details regarding the nature of the few-shot NeRF optimization. As described by the PSNR graph in Figure 1, all existing methods show a noticeable PSNR drop even with slight viewpoint changes, which indicates the tendency of NeRF to memorize the given input views. Such a tendency results in broken geometry that looks perfect in known viewpoints but progressively degenerates as the rendering view gets further away from known views. Although training with additional data directly solves this problem, obtaining high-quality images with accurate poses is extremely expensive. Instead, we notice that although images (rendered from NeRF trained with only sparse viewpoints) contain artifacts and erroneous geometry, there are reliable pixels of the image that are close to the corresponding ground truth pixels, which can be used as additional data. Figure 2: Illustration of our overall framework for applying self-training to NeRF. SE-NeRF utilizes the self-training framework to distill the knowledge of learned appearance and 3D geometry from teacher to student. The process is done iteratively as the student becomes the new teacher. To check the feasibility that using reliable pixels from the rendered images as additional data can help prevent NeRF from overfitting, we conduct an experiment of first optimizing NeRF under the identical few-shot setting. After training a teacher NeRF with three images, we train a new student NeRF with the extended set of images $S \cup S^+$ where $S^+$ is the set of rendered images. To train with only the reliable pixels of $S^+$, we define a binary reliability mask $M(r)$, which masks out pixels where the difference between the rendered color value $C(r; \theta^T)$ and its ground truth color value $C_{gt}(r)$ is above a predetermined threshold. Training the student NeRF network to follow the reliably rendered color values $\{C(r; \theta^T) \| M(r) = 1\}$ of the teacher can be seen as a weak distillation from the teacher to the student. The new student NeRF is trained with the following loss function: $$L_{photo}(\theta) + \lambda \sum_{r \in R^+} M(r)\\|C(r; \theta^T) - C(r; \theta)\\|^2_2,$$ where $R^+$ is a set of rays corresponding to each pixel in the rendered image set $S^+$, and $\lambda$ denotes the weight parameter. The result of this experiment, described in "GT Masked" of the PSNR graph in Figure 1 shows that the student trained with K-Planes (Fridovich-Keil et al., 2023) as the baseline, displays staggering improvement in performance, with unknown viewpoints showing higher PSNR values and their rendered geometry remaining highly robust and coherent. This leads us to deduce that a major cause of few-shot NeRF geometry breakdown is its tendency to memorize the given sparse viewpoints and that selected distillation of additional reliable rays is crucial to enhance the robustness and coherence of 3D geometry. Based on this observation, our concern now moves on to how to estimate the reliability mask $M$ for the rendered novel images of $S^+$ to develop a better few-shot NeRF model. 4 METHOD 4.1 TEACHER-STUDENT FRAMEWORK Teacher network optimization. A teacher network is trained naively by optimizing the standard NeRF photometric loss where the number of known viewpoints is $\|S\| < 10$. During this process, NeRF recovers accurate geometry for certain regions and inaccurate, broken geometry in other regions. The parameters of teacher network $\theta^T$ is optimized as the following equation: $$\theta^T = \arg\min_\theta L_{photo}(\theta).$$ Pseudo labeling with teacher network. By evaluating the optimized teacher NeRF representation $\theta^T$, we can generate per-ray pseudo labels $\{C(r; \theta^T) \| r \in R^+\}$ from the rendered images $S^+$ from unknown viewpoints. To accurately identify and distill the reliable regions of $S^+$ to the student model, we assess the reliability of every pseudo label in $R^+$ to acquire a reliability mask $M(r)$ using a novel reliability estimation method we describe in detail in Section 4.2. Student network optimization. The student network $\theta^S$ is then trained with the extended training set of $S \cup S^+$, with the reliability mask $M$ taken into account. In addition to the photometric loss with the initial image set $S$, the student network is also optimized with a distillation loss that encourages it to follow the robustly reconstructed parts of the teacher model in $S^+$. In the distillation process, the estimated reliability mask $M$ determines how each ray should be distilled, a process which we explain further in Section 4.3. In summary, student network $\theta^S$ is optimized by the following equation: $$\theta^S = \arg\min_{\theta} \left\{ L_{\text{photo}}(\theta) + \lambda \sum_{r \in R^+} M(r) \\| C(r; \theta^T) - C(r; \theta) \\|_2^2 \right\},$$ where $C(r; \theta^T)$ and $C(r; \theta)$ is the rendered color of the teacher and student model, respectively and $\lambda$ denotes the weight parameter. Iterative labeling and training. After the student network is fully optimized, the trained student network becomes the teacher network of the next iteration for another distillation process to a newly initialized NeRF, as described in Figure 2. We achieve improvement of the NeRF’s quality and robustness every iteration with the help of the continuously extended dataset. 4.2 Ray Reliability Estimation To estimate the reliability of per-ray pseudo labels $\{C(r; \theta^T)\mid r \in R^+\}$ from the rendered images $S^+$, we expand upon an important insight that if a ray has accurately recovered a surface location and this location is projected to multiple viewpoints, the semantics of the projected locations should be consistent except for occlusions between viewpoints. This idea has been used in previous works that formulate NeRF for refined surface reconstruction (Chibane et al., 2021), but our work is the first to leverage it for explicitly modeling ray reliability in a self-training setting. The surface location recovered by a ray $r$ corresponding to pixel $p_i$ of the viewpoint $i$ can be projected to another viewpoint $j$ with the extrinsic matrix $R_{i \rightarrow j}$, intrinsic matrix $K$, and the estimated depth $D_i$ from viewpoint $i$ with the following projection equation: $$p_{i \rightarrow j} \sim KR_{i \rightarrow j}D_i(r)K^{-1}p_i.$$ Using the projection equation, we can make corresponding pixel pairs between viewpoint $i$ and $j$ such as $(p_i, p_j)$ where $p_j = p_{i \rightarrow j}$. Similarly, if we acquire pixel-level feature maps from viewpoint $i$ and $j$ using a pre-trained 2D CNN, we can make corresponding feature pairs as $(f^i_p, f^j_p)$. In our case, by projecting the feature vector of the corresponding pseudo label $\{C(r; \theta^T)\mid r \in R^+\}$ to all given input viewpoints, we can achieve $\|S\|$ feature pairs for every pseudo label. To generate a reliability mask for each ray, if a ray has at least one feature pair whose similarity value is higher than the threshold value $\tau$, it indicates that the feature consistency of the ray’s rendered geometry has been confirmed and classify such rays as reliable. Summarized in equation, the binary reliability mask $M(r)$ for the ray $r$ rendered from viewpoint $i$ can be defined as follows: $$M(r) = \min \left\{ \sum_{j \in \|S\|} \frac{1}{\\| f^i_p \\| \\| f^j_p \\|} > \tau \right\}, 1 \right\}.$$ To prevent the unreliable rays from being misclassified as reliable, we must carefully choose the threshold $\tau$. Although using a fixed value for the $\tau$ is straightforward, we find that choosing the adequate value is extremely cumbersome as the similarity distribution for each scene varies greatly. Instead, we adopt the adaptive thresholding method, which chooses the threshold by calculating the $(1 - \alpha)^{th}$ percentile of the similarity distribution where $\alpha$ is a hyperparameter in the range $\alpha \in [0, 1]$. This enables the threshold $\tau$ to be dynamically adjusted to each scene, leading to a better classification of the reliable rays. 4.3 Reliability-based Distillation To guide the student network to learn a more robust representation of the scene, we distill the label information from the teacher to the student with two distinct losses based on the ray’s reliability. By remembering the rays evaluated in the teacher network and re-evaluating the same rays in the student network, the geometry and color information of reliable rays is directly distilled into the student network through distillation loss, while the rays classified as unreliable are regularized with nearby reliable rays for improved geometry before applying the distillation loss. Figure 3: Distillation of pseudo labels. After estimating the reliability of the rays from unknown views, we apply distinct distillation schemes for reliable and unreliable rays. Reliable rays are directly distilled to the student while we aggregate the nearby reliable rays to regularize the unreliable rays. Reliable ray distillation. Since we assume the reliable rays’ appearance and geometry have been accurately predicted by the teacher network, we directly distill their rendered color so that the student network faithfully follows the outputs of the teacher for these reliable rays. With the teacher-generated per-ray pseudo labels $ \{C(r; \theta^T) \| r \in R^+\} $ from the rendered images $ S^+ $ and the estimated reliability mask $ M $, the appearance of a reliable ray is distilled by the reformulated photometric loss $ L_c^R $: \[ L_c^R(\theta) = \sum_{r \in R^+} M(r) \\| C(r; \theta^T) - C(r; \theta) \\|_2^2. \] In addition to the photometric loss $ L_c^R $, we follow Deng et al., (2022); Roessle et al., (2022) of giving the depth-supervision together to NeRF. As the teacher network $ \theta^T $ also outputs the density $ \sigma(r; \theta^T) $ for each of the rays, we distill the density weights of the sampled points of the reliable rays to the student network. Within the same ray, we select an identical number of points randomly sampled from evenly spaced bins along the ray. This allows us to follow the advantages of injecting noise to the student as in Xie et al., (2020) as randomly sampling points from each bin induces each corresponding point to have slightly different positions, which acts as an additional noise to the student. The density distillation is formulated by the geometry distillation loss $ L_g^R $, which is L2 loss between accumulated density values of corresponding points within the teacher and student rays, with teacher rays’ density values $ \sigma^T $ serving as the pseudo ground truth labels. Therefore, for reliable rays, our distillation loss along the camera ray $ r(t) = o + td $ is defined as follows: \[ L_g^R(\theta) = \sum_{r \in R^+} \sum_{t,t' \in T} M(r) \\| \sigma(r(t); \theta^T) - \sigma(r(t'); \theta) \\|_2^2. \] where $ T $ refers to the evenly spaced bins from $ t_n $ to $ t_f $ along the ray, $ t $ and $ t' $ indicate randomly selected points from each bins. Unreliable ray distillation. In traditional semi-supervised methods, unreliable labels are ignored to prevent the confirmation bias problem. Similarly, unreliable rays must not be directly distilled as they are assumed to have captured inaccurate geometry. However, stemming from the prior knowledge that depth changes smoothly above the surface, we propose a novel method for regularizing the unreliable rays with geometric priors of nearby reliable rays, dubbed prior-based distillation. To distill the knowledge of nearby reliable rays, we calculate a weighted average of nearby reliable rays’ density distribution and distill this density to the student. As described in Figure 3, we apply a Gaussian mask to unreliable ray $ r $ to calculate per-ray weights for nearby reliable rays. The intuition behind this design choice is straightforward: the closer a ray is to an unreliable ray, the more likely it is to be that the geometry of the two rays will be similar. Based on these facts, we apply the prior-based geometry distillation loss $ L_g^P $, which is the L2 loss between the weighted-average density $ \tilde{\sigma}(r; \theta^T) $ and the student density outputs $ \sigma(r; \theta) $, is described in the following equation: \[ L_g^P(\theta) = \sum_{r \in R^+} \sum_{t,t' \in T} (1 - M(r)) \\| \tilde{\sigma}(r(t); \theta^T) - \sigma(r(t'); \theta) \\|_2^2. \] We apply the prior-based geometry distillation loss to the unreliable rays only when adjacent reliable rays exist. A more detailed explanation can be found in Appendix B.3. Table 1: Quantitative comparison on NeRF Synthetic and LLFF. \| Methods \| NeRF Synthetic Extreme \| NeRF Synthetic \| LLFF \| \|---------------\|------------------------\|----------------\|------\| \| \| PSNR↑ SSIM↑ LPIPS↓ Avg ↓ \| PSNR↑ SSIM↑ LPIPS↓ Avg ↓ \| PSNR↑ SSIM↑ LPIPS↓ Avg ↓ \| \| NeRF \| 14.85 0.73 0.32 0.27 \| 19.38 0.82 0.17 0.20 \| 17.50 0.50 0.47 0.40 \| \| K-Planes \| 15.45 0.73 0.28 0.28 \| 17.99 0.82 0.18 0.21 \| 15.77 0.44 0.46 0.41 \| \| DietNeRF \| 14.46 0.72 0.28 0.28 \| 15.42 0.73 0.21 0.20 \| 14.94 0.37 0.50 0.44 \| \| InfoNeRF \| 14.62 0.74 0.26 0.27 \| 18.44 0.80 0.22 0.12 \| 13.57 0.33 0.58 0.48 \| \| RegNeRF \| 13.73 0.70 0.30 0.30 \| 13.71 0.79 0.35 0.21 \| 19.08 0.59 0.34 0.15 \| \| SE−NeRF (NeRF)\| 17.41 0.78 0.21 0.22 \| 20.53 0.84 0.16 0.19 \| 18.10 0.54 0.45 0.38 \| \| \| (+2.56) (+0.05) (-0.11) (-0.05) \| (+1.15) (+0.02) (-0.01) (-0.01) \| (+6.60) (+0.04) (-0.02) (-0.02) \| \| SE−NeRF (K−Planes) \| 17.40* 0.78* 0.23* 0.25* \| 17.93* 0.83* 0.17* 0.26* \| 16.36* 0.49* 0.44* 0.59* \| \| \| (+2.04) (+0.05) (-0.05) (-0.04) \| (+1.94) (+0.01) (-0.01) (-0.01) \| (+0.53) (+0.05) (-0.02) (-0.02) \| Total distillation loss. Finally, our entire distillation loss can be formulated as follows: $$\theta^S = \arg\min_\theta \{L_{\text{photo}}(\theta) + \lambda_c^R L_c^R(\theta) + \lambda_g^R L_g^R(\theta) + \lambda_g^P L_g^P(\theta)\},$$ where $\lambda_c^R$, $\lambda_g^R$, and $\lambda_g^P$ denotes the weight parameters. Figure 4: Qualitative comparison on NeRF Synthetic Extreme. The results show the rendered images from viewpoints far away from the seen views. A noticeable improvement over existing models regarding artifacts and distortion removal can be observed in SE−NeRF. 5 EXPERIMENTS 5.1 Setups Datasets and metrics. We evaluate our methods on NeRF Synthetic [Mildenhall et al., 2021] and LLFF dataset [Mildenhall et al., 2019]. For the NeRF Synthetic dataset, we randomly select 4 views in the train set and use 200 images in the test set for evaluation. For LLFF, we chose every 8-th image as the held-out test set and randomly select 3 views for training from the remaining images. In addition, we find that all existing NeRF models’ performance on the NeRF Synthetic dataset is largely affected by the randomly selected views. To explore the robustness of our framework and existing methods, we introduce a novel evaluation protocol of training every method with an extreme 3-view setting (NeRF Synthetic Extreme) where all the views are selected from one side of the scene. The selected views can be found in Appendix C. We report PSNR, SSIM [Wang et al., 2004], LPIPS [Zhang et al., 2018] and geometric average [Barron et al., 2021] values for qualitative comparison. Implementation details. Although any NeRF representation is viable, we adopt $K$-Planes [Fridovich-Keil et al., 2023] as our main baseline to leverage its memory and time efficiency. Also, we conduct experiments using our framework with NeRF [Mildenhall et al., 2021] and Instant-NGP [Müller et al., 2022] to demonstrate the applicability of our framework. For our reliability estimation method, we use VGGNet [Simonyan & Zisserman, 2014], specifically VGG-19, and utilize the first 4 feature layers located before the pooling layers. We train $K$-Planes for 20 minutes on NeRF Synthetic and 60 minutes on LLFF using a single RTX 3090, and NeRF is trained for 90 minutes on NeRF Synthetic and 120 minutes on LLFF using 4 RTX 3090 GPUs for each iteration. 1For Instant-NGP, we train the model for 5 minutes on NeRF Synthetic Extreme. Hyper-parameters. We set the adaptive threshold value at $\alpha = 0.15$ for the first iteration. To enable the network to benefit from more reliable rays for each subsequent iteration, we employ a curriculum labeling approach that increases $\alpha$ by 0.05 every iteration. As images rendered from views near the initial inputs include more reliable regions, we progressively increase the range of where the pseudo labels should be generated. We start by selecting views that are inside the range of 10 degrees in terms of $\phi, \theta$ of the initial input and increase range after iterations. For the weights for our total distillation loss, we use $\lambda_c^R = 1.0$, $\lambda_g^R = 1.0$, and $\lambda_g^P = 0.005$. Table 2: Quantitative comparison per-scene on NeRF Synthetic Extreme. \| Methods \| chair \| drums \| focus \| hotdog \| lego \| maten \| ship \| mic \| \|--------------------------\|-------\|-------\|-------\|--------\|------\|-------\|------\|-----\| \| NeRF \| 15.08 \| 11.98 \| 17.16 \| 13.83 \| 16.31\| 17.31 \| 10.84\| 16.29\| \| K-Planes \| 15.61 \| 13.23 \| 18.29 \| 12.45 \| 14.67\| 16.30 \| 13.35\| 19.74\| \| Instant-NGP \| 17.66 \| 12.75 \| 18.44 \| 13.67 \| 13.17\| 16.83 \| 13.82\| 19.05\| \| DietNeRF \| 16.60 \| 8.09 \| 18.32 \| 19.00 \| 11.45\| 16.97 \| 15.26\| 10.01\| \| InfoNeRF \| 15.38 \| 12.48 \| 18.59 \| 19.04 \| 12.27\| 15.25 \| 7.23 \| 16.76\| \| RegNeRF \| 15.92 \| 12.09 \| 14.83 \| 14.06 \| 14.86\| 10.53 \| 11.44\| 16.12\| \| SE-NeRF (NeRF) \| 19.96 \| 14.72 \| 19.29 \| 16.06 \| 16.45\| 17.51 \| 14.20\| 21.09\| \| \| (+4.88)\| (+2.74)\| (+2.13)\| (+2.23)\| (+0.14)\| (+0.20)\| (+3.36)\| (+4.80)\| \| SE-NeRF (K-Planes) \| 20.54 \| 13.38 \| 18.33 \| 20.14 \| 16.65\| 17.01 \| 13.72\| 20.13\| \| \| (+4.93)\| (+0.15)\| (+0.04)\| (+7.69)\| (+1.98)\| (+0.71)\| (+0.37)\| (+0.39)\| \| SE-NeRF (Instant-NGP) \| 20.46 \| 13.34 \| 19.07 \| 18.15 \| 15.99\| 17.94 \| 14.61\| 20.23\| \| \| (+2.74)\| (+0.59)\| (+0.63)\| (+4.48)\| (+2.82)\| (+1.11)\| (+0.79)\| (+1.18)\| \| SE-NeRF (DietNeRF) \| 20.46 \| 13.34 \| 19.07 \| 18.15 \| 15.99\| 17.94 \| 14.61\| 20.23\| 5.2 Comparison Qualitative comparison. Figure 4 and Figure 5 illustrate the robustness of our model to unknown views, even when the pose differs significantly from the training views. Our model demonstrates robust performance on unknown data, surpassing the baselines. This is particularly evident in the "chair" scene, where all existing methods exhibit severe overfitting to the training views, resulting in heavy artifacts when the pose significantly changes from those used during training. RegNeRF fails to capture the shape and geometry in unknown views and although DietNeRF is capable of capturing the shape of the object accurately, it produces incorrect information, such as transforming the armrests of the chair into wood. In contrast, SE-NeRF maintains the shape of an object even from further views with less distortion, resulting in the least artifacts and misrepresentation. Quantitative comparison. Table 1 and Table 2 show quantitative comparisons of applying our framework against other few-shot NeRFs and our baseline models on NeRF synthetic and LLFF datasets. As shown in Table 1, SE-NeRF outperforms previous few-shot NeRF models in the NeRF synthetic Extreme and the conventional 4-view setting. By applying SE-NeRF, we observe an general improvement in performance over different methods and different datasets, demonstrating that our framework successfully guides networks of existing methods to learn more robust knowledge of the 3D scene. 5.3 Ablation study. Iterative training. As shown in Figure 6, which presents the quantitative results for each iteration, a significant improvement in performance can be observed after the first iteration. The performance continues to be boosted with each subsequent iteration until the convergence. Based on our experimental analysis, we find that after the simultaneous distillation of reliable rays and regularization of unreliable rays in the first iteration, there is much less additional knowledge to distill to the student in certain scenes which leads to a smaller performance gain from the second iteration. However, although the performance gain in terms of metrics is small, the remaining artifacts and noise in the images continue to disappear after the first iteration, which is important in perceptual image quality. Prior-based ray distillation. In Table 3, we conduct an ablation study on the "lego" scene of the NeRF Synthetic Extreme setting and show that using both reliable and unreliable ray distillation is crucial to guide the network to learn a more robust representation of the scene, showing the highest results in all metrics. This stands in contrast to existing semi-supervised approaches (Xie et al., 2020; Amini et al., 2023), which typically discard unreliable pseudo labels to prevent the student learning from erroneous information (Arazo et al., 2020). We show that when applying self-training to NeRF, the unreliable labels can be further facilitated by the prior knowledge that depth within a 3D space exhibits smoothness. Thresholding. In Table 4, we show the results of SE-NeRF trained on the NeRF Synthetic Extreme setting with different thresholding strategies. Following traditional semi-supervised approaches (Tur et al., 2005; Cascante-Bonilla et al., 2021; Zhang et al., 2021a; Chen et al., 2023), we conducted experiments using a predefined fixed threshold, adaptive threshold (ours), and a unified threshold which does not classify pseudo labels as reliable and unreliable but uses the similarity value to decide how much the distillation should be made from the teacher to the student. The adaptive thresholding method resulted in the most performance gain, showing the rationale of our design choice. A comprehensive and detailed analysis regarding the threshold selection process is provided in Appendix B.4. ### Table 3: Ray distillation ablation. \| Method \| PSNR ↑ \| SSIM ↑ \| LPIPS ↓ \| Average ↓ \| \|-------------------------\|--------\|--------\|---------\|-----------\| \| K-Planes \| 14.67 \| 0.68 \| 0.31 \| 0.30 \| \| K-Planes + Reliable \| 16.15 (+1.48) \| 0.72 (+0.04) \| 0.27 (-0.04) \| 0.27 (-0.03) \| \| K-Planes + Reliable/Unreliable \| 16.65 (+1.98) \| 0.75 (+0.07) \| 0.24 (-0.07) \| 0.25 (-0.05) \| ### Table 4: Thresholding ablation. \| Threshold \| PSNR ↑ \| SSIM ↑ \| LPIPS ↓ \| Avg. ↓ \| \|-----------\|--------\|--------\|---------\|--------\| \| Fixed \| 17.02 \| 0.77 \| 0.25 \| 0.25 \| \| Unified \| 15.95 \| 0.73 \| 0.28 \| 0.27 \| \| Adaptive \| 17.49 \| 0.78 \| 0.23 \| 0.24 \| ## 6 Conclusion And Limitations In this paper, we present a novel self-training framework Self-Evolving Neural Radiance Fields (SE-NeRF), specifically designed for few-shot NeRF. By employing a teacher-student framework in conjunction with our unique implicit distillation method, which is based on the estimation of ray reliability through feature consistency, we demonstrate that our self-training approach yields a substantial improvement in performance without the need for any 3D priors or modifications to the original architecture. Our approach is able to achieve state-of-the-art results on multiple settings and shows promise for further development in the field of few-shot NeRF. However, our framework also shares similar limitations to existing semi-supervised approaches. 1) Sensitivity to inappropriate pseudo labels: when unreliable labels are classified as reliable and used to train the student network, this leads to performance degradation of the student model. 2) Teacher initialization: if the initialized teacher network in the first iteration is too poor, our framework fails to enhance the performance of the models even after several iterations. Even with these limitations, our framework works robustly in most situations, and we leave the current limitations as future work. 7 REPRODUCIBILITY STATEMENT For the reproducibility of our work, we will release all the source codes and checkpoints used in our experiments. For those who want to try applying our self-training framework to existing works, we provide the pseudo codes for our reliability estimation method for the per-ray pseudo labels and the overall self-training pipeline. Algorithm 1 Reliability estimation method for per-ray pseudo labels 1: Input: Labeled Image $I$, rendered Image $I^+$, rendered depth $D^+$, threshold $\tau$ 2: Output: Mask $M$ for $I^+$ 3: $f \leftarrow \text{VGG19}(I)$ 4: $f^+ \leftarrow \text{VGG19}(I^+)$ 5: for $i \leftarrow 0$ to (Height - 1) do 6: for $j \leftarrow 0$ to (Width - 1) do 7: $(i', j') \leftarrow \text{Warp}(I^+, D^+, I, i, j)$ ▷ $I_{i,j}$ is warped to $I_{i',j'}$ using rendered depth $D^+$ 8: $S \leftarrow \text{CosineSimilarity}(f_{i,j}, f_{i',j'})$ 9: if $S > \tau$ then 10: $M_{i,j} \leftarrow 1$ 11: else 12: $M_{i,j} \leftarrow 0$ 13: end if 14: end for 15: end for Algorithm 2 Self-Training 1: Input: Teacher Network $T$, set of labeled ray $R$, set of rendered ray $R^+$ 2: Output: Teacher Network $T$ for next iteration 3: for each step do 4: Initialize $S$ ▷ Initialize Student Network 5: Loss $\leftarrow 0$ 6: for each $r$ in $R$ do 7: Loss $\leftarrow$ Loss + L2($c$, Color($S$, $r$)) 8: end for 9: for each $r$ in $R^+$ do 10: Evaluate $M(r)$ 11: if $M(r) = 1$ then 12: Loss $\leftarrow$ Loss + L2(Color($T$, $r$), Color($S$, $r$)) ▷ Reliable RGB Loss 13: Loss $\leftarrow$ Loss + L2(Weight($T$, $r$), Weight($S$, $r$)) ▷ Reliable Density Loss 14: else 15: Loss $\leftarrow$ Loss + L2(GaussianWeight($T$, $r$), Weight($S$, $r$)) ▷ Unreliable Density Loss 16: end if 17: Update $T$ with Loss 18: end for 19: end for 20: $T \leftarrow S$
uHdf9F1tY4	Currently, the authors define these protected images as those filtered using a classifier, which we will refer to as C. This approach raises concerns as the authors have not sufficiently demonstrated the quality of classifier C, and it may not accurately gauge the impact of protected training data on image generation	DiffusionShield: A Watermark for Data Copyright Protection against Generative Diffusion Models Anonymous authors Paper under double-blind review Abstract Recently, Generative Diffusion Models (GDMs) have showcased their remarkable capabilities in learning and generating images. A large community of GDMs has naturally emerged, further promoting the diversified applications of GDMs in various fields. However, this unrestricted proliferation has raised serious concerns about copyright protection. For example, artists including painters and photographers are becoming increasingly concerned that GDMs could effortlessly replicate their unique creative works without authorization. In response to these challenges, we introduce a novel watermarking scheme, DiffusionShield, against GDMs. DiffusionShield protects images from copyright infringement through encoding the ownership information into an imperceptible watermark and injecting it into the images. Its watermark can be easily learned by GDMs and will be reproduced in their generated images. By detecting the watermark from generated images, copyright infringement can be exposed with evidence. Benefiting from the uniformity of the watermarks and the joint optimization method, DiffusionShield ensures low distortion of the original image, high watermark detection performance, and the ability to embed lengthy messages. We conduct rigorous and comprehensive experiments to show the effectiveness of DiffusionShield in defending against infringement by GDMs and its superiority over traditional watermarking methods. 1 Introduction Generative diffusion models (GDMs), such as Denoising Diffusion Probabilistic Models (DDPM) [Ho et al., 2020] have shown their great potential in generating high-quality images. This has also led to the growth of more advanced techniques, such as DALL-E2 [Ramesh et al., 2022], Stable Diffusion [Rombach et al., 2022], and ControlNet [Zhang & Agrawala, 2023]. In general, a GDM learns the distribution of a set of collected images, and can generate images that follow the learned distribution. As these techniques become increasingly popular, concerns have arisen regarding the copyright protection of creative works shared on the Internet. For instance, a fashion company may invest significant resources in designing a new fashion. After the company posts the pictures of this fashion to the public for browsing, an unauthorized entity can train their GDMs to mimic its style and appearance, generating similar images and resulting in products. This infringement highlights the pressing need for copyright protection mechanisms. To provide protection for creative works, watermark techniques such as [Cox et al., 2002; Podilchuk & Delp, 2001; Zhu et al., 2018; Navas et al., 2008; Yu et al., 2021] are often applied, which aim to inject (invisible) watermarks into images and then detect them to track the malicious copy and accuse the infringement. However, directly applying these existing methods to GDMs still faces tremendous challenges. Indeed, since existing watermark methods have not specifically been designed for GDMs, they might be hard to learn for GDMs and could disappear in the generated images. Then, the infringement cannot be effectively verified and accused. As empirical evidence in Figure 1, we train two popular GDMs on a CIFAR10 dataset whose samples are watermarked by two representative watermark methods [Navas et al., 2008; Zhu et al., 2018], and we try to detect the watermarks in the GDM-generated images. The result demonstrates that the watermarks from these methods are either hardly learned and reproduced by GDM (e.g., FRQ [Navas et al., 2008]), or require a very large budget (the extent of image distortion) to partially maintain the watermarks (e.g., HiDDeN [Zhu et al., 2018]). Therefore, dedicated efforts are still greatly desired to developing the watermark technique tailored for GDMs. In this work, we argue that one critical factor that causes the inefficacy of these existing watermark techniques is the inconsistency of watermark patterns on different data samples. In methods such as [Navas et al., 2008; Zhu et al., 2018], the watermark in each image from one owner is distinct. Thus, GDMs can hardly learn the distribution of watermarks and reproduce them in the generated samples. To address this challenge, we propose DiffusionShield which aims to enhance the “pattern uniformity” (Section 3.2) of the watermarks to make them consistent across different images. We first empirically show that watermarks with pattern uniformity are easy to be reproduced by GDMs in Section 3.2. Then, we provide corresponding theoretic analysis in two examples to demonstrate that the watermarks with pattern uniformity will be learned prior to other features in Section 3.5. The theoretical evidence further suggests that if unauthorized GDMs attempt to learn from the watermarked images, they are likely to learn the watermarks before the original data distribution. To leverage pattern uniformity, DiffusionShield designs a blockwise strategy to divide the watermarks into a sequence of basic patches, and a user has a specific sequence of basic patches which forms a watermark applied on all his/her images and encodes the copyright message. The watermark will repeatedly appear in the training set of GDMs, and thus makes it reproducible and detectable. In the case with multiple users, each user will have his/her own watermark pattern based on encoded message. Furthermore, DiffusionShield introduces a joint optimization method for basic patches and watermark detector to enhance each other, which achieves a smaller budget and higher accuracy. In addition, once the watermarks are obtained, DiffusionShield does not require re-training when there is an influx of new users and images, indicating the flexibility of DiffusionShield to accommodate multiple users. In summary, with the enhanced pattern uniformity in blockwise strategy and the joint optimization, we can successfully secure the data copyright against the infringement by GDMs. 2 RELATED WORK 2.1 GENERATIVE DIFFUSION MODELS In recent years, GDMs have made significant strides. A breakthrough in GDMs is achieved by DDPM [Nichol & Dhariwal, 2021], which demonstrates great superiority in generating high-quality images. The work of [Ho & Salimans, 2022] further advances the field by eliminating the need for classifiers in the training process. [Song et al., 2020] presents Denoising Diffusion Implicit Models (DDIMs), a variant of GDMs with improved efficiency in sampling. Besides, techniques such as [Rombach et al., 2022] achieve high-resolution image synthesis and text-to-image synthesis. These advancements underscore the growing popularity and efficacy of GDM-based techniques. To train GDMs, many existing methods rely on collecting a significant amount of training data from public resources [Deng et al., 2009; Yu et al., 2015; Guo et al., 2016]. However, there is a concern that if a GDM is trained on copyrighted material and produces outputs similar to the original copyrighted works, it could potentially infringe on the copyright owner’s rights. This issue has already garnered public attention [Vincent, 2023], and our paper focuses on mitigating this risk by employing a watermarking technique to detect copyright infringements. 2.2 IMAGE WATERMARKING Image watermarking involves embedding invisible information into the carrier images and is commonly used to identify ownership of the copyright. Traditional watermarking techniques include spatial domain methods and frequency domain methods [Cox et al., 2002; Navas et al., 2008; Shih & Wu, 2003; Kumar, 2020]. These techniques embed watermark information by modifying the pixel values [Cox et al., 2002], frequency coefficients [Navas et al., 2008], or both [Shih & Wu, 2003; Kumar, 2020]. In recent years, various digital watermarking approaches based on Deep Neural Networks (DNNs) have been proposed. For example, [Zhu et al., 2018] uses an autoencoder-based network architecture, while [Zhang et al., 2019] designs a GAN for watermark. Those techniques are then further generalized to photographs [Tancik et al., 2020] and videos [Weng et al., 2019]. Figure 1: Watermark detection accuracy (%) on GDM-generated images and the corresponding budget ($l_2$ norm) of watermarks. Notably, there are existing studies focusing on watermarking generative neural networks, such as GANs (Goodfellow et al., 2020) and image processing networks (Schwag et al., 2022). Their goal is to safeguard the intellectual property (IP) of generative models and generated images, while our method is specifically designed for safeguarding the copyright of data against potential infringement by these GDMs. To accomplish their goals, the works (Wu et al., 2020; Yu et al., 2021; Zhao et al., 2023a; Zhang et al., 2020) embed imperceptible watermarks into every output of a generative model, enabling the defender to determine whether an image was generated by a specific model or not. Various approaches have been employed to inject watermarks, including reformulating the training objectives of the generative models (Wu et al., 2020), modifying the model’s training data (Yu et al., 2021; Zhao et al., 2023a), or directly applying a watermark embedding process to the output images before they are presented to end-users (Zhang et al., 2020). 3 Method In this section, we first formally define the problem and the key notations. Next, we show that the “pattern uniformity” is a key factor for the watermark of generated samples. Based on this, we introduce two essential components of our method, DiffusionShield, i.e., blockwise watermark with pattern uniformity and joint optimization, and then provide theoretic analysis of pattern uniformity. 3.1 Problem Statement In this work, we consider two roles: (1) a data owner who holds the copyright of the data, releases them solely for public browsing, and aspires to protect them from being replicated by GDMs, and (2) a data offender who employs a GDM on the released data to appropriate the creative works and infringe the copyright. On the other hand, in reality, data are often collected from multiple resources to train GDMs. Thus, we also consider a scenario where there are multiple owners to protect their copyright against GDMs by encoding the copyright information into watermarks. We start by defining the one-owner case, and then extend the discussion to the multiple-owner case: • Protection for one-owner case. An image owner aims to release $ n $ images, $ \{X_{1:n}\} $, strictly for browsing. Each image $ X_i $ has a shape of $(U, V)$ where $ U $ and $ V $ are the height and width, respectively. As shown in Figure 2, the protection process generally comprises two stages: 1) a protection stage when the owner encodes the copyright information into the invisible watermark and adds it to the protected data; and 2) an audit stage when the owner examines whether a generated sample infringes upon their data. In the following, we introduce crucial definitions and notations. 1) The protection stage happens before the owner releases $ \{X_{1:n}\} $ to the public. To protect the copyright, the owner encodes the copyright message $ M $ into each of the invisible watermarks $ \{W_{1:n}\} $, and adds $ W_i $ into $ X_i $ to get a protected data $ \tilde{X}_i = X_i + W_i $. $ M $ can contain information like texts which can signify the owners’ unique copyright. The images $ \tilde{X}_i $ and $ X $ appear similar in human eyes with a small watermark budget $ \\|W_i\\|_p \leq \epsilon $. Instead of releasing $ \{X_{1:n}\} $, the owner releases the protected $ \{\tilde{X}_{1:n}\} $ for public browsing. 2) The audit stage refers to that the owner finds suspicious images which potentially offend the copyright of their images, and they scrutinize whether these images are generated from their released data. We assume that the data offender collects a dataset $ \{X^G_{1:N}\} $ that contains the protected images $ \{\tilde{X}_{1:n}\} $, i.e., $ \{\tilde{X}_{1:n}\} \subset \{X^G_{1:N}\} $ where $ N $ is the total number of both protected and unprotected images ($ N > n $), and trains a GDM, $ G $, from scratch to generate images, $ X_G $. If $ X_G $ contains the copyright information of the data owner, once $ X_G $ is inputted to a decoder $ D $, the copyright message should be decoded by $ D $. • Protection for multiple-owner case. When there are $K$ data owners to protect their distinct sets of images, we denote their sets of images as $\{X_{k:n}^i\}$ where $k = 1, \ldots, K$. Following the methodology of one-owner case, each owner can re-use the same encoding process and decoder to encode and decode distinct messages in different watermarks, $W_i^k$, which signifies their specific copyright messages $M^k$. The protected version of images is denoted by $\tilde{X}_k^i = X_k^i + W_i^k$. Then the protected images, $\{\tilde{X}_{1:n}^i\}$, can be released by their respective owners for public browsing, ensuring their copyright is maintained. More details about the two protection cases can be found in Appendix A. ### 3.2 Pattern Uniformity In this subsection, we uncover one important factor “pattern uniformity” which could be an important reason for the failure of existing watermark techniques. Previous studies (Sehwag et al., 2022; Um & Ye, 2023; Daras et al., 2023) observe that GDMs tend to learn data samples from high probability density regions in the data space and ignore the low probability density regions. However, many existing watermarks such as FRQ (Navas et al., 2008) and HiDDeN (Zhu et al., 2018) can only generate distinct watermarks for different data samples. Since their generated watermarks are dispersed, these watermarks cannot be effectively extracted and learned. Observing the above, we formally define the “pattern uniformity” as the consistency of different watermarks injected for different samples: $$Z = 1 - \frac{1}{n} \sum_{i=1}^{n} \left\\| \frac{W_i}{\\|W_i\\|_2} - W_{\text{mean}} \right\\|_2,$$ where $W_{\text{mean}} = \frac{1}{n} \sum_{i=1}^{n} \frac{W_i}{\\|W_i\\|_2}$. We further conduct experiments to illustrate the importance of this “pattern uniformity”. In the experiment shown in Figure 3, we test DDPM’s ability in learning watermarks with different pattern uniformity. The watermarks $W_i$ are random pictures whose pixel value is re-scaled by the budget $\sigma$, and the watermarked images are $\tilde{X}_i = X_i + \sigma \times W_i$. More details about the settings for this watermark and the detector can be found in Appendix C.1. Figure 3 illustrates a positive correlation between the watermark detection rate in the GDM-generated images and the pattern uniformity, which implies that pattern uniformity improves watermark reproduction. Based on pattern uniformity, in Section 3.3 and 3.4, we introduce how to design DiffusionShield, and in Section 3.5, we provide the theoretic analysis of the pattern uniformity based the two examples to justify that the watermarks will be first learned prior to other sparse hidden features and, thus, provide an effective protection. ### 3.3 Watermarks and Decoding Watermarks In this subsection, we introduce our proposed approach, referred as DiffusionShield. This model is designed to resolve the problem of inadequate reproduction of prior watermarking approaches in generated images. It adopts a blockwise watermarking approach to augment pattern uniformity, which improves the reproduction of watermarks in generated images and enhances flexibility. Blockwise watermarks. In DiffusionShield, to strengthen the pattern uniformity in $\{W_{1:n}\}$, we use the same watermark $W$ for each $X_i$ from the same owner. The sequence of basic patches encodes the textual copyright message $M$ of the owner. In detail, $M$ is first converted into a sequence of binary numbers by predefined rules such as ASCII. To condense the sequence’s length, we convert the binary sequence into a $B$-nary sequence, denoted as $\{b_{1:m}\}$, where $m$ is the message length and $B$-nary represents different numeral systems like quaternary ($B = 4$) and octal ($B = 8$). Accordingly, DiffusionShield partitions the whole watermark $W$ into a sequence of $m$ patches, $\{w_{1:m}\}$, to represent $\{b_{1:m}\}$. Each patch is chosen from a candidate set of basic patch $\{w^{(1:B)}\}$. The set $\{w^{(1:B)}\}$ has $B$ basic patch candidates with a shape $(u,v)$, which represent different values of the $B$-nary bits. The sequence of $\{w_{1:m}\}$ denotes the $B$-nary bits $\{b_{1:m}\}$ derived from $M$. For example, in Figure 4, we have 4 patches ($B = 4$), and each of the patches has a unique pattern which represents 0, 1, 2, and 3. To encode the copyright message $M =$ “Owned by XXX”, we first convert it into binary sequence “01001111 01110111…” based on ASCII, and transfer it into quaternary sequence $\{b_{1:m}\}$, “103313131232…”. (The sequence length $m$ should be less or equal to $8 \times 8$, since there are only $8 \times 8$ patches in Figure 4.) Then we concatenate these basic patches in the order of $\{b_{1:m}\}$ for the complete watermark $W$ and add $W$ to each image from the data owner. Once the offender uses GDMs to learn from it, the watermarks will appear in generated images, serving as evidence of infringement. Decoding the watermarks. DiffusionShield employs a decoder $ D_\theta $ by classification in patches, where $ \theta $ is the parameters. $ D_\theta $ can classify $ w_i $ into a bit $ b_i $. The decoder $ D_\theta $ accepts a watermarked image block, $ x_i + w_i $, as input and outputs the bit value of $ w_i $, i.e., $ b_i = D_\theta(x_i + w_i) $. The suspect generated image is partitioned into a sequence $ \{(x + w)\}_{i=1:m} $, and then is classified into $ \{b_{1:m}\} = \{D_\theta(x_i + w_i)\}_{i=1,...,m} $ in a patch-by-patch manner. If $ \{b_{1:m}\} $ is the $ B $-nary message that we embed into the watermark, we can accurately identify the owner of the data, and reveal the infringement. Remarks. Since we assign the same watermark $ W $ to each image of one user, the designed watermark evidently has higher uniformity. Additionally, DiffusionShield shows remarkable flexibility when applied to multiple-owner scenarios since basic patches and decoder can be reused by new owners. 3.4 JOINTLY OPTIMIZE WATERMARK AND DECODER While pattern uniformity facilitates the reproduction of watermarks in GDM-generated images, it does not guarantee the detection performance of the decoder, $ D_\theta $. Therefore, we further propose a joint optimization method to search for the optimal basic patch patterns and obtain the optimized detection decoder. Ideally, the basic patches and the decoder should satisfy: \[ b^{(i)} = D_\theta(p + w^{(i)}) \quad \forall \; i \in \{1, 2, ..., B\}, \] where $ w^{(i)} $ is one of the $ B $ basic patch candidates, $ b^{(i)} $ is the correct label for $ w^{(i)} $, and $ p $ can be a random block with the same shape as $ w^{(i)} $ cropped from any image. The ideal decoder, capable of accurately predicting all the watermarked blocks, ensures that all embedded information can be decoded from the watermark. To increase the detection performance of the decoder, we simultaneously optimize the basic patches and the decoder using the following bi-level objective: \[ \min_{w^{1:B}} \min_{\theta} \mathbb{E} \left[ \sum_{i=1}^{B} L_{CE} \left( D_\theta \left( p + w^{(i)} \right), b^{(i)} \right) \right] \text{ s.t. } \\|w^{(i)}\\|_\infty \leq \epsilon, \] where $ L_{CE} $ is the cross-entropy loss for the classification. The $ l_\infty $ budget is constrained by $ \epsilon $. To reduce the number of categories of basic patches, we set $ w^{(1)} = 0 $, which means that the blocks without watermark should be classified as $ b = 1 $. Thus, the bi-level optimization can be rewritten as: \[ \begin{align} \theta^ &= \arg \min_{\theta} \mathbb{E} \left[ \sum_{i=1}^{B} L_{CE} \left( D_\theta \left( p + w^{(i)} \right), b^{(i)} \right) \right] \\ w^{(2:B),} &= \arg \min_{w^{(2:B)}} \mathbb{E} \left[ \sum_{i=2}^{B} L_{CE} \left( D_{\theta^} \left( p + w^{(i)} \right), b^{(i)} \right) \right] \text{ s.t. } \\|w^{(i)}\\|_\infty \leq \epsilon. \end{align} \] The upper-level objective aims to increase the performance of $ D_\theta $, while the lower-level objective optimizes the basic patches to facilitate their detection by the decoder. By the two levels of objectives, the basic patches and decoder potentially promote each other to achieve higher accuracy on smaller budget. To ensure basic patches can be adapted to various image blocks and increase their flexibility, we use randomly cropped image blocks as the host images in the training process of basic patches and decoder. More details about the algorithm of joint optimization can be found in Appendix D. 3.5 THEORETIC ANALYSIS OF PATTERN UNIFORMITY BASED ON TWO EXAMPLES In this subsection, we provide theoretic analysis with two examples, a linear regression model for supervised task, and a multilayer perceptron (MLP) with a general loss function (which can be a generation task), to justify that watermarks with pattern uniformity are stronger than other features, and machine learning models can learn features from watermarks earlier and more easily regardless of the type of tasks. Following the same idea, DiffusionShield provides an effective protection since GDMs have to learn watermarks first if they want to learn from protected images. For both two examples, we use the same assumption for the features in the watermarked dataset. For simplicity, we assume the identical watermark is added onto each sample in the dataset. We impose the following data assumption, which is extended from the existing sparse coding model [Olshausen & Field, 1997; Mairal et al., 2010; Arora et al., 2016; Allen-Zhu & Li, 2022]. Assumption 1 (Sparse coding model with watermark). The observed data is $ Z = MS $, where $ M \in \mathbb{R}^{d \times d} $ is a unitary matrix, and $ S = (s_1, s_2, \cdots, s_d)^\top \in \mathbb{R}^d $ is the hidden feature composed of $ d $ sparse features: \[ P(s_i \neq 0) = p, \text{ and } s_i^2 = O(1/pd) \text{ when } s_i \neq 0. \] The norm $ \\|\cdot\\| $ is $ L_2 $ norm. For $ \forall i \in [d], E[s_i] = 0 $. The watermarked data is $ \tilde{Z} = MS + \delta $, and $ \delta $ is a constant watermark vector for all the data samples because of pattern uniformity. For the linear regression task, $ Y = S^\top \beta + \epsilon $ is the ground truth label, where $ \epsilon \sim N(0, \sigma^2) $ is the noise and $ \beta_i = \Theta(1) $ so that $ Y^2 = O_p(1) $. We represent the linear regression model as $ \hat{Y} = \tilde{Z}^\top w $, using the watermark data $ \tilde{Z} $, where $ w \in \mathbb{R}^{1 \times d} $ is the parameter to learn. The mean square error (MSE) loss for linear regression task can be represented as \[ L(w) = (\tilde{Z}^\top w - S^\top \beta - \epsilon)^2. \] Given the above problem setup, we have the following result: Example 1.* Consider the initial stage of the training, i.e., $ w $ is initialized with $ w_i \overset{i.i.d.}{\sim} N(0, 1) $. With Assumption 7, the gradient, with respect to $ w $, of MSE loss for the linear regression model defined above given infinite samples can be derived as \[ E \left[ \frac{\partial L}{\partial w} \right] = E[A(S)] + E[B(\delta)], \] where $ E[A(S)] $ is the hidden feature term that contains the gradient terms from hidden features, and $ E[B(\delta)] $ is the watermark term that contains the gradient terms from the watermark. There are three observations. First, watermark is learned prior to other hidden features after initialization. If $ \\|\delta\\| \gg 1/\sqrt{d} $, then with high probability w.r.t. the initialization, $ E[B(\delta)] \gg E[A(S)] $, and $ E[B(\delta)] $ is maximized with the best uniformity. Second, since $ \\|\delta\\| \ll 1/\sqrt{pd} $, the watermark $ \delta $ will be much smaller than any active hidden feature. Finally, when the training converges, the final trained model does not forget $ \delta $. (The proof can be found in Appendix B.1.) In addition to the linear regression task, we extend our analysis to neural networks with a general loss to further explain the feasibility of the intuition for a generative task. We follow Assumption 1 and give the toy example for neural networks: Example 2. We use an MLP with $ \tilde{Z} $ as input to fit a general loss $ L(W, \tilde{Z}) $. $ L(W, \tilde{Z}) $ can be a classification or generation task. $ W $ is the parameter of it, and $ W_1 $ is the first layer of $ W $. Under mild assumptions, we can derive gradient with respect to each neuron in $ W_1 $ into hidden feature term and watermark term as Eq. 6. When $ 1/\sqrt{d} \ll \\|\delta\\| \ll 1/\sqrt{pd} $, the watermark term will have more influence and be learned prior to other hidden features in the first layer even though the watermark has a much smaller norm than each active hidden feature. (The proof can be found in Appendix B.2.) With the theoretical analysis on the two examples, we justify that the watermark with high pattern uniformity is easier/earlier to be learned than other sparse hidden features. It suggests if the authorized people use GDM to learn from the protected images, the GDM will first learn the watermarks before the data distribution. Therefore, our method can provide an effective protection against GDM. We also provide empirical evidence to support this analysis in Appendix B.3. ### 4 Experiment In this section, we assess the efficacy of DiffusionShield across various budgets, datasets, and protection scenarios. We first introduce our experimental setups in Section 4.1. In Section 4.2, we evaluate the performance in terms of its accuracy and invisibility. Then we investigate the flexibility and efficacy in multiple-user cases, capacity for message length and robustness, in Section 4.3 to 4.6 respectively. We also evaluate the quality of generated images in Appendix H. #### 4.1 Experimental Settings Datasets, baselines and GDM. We conduct the experiments using four datasets and compare DiffusionShield with four baseline methods. The datasets include CIFAR10 and CIFAR100, both with $(U, V) = (32, 32)$, STL10 with $(U, V) = (64, 64)$ and ImageNet-20 with $(U, V) = (256, 256)$. The baseline methods include Image Blending (IB) which is a simplified version of DiffusionShield without joint optimization, DWT-DCT-SVD based watermarking in the frequency domain (FRQ) (Navas et al., 2008), HiDDeN (Zhu et al., 2018), and DeepFake Fingerprint Detection (DFD) (Yu et al., 2021) (which is designed for DeepFake Detection and adapted to our data protection goal). In the audit stage, we use the improved DDPM (Nichol & Dhariwal, 2021) as the GDM to train on watermarked data. More details about baselines and improved DDPM is in Appendix C.4 and C.5, respectively. Evaluation metrics. In our experiments, we generate $T$ images from each GDM and decode copyright messages from them. We compare the effectiveness of watermarks in terms of their invisibility, the decoding performance, and the capacity to embed longer messages: - (Perturbation) Budget. We use the LPIPS (Zhang et al., 2018) metric together with $l_2$ and $l_\infty$ differences to measure the visual discrepancies between the original and watermarked images. The lower values of these metrics indicate better invisibility. - (Detection) Accuracy. Following Yu et al. (2021) and Zhao et al. (2023b), we apply bit accuracy to evaluate the correctness of detected messages encoded. To compute bit accuracy, we transform the ground truth $B$-nary message $\{b_{1:m}\}$ and the decoded $\{\hat{b}_{1:m}\}$ back into binary messages $\{b'_{1:m \log_2 B}\}$ and $\{\hat{b}'_{1:m \log_2 B}\}$. The bit accuracy for one watermark is $$\text{Bit-Acc} = \frac{1}{m \log_2 B} \sum_{k=1}^{m \log_2 B} 1 \left( b'_{1:m \log_2 B} = \hat{b}'_{1:m \log_2 B} \right).$$ The worst bit accuracy is expected to be 50%, which is equivalent to random guessing. - Message length. The length of encoded message reflects the capacity of encoding. To ensure accuracy of FRQ and HiDDeN, we use a 32-bit message for CIFAR images and 64 bits for STL10. For others, we encode 128 bits into CIFAR, 512 bits into STL10 and 256 bits into ImageNet. Implementation details. We set $(u,v) = (4,4)$ as the shape of the basic patches and set $B = 4$ for quarternary messages. We use ResNet (He et al., 2016) as the decoder to classify different basic patches. For the joint optimization, we use 5-step PGD (Madry et al., 2017) with $l_\infty < \epsilon$ to update the basic patches and use SGD to optimize the decoder. As mentioned in Section 3.1, the data offender may collect and train the watermarked images and non-watermarked images together to train GDMs. Hence, in all the datasets, we designate one random class of images as watermarked images, while treating other classes as unprotected images. To generate images of the protected class, we either 1) use a class-conditional GDM to generate images from the specified class, or 2) apply a classifier to filter images of the protected class from the unconditional GDM’s output. The bit accuracy on unconditionally generated images may be lower than that of the conditional generated images since object classifiers cannot achieve 100% accuracy. In the joint optimization, we use SGD with learning rate = 0.01 and weight decay = $5 \times 10^{-4}$ to train the decoder and we use 5-step PGD with step size to be 1/10 of the $L_\infty$ budget to train the basic patches. More details are presented in Appendix C.3. ### 4.2 Results on Protection Performance against GDM In this subsection, we show that DiffusionShield provides protection with high bit accuracy and good invisibility in Table 1. We compare on two groups of images: (1) the originally released images with watermarks (Released) and (2) the generated images from class-conditional GDM or unconditional GDM trained on the watermarked data (Cond. and Uncond.). Based on Table 1, we can see: First, DiffusionShield can protect the images with the highest bit accuracy and the lowest budget among all the methods. For example, on CIFAR10 and STL10, with all the budgets from 1/255 to 8/255, DiffusionShield can achieve almost 100% bit accuracy on released images and conditionally generated images, which is better than all the baseline methods. Even constrained by the smallest budget with an $l_\infty$ norm of 1/255, DiffusionShield can still achieve a high successful reproduction rate. On CIFAR100 and ImageNet, DiffusionShield with an $l_\infty$ budget of 4/255 achieves a higher bit accuracy in generated images with a much lower $l_\infty$ difference and LPIPS than baseline methods. For baselines, FRQ cannot be reproduced by GDM, while HiDDeN and DFD require a much larger perturbation budget over DiffusionShield (Image examples are shown in Appendix E). The accuracy of IB is much worse than the DiffusionShield with 1/255 budget on CIFAR10 and STL10. To explain IB, without joint optimization, the decoder cannot perform well on released images and thus cannot guarantee its accuracy on generated images, indicating the importance of joint optimization. Second, enforcing pattern uniformity can promote the reproduction of watermarks in generated images. In Table 1, we can see that the bit accuracy of the conditionally generated images watermarked by DiffusionShield is as high as that of released images with a proper budget. In addition to DiffusionShield, IB’s accuracy in released data and conditionally generated data are also similar. This is because IB is a simplified version of our method without joint optimization and also has high pattern uniformity. In contrast, other methods without pattern uniformity all suffer from a Table 1: Bit accuracy (%) and budget of the watermark \| \| IB \| FRQ \| HiDDeN \| DFD \| DiffusionShield (ours) \| \|----------------\|------\|------\|--------\|------\|------------------------\| \| Budget \| \| \| \| \| \| \| $l_\infty$ \| 7/255\| 13/255\| 65/255 \| 28/255\| 1/255 \| \| $l_2$ \| 0.52 \| 0.70 \| 2.65 \| 1.21 \| 0.18 \| \| LPIPS \| 0.01582 \| 0.01790 \| 0.14924 \| 0.07095 \| 0.00005 \| \| CIFAR10 \| \| \| \| \| \| \| Released \| 87.2767 \| 99.7875 \| 99.0734 \| 95.7763 \| 99.6955 \| \| Cond. \| 87.4840 \| 57.7469 \| 98.9250 \| 93.5703 \| 99.8992 \| \| Uncond. \| 81.4839 \| 55.6907 \| 97.1536 \| 89.1977 \| 93.8186 \| \| Pattern Uniformity \| 0.963 \| 0.056 \| 0.260 \| 0.236 \| 0.974 \| \| CIFAR100 \| \| \| \| \| \| \| Released \| 84.6156 \| 99.5250 \| 99.7000 \| 96.1297 \| 99.5547 \| \| Cond. \| 54.3406 \| 54.4438 \| 95.8640 \| 90.5828 \| 52.0078 \| \| Uncond. \| 52.2786 \| 55.5380 \| 77.7616 \| 77.7961 \| 52.8320 \| \| Pattern Uniformity \| 0.822 \| 0.107 \| 0.161 \| 0.180 \| 0.854 \| \| STL10 \| \| \| \| \| \| \| Released \| 92.5895 \| 99.5750 \| 97.2769 \| 94.2813 \| 99.4969 \| \| Cond. \| 96.0541 \| 54.3945 \| 96.5164 \| 94.7236 \| 95.4848 \| \| Uncond. \| 89.2259 \| 56.3038 \| 91.3919 \| 91.8919 \| 82.5841 \| \| Pattern Uniformity \| 0.895 \| 0.071 \| 0.155 \| 0.203 \| 0.924 \| \| ImageNet-20 \| \| \| \| \| \| \| Released \| - \| 99.8960 \| 98.0625 \| 99.3554 \| 99.9375 \| \| Cond. \| - \| 50.6090 \| 98.2500 \| 81.3232 \| 53.6865 \| \| Pattern Uniformity \| - \| 0.061 \| 0.033 \| 0.041 \| 0.941 \| drop of accuracy from released images to conditionally generated images, especially FRQ, which has pattern uniformity lower than 0.11 and an accuracy level on par with a random guess. This implies that the decoded information in watermarks with high pattern uniformity (e.g., IB and ours in CIFAR10 are higher than 0.95) does not change much from released images to generated images and the watermarks can be exactly and easily captured by GDM. Notably, the performance drop on CIFAR100 and ImageNet in 1/255 and 2/255 is also partially due to the low watermark rate. In fact, both a small budget and a low watermark rate can hurt the reproduction of watermarks in generated images. In Appendix E, we discuss the effectiveness when watermark rate is low. We find that in multiple user case, even though the watermark rate for each user is low and they encode different messages and do not share the pattern uniformity, our method can still performs well. 4.3 Flexibility and Efficacy in Multiple-user Case In this subsection, we demonstrate that DiffusionShield is flexible to be transferred to new users while maintaining good protection against GDMs. We assume that multiple copyright owners are using DiffusionShield to protect their images, and different copyright messages should be encoded into the images from different copyright owners. In Table 2, we use one class in the dataset as the first owner and the other classes as the new owners. The basic patches (with 4/255 $l_\infty$ budget) and decoder are optimized on the first class and re-used to protect the new classes. Images within the same class have the same message embedded, while images from different classes have distinct messages embedded in them. After reordering the basic patches for different message, transferring from one class to the other classes does not take any additional calculation, and is efficient. We train class-conditional GDM on all of the protected data and get the average bit accuracy across classes. As shown in Table 2, on both CIFAR10 and CIFAR100, when we reorder the basic patches to protect the other 3 classes or 9 classes, the protection performance is almost the same as the one class case, with bit accuracy all close to 100%. Besides flexibility, our watermarks can protect each of the multiple users and can distinguish them clearly even when their data are mixed by the data offender. 4.4 Generalization to Fine-tuning GDMs In this subsection, we test the performance of our method when generalized to the fine-tuning GDMs [Rombach et al., 2022], which is also one of common strategies for learning and generating Table 2: Average bit accuracy (%) across different numbers of copyright owners (on class-conditional GDM). \| owners \| CIFAR-10 \| CIFAR-100 \| \|--------\|----------\|-----------\| \| 1 \| 100.0000 \| 99.8000 \| \| 4 \| 99.9986 \| 99.9898 \| \| 10 \| 99.9993 \| 99.9986 \| images. Fine-tuning is a more difficult task compared the training-from-scratch setting, because fine-tuning only changes the GDM parameters in a limited extent. This change may be not sufficient to learn all the features in the fine-tuned dataset, therefore, the priority by pattern uniformity becomes even more important. To better generalize our method to the fine-tuning case, we enhance the uniformity in hidden space instead of the pixel space, and limit $l_2$ norm instead of $l_\infty$ norm. More details of fine-tuning and its experiment settings can be found in Appendix I. We assume that the data offender fine-tunes Stable Diffusion (Rombach et al., 2022) to learn the style of pokemon-blip-captions dataset (Pinkney, 2022). In Table 3, we compare the budget and bit accuracy of our method with three baselines. The observation is similar to that in Table 1. Although FRQ has smaller budget than ours, the bit accuracy on generated images are much worse. DFD has bit accuracy of 90.31%, but the budget is three times of ours. HiDDeN is worse than ours in both budget and bit accuracy. In summary, our method has the highest accuracy in both released data and generated data. ### 4.5 Capacity for Message Length The capacity of embedding longer messages is important for watermarking methods since encoding more information can provide more conclusive evidence of infringement. In this subsection, we show the superiority of DiffusionShield over other methods in achieving high watermark capacity. Figure 5 shows the bit accuracy and $l_2$ budgets of watermarks with different message lengths on the released protected images in CIFAR10. In Figure 5a, we can see that HiDDeN consistently requires a large budget across varying message lengths, and its accuracy diminishes to 77% at 128 bits. Conversely, DiffusionShield maintains nearly 100% accuracy at 128 bits, even with a much smaller budget. Similarly, in Figure 5b, ours maintains longer capacity with better accuracy and budget than DFD. This indicate that DiffusionShield has much greater capacity compared to HiDDeN and DFD and can maintain good performance even with increased message lengths. ### 4.6 Robustness of DiffusionShield Robustness of watermarks is important since there is a risk that the watermarks may be distorted by disturbances, such as image corruption due to deliberate post-processing activities during the images’ circulation, the application of speeding-up sampling methods in the GDM (Song et al., 2020), or different training hyper-parameters used to train GDM. This subsection demonstrate that DiffusionShield is robust in accuracy on generated images when corrupted. In Appendix G.1 and G.2, we show similar conclusions when sampling procedure is fastened and hyper-parameters are changed. We consider Gaussian noise, low-pass filter, greyscale and JPEG compression to test the robustness of DiffusionShield against image corruptions. Different from the previous experiments, during the protection stage, we augment our method by incorporating corruptions into the joint optimization. Each corruption is employed after the basic patches are added to the images. Table 4 shows the bit accuracy of DiffusionShield (with an $l_\infty$ budget of 8/255) on corrupted generated images. It maintains around 99.8% accuracy under Greyscale and low-pass filter, nearly matching the accuracy achieved without any corruption. In other corruptions, our method performs better than baselines except HiDDeN in Gaussian noise. In contrast, DFD has a significant reduce in Gaussian noise, Greyscale and JPEG compression, and HiDDeN shows a poor performance under low-pass filter and JPEG Compression. From these results, we can see that DiffusionShield is robust against image corruptions. ### 5 Conclusion In this paper, we introduce DiffusionShield, a watermark to protect data copyright, which is motivated by our observation that the pattern uniformity can effectively assist the watermark to be captured by GDMs. By enhancing the pattern uniformity of watermarks and leveraging a joint optimization method, DiffusionShield successfully secures copyright with better accuracy and a smaller budget. Theoretic analysis and experimental results demonstrate the superior performance of DiffusionShield. \| Table 3: Bit accuracy (%) in fine-tuning \| Table 4: Bit accuracy (%) under corruptions \| \|-----------------------------------------\|-------------------------------------------\| \| FRQ \| DFD \| HiDDeN \| Ours \| DFD \| HiDDeN \| Ours \| \| $l_2$ \| Released \| Generated \| \| \| \| \| \| 8.95 \| 88.86 \| 57.13 \| 61.30 \| 99.50 \| 92.88 \| 93.57 \| \| 63.40 \| 99.20 \| 90.31 \| 83.59 \| 81.05 \| 99.86 \| 98.93 \| \| 21.22 \| 89.48 \| 60.16 \| 81.93 \| 99.86 \| 99.81 \| 99.99 \| \| \| \| \| 50.82 \| 74.84 \| 94.45 \| Figure 5: Bit acc. and $l_2$ of different message lengths (a) HiDDeN & ours (1/255) (b) DFD & ours (1/255) REFERENCES Zeyuan Allen-Zhu and Yuanzhi Li. Feature purification: How adversarial training performs robust deep learning. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 977–988. IEEE, 2022. 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EGjvMcKrrl	Can the authors clarify whether, in Table 2, w/o (8, 9) corresponds to the original S4 model? The numbers are slightly lower than in the original paper, and I am trying to clarify whether these numbers are like-for-like within the table, and how comparable to S4 they are.	FROM GENERALIZATION ANALYSIS TO OPTIMIZATION DESIGNS FOR STATE SPACE MODELS Anonymous authors Paper under double-blind review ABSTRACT A State Space Model (SSM) is a foundation model in time series analysis, which has recently been shown as an alternative to transformers in sequence modeling. In this paper, we theoretically study the generalization of SSMs and propose improvements to training algorithms based on the generalization results. Specifically, we give a data-dependent generalization bound for SSMs, showing an interplay between the SSM parameters and the temporal dependencies of the training sequences. Leveraging the generalization bound, we (1) set up a scaling rule for model initialization based on the proposed generalization measure, which significantly improves the robustness of the output value scales on SSMs to different temporal patterns in the sequence data; (2) introduce a new regularization method for training SSMs to enhance the generalization performance. Numerical results are conducted to validate our results. 1 INTRODUCTION Sequence modeling has been a long-standing research topic in many machine learning areas, such as speech recognition [Hinton et al., 2012], time series prediction [Li et al., 2019], and natural language processing [Devlin et al., 2019]. Various machine learning models have been successfully applied in sequence modeling to handle different types of sequence data, ranging from the (probabilistic) Hidden Markov model [Baum & Petrie, 1966] to deep learning models, e.g., Recurrent Neural Networks (RNNs), Long Short-Term Memory units [Hochreiter & Schmidhuber, 1997], Gated Recurrent Unit [Chung et al., 2014], and transformers [Vaswani et al., 2017]. In this paper, we focus on the state space model (SSM), which has a simple mathematical expression: \[ h'(t) = Ah(t) + Bx(t), \quad y(t) = Ch(t) + Dx(t) \] where $ h(t) $ is the hidden state, $ x(t) $ is the input sequence, $ y(t) $ is the output sequence and $ A, B, C, D $ are trainable parameters. Recent studies have demonstrated the power of SSMs in deep learning. For example, it was shown in [Gu et al., 2022a] that by a new parameterization and a carefully chosen initialization, the structured state space sequence (S4) model achieved strong empirical results on image and language tasks. Following the S4 model, more variants of SSMs are proposed, e.g., the diagonal SSM [Gu et al., 2022b], Gupta et al., 2022, the S5 model [Smith et al., 2023], the H3 model [Fu et al., 2023], the GSS model [Mehta et al., 2023], and the Hyena Hierarchy [Poli et al., 2023]. Theoretical analysis and understanding of the approximation and optimization of SSMs are well studied in the literature such as [Li et al., 2021, 2022, Gu et al., 2022a, 2023]. Since the SSM can be regarded as a continuous linear RNN model [Li et al., 2022], most generalization analysis of SSMs is based on the generalization theory of RNNs [Zhang et al., 2018, Chen et al., 2019, Tu et al., 2019]. However, these previous works did not study the effects of the temporal dependencies in the sequence data on the SSM generalization (See more details on the comparison in Section 4.1). As an attempt to understand the relationship between the temporal dependencies and the generalization performance, this paper aims to provide a generalization bound that connects the memory structure of the model with the temporal structure of the data. We can, in turn, use the proposed bound to guide us in designing new algorithms to improve optimization and generalization. Specifically, we discover two roles for the proposed generalization measure: (1) generalization bound as an initialization scheme; (2) generalization bound as a regularization method. The common initialization method for the S4 model and its variants follows from the HiPPO framework [Gu et al., 2022a]. $^{1}$To simplify the analysis, we omit the skip connection by letting $ D = 0 $ which is based on the prerequisite that the training sequence data is stable. To improve the robustness of the output value scales on SSMs to different temporal patterns in the sequence data, we consider to rescale the initialization of SSMs with respect to the generalization measure. This new initialization scheme makes the SSMs more resilient on their initial output value scales to variations in the temporal patterns of the training data. Except for the initialization setup, our generalization bound can also be served as a regularizer. Regularization methods like weight decay and dropout are widely applied to training SSMs, but the hidden state matrix $A$ is not regularized because its imaginary part controls the oscillating frequencies of the basis function $e^{At}B$ (Gu et al., 2022b). By taking into account the interaction between the SSM structure and the temporal dependencies, we introduce a new regularization method based on our bound, and it can be applied to the hidden state space to improve the generalization performance. When combining the initialization scheme and the regularization method, our method is applicable to various tasks, ranging from image classification to language processing, while only introducing a minimal computational overhead. To summarize, our contributions are as follows: - We provide a data-dependent generalization bound for SSMs by taking into account the temporal structure. Specifically, the generalization bound correlates with the memory structure of the model and the (auto)covariance process of the data. It indicates that instead of the weight or the data norm, it is the interplay between the memory structure and the temporal structure of the sequence data that influences the generalization. - Based on the proposed generalization bound, we setup an initialization scaling rule by adjusting the magnitude of the model parameters with respect to the generalization measure at initialization. This scaling rule improves the robustness of the initial output value scales on SSMs across different temporal patterns of the sequence data. - Apart from the initialization scheme, we design a new regularizer for SSMs. Unlike weight decay, our regularizer does not penalize the parameter norm but encourages the model to find a minimizer with lower generalization bound to improve the generalization performance. 2 RELATED WORKS Since a SSM is also a continuous linear RNN, there are three lines of research that are related to our work: generalization of RNNs, temporal structure analysis on RNNs, and optimization of SSMs. Generalization of RNNs. Existing works on the generalization of RNNs focus on the generalization error bound analysis. Specifically, in the early two works of Dasgupta & Sontag (1995) and Koiran & Sontag (1998), VC dimension-based generalization bounds were provided to show the learnability of RNNs. In recent studies, Zhang et al. (2018); Chen et al. (2019); Tu et al. (2019) proved norm-based generalization bounds, improving the VC dimension-based bounds by the Rademacher complexity technique (Bartlett & Mendelson 2002) under the uniform-convergence framework. In the overparameterization settings, it was shown in Allen-Zhu & Li (2019) that RNNs can learn some concept class in polynomial time given that the model size is large enough. These generalization bounds, however, do not take into account the temporal dependencies and their effects on generalization. In this work, we provide a new generalization bound by combining the memory structure of the model and the temporal structure of the data. Temporal structure analysis on RNNs. Sequence data has long-range temporal dependencies across the time domain, which notably set it apart from non-sequence data. Recent studies have studied the effects of such temporal dependencies on the approximation and optimization of RNNs. For example, in the two works of Li et al. (2021; 2022), a “curse of memory” phenomenon was discovered when using linear RNNs to model the temporal input-output relationships. Particularly, when the target relationship between the input and output has a long-term memory, then both approximation and optimization become extremely challenging. In Wang et al. (2023), the “curse of memory” phenomenon on approximation and optimization was extended to non-linear RNNs based on the temporal relationships. In this paper, we conduct a fine-grained analysis on the effects of the temporal structure analysis on the generalization of RNNs. Optimization of SSMs. RNN optimization is known for two issues: training stability and computational cost (Bengio et al., 1994; Pascanu et al., 2013). To address these two issues and capture the long dependencies more efficiently in sequence modeling, the S4 model was proposed by in- troducing new parameterization, initialization and discretization (Gu et al., 2022a). Recent variants for the S4 model simplified the hidden state matrix by a diagonal matrix to enhance computational efficiency (Gu et al., 2022b; Gupta et al., 2022; Smith et al., 2023; Orvieto et al., 2023). Regularization methods are also applied for SSMs to prevent overfitting, such as dropout, weight decay and the data continuity regularizer (Qu et al., 2023). However, the principled way to regularize and initialize the parameters still remains to be explored. In this study, we design a new regularization and initialization scheme to improve both optimization and generalization. 3 PRELIMINARIES In this section, we briefly introduce the SSM in Section 3.1 and the motivation for optimization designs based on the generalization analysis in Section 3.2. 3.1 INTRODUCTION TO SSMs In this paper, we consider the following single-input single-output SSM, \[ h'(t) = Ah(t) + Bx(t), \quad y(t) = Ch(t), \quad t \geq 0 \] where $ x $ is the input from an input space $ \mathcal{X} := C_0(\mathbb{R}_{\geq 0}, \mathbb{R}) $; $ y(t) \in \mathbb{R} $ is the output at time $ t $; $ h(t) \in \mathbb{R}^m $ is the hidden state with $ h(0) = 0 $; $ A \in \mathbb{R}^{m \times m}, B \in \mathbb{R}^{m \times 1}, C \in \mathbb{R}^{1 \times m} $ are trainable parameters. Then (1) has an explicit solution $ y(t) = \int_0^t \rho_\theta(s)x(t-s)ds $, where $ \rho_\theta(s) := Ce^{As}B $ with $ \theta = (C, A, B) $. The function $ \rho_\theta(s) $ captures the memory structure of the model and the temporal input-output relationship (Li et al., 2022). For the S4 model and its variants (Gu et al., 2022a,b; Gupta et al., 2022; Gu et al., 2023), (1) is usually discretized by the Zero-Order Hold method, i.e., given a timescale $ \Delta \in \mathbb{R}_+ $, $ h_{k+1} = Ah_k + Bx_k, \quad y_k = Ch_k, \quad k = 0, 1, \ldots $, where $ \tilde{A} = e^{\Delta A}, \tilde{B} = (\tilde{A} - I_m)\tilde{A}^{-1}B, \tilde{C} = C $. Then, $ y_k = \tilde{C}\tilde{A}^kBx_0 + \tilde{C}\tilde{A}^{k-1}Bx_1 + \ldots + \tilde{C}\tilde{B}x_k = [\tilde{K} * x]_k $ where $ \tilde{K} = (\tilde{CB}, \tilde{CAB}, \ldots, \tilde{CA}^kB) $ and $ * $ represents convolution. 3.2 MOTIVATION: A LINEAR REGRESSION MODEL In this subsection, we use a linear regression model on non-sequential data as an example to illustrate the connection between the generalization analysis and the optimization designs. This example then motivates us to extend the connection to SSMs on sequential data. Linear regression. We consider a simple linear model $ y = \theta^\top x $ with input $ x \in \mathbb{R}^d $, output $ y \in \mathbb{R} $ and parameter $ \theta \in \mathbb{R}^d $. Let the training data $ \{(x_i, y_i)\}_{i=1}^n $ be i.i.d. sampled from a distribution $ D $ such that $ \\|x_i\\|_2 = r, \\|y_i\\| \leq 1 \forall i \in [1:n] $. Define the empirical risk $ L_n(\theta) := \frac{1}{n} \sum_{i=1}^n (\theta^\top x_i - y_i)^2 $ and the population risk $ L_D(\theta) := \mathbb{E}_{x,y}[(\theta^\top x - y)^2] $. Then given a norm-constrained space $ \Theta := \{\theta \in \mathbb{R}^d : \\|\theta\\|_2 \leq R\} $, \[ \sup_{\theta \in \Theta} \|L_n(\theta) - L_D(\theta)\| \leq (rR + 1)^2 \cdot O(\sqrt{\log(1/\delta)/n}). \] This is a well-known norm-based generalization bound based on the Rademacher theory (Mohri et al., 2012), and we provide a proof in Appendix B for completeness. Notice that the key term $ r^2R^2 $ in the generalization bound (2) is also an upper bound for the magnitude of the linear model output, i.e., $ \sup_{\theta \in \Theta} (\theta^\top x_i)^2 \leq r^2R^2 $. Thus, we connect the model stability with the generalization bound stability, and this connection induces an initialization scheme for the initialization $ \theta^{(0)} $ by setting $ \\|\theta^{(0)}\\|_2 \sim O(1/r) $. In particular, if we normalize each input $ x_i $ such that $ r $ is also $ O(1) $, then $ \\|\theta^{(0)}\\|_2 \sim O(1) $. Since $ \theta^{(0)} \in \mathbb{R}^d $, one possible initialization scheme is that $ \theta^{(0)} $ follows a Uniform distribution $ U[-1/\sqrt{d}, 1/\sqrt{d}] $, which corresponds to the Kaiming initialization (up to some constant) (He et al., 2015). When treating the term $ r^2R^2 $ as a regularizer to improve the generalization, we get the weight decay method, i.e., the $ \ell_2 $ regularization w.r.t. $ \\|\theta\\|_2^2 $. We summarize the above logic chain that connects the generalization analysis with optimization designs in Figure 1. Now for SSMs, we extend the generalization analysis from non-sequential data to sequential data by taking into account the temporal structure of the data. This linear regression example motivates us to apply the same logic diagram (Figure 1) to the SSMs, and this is exactly what we are going to present in the following part of this paper. --- 2A linear space of continuous functions from $ \mathbb{R}_{\geq 0} $ to $ \mathbb{R} $ that vanishes at infinity. 4 MAIN RESULTS In this section, we first give a generalization bound for SSMs in Section 4.1, then we design a new initialization scheme in Section 4.2 based on this proposed bound. Apart from the initialization scheme, we introduce a new regularization method in Section 4.3. Finally, we conduct experiments to test the initialization scheme and the regularization method in Section 4.4. 4.1 A GENERALIZATION BOUND OF SSMs In this section, we present a generalization bound for the SSM (1) and reveal the effects of the temporal dependencies on the generalization performance. We show that our bound gives a tighter estimate compared with previous norm-based bounds through a toy example. Following the same notation in Section 3.1, we define the empirical risk $R_n(\theta)$ and the population risk $R_x(\theta)$ as $$R_n(\theta) := \frac{1}{n} \sum_{i=1}^{n} \left\| \int_0^T \rho_\theta(T-s)x_i(s)ds - y_i \right\|^2,$$ $$R_x(\theta) := \mathbb{E}_x \left[ \int_0^T \rho_\theta(T-s)x(s)ds - y \right]^2,$$ where $T > 0$ is some finite terminal time, the training sequence data $\{x_i(t)\}_{i=1}^{n}$ are independently sampled from a stochastic process with mean $\mathbb{E}[x(t)] := \mu(t)$ and covariance $\mathbb{E}[(x(s)-\mu(s))(x(t)-\mu(t))] := K(s,t)$, and the label $y$ is generated by some underlying functional $H_T : X \rightarrow \mathbb{R}$, i.e., $y = H_T(x)$. We assume that $\|y\| \leq 1$ for any $x \in X$, otherwise, we truncate the value of the label to 1. In the next, we make an assumption on the normalized process $\tilde{x}(t) := (x(t) - \mu(t))/\sqrt{K(t,t)}$: Assumption 1. The normalized process $\tilde{x}(t)$ is (1): almost surely Hölder continuous, i.e., $\exists L,H > 0, s.t.\forall s,t \in [0,T], \|\tilde{x}(s) - \tilde{x}(t)\| \leq L\|s-t\|^H a.s.;$ (2): is $\sigma^2$-sub-Gaussian for every $t \in [0,T]$, i.e., $\exists \sigma > 0, s.t.\forall u > 0, P(\|\tilde{x}(t)\| \geq u) \leq 2\exp(-u^2/2\sigma^2)$ for any $t \in [0,T]$. We leave the discussion of the assumption after the statement of the main theorem. Now we proceed to bound generalization gap $\|R_x(\theta) - R_n(\theta)\|$ by establishing uniform convergence of the empirical risk to its corresponding population risk, as stated in following theorem: Theorem 1. For a SSM $\int_0^T \rho_\theta(T-s)x(s)ds$, following notations and settings in Section 3.1 & 4.1 we define $\psi(\Theta) := \sup_{\theta \in \Theta} \int_0^T \|\rho_\theta(T-s)\|\sqrt{K(s,s)}ds + \sup_{\theta \in \Theta} \int_0^T \rho_\theta(T-s)\mu(s)ds$. Then under Assumption 1, given a parameter space $\Theta$ for $\theta$, for any $\delta \in (0,1)$, with probability at least $1-\delta$ over the training sequences, $$\sup_{\theta \in \Theta} \|R_x(\theta) - R_n(\theta)\| \leq (\psi(\Theta) + 1)^2 \cdot O(\log^{3/2}(Tn/\delta)/\sqrt{n}).$$ Where $O$ hides a constant that depends on $\sigma,L,H$. The proof is given in Appendix E. We see that this bound decreases to zero as the sample size $n \rightarrow \infty$, provided that the terminal time $T$ is finite and the supremum term in (3) is bounded. Theorem 1 captures the temporal dependencies of the sequence data on the SSM generalization, yielding that the mean and variance at each length position together play important roles in generalization analysis. Specifically, as long as $\psi(\Theta)$ is small, the generalization gap is small. Since the function $\rho_\theta(s)$ is exponentially decay, we do not require the mean and variance to be uniformly small along the time $t$ to get a small generalization gap. Proof sketch. The proof is based on Rademacher theory (Bartlett & Mendelson, 2002). The main difficulty is to bound the Rademacher complexity of the SSM function $\int_0^T \rho_\theta(T-s)x(s)ds$ for a stochastic process $x(s)$. We first use the Hölder inequality to get an upper bound for the Rademacher complexity w.r.t. the normalized process $\tilde{x}(s)$, then combining Hölder continuity and the heavy-tail property in Assumption 1, we show the finiteness of $\sup_{s \in [0,T]} \tilde{x}(s)$. Finally we use an $\varepsilon$-net argument to give an explicit bound for the Rademacher complexity, which then finishes the proof. Discussions of Assumption 1. This assumption contains two parts. Hölder continuity is used to bound $ \sup_{s \in [0,T]} \tilde{x}(s) $ and the Rademacher complexity of the SSM function class. By the Kolmogorov continuity theorem (Stroock & Varadhan, 1997), Hölder continuity covers a wide range of random processes that satisfy certain inequalities for its moments. For the sub-Gaussian property, it ensures $ \tilde{x}(s) $ is bounded in a finite time set with high probability. Sub-Gaussian random variables include Gaussian and any bounded variables. Specifically, for image classification tasks with flattened image pixels, if the range of the pixel values is a finite class, then the Hölder continuity condition can be dropped. We leave more detailed discussions and provide some concrete examples that satisfy Assumption 1 in Appendix C. Comparison to previous bounds. Since a SSM is also a continuous linear RNN model, we compare (3) with previous bounds for linear RNNs. In Chen et al. (2019), a generalization bound $ O(\\|x\\|_2 \\|B\\|_2 \\|C\\|_2 \\|A\\|_2 / \sqrt{n}) $ is provided, where $ \\|x\\|_2 $ is the 2-norm of the discrete input sequence. In the continuous case, $ \\|x\\|_2 $ corresponds to the $ L^2 $ norm w.r.t. a Dirac measure. By changing the matrix 2-norm to matrix 1-norm, Tu et al. (2019) shows another similar generalization bound. These bounds separate the data complexity and the model complexity by the data norm and the model parameter norm individually, and do not account for the temporal dependencies across the time domain. In this work, instead, we incorporate the temporal dependencies via the sequence statistics (mean and variance) to get a generalization bound. Next, we use a toy example to illustrate that our bound gives a tighter estimation. Given a stochastic process $ \{x(t)\}_{t \in [0,T]} $ with mean $ \mu(t) $ and covariance $ K(s,t) $, we consider the following two upscale transformations (by increasing $ T $ to $ 2T $): 1. left zero padding: $ x_1(t) = 0, \ t \in [0,T); \quad x_1(t) = x(t-T), \ t \in [T,2T] $ 2. right zero padding: $ x_2(t) = x(t), \ t \in [0,T]; \quad x_2(t) = 0, \ t \in (T,2T] $ Then the two SSM outputs are given by $ y_i(2T) = \int_0^{2T} \rho_\theta(2T-s)x_i(s)ds $ for $ i = 1,2 $. Hence, \[ y_1(2T) = C \int_0^T e^{A(T-s)} B x(s) ds, \quad y_2(2T) = Ce^{AT} \int_0^T e^{A(T-s)} B x(s) ds. \] We see that the magnitude of $ y_1(2T) $ and $ y_2(2T) $ differs with an exponential factor $ e^{AT} $. Since all the eigenvalues of $ A $ have negative real part, $ y_2(2T) \to 0 $ as $ T $ increases. Hence, the right zero padding transformation degenerates the SSM function class to a zero function class for large $ T $, inducing a minimal generalization gap that only contains the statistical sampling error (see (3) by letting $ K(s,s) = \mu(s) = 0 $). Therefore, a desired generalization bound should reflect such a difference caused by the different temporal dependencies. However, previous norm-based generalization bounds do not capture such a difference for these two transformations as they produce the same $ L^2 $ norm for the input sequence. Let us see what happens for our proposed generalization measure. For the left zero padding, the key term in (5) becomes \[ \int_0^T \left\| Ce^{A(T-s)} B \right\| \sqrt{K(s,s)} ds + \left\| \int_0^T Ce^{A(T-s)} B \mu(s) ds \right\| + 1 \] For the right zero padding, the key term in (5) becomes \[ \int_0^T \left\| Ce^{AT} e^{A(T-s)} B \right\| \sqrt{K(s,s)} ds + \left\| \int_0^T Ce^{AT} e^{A(T-s)} B \mu(s) ds \right\| + 1 \] The detailed derivations are given in Appendix D. By the same argument, our bound (3) indeed captures the difference on the magnitude of the generalization performance for these two sequence transformations. In particular, as $ T \to \infty $, (6) reduces to 1, which yields a minimal generalization gap as expected for the zero function class. In that sense, we get a tighter bound for the SSMs. Zero shot transferability. A benefit of SSMs is the zero-shot transfer to other sampling frequencies (i.e., the timescale measure in continuous case). For example, for a SSM function $ y_T = \int_0^T \rho_\theta(T-s)x(s)ds $, if we downscale the input sequence $ x(s) $ by half of the sampling frequency, then the SSM output becomes $ y_T = \int_0^{T/2} \rho_\theta(T-2s)x(2s)ds $, which equals to $ \int_0^T \frac{1}{2} \rho_\theta(T-s)x(s)ds $. Now for a new SSM parameter $ \theta = (2C,A,B) $, we have $ \rho_\theta(s) = 2\rho_\theta(s) $, indicating that by simply modifying the SSM parameters, one can transfer the model to half the sampling frequency while keeping the output invariant. One advantage for our generalization measure is that it is also zero shot transferable. To see this, we use the same example here. Under the downscale sampling, both $ \int_0^T \|\rho_\theta(T-s)\| \sqrt{K(s,s)} ds $ and $ \left\| \int_0^T \rho_\theta(T-s)\mu(s)ds \right\| $ remain invariant for the new parameter $ \tilde{\theta} $ because $ \sqrt{K(s,s)} $ and $ \mu(s) $ have the same scaling as $ x(s) $. Similarly, other sampling frequencies are also zero shot transferable for our generalization measure by simply adjusting the SSM parameters. 4.2 Generalization bound as an initialization scheme In this section, we design a scaling rule for the SSM parameters at initialization based on the generalization bound (3). This new initialization scheme improves the robustness of the initial output value scales on SSMs across different temporal patterns of the sequence data. Our proposed initialization scheme is built on the HiPPO based initialization [Gu et al., 2023], which is a data independent initialization method. Specifically, the HiPPO framework initializes the hidden state matrices $ A, B $ to produce orthogonal basis functions, and the matrix $ C $ to be standard normal for training stability. However, the argument for the training stability relies on the prerequisite that the input sequence is constant along the length ([Gu et al., 2023, Corollary 3.4]), which is restricted as the long-range dependencies may lead to very different temporal patterns on the input sequence. As the dashed lines in the left and the right part of Figure 2 show, the SSM output value scale and the loss value scale under the HiPPO based initialization vary much across different temporal dependencies, making the loss values inconsistent during training. To address this issue, we follow the logic diagram in Figure 1 by adjusting the generalization complexity to be $ O(1) $. Specifically, we extract the dominant term in the generalization bound (3): \[ \tau(\theta) := \left( \int_0^T \|\rho_\theta(T-s)\| \sqrt{K(s,s)} ds + \left\| \int_0^T \rho_\theta(T-s)\mu(s)ds \right\| \right)^2. \] Notice that $ \rho_\theta(s) = Ce^{As}B $, if we rescale $ C $ to $ \xi C $ for some $ \xi \in \mathbb{R} $, we have $ \tau(\tilde{\theta}) = \xi^2 \cdot \tau(\theta) $ for $ \tilde{\theta} = (\xi C, A, B) $. This induces a new initialization scheme, i.e., once the parameters $ \theta = (C, A, B) $ are initialized by the HiPPO method, we rescale $ C $ to $ \tilde{C} $ such that \[ \tilde{C} = \frac{1}{\sqrt{\tau(\theta)}} \cdot C = \frac{1}{\int_0^T \|\rho_\theta(T-s)\| \sqrt{K(s,s)} ds + \left\| \int_0^T \rho_\theta(T-s)\mu(s)ds \right\|} \cdot C. \] This rescaling method guarantees the SSM output value to be bounded at initialization for any stochastic process that satisfies Assumption 1, ensuring the robustness of the initial loss value scales on SSMs across different temporal dependencies. We formalize the statement in Proposition 1. Proposition 1. Consider a SSM $ \int_0^T \rho_\theta(T-s)x(s)ds $ with $ \theta = (C, A, B) $, for any stochastic process $ x(s) $ that satisfies Assumption 1, let $ \tilde{C} $ given by the rescaling method (8), then for $ \tilde{\theta} := (\tilde{C}, A, B) $, we have $ \mathbb{E}_x \left[ \int_0^T \rho_{\tilde{\theta}}(T-s)x(s)ds \right] \leq O(\sqrt{\log T}) $. The proof is provided in Appendix F. Proposition 1 shows that the SSM output values are uniformly bounded over all the stochastic processes that satisfy Assumption 1 even when the input sequence is not almost surely bounded. This improves the robustness of the output value scales on SSMs in the sense that the scale of the output value does not depend on the variations of the temporal structures. It is worth noting that different from the data normalization methods such as min-max normalization and standardization, our rescaling method only changes the model parameters. This is important because normalization on the data numerical values in language tasks can lead to loss of crucial information. For example, mathematical expressions like "$ \max(1,9) = 9 $" have a contextual meaning where normalizing could result in the loss of structured information essential to understand. Implementation. In the practical training, the SSMs used for tasks such as image classification or language processing are usually deep and high dimensional ($ d > 1 $), while our initialization scheme (8) is designed based on the one-dimensional shallow SSM. To extend to high-dimensional SSMs, we empirically treat all features to be independent and calculate $ \tau(\theta) $ by its average along the feature dimension. For a $ k $-layer SSM with the initial matrix $ C_1, \ldots, C_k $ at each layer, we first calculate the complexity measure $ \tau_1(\theta) $ for the first layer and rescale $ C_1 $ by $ C_1/\sqrt{\tau_1(\theta)} $. Then we calculate the complexity measure $\tau_2(\theta)$ for the second layer for the updated input sequence of layer 2 and rescale $C_2$ by $C_2/\sqrt{\tau_2(\theta)}$. We repeat this process until the last layer. We describe the complete procedures for one-layer SSMs in Algorithm 1, where the $\|\cdot\|$ and $\sqrt{\cdot}$ in Line 5 represent to element-wise absolute value and element-wise square root respectively. $[\cdot]_L$ extracts the last position of an element obtained from the convolution. The $\text{Mean}(\cdot)$ operation in Line 6 calculates the mean value of a vector. Algorithm 1 Training one-layer SSMs with the initialization scheme (8) Input: Training sequences $x_1, \ldots, x_n \in \mathbb{R}^{L \times d}$ with length $L$ and dimension $d$, a SSM initialization $\theta_0 = (C, A, B)$, a SSM kernel function $k(\theta) \in \mathbb{R}^{L \times d}$, number of epochs $s$ 1: for $i = 0$ to $s - 1$ do 2: if $i = 0$ then 3: Sample a minibatch sequence $x = (x^{(1)}, \ldots, x^{(B)}) \in \mathbb{R}^{B \times L \times d}$ 4: Compute the mean $\mu \in \mathbb{R}^{L \times d}$ and variance $K \in \mathbb{R}^{L \times d}$ for $x$ along the batch dimension 5: Compute $\tau(\theta_i)$ via convolution: $\tau(\theta_i) \leftarrow \left[\|k(\theta_i)\| * \sqrt{K} + \|k(\theta_i) * \mu\|\right]_L \in \mathbb{R}^d$ 6: Average over the feature dimension: $\tau(\theta_i) \leftarrow \text{Mean}^2(\tau(\theta_i))$ 7: Rescale by the initialization scheme (8): $\hat{C} \leftarrow C/\sqrt{\tau(\theta_i)}$ 8: Start to train with the updated initialization $(\hat{C}, A, B)$ 9: end if 10: Regular training procedure 11: end for Output: Updated model parameter $\theta_s$ ### 4.3 Generalization Bound as a Regularization Method In addition to its role as an initialization scheme, the generalization measure can also be regarded as a regularizer. In this section, we utilize the bound (5) to design a regularization method to improve the generalization performance, and simultaneously bring a little extra computational cost. For the generalization bound (3), we consider to use the dominant term (for large $T$) $\tau(\theta)$ defined in (7) as a regularizer. Then, the new empirical risk with regularization is given by $$\tilde{R}_n(\theta) := R_n(\theta) + \lambda \cdot \tau(\theta),$$ where $\lambda \geq 0$ is the regularization coefficient. When training multi-layer SSMs, we calculate the complexity $\tau(\theta)$ in (9) at each layer and add them together as a total regularization. We describe the training procedures for one-layer SSMs in Algorithm 2, where the notations follow Algorithm 1. Algorithm 2 Training one-layer SSMs with the regularization method (9) Input: Training sequences $x_1, \ldots, x_n \in \mathbb{R}^{L \times d}$ with length $L$ and dimension $d$, a SSM initialization $\theta_0$, a SSM kernel function $k(\theta) \in \mathbb{R}^{L \times d}$, loss function $\tilde{R}(\cdot, \cdot) : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$, regularization coefficient $\lambda$, optimizer $\text{OPT}$, number of epochs $s$ 1: for $i = 0$ to $s - 1$ do 2: Sample a minibatch input $x = (x^{(1)}, \ldots, x^{(B)}) \in \mathbb{R}^{B \times L \times d}$ with labels $(y^{(1)}, \ldots, y^{(B)})$ 3: Calculate the mean $\mu \in \mathbb{R}^{L \times d}$ and variance $K \in \mathbb{R}^{L \times d}$ for $x$ along the batch dimension 4: Compute the SSM output via convolution: $y \leftarrow [k(\theta_i) * x]_L \in \mathbb{R}^{B \times d}$ 5: Compute the regularization via convolution: $\tau(\theta_i) \leftarrow \left[\|k(\theta_i)\| * \sqrt{K} + \|k(\theta_i) * \mu\|\right]_L \in \mathbb{R}^d$ 6: Average over the feature dimension: $\tau(\theta_i) \leftarrow \text{Mean}^2(\tau(\theta_i))$ 7: Compute the total loss $L \leftarrow \frac{1}{B} \sum_{i=1}^{B} \tilde{R}(y_i, y^{(i)}) + \lambda \cdot \tau(\theta_i)$ 8: Parameters update: $\theta_{i+1} \leftarrow \text{OPT}(\theta_i, L)$ 9: end for Output: Updated model parameter $\theta_s$ ### Computational Cost Analysis From the training procedures in Algorithm 2, we can see that the newly introduced training complexity mainly comes from the calculation for the convolution between the SSM kernel and the sequence statistics ($\mu, K$). Since the convolution can be conducted by the fast Fourier transform (Gu et al., 2022a) with complexity $O(BdL \log L)$, then the new complexity for Algorithm 2 becomes $O((B + 2)dL \log L)$, which is acceptable in the practical training. Figure 2: Effects of the initialization scheme (8) on the model output, the gradient norm and the optimization under different temporal dependencies. (Left) The output $\mathbb{E}_x[\|y_L\|]$ at initialization w.r.t. the Gaussian white noise sequence $(x_1, \ldots, x_L)$ for length $L$ from 1 to 1000; (Middle) The gradient norm $\|\nabla R_n(\theta)\|$ at initialization w.r.t. the mean squared error (MSE) for varied sequence length; (Right) The training MSE curve for the Gaussian white noise with length $L = 1000$. Table 1: Training and test loss on the Gaussian white noise sequences with different coefficients $b$ after convergence. By adding the initialization scheme (8), SSMs achieve better optimization performance and are more robust on the final training loss value across different temporal dependencies. By adding the regularization term (9), SSMs get better generalization performance. 4.4 Experiments This section contains experiments to demonstrate the effectiveness of the proposed initialization scheme (8) and the regularization method (9). We use a synthetic sequence dataset and the Long Range Arena (LRA) benchmark (Tay et al., 2021) for numerical validations. To simplify the notation, we use w/o (8, 9), w (8), w (9) and w (8, 9) to represent the original base model, model trained with rescaling, model trained with regularization and model trained with both methods respectively. A synthetic dataset. We consider a synthetic sequence dataset generated by a centered Gaussian white noise with the covariance function $K(s, t) = \frac{1}{\|b\|\sqrt{\pi}} e^{-((s-t)/b)^2}$, which is a stationary Gaussian process and satisfies Assumption I (ref Section C). Then we can get different temporal dependencies by varying the coefficient $b$, i.e., as the magnitude of $b$ decreasing, the temporal dependence of the corresponding Gaussian white noise decreases as well. In particular, as $b \to 0$, $\frac{1}{\|b\|\sqrt{\pi}} e^{-(x/b)^2}$ becomes a delta function $\delta(x)$, entailing a zero temporal dependence for the sequence data. In the following experiment, we generate the sequence data by the Gaussian white noise with $b = [1, 0.1, 0.01]$. For each input sequence $(x_1, \ldots, x_L)$, its corresponding label is obtained by $\sin(x_{L/2})$, i.e., the sine value of the time-lagged input. We use the unidirectional S4-Legs model (Gu et al., 2022a) (that only contains the convolution layer) to train the sequence data. More details about the experiment setup are provided in Appendix A.1. In Figure 2, we plot the model output $\mathbb{E}_x[\|y_L\|]$ and the gradient norm $\|\nabla R_n(\theta)\|$ at initialization, and the training loss (w (8)) with different temporal patterns by varying the Gaussian white noise parameter $b$. We see that the initialization scheme (8) enhances the robustness of the output value scales (matches with Proposition 1), gradient norm at initialization and also the training loss value across different temporal structures. By comparing the final training loss with and without (8) in Table 1 (w/o (8, 9) vs w (8) and w (9) vs w (8, 9)), we see that adding the rescaling (8) also improves the training performance and makes the final training error more robust on different temporal dependencies (by varying $b$). For the regularization method (9), we compare the final test loss with and without (9) in Table 1 (w/o (8, 9) vs w (9) and w (8) vs w (8, 9)). We can see that our regularization method improves the generalization performance. Moreover, combining (8) and (9), the model get the best test performance across various temporal structures of the sequence data. \| \| LastOps \| Text \| Retrieval \| Image \| Pathfinder \| PathX \| \|------------------\|---------\|------\|-----------\|-------\|------------\|-------\| \| unidirectional S4-Legs \| \| \| \| \| \| \| \| w/o \| 59.45 \| 79.27\| 88.28 \| 87.39 \| 87.84 \| \| \| w \| 60.30 \| 81.44\| 89.38 \| 88.11 \| 87.95 \| \| \| w (9) \| 60.65\| 81.45\| 89.21\| 87.79\| 90.36\| \| \| w (8, 9) \| 60.40 \| 82.56\| 90.13\| 88.28\| 90.03 \| \| \| Time / epoch, w \| 2min 57s\| 2min 4s\| 2min 5min\| 2min 5min\| 2min 2min\| 2min 2min\| \| Time / epoch, w (9) \| 2min 35s\| 1min 45s\| 2min 15s\| 2min 15s\| 2min 15s\| 2min 15s\| \| bidirectional S4-Legs \| \| \| \| \| \| \| \| w/o \| 62.45 \| 89.09\| 91.21 \| 89.32 \| 95.75 \| 89.17±1.71 \| \| w \| 61.90 \| 88.90\| 91.44 \| 89.52 \| 95.43 \| 89.67±0.18 \| \| w (9) \| 61.09 \| 89.27\| 91.32\| 89.93\| 95.80\| 90.21±0.16 \| \| w (8, 9) \| 61.79 \| 89.19\| 91.46\| 89.80\| 95.86\| 89.85±0.72 \| \| Time / epoch, w \| 2min 46s\| 3min 06s\| 18min 20s\| 2min 12s\| 4min 06s\| 4min 06s\| \| Time / epoch, w (9) \| 2min 18s\| 3min 34s\| 20min 30s\| 2min 46s\| 4min 20s\| 4min 40s\| \| bidirectional S4D-Legs \| \| \| \| \| \| \| \| w/o \| 57.80 \| 83.91\| 90.84 \| 86.47 \| 87.26 \| 90.19±0.78 \| \| w \| 57.25 \| 84.79\| 91.01 \| 86.34 \| 88.35 \| 90.25±0.15 \| \| w (9) \| 57.50 \| 84.52\| 91.08\| 87.33\| 87.21\| 90.19±0.34 \| \| w (8, 9) \| 58.45\| 85.75\| 91.03\| 87.28\| 88.36\| 89.40±1.21 \| \| Time / epoch, w \| 2min 19s\| 2min 19s\| 18min 15s\| 1min 50s\| 1min 50s\| 1min 50s\| \| Time / epoch, w (9) \| 2min \| 2min 35s\| 22min 36s\| 1min 50s\| 1min 50s\| 1min 11s\| Table 2: Test accuracy and running time (per epoch in A100 GPU) on the LRA benchmark under different settings for different models. The unidirectional model processes a sequence in one direction while the bidirectional model consists of two separate layers that process the sequence in opposite directions. Mean and standard error for the PathX results are reported based on 3 independent runs. LRA benchmark. We investigate the effects of the initialization scheme (8) and the regularization method (9) on the LRA benchmark. We consider three base models: unidirectional S4-Legs (Gu et al., 2022a), bidirectional S4-Legs (Goel et al., 2022) and bidirectional S4D-Legs (Gu et al., 2022b). Among these three models, the unidirectional S4-Legs is the one that is closest to our model setting (1), however, it performs poorly in challenge datasets. Thus, we do not use the unidirectional S4-Legs to train the PathX. We follow the training rules as described by Gu et al. (2023), but with adjustments to the model size. For example, the model sizes used to train the PathX for both S4-Legs and S4D-Legs are relatively small compared with the ones used in Gu et al. (2023) to save training time. More details on the dataset description and the experiment setup are given in Appendix A.2. By comparing the test accuracy for w/o (8, 9) vs w (9) and w (8) vs w (8, 9) in Table 2, we see that adding the regularization (9) enhances the generalization performance in most cases for all three models. In particular, when combining the initialization scheme (8) and the regularization (9), one gets the best test performance in half of tasks, indicating that our proposed optimization designs effectively improve the generalization performance. We also compare the running time without or with the proposed optimization designs. Since (8) is conducted before training which will not introduce additional training complexity, we report the running time for w/o (8, 9) and w (9) in Table 2. The results show that the regularization brings a little extra computational cost, matching the computational cost analysis in Section 4.3. We include an ablation study for the hyperparameter λ and add more experiment results in Appendix A.2. 5 DISCUSSION In this work, we study the optimization and the generalization for SSMs. Specifically, we give a data-dependent generalization bound, revealing an effect of the temporal dependencies of the sequence data on the generalization. Based on the bound, we design two algorithms: a new initialization scheme and a regularization method, to improve the optimization and generalization for SSMs. There are still some gaps between the theory and the methodologies in this paper. The first one is that the skip connection matrix $D$ is omitted in our defined model (1). This will not affect our generalization bound because we may express the explicit solution for (1) as $y(t) = \int_0^t (\rho_\theta(s) + D\delta(s))x(t-s)ds$ where $\delta(\cdot)$ is a delta function, which is still a convolution model with a new kernel $\rho_\theta(s)+D\delta(s)$. However, the initialization scheme (8) only adjusts $C$ and requires the kernel function to be linear in $C$. Hence, (8) may not work well when $Dx(t)$ is much larger than $\int_0^t \rho_\theta(s)x(t-s)ds$. The second gap is that our theory is for single-layer linear SSMs. When nonlinearities are added, our generalization bound still works for single-layer SSMs if the nonlinearity does not affect the Hölder condition and the sub-Gaussian property (Assumption 1). For Lipschitz (also Hölder continuous) functions, there are some known examples (see Appendix G) where the sub-Gaussian condition remains after the nonlinearity. The extension of our theory to the multi-layer case is an interesting direction, which we leave for future work. Reproducibility The generalization bound (2) for linear regression is proved in Appendix B. The proof for Theorem 1 is provided in Appendix E. The derivations for (5) and (6) in Section 4.1 are given in Appendix D. The proof for Proposition 1 is in Appendix F. The details for the experiment settings are shown in Appendix A.1 and Appendix A.2. REFERENCES Zeyuan Allen-Zhu and Yuanzhi Li. Can sgd learn recurrent neural networks with provable generalization? Advances in Neural Information Processing Systems, 32, 2019. Ehsan Azmoodeh, Tommi Sottinen, Lauri Viitasaari, and Adil Yazigi. Necessary and sufficient conditions for hölder continuity of gaussian processes. Statistics & Probability Letters, 94:230–235, 2014. Peter L Bartlett and Shahar Mendelson. Rademacher and gaussian complexities: Risk bounds and structural results. 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Y0wAim2F8A	Does the average performance shown in Figure 5 of RMA and the values reported in table 2 correspond to RMA trained with 2 million steps worth of data? Was the adaptation module/extrinsics estimator trained using the same data?	PRIVILEGEDDREAMER: EXPLICIT IMAGINATION OF PRIVILEGED INFORMATION FOR ADAPTATION IN UNCERTAIN ENVIRONMENTS Anonymous authors Paper under double-blind review ABSTRACT Numerous real-world control problems involve dynamics and objectives affected by unobservable hidden parameters, ranging from autonomous driving to robotic manipulation. To represent these kinds of domains, we use Hidden-parameter Markov Decision Processes (HIP-MDPs), which model sequential decision problems where hidden variables parameterize transition and reward functions. Existing approaches, such as domain randomization, domain adaptation, and meta-learning, simply treat the effect of hidden parameters as additional variance and often struggle to effectively handle HIP-MDP problems, especially when rewards are parameterized by hidden variables. To address this, we introduce Privileged-Dreamer, a model-based reinforcement learning framework that extends Dreamer, a powerful world-modeling approach, by incorporating an explicit parameter estimation module. We introduce a novel dual recurrent architecture that explicitly estimates hidden parameters from limited historical data and enables us to condition the model, actor, and critic networks on these estimated parameters. Our empirical analysis on five diverse HIP-MDP tasks demonstrates that it outperforms state-of-the-art model-based, model-free, and domain adaptation learning algorithms. Furthermore, we also conduct ablation studies to justify our design decisions. 1 INTRODUCTION The Markov Decision Process (MDP) has been a powerful mathematical framework for modeling a spectrum of sequential decision scenarios, from computer games to intricate autonomous driving systems; however, they often assume fixed transition or reward functions. In many real-world domains, there exists a family of related problems characterized by the presence of hidden or uncertain parameters that play a significant role in their dynamics or reward functions, which is referred to as a hidden-parameter MDP (HIP-MDP) (Doshi-Velez & Konidaris, 2016). For instance, autonomous driving must deal with diverse vehicles with distinctive dynamic attributes and properties for better driving experience, while the agricultural industry sorts produce that varies in weight. Consequently, research endeavors have explored diverse algorithmic approaches, including domain randomization (Tobin et al., 2017), domain adaptation (Peng et al., 2020), and meta-learning (Wang et al.), to address these challenges effectively. We approach these HIP-MDP problems using model-based reinforcement learning because a world model holds significant promise in efficiently capturing these dynamic behaviors characterized by hidden parameters, ultimately resulting in improved policy learning. Particularly, we establish our framework based on Dreamer (Hafner et al., 2019), which has been effective in solving multiple classes of problems, including DM control suite (Tassa et al., 2018), Atari (Hafner et al., 2020), and robotic control (Wu et al., 2022). Our initial hypothesis was that the Dreamer framework may be able to capture parameterized dynamics accurately by conditioning the model on latent variables, leading to better performance at the end of learning. However, Dreamer is designed to predict action-conditioned dynamics in the observation space and does not consider the effect of hidden parameters. This paper presents PrivilegedDreamer to solve HIP-MDPs via explicit prediction of hidden parameters. Our key intuition is that a recurrent state space model (RSSM) of model-based RL must be explicitly conditioned on hidden parameters to capture the subtle changes in dynamics or rewards. However, a HIP-MDP assumes that hidden variables are not available to agents. Therefore, we introduce an explicit module to estimate hidden parameters from a history of state variables via a Long short-term memory (LSTM) network, which can be effectively trained by minimizing an additional reconstruction loss. This dual recurrent architecture allows accurate estimation of hidden parameters from a short amount of history. The estimated hidden parameters are also fed into the transition model, actor, and critic networks to encourage their adaptive behaviors conditioned on hidden parameters. We evaluate our method in five HIP-MDP environments, where two of them have parameter-conditioned reward functions. We compare our method against several state-of-the-art baselines, including model-based (DreamerV2 [Haftner et al., 2020]), model-free (Soft Actor Critic [Haarnoja et al., 2018]) and Proximal Policy Optimization (Schulman et al., 2017), and domain adaptation (Rapid Motor Adaptation [Kumar et al., 2021]) algorithms. Our PrivilegedDreamer achieves 41% higher average rewards over five tasks, particularly on HIP-MDPs with parameterized reward functions. We further analyze the behaviors of the learned policies to investigate how rapid estimation of hidden parameters affects the final performance and also to justify the design decisions of the framework. Finally, we outline a few interesting future research directions. 2 RELATED WORK World Models Model-based RL improves sample efficiency over model-free RL by learning an approximate model for the transition dynamics of the environment, allowing for policy training without interacting with the environment itself. However, obtaining accurate world models is not straightforward because the learned model can easily accumulate errors exponentially over time. To alleviate this issue, Chua et al. (2018) designs ensembles of stochastic dynamics models to attempt to incorporate uncertainty. The Dreamer architecture (Haftner et al., 2019, 2020, 2023) models the environment using the recurrent state space machine, which also includes the recurrent GRU network (Cho et al., 2014) and the VAE (Kingma & Welling, 2013), via reconstructing the input from a latent space. With this generative world model, the policy is trained with imagined trajectories in this learned latent space. Robine et al. (2023) and Micheli et al. (2022) leverage the Transformer architecture (Vaswani et al., 2017) to autoregressively model the world dynamics and similarly train the policy in latent imagination. Our work is built on top of the Dreamer architecture, but the idea of explicit modeling of hidden parameters has the potential to be combined with other architectures. Randomized Approaches without Explicit Modeling One of the most popular approaches to deal with uncertain or parameterized dynamics is domain randomization (DR), which aims to improve the robustness of the policy by exposing the agent to randomized environments. It has been effective in many applications, including manipulation (Peng et al., 2018; Tobin et al., 2017; Zhang et al., 2016; James et al., 2017), locomotion (Peng et al., 2020; Tan et al., 2018), autonomous driving (Tremblay et al., 2018), and indoor drone flying (Sadeghi & Levine). Domain randomization has also shown great success in deploying trained policies on actual robots, as in Tan et al. (2018), which used it for sim-to-real transfer for a quadrupedal robot, and Peng et al. (2018), which used it to improve performance for a robotic manipulator. While DR works very well in many situations, it tends to lead to an overly conservative policy that is suboptimal for challenging problems with a wide range of transition or reward functions. Domain Adaptation Another common strategy for dealing with variable environments is to incorporate the hidden environmental parameters into the policy for adaptation. This privileged information of the hidden parameters can be exploited during training, but at test time, system identification must occur online. For model-free RL, researchers typically train a universal policy conditioned on hidden parameters and estimate them at testing time by identifying directly from a history of observations (Yu et al., 2017; Kumar et al., 2021; Nahrendra et al., 2023). Another option is to improve state estimation while training in diverse environments, which similarly allows for adaptation without needing to perform explicit system identification (Ji et al., 2022). For model-based RL, the problem of handling variable physics conditions is handled in multiple ways. A few research groups Nagabandi et al. (2018); Sæmundsson et al. (2018) propose using meta-learning to rapidly adapt to environmental changes online. Wang & van Hoof (2021) uses a graph-based meta RL technique to handle changing dynamics. Ball et al. (2021) used data augmentation in offline RL to get zero-shot dynamics generalization. The most applicable methods for our work are the problems that use a learned encoder to estimate a context vector that attempts to capture the environmental information and is used to condition the policy and for forward prediction, as in Wang et al. (2022); Lee et al. (2020); Huang et al. (2021); Seo et al. (2020). 3 PRIVILEGED DREAMER: ADAPTATION VIA EXPLICIT IMAGINATION 3.1 BACKGROUND Hidden-parameter MDP A Markov decision process (MDP) formalizes a sequential decision problem, which is defined as a tuple $(S, A, T, R, p_0)$, where $S$ is the state space, $A$ is the action space, $T$ is the transition function, $R$ is the reward function, and $p_0$ is the initial state distribution. For our work, we consider the hidden-parameter MDP (HIP-MDP), which generalizes the MDP by conditioning the transition function $T$ and/or the reward function $R$ on an additional hidden latent variable $\omega$ sampled from a distribution $p_\omega$ (Doshi-Velez & Konidaris, 2016). Without losing generality, $\omega$ can be a scalar or a vector. In the setting of continuous control, which is the primary focus of this work, this latent variable represents physical quantities, such as mass or friction, that govern the dynamics but are not observable in the state space. Dreamer For our model, we build upon the DreamerV2 model of Hafner et al. (2020). DreamerV2 uses a recurrent state space model (RSSM) to model dynamics and rewards. This RSSM takes as input the state $x_t$ and the action $a_t$ to compute a deterministic recurrent state $h_t = f_\phi(h_{t-1}, z_{t-1}, a_{t-1})$ using a GRU $f_\phi$ and a sampled stochastic state $z_t \sim q_\phi(z_t\|h_t, x_t)$ using an encoder $q_\phi$. The combination of these deterministic and stochastic states is used as a representation to reconstruct the state $\hat{x}_t \sim p_\phi(\hat{x}_t\|h_t, z_t)$, and to also predict the reward $\hat{r}_t \sim p_\phi(\hat{r}_t\|h_t, z_t)$ and the discount factor $\hat{\gamma}_t \sim p_\phi(\hat{\gamma}_t\|h_t, z_t)$. The final component of the RSSM is the transition predictor $\hat{z}_t \sim p_\phi(\hat{z}_t\|h_t)$. This computes the stochastic state $z_t$ using only the deterministic state $h_t$, which is necessary for training in imagination where the state $x_t$ is not available. For policy learning, Dreamer adopts an actor-critic network, which is trained via imagined rollouts. For each imagination step $t$, the latent variable $\hat{z}_t$ is predicted using only the world model, the action is sampled from the stochastic actor: $a_t \sim \pi_\theta(a_t\|\hat{z}_t)$, and the value function is estimated as: $v_\psi \approx \mathbb{E}_{p_\phi, p_\theta}[\sum \gamma^{\tau-t}\hat{r}_\tau]$, where $\hat{r}_\tau$ is computed from the reward predictor above. The actor is trained to maximize predicted discounted rewards over a fixed time horizon $H$. The critic aims to accurately predict the value from a given latent state. The actor and critic losses are: \[ \text{Actor loss: } L = \mathbb{E}_{p_\phi, p_\theta} \left[ \sum_{t=1}^{H-1} - \ln \pi_\theta(\hat{a}_t\|\hat{z}_t) sg(V^\lambda_t - v_\psi(\hat{z}_t)) - \eta H[a_t\|\hat{z}_t] \right] \] \[ \text{Critic loss: } L = \mathbb{E}_{p_\phi, p_\theta} \left[ \sum_{t=1}^{H-1} \frac{1}{2}(v_\psi(\hat{z}_t) - sg(V^\lambda_t))^2 \right] \] 3.2 ALGORITHM While the original DreamerV2 layout works effectively for many tasks, in the HIP-MDP domain, it falters, especially in the case where the reward explicitly depends on the hidden latent variable. Even though the RSSM has memory to determine the underlying dynamics, prior works such as Seo et al. (2020) have shown that this hidden state information is poorly captured implicitly and must be explicitly learned. Explicit Imagination via LSTM To help remedy this, we incorporate an additional independent module for estimating the privileged information from the available state information. This dual recurrent architecture allows us to effectively estimate the important hidden parameters in the first layer and model other variables conditioned on this estimation in the second layer. Our estimation module $\tilde{\omega}_t \sim \eta_\phi(\tilde{\omega}_t\|x_t, a_{t-1})$ takes the state $x_t$ and previous action $a_{t-1}$ as inputs and predicts the intermediate hidden parameter $\hat{\omega}_t$. It is still parameterized by $\phi$ because we treat it as part of the world model. The estimation module is comprised of an LSTM (Hochreiter & Schmidhuber) followed by MLP layers to reshape the output to that of the privileged data. We use an LSTM because its recurrent architecture is more suitable to model subtle and non-linear relationships between state and hidden variables over time. However, the choice of the architecture was not significant to the performance. In our experience, LSTM and GRU demonstrated similar performance. Note that we use $\hat{\omega}_t$ to make the recurrent world model conditioned on the estimated hidden variable. For the actor and critic, we feed the value from the prediction head, $\hat{\omega}_t$ which will be described in the next paragraph. Additional Prediction Head We also added an additional prediction head $p_\phi(\hat{\omega}_t\|h_t, z_t)$, which is similar to the reward or state prediction heads. While the previous LSTM estimation $\eta$ predicts the intermediate parameter $\hat{\omega}_t$ to make the model conditioned on the hidden parameter, this additional prediction head offers two major improvements: 1) encouraging the RSSM state variables $h_t$ and $z_t$ to contain enough information about the hidden parameter and 2) improving the prediction accuracy. Hidden Variable Loss We design an additional loss to train the estimation module, which is similar to the other losses of the DreamerV2 architecture. We do not use the discount predictor from the original DreamerV2 architecture as all of our tests are done in environments with no early termination. We group the other Dreamer losses all under $L_{Dreamer}$ to highlight our differences. This makes the total loss for the world model: $$L(\phi) = L_{Dreamer} + \mathbb{E}_{q_\phi(z_1:T\|a_1:T, x_1:T, \omega_1:T)} \left[ \sum_{t=1}^{T} - \ln \eta_\phi(\hat{\omega}_t\|x_t, a_{t-1}) - \ln p_\phi(\hat{\omega}_t\|h_t, z_t) \right].$$ where the first loss is to compute an intermediate estimate $\hat{\omega}$ for the hidden parameter $\omega$ using the environment states $x$ and actions $a$ and the second term is the world model reconstruction loss for $\hat{\omega}$ based on the RSSM latent variables $h$ and $z$. It is important to highlight that relying solely on this hidden parameter loss term is not sufficient. Theoretically, it seems like the loss encourages the recurrent state variables $h_t$ and $z_t$ to encapsulate all relevant information and increase all the model, actor, and critic networks’ awareness of hidden parameters. However, in practice, this privileged information remains somewhat indirect to those networks. Consequently, this indirect access hinders their ability to capture subtle changes and results in suboptimal performance. Hidden parameter conditioned Networks (ConditionedNet) Once we obtain the estimated hidden parameter $\omega_t$, we feed this information to the networks. This idea of explicit connection has been suggested in different works in reinforcement learning, such as rapid motor adaptation (RMA) (Kumar et al., 2021) or meta strategy optimization (MSO) (Yu et al., 2020). Similarly, we augment the inputs of the representation model $z_t \sim q_\phi(z_t\|h_t, x_t, \hat{\omega}_t)$, the critic network $v_\psi$, and the actor network $\pi_\theta$ to encourage them to incorporate the estimated $\omega_t$ and $\hat{\omega}_t$. Additional Proprioceptive State as Inputs In our experience, it is beneficial to provide the estimated state information as additional inputs to the actor and critic networks. We hypothesize that this may be because the most recent state information $x_t$ is highly relevant for our continuous control tasks. On the other hand, the RSSM states $h_t$ and $z_t$ are indirect and more suitable for establishing long-term plans. Summary On top of DreamerV2, Our PrivilegedDreamer includes the following components: - Recurrent hidden parameter predictor: $\hat{\omega}_t \sim \eta_\phi(\hat{\omega}_t\|h_t, z_t)$ - HIP-conditioned representation model: $z_t \sim q_\phi(z_t\|h_t, x_t, \hat{\omega}_t)$ - HIP prediction head: $\hat{\omega}_t \sim p_\phi(\hat{\omega}_t\|h_t, z_t)$ - HIP-conditioned critic: $v_t \sim v_\psi(v_t\|h_t, z_t, x_t, \hat{\omega}_t)$ - HIP-conditioned actor: $\hat{a}_t \sim \pi_\theta(a_t\|h_t, z_t, x_t, \hat{\omega}_t)$ Figure 1: Architecture of the PrivilegedDreamer. Compared to the default DreamerV2 model (top), our architecture (bottom) adopts an explicit parameter estimation model $\eta$ to predict the hidden parameters $\omega_t$ from a history of states. Then, the estimated parameters $\hat{\omega}_t$ are fed into the model to establish the explicit dependency. We omit the unchanged components from DreamerV2, such as input and reward predictors, for brevity. A schematic of the model architecture used for training the world model itself can be seen in Figure 1. This setup trains the encoder network, decoder network, and the latent feature components $z$ and $h$. The estimation module $\eta$ that initially estimates the value of $\hat{\omega}_t$ is also trained here. For training the policy network in imagination, we use the structure in Figure 2. When training the policy, we start with a seed state sampled from the replay buffer and then proceed in imagination only, as in the original DreamerV2. Via this setup, the actor and critic networks are trained to maximize the estimated discounted sum of rewards in imagination using a fixed world model. However, the key difference is that both the actor and critic networks take the estimated parameter $\hat{\omega}_t$ from the prediction head as an additional input, as well as the reconstructed state $\hat{x}_t$. Because the entire model can learn the parameter estimation much faster than the world model, this new connection works almost the same as providing the ground-truth hidden parameter for the majority of the learning time. We will discuss this behavior in the discussion section. 4 EXPERIMENTS We evaluate PrivilegedDreamer on several HIP-MDP problems to answer the following research questions: 1. Can our PrivilegedDreamer solve HIP-MDP problems more effectively than the baseline RL and domain adaptation algorithms? 2. Can the estimation network accurately find ground-truth hidden parameters? 3. What are the impacts of the HIP reconstruction loss and hip-conditioned policy? \| Task \| Physics Randomization Target \| Range \| Reward \| \|-----------------\|---------------------------------------\|---------------\|--------------\| \| Walker Run \| Contact Friction \| [0.05 - 4.5] \| Fixed \| \| Pendulum Swingup\| Mass Scaling Factor of Pendulum \| [0.1 - 2.0] \| Fixed \| \| Throwing \| Mass Scaling Factor of Ball \| [0.2 - 1.0] \| Fixed \| \| Sorting \| Mass Scaling Factor of Arm \| [0.2 - 1.0] \| Parameterized\| \| Pointmass \| X/Y Motor Scaling Factor \| X [1 - 2] \| Parameterized\| \| \| \| Y [1 - 2] \| \| Table 1: Parameter randomization applied for each task. ![Figure 3](image.png) Figure 3: Five HIP-MDP tasks used in our experiments. ### 4.1 HIP-MDP Tasks We evaluate our model on a variety of continuous control tasks from the DeepMind Control Suite (Tassa et al., 2018), along with some tasks developed in MuJoCo (Todorov et al., 2012). All tasks involve operating in a continuous control environment with varying physics. The tasks are as follows: - DMC Walker Run - Make the Walker run as fast as possible in 2D, where the contact friction is variable. - DMC Pendulum Swingup - Swing a pendulum to an upright position, where the pendulum mass is variable. - Throwing - Control a paddle to throw a ball into a goal, where the ball mass is variable. - Sorting - Move an object to a desired location, where the object mass is variable and the target location depends on the mass: heavier objects to the left and lighter objects to the right. - DMC Pointmass - Move the point mass to the target location, where the x and y motors are randomly scaled. The target location depends on the motor scaling: away from the center for high motor scaling and towards the center for lower motor scaling. When we design these tasks, we start by simply introducing randomization to the existing two tasks, DMC Walker Run and DMC Pendulum Swingup. Then, we purposely design the last two tasks, Sorting and DMC Pointmass, to incorporate a reward function that depends on their hidden parameters. Throwing also implicitly necessitates a policy for identifying the ball’s mass and adjusting its trajectory. However, its reward function is not explicitly parameterized. All the environments are visualized in Figure 3 and their randomization ranges are summarized in Table 1. A full description of all the environments used is in Section A in the appendix. ### 4.2 Baseline Algorithms The baseline algorithms that we compare against are as follows: - DreamerV2: original DreamerV2 model proposed by Hafner et al. (2020). Table 2: Model performance after 2 million timesteps of training \| Method \| Walker \| Pendulum \| Throwing \| Sorting \| Pointmass \| Mean \| \|-------------------------\|--------------\|--------------\|--------------\|--------------\|--------------\|--------------\| \| PrivilegedDreamer \| 766.20 ± 20.19 \| 563.14 ± 147.44 \| 788.59 ± 45.66 \| 554.68 ± 26.25 \| 670.23 ± 13.93 \| 668.56 ± 70.87 \| \| Dreamer + Decoder + ConditionedNet \| 576.89 ± 96.68 \| 329.80 ± 37.10 \| 785.78 ± 64.18 \| 180.85 ± 46.55 \| 492.77 ± 17.82 \| 473.22 ± 58.87 \| \| Dreamer + Decoder \| 671.85 ± 10.46 \| 259.84 ± 26.06 \| 707.51 ± 20.63 \| 87.74 ± 43.24 \| 480.96 ± 29.91 \| 441.58 ± 28.21 \| \| DreamerV2 (Hafner et al., 2020) \| 715.57 ± 39.95 \| 289.43 ± 214.12 \| 706.09 ± 26.24 \| 167.61 ± 33.38 \| 488.41 ± 3.60 \| 473.42 ± 99.26 \| \| SAC (Haarnoja et al., 2018) \| 475.22 ± 13.02 \| 454.67 ± 268.98 \| 945.65 ± 17.02 \| 74.85 ± 88.03 \| 393.49 ± 210.47 \| 468.78 ± 158.03 \| \| PPO (Schulman et al., 2017) \| 79.73 ± 10.95 \| 470.04 ± 324.05 \| 707.03 ± 115.63 \| 229.93 ± 181.12 \| 545.86 ± 72.22 \| 406.52 ± 176.93 \| \| RMA (Kumar et al., 2021) \| 75.28 ± 11.31 \| 516.83 ± 386.43 \| 624.57 ± 118.70 \| 82.33 ± 416.57 \| 545.31 ± 357.86 \| 368.86 ± 305.00 \| Figure 4: Learning curves for all tasks. PrivilegedDreamer shows the best performance against all the baseline algorithms, except for the throwing task that requires a very long horizon prediction. - Proximal Policy Optimization (PPO): model-free, on-policy learning algorithm proposed by Schulman et al. (2017) using the implementation from Raffin et al. (2021). - Soft Actor Critic (SAC): model-free, off-policy learning algorithm proposed by Haarnoja et al. (2018) using the implementation from Yarats & Kostrikov (2020). - Rapid Motor Adaptation (RMA): model-free domain adaptation algorithm proposed by Kumar et al. (2021), which estimates hidden parameters from a history of states and actions. We train an expert PPO policy with $\omega$ as input and compare to the student RMA policy, which is trained with supervised learning to match $\omega$ using a history of previous states. We select our baseline to cover all the state-of-the-art in model-based/model-free, on-policy/off-policy, domain randomization/adaptation algorithms. All models were trained for 2 million timesteps in each environment randomized as specified in Table 1. To validate our design choices, we further evaluate the following intermediate versions of the algorithm. - Dreamer + Decoder: This version only trains a decoder $\hat{\omega}_t \sim p_\phi(\hat{\omega}_t\|h_t, z_t)$ by minimizing the hidden variable loss without an estimation module $\eta$. Also, $\hat{\omega}_t$ is not provided to the actor and critic and $h_t$ and $z_t$ are expected to contain all the information about the hidden parameter $\omega_t$. - Dreamer + Decoder + ConditionedNet: This version is similar to the previous Dream + Decoder, but the estimated $\hat{\omega}_t$ is given to the actor and critic networks. Note that the proposed PrivilegedDreamer can be viewed as the combination of Dreamer, an external estimation module, and conditioned networks trained with the hidden variable loss (PrivilegedDreamer = Dreamer + ExternalEstimation + Decoder + ConditionedNet). Figure 5: Hidden parameter reconstruction error during learning. Figure 6: Online parameter estimation within an episode. The two estimated values for the Pointmass model are shown in separate plots to improve readability. 4.3 Evaluation Performance To evaluate the effectiveness of the proposed method, we first compare the learning curves and the final performance of all the learned models. Learning curves for all models are shown in Figure 4, where the means and standard deviations are computed over three random seeds. Since RMA is trained in a supervised fashion using an expert policy and is not trained using on-policy environment interactions, we do not have a comparable learning curve, so we display the average performance as a horizontal line for comparison. Table 2 shows the average reward over 100 runs for each seed. We also report the average performance over five tasks in both Figure 4 and Table 2. Overall, the proposed PrivilegedDreamer achieves the best average reward over five tasks. It shows a significant performance improvement over the second best model, vanilla DreamerV2, in both standard DM Control Suite tasks (Walker, Pointmass, Pendulum) as well as tasks we created ourselves (Sorting, Throwing). Performance margins are generally larger in the Sorting and DMC Pointmass tasks, where PrivilegedDreamer is the only model tested that does appreciably better than random. This is likely because the reward for these tasks explicitly depends on $\omega$ and DreamerV2 only implicitly adapts its behaviors to the hidden parameters. This indicates that a novel architecture of PrivilegedDreamer is effective for solving HIP-MDPs, particularly when the reward function is parameterized. We suspect that RMA and PPO do especially poorly on the Walker task because the 2 million timestep training limit is insufficient for on-policy algorithms. Similarly, we suspect that the small training size affects the ability of RMA to effectively adapt, and that it would be more competitive with our method with a larger training dataset, which our method does not need due to its better sample efficiency. One notable outlier is the great performance of SAC on the Throwing task. We suspect that the nature of the problem makes it difficult for model-based RL algorithms, both PrivilegedDreamer and DreamerV2. In this task, a policy only has a few steps to estimate its hidden parameters and predict the ball’s trajectory, which can easily accumulate model errors over a long time horizon. On the other hand, SAC, a model-free RL algorithm, efficiently modifies its behaviors in a model-free fashion without estimating a ball trajectory. On-policy algorithms, PPO and RMA, are not sample-efficient enough to achieve good performance within two million steps. Hidden Parameter Estimation PrivilegedDreamer is based on the assumption that estimating hidden parameters is crucial for solving HIP-MDPs. Figure 5 illustrates the reconstruction errors during the learning process for the Pendulum, Throwing, and Pointmass tasks. In all cases, our PrivilegedDreamer exhibits faster convergence, typically within less than 0.5 million environmental steps, resulting in more consistent learning curves. Additionally, Figure 6 displays the real-time estimation of hidden parameters during episodes. Our model accurately predicts these parameters within just a few steps, enhancing the performance of the final policies. These findings justify the effectiveness of an external LSTM-based hidden parameter estimation module. ### 4.4 Ablation Studies Comparing our full PrivilegedDreamer model to the ablations, we see that our model is superior and each component is necessary for optimal performance. From Figure 5, we see that our full model is significantly better at reconstructing the hidden variable $\omega$ than Dreamer + Decoder + ConditionedNet, which is already better than Dreamer + Decoder. With this low reconstruction error, online estimation of $\omega$ is very effective, as shown in Figure 6, which shows that our method rapidly converges within 5% of the real value, while the ablated versions take longer to converge to a lower quality estimate. Specifically, our agents find near-correct hidden parameters at the beginning of the episodes within a few environmental steps in all scenarios, while the other baselines take more than 500 steps (Dreamer+Decoder+ConditionedNet in Pointmass) or converge to wrong values (Dreamer+Decoder in Pendulum and Pointmass). Using this high quality estimate of $\omega$ within our ConditionedNet, Figure 4 and Table 2 demonstrate that our method greatly outperforms the ablations. This validates our hypothesis that incorporating a good estimate of $\omega$ into the world model and policy networks improves the performance of a RL policy operating in an environment with variable $\omega$. ## 5 Conclusion This paper presents a novel architecture for solving problems where the dynamics are dictated by hidden parameters. We model these problems with the Hidden parameter Markov Decision Process (HIP-MDP) and solve them using model-based reinforcement learning. We introduce a new model PrivilegedDreamer, based on the DreamerV2 world model, that handles the HIP-MDP problem via explicit prediction of these hidden variables. Our key invention consists of an external recurrent module to estimate these hidden variables to provide them as inputs to the world model itself. We evaluate our model on five HIP-MDP tasks, including both DeepMind Control Suite tasks and tasks we manually created where the reward explicitly depends on the hidden parameter, and found our model significantly outperforms the DreamerV2 model, as well as the other baselines we tested against. Our research opens up several intriguing agendas for future investigation. Firstly, this paper has concentrated our efforts on studying hidden parameter estimation within proprioceptive control problems, intentionally deferring the exploration of visual control problems like Atari games or vision-based robot control for future works. We believe that the same principle of explicitly modeling hidden parameters can be effectively applied to these visual control challenges with minor adjustments to the neural network architectures. Furthermore, we plan to investigate more complex robotic control problems, such as legged locomotion (Wu et al., 2022), where real-world dynamics may be too sensitive to be precisely replicated by any of the hidden parameters. In such cases, we anticipate the need to devise better approximation methods. Lastly, we plan to delve into multi-agent scenarios in which these hidden parameters have an impact on the AI behavior of other agents. These subsequent research directions promise to extend the scope and impact of the original paper. REFERENCES Philip J. Ball, Cong Lu, Jack Parker-Holder, and Stephen Roberts. Augmented world models facilitate zero-shot dynamics generalization from a single offline environment. 4 2021. URL http://arxiv.org/abs/2104.05632. Kyunghyun Cho, Bart van Merriënboer, Dzmitry Bahdanau, and Yoshua Bengio. On the properties of neural machine translation: Encoder-decoder approaches. 9 2014. URL http://arxiv.org/abs/1409.1259. 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K1mcPiDdOJ	In Table 3 you have an interesting result. In terms of RMSE, it can be observed that mean imputation (one of the most naive approaches) outperforms any baseline, even state-of-the-art methods. What do you think about this result? As observed in [2], sometimes error metrics are inconclusive when evaluating temporal scenarios. Did you find this behaviour in other datasets? Did you think about the possibility of instead of using missing data randomly or following other mechanisms (which you withdrew from GPVAE and other papers I believe) using missing data in burts or sequences, where actually it is the standard scenario you would find in a temporal scenario such as in healthcare? This could be an interesting point to be analyzed here: check whether the solutions from baselines and proposed methods are more correlated.	Conditional Information Bottleneck Approach for Time Series Imputation MinGyu Choi Massachusetts Institute of Technology, USA chemgyu@mit.edu Changhee Lee Chung-Ang University, Korea changheelee@cau.ac.kr Abstract Time series imputation presents a significant challenge because it requires capturing the underlying temporal dynamics from partially observed time series data. Among the recent successes of imputation methods based on generative models, the information bottleneck (IB) framework offers a well-suited theoretical foundation for multiple imputations, allowing us to account for the uncertainty associated with the imputed values. However, direct application of IB framework to time series data without considering their temporal context can lead to a substantial loss of temporal dependencies, which, in turn, can degrade the overall imputation performance. To address such a challenge, we propose a novel conditional information bottleneck (CIB) approach for time series imputation, which aims to mitigate the potentially negative consequences of the regularization constraint by focusing on reducing the redundant information conditioned on the temporal context. We provide a theoretical analysis of its effect by adapting variational decomposition. We use the resulting insight and propose a novel deep learning method that can approximately achieve the proposed CIB objective for time series imputation as a combination of evidence lower bound and novel temporal kernel-enhanced contrastive optimization. Our experiments, conducted on multiple real-world datasets, consistently demonstrate that our method significantly improves imputation performance (including both interpolation and extrapolation), and also enhances prediction performance based on the imputed values. 1 Introduction Multivariate time series data often includes missing features, with diverse missing ratios and patterns depending on distinct sampling periods or measurement strategies (Johnson et al., 2016). Since these missing features can significantly impair the performance of downstream tasks and comprehension of the temporal dynamics, time series imputation, which aims to reconstruct the missing features, has become a pivotal and pervasive topic across numerous practical domains, including healthcare, environmental science, and various other fields. What makes time series imputation challenging is that an imputation method must satisfy two essential requirements: i) it must account for underlying temporal dependencies, and ii) it should allow for multiple imputations to facilitate uncertainty quantification for real-world decision-making. Generative models, particularly variational autoencoders (VAEs) (Kingma & Welling, 2014), have been employed in the context of multiple imputation tasks due to their capability to generate samples in a probabilistic manner. VAE-based imputation methods primarily focus on defining the evidence lower bound, where the reconstruction error is computed only over the observed part of the incomplete data (Sohn et al., 2015; Nazabal et al., 2020). These methods can be naturally interpreted under the information bottleneck (IB) principle (Tishby & Zaslavsky, 2015), providing an information-theoretic understanding of what constitutes an imputation-relevant representation. This understanding is based on the fundamental trade-off between maintaining a concise representation (i.e., regularization) and preserving good representation power (i.e., reconstruction) (Voloshynovskiy et al., 2019). However, a direct application of the IB principle to time series imputation struggles with capturing the underlying temporal dependencies. Our motivating examples in Figure 1(B) show that imputation methods trained with the conventional IB framework lose a significant amount of information about temporal dynamics relevant for imputing missing values. In this paper, we theoretically analyze that... the overly strict regularization in the conventional IB may force the encoder to rely solely on the observed features at a particular time point, rather than learning the underlying temporal dependencies from the remaining observations from other time steps. To overcome such an issue, we propose a novel conditional information bottleneck (CIB) framework for time series imputation. Our framework adopts the reconstruction-regularization structure of the IB principle while preserving temporal information through conditional regularization, allowing us to circumvent the strict regularization constraints of the conventional IB. Throughout the experiments conducted on multiple real-world datasets including image sequences, weather measurements, and electrical health records, our proposed method consistently outperforms the state-of-the-art imputation methods with respect to both imputation performance and prediction performance based on the imputed values. 2 Preliminaries: Information Bottleneck Approach to Imputation In this section, we first formally describe the information bottleneck (IB) principle (Tishby & Zaslavsky, 2015; Alemi et al., 2017), which provides an information-theoretic understanding of what a task-relevant representation is in terms of the fundamental trade-off between having a concise representation and good representative power. Then, we present a generative model for imputing missing features under the IB principle. Let $ X $ and $ Y $ be random variables for the input feature and the target label, respectively. The IB principle aims to find the bottleneck random variable $ Z $ that compresses the information in $ X $ while keeping the information relevant for predicting $ Y $ as the following (Tishby & Zaslavsky, 2015), \[ \min_{\phi, \theta} I_\phi(Z; X) - \beta I_\theta(Y; Z) \] (1) where $ \beta \in \mathbb{R} $ is a Lagrangian multiplier that balances the two mutual information terms, and $ \phi $ and $ \theta $ correspond to learnable parameters that define probabilistic mappings $ q_\phi(Z\|X) $ and $ q_\theta(Y\|Z) $, respectively. The core motivation of (1) is to find the optimal distribution of latent representation $ Z $ and the corresponding inference model parameters $ \phi $ that removes label-irrelevant information from $ X $ while preserving the information about the class label $ Y $. This offers an information-theoretic perspective on generative model-based imputation methods which generate missing observations from the observed features. Definition 1. (Imputation) Let $ X^o $ and $ X^m $ be random variables for the partially observed features and missing features of $ X $, respectively, such that $ X = X^o \cup X^m $. Then, we define imputation as an unsupervised IB as follows: \[ \min_{\phi, \theta} I_\phi(Z; X^o) - \beta I_\theta(X; Z) \] (2) where $ \beta \in \mathbb{R}_+ $ is a Lagrangian multiplier, and $ \phi $ and $ \theta $ correspond to learnable parameters that define probabilistic mappings $ q_\phi(Z\|X^o) $ and $ q_\theta(X\|Z) $, respectively. The above definition in (2) aims at finding the distribution of latent representation $ Z $ and the corresponding parameters $ \phi $ that preserves the core information for accurately reconstructing the (complete) original input $ X $ while suppressing redundant information from its incomplete observation, $ X^o $. 3 Method 3.1 Problem Formulation We consider a general temporal dynamics setting in which each instance of a (discrete) time series input comprises a sequence of measurements (i.e., observations), denoted as $ x_{1:T} \equiv [x_1, \ldots, x_T] $, collected during the time interval $[0, \tau_T)$. Here, $ x_t \in \mathbb{R}^d $ is the complete input vector measured at time $ t \in [\tau_{t-1}, \tau_t) $ and is a realization of a random variable $ X_t $.$^1$ However, in practice, time series data often contains missing features with arbitrary patterns such that $ x^l_t $ is not observed during $[\tau_{t-1}, \tau_t)$ for any feature $ l \in \{1, \ldots, d\} $ at any time step $ t \in \{1, \ldots, T\} $. This phenomenon is particularly common in domains such as healthcare (Johnson et al., 2016) where each feature may $^1$Throughout the paper, we will often use upper-case letters to represent random variables and lower-case letters to represent their corresponding realizations. Please refer to Appendix E for a notation table. Figure 1: (A) Conceptual illustration of the IB and CIB principles. By conditioning regularization on the remaining input time steps, the latent representation can better preserve the underlying temporal dependency. (B) Motivating experimental results on interpolation (left) and extrapolation (right). Because features in a single time step are completely missing, a model must collect information from other time steps. The conventional IB approach (HI-VAE) shows deteriorating performance in both cases. Another IB approach (GP-VAE) using a Gaussian process prior demonstrates enhanced performance for interpolation but often significantly loses time series characteristics for extrapolation (i.e., the writing style is corrupted). The CIB approach (Ours) exhibits improved imputation performance for both cases. Complete quantitative results are available in Table I. have a distinct sampling period or when non-uniform sampling strategies are employed. To denote missing observations, we partition the input vector $ x_t $ at each time step into observed features $ x^o_t $ and missing features $ x^m_t $, such that $ x_t = x^o_t \cup x^m_t $. Objective. Our aim is to reconstruct the complete time series input $ x_{1:T} $ by filling in the missing features from the observed time series input $ x^o_{1:T} $. Formally, we seek to generate $ x^m_t $ from the conditional distribution $ p(X^m_t \| X^o_{1:T}) $. By modeling the conditional distribution $ p(X^m_t \| X^o_{1:T}) $ instead of using a deterministic mapping, we can generate multiple imputations, allowing us to capture the uncertainty associated with the imputed values. What makes this problem challenging is that we must account for the underlying temporal dynamics represented by $ x^o_{1:T} $ when imputing missing features $ x^m_t $ for $ t \in \{1, \ldots, T\} $. We can straightforwardly apply the unsupervised IB described in (2) to obtain latent representations $ Z_t $ by discarding information from the observed time series input $ X^o_{1:T} $ that is redundant for reconstructing $ X_t $. Formally, this can be achieved by minimizing $ I_\phi(Z_t; X^o_{1:T}) - \beta I_\theta(X_t; Z_t) $ with a comprehensive encoder (e.g., RNN or Transformer) capable of effectively modeling the temporal dependencies within the observed time series observations, i.e., $ q_\phi(Z_t \| X^o_{1:T}) $. However, enforcing such strict regularization constraints on the encoder may lead to a significant loss of information regarding the temporal context that can be achieved by observations at different time steps, which we denote as $ X^o_{\backslash t} \overset{\text{def}}{=} \{X^o_\tau : \tau \in \{1, \ldots, T\} \setminus t\} $. This may cause the imputation of $ X^m_t $ at time step $ t $ to heavily rely on the observed features at that particular time point, i.e., $ X^o_t $, rather than being able to learn from temporal dependencies present in other observations, i.e., $ X^o_{\backslash t} $ (as shown in Figure 1(B)). To tackle this issue, we alleviate the potentially negative consequences of the regularization constraint by directing our attention to the redundant information of the observed input at time step $ t $ when it is conditioned on its temporal context represented by the remaining observed time series $ X^o_{\backslash t} $. This offers a novel information-theoretic rationale for time series imputation, as defined below: Definition 2. (Time Series Imputation) Let $ X^o_t $ and $ X^m_t $ be random variables for the partially observed features and missing features of $ X_t $ at time step $ t $. Then, given the observed time series input $ X^o_{1:T} $, we define time series imputation at time step $ t $ as an unsupervised CIB as follows: \[ \min_{\phi, \theta} \underbrace{I_\phi(Z_t; X^o_t \| X^o_{\backslash t})}_{\text{Conditional Regularization}} - \beta I_\theta(X_t; Z_t) \tag{3} \] where $ X^o_{\backslash t} $ represents the random variables for the remaining input observations, excluding $ X^o_t $. By conditioning on $ X^o_{\backslash t} $, (3) guides us to find latent representations $ Z_t $ and the corresponding inference model parameter $ \phi $ which encompass all retrievable information from the entire observed input time series $ X^o_{1:T} $ (reconstruction), while discarding information that is redundant for capturing $ X^m_t $ given the available temporal context from the remaining observed time series $ X^o_{\backslash t} $ (conditional regularization). Overall, the proposed objective in (3) enables us to more effectively utilize information... from $ X_{o \setminus t} $ for imputing $ X_t^o $ compared to other IB-related alternatives, whose conceptual illustration can be seen in Figure 1(A). ### 3.2 Deep Variational Conditional Information Bottleneck on Time Series In this subsection, we will transform our objective (3) into a learnable form by utilizing variational decomposition. (3) is represented as a combination of the traditional ELBO with mutual information along the time axis, which can be approximately achieved by minimizing the contrastive loss. #### 3.2.1 Maximizing Reconstruction: $ \min_{\phi, \theta} -I(X_t; Z_t) $ Following the derivations introduced in (Voloshynovskiy et al., 2019), we can find a lower bound of the reconstruction term as the following: \[ I_\theta(x_t; Z_t) = H(x_t) + D_{KL}(p(x_t\|z_t)\|\|p_\theta(x_t\|z_t)) + E_{x_{1:T} \sim p_{data}} \left[ E_{z_t \sim q_\phi(z_t\|x_{1:T})} [\log p_\theta(x_t\|z_t)] \right] \] \[ \geq E_{x_{1:T} \sim p_{data}} \left[ E_{z_t \sim q_\phi(z_t\|x_{1:T})} [\log p_\theta(x_t\|z_t)] \right] \overset{\text{def}}{=} -L_{\phi, \theta} \] (4) where $ H(\cdot) $ is the entropy and the last inequality holds due to the non-negativity of entropy and KL-divergence. Here, we introduce a feature estimator, denoted as $ p_\theta(X_t\|Z_t) $, as a variational approximation of $ p(X_t\|Z_t) $. We model the feature estimator as an isotropic Gaussian, i.e., $ p_\theta(X_t\|Z_t) = N(\mu_\theta(Z_t), \text{diag}(\sigma_\theta(Z_t))) $ where $ \mu_\theta(\cdot) $ and $ \sigma_\theta(\cdot) $ are implemented by neural networks parameterized by $ \theta $. In many practical scenarios, the ground-truth values for missing features are unknown during training. Thus, to accurately learn the reconstruction process given the latent representation of the observed time series, we apply (4) only to the features observed at each time point, similar to the approach in (Nazabal et al., 2020). Discussion on the Conditional Reconstruction. One might question why the reconstruction term is not conditioned on $ X_{o \setminus t} $, as given by an alternative form of the CIB, i.e., $ \min_{\phi, \theta} I_\phi(Z_t; X_t^o\|X_{o \setminus t}) - \beta I(X_t; Z_t\|X_{o \setminus t}) $. Applying the chain rule of mutual information$^2$ decomposes the conditional reconstruction as the follows: $ I(X_t; Z_t\|X_{o \setminus t}) = I(X_t; Z_t; X_{o \setminus t}) - I(X_t; X_{o \setminus t}) $. It turns out that the first term can be bounded by a mathematically equivalent expression as shown in (4), suggesting that this term encourages mitigating constraints on temporal context for reconstruction (see Appendix A.1). However, minimizing the second term attempts to eliminate information about the target $ X_t $ at time point $ t $, which can be achieved from the observation other than time point $ t $, i.e., $ X_{\setminus t} $. This contradicts the goal of time series imputation where we aim to capture temporal context from the remaining observed time steps. Our empirical results also support that minimizing $ I(X_t; X_{o \setminus t}) $ deteriorates the model performance (see Appendix A.2 for derivation, B.2 for experimental results).$^3$ #### 3.2.2 Minimizing Conditional Regularization: $ \min_{\phi, \theta} I_\phi(Z_t; X_t^o\|X_{o \setminus t}) $. We employ the chain rule for mutual information on the conditional regularization term as follows: \[ \min_{\phi, \theta} I(Z_t; X_t^o\|X_{o \setminus t}) = \min_{\phi, \theta} I(Z_t; X_{1:T}) - I(Z_t; X_{o \setminus t}). \] (5) It is worth highlighting that the application of the chain rule decomposes the conditional regularization into two components: (i) minimizing the information between the latent representation $ Z_t $ and the entire observed time series input $ X_{1:T} $ that encourages the latent representation to be concise, while (ii) maximizing the information from $ X_{o \setminus t} $ to capture the underlying temporal dynamics provided by the observations at the remaining time steps. This prevents a significant loss of temporal context in the IB and, in turn, enhances the utilization of temporal dependencies from the remaining time steps. Minimizing $ I(Z_t; X_{1:T}) $. The first term in (5) can be bounded as follows (see Appendix A.3): \[ I(Z_t; X_{1:T}) \leq E_{x_{1:T} \sim p_{data}} [D_{KL}(q_\phi(z_t\|x_{1:T})\|\|p(z_t))] \overset{\text{def}}{=} L_\phi \] (6) $^2$Let $ V, W, $ and $ Y $ be random variables, then the chain rule gives $ I(Y; W\|V) = I(Y; W, V) - I(Y; V) $. $^3$Conditional reconstruction can be appropriate for capturing information that exclusively depends on the corresponding input, as introduced in (Fischer, 2020; Lee et al., 2023). where we utilize the unit isotropic Gaussian as the prior distribution, i.e., $ p(Z_t) = \mathcal{N}(0, I) $. We model the stochastic encoder as a multivariate Gaussian distribution defined as $ q_\phi(Z_t \| X_{1:T}^o) = \mathcal{N}(\mu_\phi(X_{1:T}^o), \text{diag}(\sigma_\phi(X_{1:T}^o))) $, where $ \mu_\phi(\cdot) $ and $ \sigma_\phi(\cdot) $ are implemented as neural networks parameterized by $ \phi $. This explains why the (unconditional) IB struggles with modeling temporal dynamics, as discussed in Section 3.1. That is, (6) forces the encoder mappings that depend on time series inputs to converge to the unit Gaussian, imposing overly strict regularization. This hinders capturing of temporal dependencies present in other observations, motivating us to explicitly capture temporal dynamics by maximizing $ I(Z_t; X_{\backslash t}^o) $ rather than relying solely on the reconstruction signal in (4). Maximizing $ I(Z_t; X_{\backslash t}^o) $. To bound the second term in (5), we adopt the InfoNCE minimization from the contrastive learning on latent representations that approximately achieves maximizing the corresponding mutual information (Oord et al., 2018; Tian et al., 2020). Suppose we have the latent representation at time step $ t $ given an observed time series $ x_{1:T}^o $ as a reference, i.e., $ z_t \sim q_\phi(Z_t \| x_{1:T}^o) $. Since our aim is to maximize information from $ X_{\backslash t}^o $, we intentionally make missing observations at time step $ t $ from the reference, and employ latent representations of time steps other than $ t $ as positive pairs. For negative pairs, we consider latent representations at different time steps from other time series observations. Finally, we define our novel contrastive learning loss with cosine similarity of latent representations along the time axis as follows (see Appendix A.4): \[ I(Z_t; X_{\backslash t}^o) \geq \mathbb{E}_{x_{1:T}^o \sim p_{\text{data}}} \left[ \log \left( \frac{\sum_{t' \in \{1,...,T\} \setminus t} \exp(z_t^T \tilde{z}_{t'}/\tau)}{\sum_{x_{1:T}^o \in \mathcal{X}_{1:T}} \sum_{t' \in \{1,...,T\}, z_{t'} \sim q_\phi(z_{t'} \| x_{1:T})} \exp(z_t^T \tilde{z}_{t'}/\tau)} \right) \right] \overset{\text{def}}{=} -L_3^\phi \tag{7} \] where $ \tau $ is the temperature parameter. Here, $ \tilde{z}_{t'} \sim q_\phi(\tilde{Z}_{t'} \| x_{\backslash t}^o) $ denotes the positive pair obtained by masking the reference time series, such that $ x_{\backslash t}^o $ is created by replacing $ x_t^o $ with zeros from $ x_{1:T}^o $. We regard such positive pairs as augmentations of a given time series since latent representations with missing values at time step $ t $ share task-relevant information about the underlying temporal dynamics of a given time series. We denote $ \mathcal{X}_{1:T} $ a set of negative samples comprising other time series in the same mini-batch, where $ x_{1:T} $ indicates an observed time series from $ \mathcal{X}_{1:T} $. This makes our encoder capture time series-level semantics – such as underlying disease progression patterns that can be distinguished from others – by pushing these samples from the reference. Such an attribution is necessary for reconstructing missing values (and associated downstream tasks in the experiments) specific to the input time series. Please refer to Appendix C for implementation details. ### 3.2.3 Optimization Now, we introduce a novel imputation method, which we refer to as Time-series Imputation using Conditional Information Bottleneck (TimeCIB), that consists of the stochastic encoder, $ q_\phi $, and the feature estimator, $ p_\theta $, introduced above. Please see Figure C1 for a schematic illustration of our framework. Overall, we optimize our method based on the following objective by combining all the loss functions that allows us to approximately achieve time series imputation defined in (3): \[ \min_{\phi, \theta} \beta L_{\phi,\theta}^1 + L_\phi^2 + \gamma L_\phi^3 \tag{8} \] where $ \gamma \in \mathbb{R}_{\geq 0} $ is a balancing coefficient that trades off the impact of $ L_\phi^3 $. We provide sensitivity analysis on $ \beta, \gamma $ in Appendix B.4. ### 3.3 Introducing Inductive Bias about Temporal Dynamics Now, we illustrate how we can inject inductive bias about the underlying temporal dynamics by employing temporal kernels to further improve the expressive power of TimeCIB. The alignment of the latent representation (Wang & Isola, 2020), attained through contrastive learning based on (7), renders the similarity between latent representations at two adjacent time points indistinguishable from the similarity between those at two distant time points. This phenomenon appears to contradict real-world temporal dynamics, such as gradually deteriorating or periodic behavior of disease progression patterns. To address this, we employ conditional alignment (Dufumier et al., 2021) that introduces inductive bias about the underlying temporal dynamics with temporal kernels as the following: \[ I(Z_t; X_{\backslash t}^o) \geq \mathbb{E}_{x_{1:T}^o \sim p_{data}} \left[ \log \left( \frac{\sum_{t' \in \{1,...,T\} \backslash t} c_{t,t'} \exp (z_t^T z_{t'}/\tau)}{\sum_{x_{1:T} \in \mathcal{X}_{1:T}} \sum_{t' \in \{1,...,T\}, z_{t'} \sim q_\phi(z_{t'}\|x_{1:T})} \exp (z_t^T z_{t'}/\tau)} \right) \right] \overset{\text{def}}{=} -L_{3\phi}^\tau \tag{9} \] where $ c_{t,t'} \in \mathbb{R} $ is a kernel coefficient as a function of two time points $ t $ and $ t' $. Incorporating prior knowledge of underlying similarity into contrastive learning is not a novel concept. Several previous works have leveraged supervisory information to adapt contrastive learning, including semantic similarities for text classification (Suresh & Ong, 2021), angle similarity for gaze estimation (Wang et al., 2022), and Gaussian priors for time series and video representation learning (Tonekaboni et al., 2021; Chen et al., 2022). In this paper, we evaluate the following two representative temporal kernels to evaluate our method on data with different temporal behaviors and will leave the choice of the kernel as a hyperparameter: \[ c_{\text{cauchy}}(\tau, \tau') = \sigma^2 \left( 1 + \frac{(\tau - \tau')^2}{l^2} \right)^{-1}, \quad c_{\text{periodic}}(\tau, \tau') = \sigma^2 \exp \left( -\frac{2 \sin^2 (\pi(\tau - \tau')/p)}{l^2} \right) \tag{10} \] - Cauchy Smoothing. Under the assumption that two nearby time points should be more similar than those far away, we smooth the latent representations of time series by assigning higher weights to nearby time points when pulling the representations, utilizing the Cauchy kernel defined as $ c_{\text{cauchy}}(\tau, \tau') $ in (10) which is the mixture of infinite RBF kernels with different time scales (Rasmussen, 2004). This is a generalized form that reduces to uniform weights as in (7), i.e., $ l = \infty $ gives $ c(\tau, \tau') = \sigma^2 $. - Periodic Smoothing. Unfortunately, Cauchy smoothing may not be appropriate when the underlying temporal dynamics exhibit periodic behavior (e.g., seasonality). To incorporate our domain knowledge about periodic time series data, we utilize the exponentiated sine-squared kernel given as $ c_{\text{periodic}}(\tau, \tau') $ in (10) where $ l \in \mathbb{R} $ corresponds to the length scale and $ p \in \mathbb{R} $ reflects the periodicity. ### 4 RELATED WORKS Time Series Imputation: Predictive Methods. Earlier works on time series imputation have been proposed for single imputation utilizing predictive models. M-RNN (Yoon et al., 2018b) and BRITS (Cao et al., 2018) are representative RNN-based methods that predict missing observations by employing bidirectional RNNs to capture both past and future temporal dependencies. Inspired by the recent success of transformers in modeling time series data over conventional RNN architectures, recent predictive methods for time series imputation employ a self-attention mechanism to enhance imputation performance (Bansal et al., 2021; Du et al., 2023; Shan et al., 2023). However, these methods cannot provide multiple imputations and, therefore, fail to incorporate the uncertainty associated with imputed values. Time Series Imputation: Generative Methods. Two main strands of generative models – VAEs (Kingma & Welling, 2014) and generative adversarial networks (GANs) (Makhzani et al., 2015) – have been introduced for multiple imputation due to their ability to stochastically generate samples. Here, we describe VAE-based methods. See Appendix D.3 for more related works. VAE-based imputation methods primarily focus on defining the evidence lower bound, where the reconstruction error is computed only over the observed part of the incomplete data while missing values are filled with arbitrary values (e.g., zeros) during inference (Sohn et al., 2015; Nazabal et al., 2020). Fortuin et al. (2020) proposed GP-VAE which adopts a similar approach to efficiently handle incomplete (missing) data in a temporal setting by assuming that the latent representation of input time series evolves smoothly over time according to a Gaussian process (GP). While introducing the GP prior improves the ability to capture the underlying temporal dynamics, GP-VAE still cannot capture shared temporal structures across time series data, as it employs an independent GP prior for each time series. More recently, L-VAE (Ramchandran et al., 2021) and its conditional extension (Ramchandran et al., 2022) further improve the GP prior of GP-VAE by utilizing auxiliary covariates information. We focus our comparison on VAE-based models since these models can be information-theoretically interpreted as optimizing the IB (Voloshynovskiy et al., 2019). It is worth clarifying that TimeCIB can be distinguished from GP-VAE in terms of how they achieve time series imputation and leverage temporal kernels. Under the IB framework, GP-VAE models the smooth temporal evolution of latent variables by replacing the traditional unit Gaussian prior with a GP prior specified by temporal kernels. However, TimeCIB is motivated by the inherent limitation of the IB in discarding temporal information (see Section 3.1) and proposes a novel CIB principle that alleviates the strict regularization of IB. Temporal kernels are optionally adopted to introduce an inductive bias to the underlying temporal dynamics. Information Bottleneck with Conditional Information. Several works have tailored the IB principle (Tishby & Zaslavsky, 2015; Alemi et al., 2017) by introducing conditional reconstruction or applying conditional regularization to extract information in alignment with their specific objectives. (Gondek & Hofmann, 2003) proposed conditional reconstruction to discover a new meaningful set of clusters that is orthogonal to the known class labels. (Fischer, 2020; Tezuka & Namekawa, 2021) introduced conditional regularization for supervised learning, which minimizes only redundant information given label information, thereby preventing the loss of label-related information due to overly strict regularization in the conventional IB principle. More recently, (Lee et al., 2023) utilized both conditional reconstruction and regularization to discover a label-related core subgraph from a pair of two molecular graphs. From this perspective, our work aligns with conditional regularization approaches. While previous works have primarily focused on mitigating regularization concerning target label information, our method aims to alleviate overly strict regularization that can hinder the learning of underlying temporal dynamics. Moreover, to the best of our knowledge, this is the first work that presents an information-theoretic definition for time series imputation and proposes a novel conditional IB that can effectively preserve temporal dynamics for better imputation. ## 5 EXPERIMENTS ### 5.1 EXPERIMENTAL SETUP Evaluation Metrics. We evaluate the imputation performance from two perspectives: i) Imputation performance which measures feature-wise (pixel-wise) reconstruction. Specifically, we assess the negative log-likelihood (NLL) and mean squared error (MSE) of the imputed values on artificially missing features. ii) Prediction performance, which indirectly measures how well the imputed values preserve task-relevant information, which is a crucial aspect of imputation methods in practice. Following the experimental setup in (Fortuin et al., 2020) and (Yoon et al., 2019), we train separate classifiers or predictors with imputed values to predict the target labels. Then, we evaluate the area under the receiver operating characteristic (AUROC) for classification tasks and the MSE of the forecast (ForecastMSE) for forecasting tasks to measure the discriminative and predictive performance of imputation methods, respectively. Baseline Models. We focus our comparison on VAE-based models since these models can be interpreted under the IB principle as suggested in (Voloshynovskiy et al., 2019). Moreover, these multiple imputation methods can provide uncertainty of the imputed values, which is often crucial to support decision-making processes such as clinical interventions in healthcare. Hence, for baseline models, we compare our proposed method with the following: i) GP-VAE (Fortuin et al., 2020) which utilizes the Gaussian process (GP) prior to model time dependency, ii) HI-VAE (Nazabal et al., 2020) and iii) VAE (Kingma & Welling, 2014), both of which use an autoencoder architecture and are capable of imputing values at each time step. In addition to the above baselines, we compare with cutting-edge predictive methods: an RNN-based method, BRITS (Cao et al., 2018), and a transformer-based methods, SAITS (Du et al., 2023). Moreover, we also compare with state-of-the-art generative imputation methods, attention-based autoencoder approach, mTANs (Shukla & Marlin, 2021), and diffusion-based approach, CSDI (Tashiro et al., 2021). For a fair comparison, the magnitudes of the number of parameters are set to be equivalent among the evaluated methods. More detailed explanations about baseline models are provided in Appendix D. --- 4We compare NLL only with two VAE-based methods, since NLL cannot be measured for predictive methods; mTAN is probabilistic but uses fixed variance; and NLL of CSDI affected by the noise schedule. (please refer to Appendix F.2 of Tashiro et al. (2021)). We mark asterisks () in Table 2 and Table 3 to specify this. Table 1: Imputation and prediction performance on the image sequence datasets. \| Methods \| HealingMNIST (missing with MNAR pattern) \| RotatedMNIST (interpolation & extrapolation) \| \|---------------\|-----------------------------------------\|---------------------------------------------\| \| \| NLL(↓) \| MSE(↓) \| AUROC(↑) \| NLL(↓) \| MSE(↓) \| \| No Imp. \| - \| 0.293 ± 0.000 \| 0.920 ± 0.000 \| - \| 0.133 ± 0.000 \| \| Mean Imp. \| - \| 0.168 ± 0.000 \| 0.938 ± 0.000 \| - \| 0.085 ± 0.000 \| \| Forward Imp. \| - \| 0.177 ± 0.000 \| 0.946 ± 0.000 \| - \| 0.080 ± 0.000 \| \| VAE \| 0.480 ± 0.002 \| 0.232 ± 0.000 \| 0.922 ± 0.000 \| 1.773 ± 0.127 \| 0.133 ± 0.000 \| \| HI-VAE \| 0.290 ± 0.001 \| 0.134 ± 0.003 \| 0.962 ± 0.001 \| 0.207 ± 0.007 \| 0.087 ± 0.001 \| \| GP-VAE \| 0.261 ± 0.001 \| 0.114 ± 0.002 \| 0.960 ± 0.002 \| 0.190 ± 0.001 \| 0.080 ± 0.004 \| \| Ours(Uniform) \| 0.204 ± 0.002 \| 0.090 ± 0.001 \| 0.967* ± 0.001 \| 0.184 ± 0.001 \| 0.077 ± 0.001 \| \| Ours(Cauchy) \| 0.202 ± 0.004 \| 0.088 ± 0.002 \| 0.967 ± 0.000 \| 0.184 ± 0.001 \| 0.076 ± 0.002 \| (a) Robustness on missing patterns. (b) Robustness on missing ratios. Figure 2: Robustness analysis for missing patterns and missing ratios on HealingMNIST. 5.2 Main Results Imputation on image sequences. To evaluate imputation performance on diverse missing scenarios, we assess imputation performance on two MNIST sequence benchmarks with various missing patterns. HealingMNIST (Krishnan et al., 2015) has approximately 60% of missing pixels under a missing-not-at-random (MNAR) pattern on every time step, where the missing probability of white pixels is twice larger than that of black pixels. Given that the model is not provided with information about the underlying missing mechanism, this task is particularly challenging, yet it mirrors many practical scenarios. For example, in healthcare, patients with depression are more likely to refuse answers about the severity of their condition (Gliklich et al., 2014). RotatedMNIST (Ramchandran et al., 2021) evaluates performance on interpolation and extrapolation, where all features at an arbitrary time step are completely missing. This makes imputation more challenging since the model must reconstruct all the missing values at a given time step solely based on the temporal dependency. Table 1 demonstrates that TimeCIB provides state-of-the-art imputation and prediction performance on both datasets, and the application of the Cauchy kernel (10) can further improve the performance. We further evaluate the robustness of our model on HealingMNIST with four additional missing patterns with 60% of missing ratio (Figure B2a) and that with different missing ratios ranging from 10% to 90% with the random missing pattern (Figure B2b). To assess the robustness of missing patterns, we employ Random, Spatial (i.e., neighboring pixels have correlated missing probabilities), and Temporal+/- (i.e., positive/negative temporal correlation). The imputation performance of our proposed method is most robust on diverse missing patterns and missing ratios. Please refer to Appendix B for more experimental details and results. Imputation for weather forecasting. Weather forecasting is one of the representative fields where we can observe diverse scales of seasonality – such as daily, weekly, monthly, or yearly basis – which is one of the most important characteristics of time series data. In this experiment, we focused on two weather forecasting datasets - Beijing Air Quality (Zhang et al., 2017) and US Local$^5$ whose time series measurements are collected every hour. Our model is capable of using the prior knowledge on the periodicity of the target data, by applying conditional alignment with temporal kernels (Section 3.3); we assume daily periodicity by setting $p = 24$. Inspired by the experiments outlined in (Yoon et al., 2019), we also evaluate the utility of imputation methods on forecasting by assessing the prediction performance (ForecastMSE) of a separately trained LSTM using time series data with imputed values. As shown in Table 2, TimeCIB outperforms the benchmarks on both weather forecasting datasets in terms of both imputation and prediction performance. The $^5$https://www.ncei.noaa.gov/data/local-climatological-data/ Table 2: Imputation and prediction performance on the weather forecasting datasets. \| Methods \| Beijing (T=24) \| US Local (T=168) \| \|------------------\|----------------\|------------------\| \| \| NLL(↓) MSE(↓) \| ForecastMSE(↓) \| NLL(↓) MSE(↓) \| ForecastMSE(↓) \| \| No Imp. \| - \| 1.015 ± 0.000 \| 0.539 ± 0.000 \| - \| 1.113 ± 0.000 \| 0.610 ± 0.000 \| \| Mean Imp. \| - \| 0.460 ± 0.000 \| 0.517 ± 0.000 \| - \| 0.509 ± 0.000 \| 0.432 ± 0.000 \| \| Forward Imp. \| - \| 0.399 ± 0.000 \| 0.502 ± 0.000 \| - \| 0.391 ± 0.000 \| 0.401 ± 0.000 \| \| BRITS \| - \| 0.396 ± 0.002 \| 0.490 ± 0.005 \| - \| 0.384 ± 0.001 \| 0.398 ± 0.027 \| \| SAITS \| - \| 0.283 ± 0.013\| 0.450 ± 0.005 \| - \| 0.275 ± 0.002 \| 0.350 ± 0.067 \| \| mTANs \| * \| 0.287 ± 0.005 \| 0.436 ± 0.005 \| * \| 0.268 ± 0.018 \| 0.357 ± 0.033 \| \| CSDI(n=5) \| * \| 0.287 ± 0.003 \| 0.423 ± 0.003 \| * \| 0.378 ± 0.001 \| 0.364 ± 0.036 \| \| CSDI(n=25) \| * \| 0.276 ± 0.001\| 0.423 ± 0.006 \| * \| 0.378 ± 0.001 \| 0.347 ± 0.036 \| \| VAE \| * \| 1.427 ± 0.001 \| 1.01 ± 0.002 \| 1.462 ± 0.002 \| 1.080 ± 0.004 \| 0.677 ± 0.041 \| \| HI-VAE \| 1.081 ± 0.003 \| 0.321 ± 0.008 \| 0.464 ± 0.008 \| 1.078 ± 0.005 \| 0.317 ± 0.010 \| 0.380 ± 0.060 \| \| GP-VAE \| 1.077 ± 0.006 \| 0.316 ± 0.011 \| 0.463 ± 0.008 \| 1.078 ± 0.005 \| 0.318 ± 0.010 \| 0.385 ± 0.051 \| \| Ours(Uniform) \| 1.063 ± 0.001 \| 0.293 ± 0.004 \| 0.445 ± 0.003 \| 1.052 ± 0.001 \| 0.265 ± 0.002 \| 0.351 ± 0.060 \| \| Ours(Periodic) \| 1.060 ± 0.002 \| 0.283 ± 0.004\| 0.443 ± 0.004 \| 1.049 ± 0.002 \| 0.260 ± 0.003 \| 0.327 ± 0.022 \| Figure 3: Comparison of the imputed values for examples in (a) Beijing, (b) US Local, and (c) Physionet2012 datasets, highlighting that TimeCIB provides more accurate imputations by considering temporal dependencies. Dots and crosses are observed and missing ground-truth values, respectively. The performance of our method is further enhanced when equipped with a temporal periodic kernel (10), highlighting our model’s ability to incorporate the correct inductive bias. Imputation for electrical health records. Time series imputation is of special importance in healthcare where each feature may have a distinct sampling period and strategies. In this context, we evaluate imputation methods on Physionet2012 – Mortality Prediction Challenge (Silva et al., 2012), which aims to predict in-hospital mortality of intensive care unit (ICU) patients from 48 hours of records with roughly 80% of missing features. Furthermore, we conduct additional evaluations to assess whether the imputation methods preserve the critical characteristics of a given time series – i.e., whether a patient’s status is deteriorating or not – after replacing the missing features with imputed values. Table 3 shows that TimeCIB provides imputation performance comparable to the best benchmark while outperforming the VAE-based methods by a great margin. Furthermore, it achieves the best classification performance, successfully capturing information about the temporal dynamics of patients’ status. Note that while the mean imputation provides better imputation performance, the imputed values drastically lose the crucial information for discriminating patient’s status. Table 3: Imputation and prediction performance on the clinical dataset. \| Methods \| Physionet2012 (mortality prediction) \| \|------------------\|--------------------------------------\| \| \| NLL(↓) MSE(↓) \| AUCROC(↑) \| \| No Imp. \| - \| 0.962 ± 0.000 \| 0.692 ± 0.000 \| \| Mean Imp. \| - \| 0.511 ± 0.000 \| 0.703 ± 0.000 \| \| Forward Imp. \| - \| 0.613 ± 0.000 \| 0.710 ± 0.000 \| \| BRITS \| - \| 0.529 ± 0.004 \| 0.700 ± 0.005 \| \| SAITS \| - \| 0.501 ± 0.024 \| 0.713 ± 0.007 \| \| mTANs \| * \| 0.499 ± 0.008 \| 0.721 ± 0.004 \| \| CSDI(n=5) \| * \| 0.548 ± 0.014 \| 0.705 ± 0.005 \| \| CSDI(n=25) \| * \| 0.478 ± 0.002 \| 0.683 ± 0.033 \| \| VAE \| - \| 1.400 ± 0.000 \| 0.962 ± 0.000 \| 0.691 ± 0.001 \| \| HI-VAE \| - \| 1.345 ± 0.009 \| 0.852 ± 0.018 \| 0.696 ± 0.004 \| \| GP-VAE \| - \| 1.227 ± 0.007 \| 0.616 ± 0.013 \| 0.730 ± 0.006 \| \| Ours(Uniform) \| 1.183 ± 0.007 \| 0.528 ± 0.014 \| 0.744 ± 0.009 \| \| Ours(Cauchy) \| 1.179 ± 0.006 \| 0.521 ± 0.012 \| 0.744 ± 0.009 \| 6 CONCLUSION In this paper, we have presented TimeCIB, a novel information-theoretic approach for time series imputation. While inheriting the multiple imputation and uncertainty measurement properties of the IB, TimeCIB addresses the limitation of the IB principle in capturing underlying temporal dynamics by replacing conventional regularization with conditional regularization. Our variational decomposition showed that CIB could be approximated by optimizing the evidence lower bound (ELBO) and the contrastive objective. We also demonstrated that introducing inductive bias based on temporal kernels can further enhance expressive power, acting as a form of conditional alignment. Our empirical results on image sequences, weather forecasting, and electrical health records prove that TimeCIB is effective in a wide range of practical cases. ACKNOWLEDGMENTS We thank anonymous reviewers for many insightful comments and suggestions. CL was supported through the IITP grant funded by the Korea government(MSIT) (No. 2021-0-01341, AI Graduate School Program, CAU). CODE AVAILABILITY Codebase used in this paper is available at https://github.com/Chemgyu/TimeCIB. REFERENCES Alexander A. Alemi, Ian Fisher, Joshua V. Dillon, and Kevin Murphy. Deep Variational Information Bottleneck. 2017. Parikshit Bansal, Prathamesh Deshpande, and Sunita Sarawagi. Missing Value Imputation on Multidimensional Time Series. Proceedings of the VLDB Endowment, 14(11):2533–2545, 2021. Wei Cao, Dong Wang, Jian Li, Hao Zhou, Lei Li, and Yitan Li. BRITS: Bidirectional Recurrent Imputation for Time Series. Advances in neural information processing systems, 31, 2018. Minghao Chen, Fangyun Wei, Chong Li, and Deng Cai. Frame-wise Action Representations for Long Videos via Sequence Contrastive Learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13801–13810, 2022. Wenjie Du, David Côté, and Yan Liu. SAITS: Self-attention-Based Imputation for Time Series. Expert Systems with Applications, 219:119619, 2023. 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Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-VAE: Learning Basic Visual Concepts with a Constrained Variational Framework. In International conference on learning representations, 2017. Alistair EW Johnson, Tom J Pollard, Lu Shen, Li-wei H Lehman, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G Mark. MIMIC-III, a Freely Accessible Critical Care Database. Scientific data, 3(1):1–9, 2016. Prannay Khosla, Piotr Teterwak, Chen Wang, Aaron Sarna, Yonglong Tian, Phillip Isola, Aaron Maschinot, Ce Liu, and Dilip Krishnan. Supervised Contrastive Learning. Advances in neural information processing systems, 33:18661–18673, 2020. Diederik P Kingma and Max Welling. Auto-encoding Variational Bayes. In Proceedings of the International Conference on Learning Representations (ICLR), volume 1, 2014. Rahul G Krishnan, Uri Shalit, and David Sontag. 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3ukT8oODY0	Why DDQS works? It seems that DDQS will select the actions that have the high values of $Q^{max}(s,a)$ and $Q^{min}(s,a)$. Why DDQS is better than other definitions based on $Q^{max}(s,a)$ and $Q^{min}(s,a)$, such as $\sum_{a}(Q^{max}(s,a)$+ $Q^{min}(s,a))/ \sum_{a}Q^{max}(s,a)$.	CAREFUL AT ESTIMATION AND BOLD AT EXPLORATION FOR DETERMINISTIC POLICY GRADIENT ALGORITHM Anonymous authors Paper under double-blind review ABSTRACT Exploration strategies within continuous action spaces often adopt heuristic approaches due to the challenge of dealing with an infinite array of possible actions. Previous research has established the advantages of policy-based exploration in the context of deterministic policy reinforcement learning (DPRL) for continuous action spaces. However, policy-based exploration in DPRL presents two notable issues: unguided exploration and exclusive policy, both stemming from the soft policy learning schema, which is famous for DPRL policy learning. In response to these challenges, we introduce a novel approach called Bold Actor Conservative Critic (BACC), which leverages Q-value to guide out-of-distribution exploration. We extend the dynamic Boltzmann softmax update theorem to the double Q function framework, incorporating modified weights and Q values. This extension enables us to derive an exploration policy directly for policy exploration, which is constructed with the modified weights. Furthermore, we explicitly utilize the minimum Q value as an intermediate step in policy gradient computation, which derives from a conservative policy. In practice, we construct such an exploration policy with a limited set of actions and train a parameterized policy by minimizing the expected KL-divergence between the target policy and a policy constructed based on the minimum Q value. To evaluate the effectiveness of our approach, we conduct experiments on the Mujoco and Roboschool benchmarks. Notably, our method excels in the highly complex Humanoid environment, demonstrating its efficacy in tackling challenging continuous action space exploration problems. 1 INTRODUCTION Deep reinforcement learning (RL) has attracted much attention in recent years. It has achieved massive success in many fields, such as DQN (Mnih et al., 2015) in simple RGB games, AlphaStar (Vinyals et al., 2019), and OpenaiFive (OpenAI et al., 2019) in multi-player combat games, chat-GPT (OpenAI, 2022) in natural language processing. When applying deep RL in continuous action control, such as robotic control, higher demands exist on the robustness of reinforcement learning policy (Haarnoja et al., 2017). Algorithms based on the maximum entropy framework (Ziebart, 2010) are more robust due to the diverse action selection, which augments the standard reward with the policy entropy, to some extent, encourages exploration in training and finally derives a robust policy. The intuitive reason for taking exploratory actions is that other actions with lower predicted rewards may be better. Moreover, the method used to select actions directly affects the rate at which the RL algorithm will converge to an optimal policy. Ideally, the algorithm should perform a non-greedy action if it lacks confidence in the current prediction and perform a bold exploration once we gather more information about the prediction result. Although various exploration methods, such as ε-greedy, Softmax, UCB-1 (Auer et al., 2002), have been suggested for use in discrete action space, these kinds of explorations are not the same thing as the exploration in the continuous action space, due to the infinite actions. Since the actions in continuous space are uncountable, the exploration strategy is heuristic, such as adding the Gaussian perturbation (Silver et al., 2014; Van Hasselt et al., 2016; Fujimoto et al., 2018; Haarnoja et al., 2018). Intuitively, this kind of unguided exploration should not be an efficient exploration strategy. It will slow down learning the optimal policy due to its randomness. However, such approaches still yield significant improvements compared to methods not incorporating exploration. For instance, in the TD3 (Fujimoto et al., 2018) algorithm, random noise is added to actions for exploration purposes. Actions that exceed bounds are clipped to ensure their validity. Then, policy-based noise is considered in the SAC (Haarnoja et al., 2018) algorithm, which also achieved good results compared to the previous method. SAC considers a stochastic actor to explore and minimize the reverse KL-divergence to learn a policy for computation consideration. However, this reverse KL-divergence leads to another issue: exclusive policy, which hinders exploring the optimal policy. This issue lies in the Q value-based policy gradient method since the DPG (Silver et al., 2014) algorithm is proposed. That is, policy learning and Q-function learning are separate processes. Policy learning lags behind Q-function learning, meaning that actions collected from the policy have relatively lower Q-values. As we assume the Q function is multimodal, the policy may be sub-optimal due to the exclusive reverse KL-divergence, preferring an unimodal approximation. To overcome the unguided exploration issue, the OAC (Ciosek et al., 2019) algorithm uses the Q-value-based policy gradient to predict Gaussian mean offset, then uses offset compensation Gaussian policy to explore. However, in practice, most of these predictions are inaccurate regarding high-dimensional action space, which leads to inefficient exploration. A natural and straightforward idea is using Q-value to guide exploration. Suppose the high Q-value actions far from the policy can also be sampled. In this case, the exclusive policy issue can also be avoided. In this paper, we propose Bold Actor Conservative Critic (BACC) algorithm to achieve such Q-value-guided out-of-distribution (OOD) exploration. Specifically, we initially introduce the DDQS operator based on the Double Q-function framework. This operator is an extension derived from the DBS operator. Within the Double Q-function framework, our primary modifications involve the weights and values associated with computing expected state values of the DBS operator. For given state-action pairs, we softmax the maximum value (greedy Q value) of two Q functions as weights and take the minimum value (conservative Q value) of the two Q functions as values. Subsequently, we provide proof of the convergence of the DDQS operator (Theorem 1), which serves as an assurance of the feasibility of the exploration method we propose. Following this, we extract weights to construct an exploration policy to guide action exploration, which is similar in approach to the SARSA (Rummery & Niranjan, 1994) algorithm. However, considering sample efficiency, we aim to separate action exploration from policy learning. Therefore, we superficially softmax the conservative Q values to construct an optimization policy. We then iteratively minimize the KL-divergence between the target policy and the conservative policy to guide target policy learning. According to Theorem 1, both two Q functions will eventually converge to the optimal Q-function. Therefore, the exploration policy and the optimization policy will also ultimately be the same, which is the optimal policy. Consequently, minimizing the KL-divergence can lead to obtaining the actual optimal policy. The use of conservative Q-values in the optimization policy wasn’t our initial contribution, but in this paper, it is the first time it is explicitly related to the conservative policy, to achieve stable policy learning. We evaluate our proposed method on Mujoco (Todorov et al., 2012) benchmarks and verify that the proposed method outperforms the previous state-of-the-art in various environments, particularly the most complex Humanoid environment. We achieved about 8k scores in 3 million steps, a massive improvement over previous methods. We also tested our method in the Roboschool (OpenAI, 2017) environments HumanoidFlagrun and HumanoidFlagrunHarder. The results indicate that our exploration method is more robust than the OAC algorithm in complex environments. 2 PRELIMINARY We first introduce notation and the maximum entropy objective, then summarize the Soft Policy Learning method. Notation. In this paper, we consider deterministic policy reinforcement learning method for continuous action space. Consider a discounted infinite-horizon Markov decision process (MDP), defined by the tuple $(S, A, p, r, \gamma)$, where the state space $S$ and the action space $A$ are continuous, and the state transition probability $p : S \times A \times S \rightarrow [0, \infty)$ represents the probability density of the next state. Given the state $s_t \in S$ and action $a_t \in A$ at time-step $t$, we can get the probability density of $s_{t+1} \in S$. The environment emits a bounded reward $r : S \times A \rightarrow [r_{\text{min}}, r_{\text{max}}]$ on for specific state and action pair. $\gamma$ is the discount factor, and its value is in the range $[0, 1)$, which makes the infinite accumulated reward finite in mathematics. Maximum entropy objective. Standard RL algorithm maximizes the expected sum of rewards $\sum_t \mathbb{E}_{(s_t, a_t) \sim \rho_\pi}[r(s_t, a_t)]$. $\rho_\pi(s_t, a_t)$ denotes state-action marginals of the trajectory distribution induced by a policy $\pi(a_t\|s_t)$. Maximum entropy objective augment the expectation with the expected entropy of the policy over $\rho_\pi(s_t)$: $$J(\pi) = \mathbb{E}_\pi \left[ \sum_{t=0}^{\infty} r(s_t, a_t) + \alpha H(\pi(\cdot\|s_t)) \right].$$ The temperature parameter $\alpha$ balance the relative importance of the entropy term and the reward, and this entropy term influence the exploration of the policy, which in result to a more stochastic optimal policy ideally. Soft policy learning. Soft policy maximizes the maximize entropy objective and modifies the Q value function using the standard Q value function minus the current action’s log probability, this Q value is called Soft Q value. Considering the discount factor in practice algorithm, the standard Q value function is $\mathbb{E}_{(s_t, a_t) \sim \rho_\pi} \left[ \sum_{t=0}^{\infty} \gamma^t r(s_t, a_t) \right]$. The soft Q value is $\sum_{t=0}^{\infty} \mathbb{E}_{(s_t, a_t) \sim \rho_\pi} \left[ \gamma^t r(s_t, a_t) + \alpha \gamma^{t+1} H(\pi(\cdot\|s_{t+1})) \right]$. For a fixed policy, the soft Q value can be computed iteratively, starting from any function $Q : S \times A \rightarrow \mathbb{R}$ and repeatedly applying the modified Bellman backup operator $T^\pi$ given by $$T^\pi Q(s_t, a_t) \triangleq r(s_t, a_t) + \gamma \mathbb{E}_{(s_{t+1}, a_{t+1}) \sim \rho_\pi} \left[ Q(s_{t+1}, a_{t+1}) - \alpha \log \pi(a_{t+1}\|s_{t+1}) \right],$$ then improve the policy by minimizing following formula $$\pi' = \arg \min_{\pi \in \Pi} D_{KL} \left( \pi(\cdot\|s_t) \parallel \frac{\exp(Q(s_t, \cdot))}{Z(s_t)} \right),$$ where $Z(s_t) = \sum_a Q(s_t, a_t)$ normalizes the distribution. ### Problems and Our Solution to Previous Work In this section, we provide illustrations to explain inefficient exploration. Then, we demonstrate why the method proposed in this paper is effective. For a fixed state, the Q-values and policy regarding one-dimensional actions are approximately as shown in Fig. 1. It is typically assumed that the Q-function is multimodal, and the policy is modeled as a Gaussian distribution. ![Figure 1](image) (a) Unguided exploration and exclusive policy (b) Q-value-guided OOD exploration Figure 1: Left: Exploration typically occurs around the policy $\pi$. However, due to the exclusiveness of KL-divergence and the delay in policy learning, there is a high likelihood that the policy could become stuck at a suboptimal state. The OAC algorithm predicts an offset to allow better exploration, however, this offset cannot be accurately predicted in high-dimensional action spaces. Right: Exploring with $\pi_Q$, constructed by softmaxing the Q-value, can help avoid sub-optimality. In contrast to the policy $\pi$, it represents a form of out-of-distribution exploration. Unguided exploration. refers to a form of exploration in which an agent takes random actions without a clear goal. This type of exploration can be inefficient and time-consuming, as the agent may spend significant amounts of time exploring unimportant or irrelevant areas of the environment. Exploration that relies solely on the current policy is limited by the quality of the policy initialization and the difficulty of improving the policy. Exclusive policy. occurs in soft policy learning methods, where reverse KL divergence is used as the objective function. We assume that the Q-function is multimodal, and in combination with policy learning lagging behind, this leads to policy convergence towards a suboptimal distribution when minimizing reverse KL divergence. As shown in Fig. 1(a), when the current policy is poorly initialized and far away from the optimal policy, exploring the optimal policy without a specific objective in mind can be challenging. This unguided exploration is inefficient and leads to poor performance, as the agent may fail to discover important states or actions necessary for achieving its objectives. The OAC algorithm attempts to combine Q-valued policy gradient information to guide exploration. It predicts an offset to allow better exploration, as shown in the figure; however, this offset cannot be accurately predicted in high-dimensional action spaces, both its value and direction. OOD exploration. Therefore, it is beneficial to construct a policy that can guide exploration. Guided exploration is similar to performing a breadth-first policy search at a state, which can help address the issues associated with the policy-based approach. A natural idea is to conduct out-of-distribution(OOD) exploration, which is more robust for policy learning compared to OOD optimization, as it is often associated with instability. However, the fundamental issue is that we still need some guidance, as random OOD exploration doesn’t guarantee good results. Since we employ soft policy learning to train the policy, with Q-function learning leading the way, we can effectively use Q-values to guide excessive exploration. As depicted in Figure 1(b), Q-value-guided out-of-distribution (OOD) exploration is effective in preventing suboptimal policies and facilitates the sampling of actions with high Q-values. ### 4 Improving Exploration in Soft Policy Learning In this section, we will begin by presenting a novel Q-value update method. Following that, we will develop an effective exploration strategy and integrate the value update and action exploration based on a specific premise. Finally, we will illustrate how to meet this premise and learn an effective policy. #### 4.1 Dynamic Double Q Softmax Update We introduce the Dynamic Double Q Softmax (DDQS) operator for updating Q-values. This operator is grounded in the double Q-function framework, where two distinct Q-functions are independently trained to estimate the value of state-action pairs. As described in the introduction, the definitions of the greedy Q-function and the conservative Q-function are as follows, \[ Q_{\text{max}}(s, a) = \max\{Q^1(s, a), Q^2(s, a)\}, \quad Q_{\text{min}}(s, a) = \min\{Q^1(s, a), Q^2(s, a)\}, \] we denote $ Q_{\text{max}} $ as the greedy Q-function and $ Q_{\text{min}} $ as the conservative Q-function. The DDQS operator is defined as follows: for all $ s $ in the state space $ S $, \[ ddqs_{\beta_t}(Q(s, \cdot)) = \frac{\sum_{a \in A} e^{\beta_t Q_{\text{max}}(s, a)} Q_{\text{min}}(s, a)}{\sum_{a \in A} e^{\beta_t Q_{\text{max}}(s, a)}}. \] Here, $ \beta_t $ represents a dynamically increasing hyper-parameter during the training iteration. We will now provide a theoretical analysis of the proposed DDQS operator and demonstrate that it offers a convergence guarantee. A modified Bellman backup operator $ T^\pi $ given by \[ T^\pi Q(s_t, a_t) \triangleq r(s_t, a_t) + \gamma \mathbb{E}_{s_{t+1} \sim p}[V(s_{t+1})], \] where \[ V(s_t) = ddqs_{\beta_t}(Q(s_t, \cdot)) \] Theorem 1 (Convergence of value iteration with the DDQS operator). For any dynamic double Q softmax operator $ \text{ddq}_{\beta_t} $, if $ \beta_t $ approaches $ \infty $ after $ t $ iterations, the value function $ Q_t $ converges to the optimal value function $ Q^* $. The proof is deferred to Appendix A.1. We expand upon the utilization of the DBS operator, as introduced in (Pan et al., 2020), within the double Q-function framework. This approach is less susceptible to overestimation. The motivation for this approach lies in the challenges posed by overestimation when learning the value function in continuous action spaces. Therefore, our goal is not solely to construct a policy $ \pi_Q $, but rather to consider the combination of the two Q-functions to effectively address and mitigate overestimation issues. 4.2 Exploration with Greedy Q Value Inspired by the results of the above theorems, we employ the double Q-function framework and utilized the greedy Q-value to construct a novel exploration policy, denoted as $ \pi_E $. We first define the exploration policy $ \pi_E $: \[ \pi_E(\cdot \| s_t) = \frac{e^{\beta_t Q_{\max}(s_t, \cdot)}}{\sum_{a \in A} e^{\beta_t Q_{\max}(s_t, a)}}. \] Based on the results of Theorem 1, we can utilize the following formula to update the target Q-value: \[ r(s_t, a_t) + \gamma \mathbb{E}_{s_{t+1} \sim p, a_{t+1} \sim \pi_E(\cdot \| s_{t+1})} [Q(s_{t+1}, a_{t+1})]. \] (1) Nevertheless, calculating the target values for the next state can be computationally intensive. It involves sampling across all possible states and actions and subsequently computing the corresponding Q-values. Drawing inspiration from the SARSA method, we can sample two consecutive $(s, a)$ pairs to estimate the expectation of the Q-value: \[ \mathbb{E}_{s_t \sim p, a_t \sim \pi_E(\cdot \| s_t), s_{t+1} \sim p, a_{t+1} \sim \pi_E(\cdot \| s_{t+1})} [r(s_t, a_t) + \gamma Q(s_{t+1}, a_{t+1})]. \] (2) The key distinction here is that we can estimate the expectation of the Q-value with finite sampling. The target Q-value (Eq. 1) necessitates evaluating the next Q-value across the entire state and action space. In contrast, consecutive pairs involve computation in an on-policy form. In continuous action RL tasks, we learn the policy separately from the Q-function. If we can sample actions from the policy $ \pi $ as follows: \[ \mathbb{E}_{(s_t, a_t, s_{t+1}) \sim (p, \pi_E(\cdot \| s_{t+1}), p)} [r(s_t, a_t) + \gamma \mathbb{E}_{a_{t+1} \sim \pi(\cdot \| s_{t+1})} [Q(s_{t+1}, a_{t+1})]], \] (3) then, we can employ this equation to update the Q-function in an off-policy manner. It’s evident that we have devised a new exploration strategy through this approach. Figure 2: The state $ s $ is fixed. Left: The two Q functions are in an energy-based form, which is the optimal solution for the maximum-entropy objective. Right: Greedy Q function take the maximum value of these two Q functions over the action space. The probability of reaching the red point increases significantly when we sample actions according to the value of the Greedy Q instead of $ Q^1 $. This strategy is more effective for escaping from sub-optimal states. As depicted in Fig. 2, our proposed greedy Q exploration strategy offers several advantages: 1) This strategy proves superior for exploration compared to relying solely on any single Q-function or the $ \pi_E $ is constructed using the greedy Q-value from the double Q-framework, while $ \pi_Q $ is constructed using the Q-value from a single Q-network. policy. As illustrated by the black and red points in the figure, the number of actions better than the suboptimal action increases, and their relative range expands. 2) The use of the max operator in our method is a form of overestimation. Overestimation can be problematic for Q-value updates, but it exhibits more favorable properties when employed for exploration purposes. 3) Although our method is named ‘Bold,’ it actually promotes exploration by diminishing the likelihood of selecting the action with the highest value. This is accomplished through overestimating the values of all available actions. We also address the prerequisites for transitioning from Eq. 2 to Eq. 3. This transition necessitates that the actions sampled from the $ \pi $ are as consistent as possible with the actions sampled from the $ \pi_E $. In other words, ensuring that these two policies are as aligned as possible. Subsequently, we delve into the discussion of how to learn this policy $ \pi $. ### 4.3 Policy Learning Drawing inspiration from soft policy learning methods, we begin by defining a conservative policy for optimization, which is defined as follows: \[ \pi_O(\cdot \| s_t) = \frac{e^{Q_{\text{min}}(s_t, \cdot)}}{\sum_{a \in A} e^{Q_{\text{min}}(s_t, a)}}, \] then we can let the policy directly learn from the target policy like the soft policy learning as follows: \[ \pi' = \arg \min_\pi D_{KL} (\pi(\cdot \| s_t) \|\| \pi_O(\cdot \| s_t)), \] Now consider the neural network parameterized $ Q_\theta $ function and policy $ \pi_\phi $, thus, \[ Q^{\max}(s_t, a_t) = \max\{Q_{\theta_1}(s_t, a_t), Q_{\theta_2}(s_t, a_t)\}, \quad Q^{\min}(s_t, a_t) = \min\{Q_{\theta_1}(s_t, a_t), Q_{\theta_2}(s_t, a_t)\} \] Next, to minimize the expected KL-divergence policy objective, \[ J_\pi(\phi) = \mathbb{E}_{s_t \sim D} [D_{KL} (\pi_\phi(\cdot \| s_t) \|\| \pi_O(\cdot \| s_t))] \] \[ = \mathbb{E}_{s_t \sim D} [\log \pi_\phi(a_t \| s_t) - \log Z_\theta(s_t)] \] where $ Z_\theta(s_t) $ is a constant for given state, $ D $ is a replay buffer and Eq. 5 requires sampling action from the policy $ p_\phi $. To make the policy trainable, which means that policy parameters are differentiable, the action is reparameterized as follows: \[ a_t = f_\phi(\epsilon_t; s_t), \quad \epsilon_t \sim \mathcal{N}(\mu, \sigma^2). \] The gradient of $ J_\pi(\phi) $ with respect to $ \phi $ as follows: \[ \nabla_\phi J_\pi(\phi) = \nabla_\phi \mathbb{E}_{s_t \sim D, \epsilon_t \sim \mathcal{N}} [\log \pi_\phi(a_t \| s_t) - Q^{\min}(s_t, a_t)] \] since we utilize a neural network to parameterize both the policy and Q-function, we can employ a deep learning framework to perform the forward computations for the two terms in Eq. 7. The automatic gradient mechanism inherent to the framework will handle the backpropagation automatically. We can then derive an unbiased estimation of Eq. 7 using the following equation: \[ \hat{\nabla}_\phi J_\pi(\phi) = \nabla_\phi \log \pi_\phi(a_t \| s_t) - \nabla_\phi Q^{\min}(s_t, a_t)\|_{a_t = f_\phi(\epsilon_t; s_t)}. \] Here, refer to Eq. 3, we write the Q learning objective: \[ J_Q(\theta) = \mathbb{E}_{(s_t, a_t) \sim D} \left[ \frac{1}{2} (Q_\theta(s_t, a_t) - \hat{Q}(s_t, a_t))^2 \right], \] where $ \hat{Q}(s_t, a_t) = r(s_t, a_t) + \gamma \mathbb{E}_{s_{t+1} \sim N} [Q^{\min}(s_{t+1}, a_{t+1}) - \log \pi(a_{t+1} \| s_{t+1})] $, and $ a_{t+1} = f_\phi(\epsilon_{t+1}; s_{t+1}) $, the $ - \log \pi(a_{t+1} \| s_{t+1}) $ term is due to the computation is based on the maximum entropy framework and the Q function is also in an energy-based form. The transition from replay buffer $ D $ is generated from the interaction of the policy $ \pi_E $ and the environment. Then the gradient of the Q learning objective(Equation 9) can be estimated with an unbiased estimator \[ \hat{\nabla}_\theta J_Q(\theta) = \nabla_\theta Q_\theta(a_t, s_t) (Q_\theta(s_t, a_t) - r(s_t, a_t) - \gamma Q^{\min}(s_{t+1}, a_{t+1}) + \gamma \log \pi(a_{t+1} \| s_{t+1})) . \] $ \pi_\phi $ contains parameters that need to be learned, while $ \pi_E $ and $ \pi_O $ do not contain parameters. 4.4 THE BOLD EXPLORATION ALGORITHM In the Bold Actor Conservative Critic (BACC) algorithm (refer to Algorithm 1 in the appendix), several key steps are followed: 1) Dynamic Increase of $\beta_t$ (line 3). This is done to ensure the convergence of the Q-value, as described in section 4.1. 2) Action Sampling from Exploration Policy (line 5): Actions are sampled from the exploration policy to interact with the environment, as detailed in section 4.2. 3) Storing Transitions in Memory Buffer: The resulting transitions are stored in a memory buffer; 4) Updating the Q-function (line 12) and Actor (line 13): BACC samples transitions from the memory buffer to update both the Q-function and the actor, as explained in section 4.3. In more detail, the policy network outputs both $\mu$ and $\sigma$ for Equation (6). We uniformly sample $s_n$ actions from the range $[\mu - s_r \ast \sigma, \mu + s_r \ast \sigma]$ to evaluate the sampled actions and construct the $\pi_E$ distribution. The hyperparameters $s_r$, $s_n$, and $\beta_t$ are three key parameters in our algorithm. Further details regarding the parameters and time cost will be discussed in Appendix E. 4.5 RELATED WORK Exploration. Classical exploration methods in reinforcement learning encompass $\epsilon$-greedy and UCB-1 (Auer et al., 2002). In policy gradient methods, policy-based exploration can be facilitated by leveraging information from the policy itself, such as entropy regularization. The deterministic policy gradient method (Silver et al., 2014) introduced the concept of separating policy learning from Q-function learning. Subsequently, deterministic-policy-based methods began exploring by randomly sampling actions around the policy, leading to the development of the Optimistic Actor-Critic (OAC) method (Ciosek et al., 2019). OAC predicts an offset of the Gaussian policy mean to encourage out-of-distribution (OOD) exploration. In the context of continuous RL tasks, several heuristic value-based exploration methods have emerged. For instance, the Coherent Exploration algorithm (Zhang & Van Hoof, 2021) directly modifies the last layer parameters of the policy network to enhance the policy’s exploratory nature. The DOIE algorithm (Lobel et al., 2022) explores using a modified Q-function, assigning an optimistic value to transitions that lie significantly beyond the agent’s prior experience. The RRS algorithm (Sun et al., 2022) directly alters Q-values, effectively adjusting the initialization parameters of the Q-network, but this requires prior knowledge, such as auxiliary rewards. In contrast, our method does not necessitate prior knowledge. In specific practical applications, the AW-OPT (Lu et al., 2022) algorithm uses both policy and Q value for exploration, assigning different weights to these two exploration approaches to achieve better control. Overestimation. The concept of overestimation was first introduced in the paper by Thrun and Schwartz (Thrun & Schwartz, 1993), discussing the positive approximation error in the function approximation for RL. Then the MCQ-L (Rummery & Niranjan, 1994) method (famous with the name “SARSA” (Sutton & Barto, 2018)) mentioned that the argmax operator is impractical in training. They estimate the Q value with the consequent two-state-action pairs (in an online form). The Double Q learning (Hasselt, 2010) updates the Q value with two estimators to avoid overestimation. Then the Double DQN (Van Hasselt et al., 2016) is proposed, which parameterizes the Q function with a neural network. Inspired by the Double DQN, TD3 (Fujimoto et al., 2018) algorithm is proposed to handle overestimation for continuous action space. Regarding the stable Q-value updates, our method follows the TD3 approach, using the minimum Q-value from the two delayed-updated Q-functions to estimate the target Q-value, thereby minimizing overestimation. As for exploration, we harness overestimation to encourage bolder exploration strategies. Policy learning. Stochastic policy gradient methods, such as A3C (Mnih et al., 2016), TRPO (Schulman et al., 2015), and PPO (Schulman et al., 2017), are applicable for policy learning in continuous action spaces. However, optimizing stochastic policy gradients in continuous action spaces can be challenging, and deterministic policy gradient methods based on value functions often yield better results. In the DPG (Silver et al., 2014) algorithm, the policy parameters are optimized to maximize the Q-function. The DDPG (Lillicrap et al., 2016) algorithm is a neural network-based variant of the DPG algorithm. The SQL (Haarnoja et al., 2017) algorithm assumes an energy-based form for the Q-function. The TD3 (Fujimoto et al., 2018) algorithm introduces conservative policy parameter updates using the minimum value of the two Q-functions. Lastly, the SAC (Haarnoja et al., 2018) algorithm employs a stochastic actor for policy exploration. Previous methods empirically calculate policy gradient with the minimum Q-value. In this paper, we explicitly define the learning scheme, minimizing the KL-divergence between the target policy and the conservative policy. 5 EXPERIMENTS We conducted experiments using the Mujoco physics engine (Todorov et al., 2012), which is currently freely available and maintained by DeepMind. Additionally, we utilized the PyBullet physics engine (Coumans & Bai, 2016–2021), which offers challenging environments for simulating robot control tasks. These simulations were interacted with through the Python API provided by OpenAI Gym (Brockman et al., 2016) for ease of use and interaction. In the following sections, we will present the primary experimental results along with their corresponding analyses. More specific and detailed results can be found in the appendix E. Unless otherwise specified, the units on the horizontal axis of the graph represent 1M steps. ![Figure 3: Results of BACC and four baseline algorithms in the six continuous environments](image) General results on MuJoCo benchmark. We compare BACC to OAC(2019) (Ciosek et al., 2019), SAC(2018) (Haarnoja et al., 2018), TD3(2018) (Fujimoto et al., 2018) and RRS(2022) (Sun et al., 2022), four recent model-free RL methods that achieve state-of-the-art performance. All methods run with six random seeds. The policy network and the Q network are the same for all methods. BACC uses three hyper-parameter related to exploration, which has been introduced in section 4.4. We provide the value of all hyper-parameter in the appendix C. The results are organized based on the complexity of the environment, ranging from complex to simple, as illustrated by Figure 3(a) through Figure 3(f). The Humanoid environment is the most complex, and the Swimmer environment is the simplest. The state dim and action dimension are summarized in the appendix D. As shown in Figure 3, our method achieves promising results on this benchmark. On Humanoid-v2, BACC achieves state-of-the-art performance and is sample efficient than previous algorithms. On Ant-v2, BACC works slightly worse than the RRS algorithm in the final performance. On Halfcheetah-v2, our method get better sample efficient. On Walker2d-v2 and Hopper-v2, our method get similar results with others. On Walker2d-v2, our method work better in the early learning stage. Assessment on exploration. When evaluating the quality of exploration solely based on the results in Fig. 3, it might not provide a clear understanding of what is happening. Therefore, we conducted an analysis of the rewards obtained during each exploration. Comparing Fig. 3 and Fig. 4, we have the following observations: In the humanoid environment, the differences in exploration rewards are not very significant among the algorithms. However, there is a substantial difference in the learned policies. In the hopper environment, OAC’s exploration seems to have become ineffective, but the algorithm continues to improve its policy. These observations reveal important insights: The first observation suggests that high-quality exploration has a significant impact on improving policy learning. The second observation indicates that off-policy algorithms are more robust to suboptimal exploration results. In other words, poor exploration outcomes do not necessarily have a fatal impact on policy learning if the policy has been optimal (at about 0.8M steps). Additionally, this graph provides a more direct illustration of the effectiveness of our exploration strategy. Figure 5: (a-b) More results on RoboSchool, Flagrun-v1 and FlagrunHarder-v1.(c) We compare the exploration result of $\pi_Q$ vs $\pi_E$ in the Humanoid-v2 environment. (d) We added additional Q-functions to see if the results would improve. Additional discussion. We conducted experiments on the Roboschool simulation platform. We add comparative experiments in HumanoidFlagrun-v1 and HumanoidFlagrunHarder-v1 on Roboschool simulation. In the HumanoidFlagrun environment, the robot must run toward a randomly generated flag. In the HumanoidFlagrunHarder environment, the robot will be constantly bombarded by white cubes. In these two environments, the position of the flag is changed randomly, so the performances of the off-policy algorithm are pretty poor. In Fig. 5(a), it can be seen that using policy for exploration is better than using $Q$ for exploration. In Fig. 5(b), since the environment is more difficult so that most of the interactions occur in the early stage, our method can be better when the information of $Q$ can be properly utilized. In Fig. 5(c), we show that exploring with $\pi_E$ gives better results than $\pi_Q$, it indicates that our utilization of overestimation is effective. If the difference between the two Q-functions is larger, it should be more efficient. We are curious about the impact of adding more Q-functions on the results, so we tested the effect of three Q functions. In Fig. 5(d), it can be found that triple Q functions gives better and more stable results. 6 CONCLUSION In this paper, we have developed a practical policy-based exploration strategy for deterministic policy reinforcement learning in continuous action spaces, realizing Q-value-guided out-of-distribution exploration. We conducted experiments on the Mujoco and RoboSchool benchmarks. In comparison to prior methods, our approach achieves more effective action exploration and demonstrates substantial improvements over previous approaches in the most complex Humanoid-v2 environments. Reproducibility statement. We have included a detailed proof of the proposed theorem in Appendix A, a comprehensive algorithm description in Appendix B, and the experimental hyperparameters in Appendix C. Additionally, we provide our code in the supplementary material to facilitate the replication and verification of our results. REFERENCES Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. 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RjYKTQ0L0W	The paper argues about the method being more cost-effective than traditional crowd-sourced dataset curation, which might exceed $1M. Yet, directly sourcing text from the web can be free, and existing filtering methods can be used to ensure data quality.	GENIE: ACHIEVING HUMAN PARITY IN CONTENT-GROUNDED DATASETS GENERATION Asaf Yehudai ♦ , Boaz Carmeli ♦, Yosi Mass ♦, Ofir Ariv ♦, Nathaniel Mills ♦, Assaf Toledo ♦, Eyal Shnarch ♦, Leshem Choshen ♦ ♠ IBM Israel Research Lab ♦, Hebrew University of Jerusalem ♠, MIT ♠ {Asaf.Yehudai, leshem.choshen}@ibm.com ABSTRACT The lack of high-quality data for content-grounded generation tasks has been identified as a major obstacle to advancing these tasks. To address this gap, we propose Genie, a novel method for automatically generating high-quality content-grounded data. It consists of three stages: (a) Content Preparation, (b) Generation: creating task-specific examples from the content (e.g., question-answer pairs or summaries). (c) Filtering mechanism aiming to ensure the quality and faithfulness of the generated data. We showcase this methodology by generating three large-scale synthetic data, making wishes, for Long-Form Question-Answering (LFQA), summarization, and information extraction. In a human evaluation, our generated data was found to be natural and of high quality. Furthermore, we compare models trained on our data with models trained on human-written data – ELI5 and ASQA for LFQA and CNN-DailyMail for Summarization. We show that our models are on par with or outperforming models trained on human-generated data and consistently outperforming them in faithfulness. Finally, we applied our method to create LFQA data within the medical domain and compared a model trained on it with models trained on other domains. 1 INTRODUCTION Content-grounded generation is needed in various tasks, such as Retrieval-Augmented Generation (RAG), and content-based virtual assistants. In such tasks, the model is expected to generate a response based on a given content (i.e., information). For example, answer a question given a document that includes information needed for the answer. Zheng et al. (2023) found those types of tasks to be the second most common use cases of Language Models. Creating datasets with elaborate responses, which rely on long content, requires an expensive and demanding manual process. This may explain why such datasets are scarce even for popular tasks such as question-answering generation. Moreover, most existing datasets were collected from noisy available resources, such as news providers (Hermann et al., 2015) and Reddit user posts (Fan et al., 2019). This lack of high-quality content-grounded data has been identified as one of the obstacles to advancing long-form QA (Stelmakh et al., 2022) and domain-specific summarization (Zhu et al., 2020), among other content-based tasks. To address this gap, we suggest Genie, Generate information & elucidate, a method for creating synthetic training data for any domain and any content-grounded task. We propose a three-step process: (a) Content Preparation, (b) Generating, and (c) Filtering. Preparation is fairly straightforward, as the data may be noisy; it is best to clean it. The generation is done using a few-shot prompting approach with a large language model (LLM). See an example in App. D. Finally, since the generation is automatic, we filter its outputs to ensure their faithfulness, well-formedness, and overall quality. Genie offers flexibility and can generate synthetic data for different domains and content-grounded generation tasks. We apply it to the tasks of long-form QA (LFQA), summarization (§ 3), and information extraction (IE) (App. C) by creating wish-QA, wish-summarization, and wish-IE. We --- We wished for a cool name and that is what we’ve got. then show in a manual evaluation that it generates high-quality data that is natural, faithful, and lexically diverse (§4). For the task of LFQA, we compare the performance of models that were trained with wish-QA generated by Genie to those trained with the same amount of data, generated by humans (§5). We show that the former models outperform or are on par with the latter models. Additionally, faithfulness scores show that models trained on our synthetic data are more faithful to the grounding content. Those results show the overall efficacy and faithfulness of our data as training data compared to that of human-generated data. We replicate our success with summarization, showcasing the generality of the method. We publicly release all three wishes datasets. 2 AUTOMATICALLY CURATING DATASET FOR CONTENT-GROUNDED TASKS In Figure 1, we illustrate Genie’s three steps for automatically curating high-quality content-grounded datasets: Content Preparation, Generation, and Filtering. We refer to a content-grounded data point, like a question-answer pair or a summary, as an example. 2.1 CONTENT PREPARATION In the preparation step, we obtain the grounding content by extracting passages from raw documents using rule-based methods. This step is the least general of our approach as it relies on the specific format in which the data is stored. If the data already exists in easy-to-use passages, it can be used as is. For example, broken by lines, found in a table, or conveniently extracted from another dataset. As the general case, we describe the extraction of content passages directly from web pages. Implementation details. We crawled Wikipedia pages using browser emulation to allow dynamic content to be retrieved. Then we pass the full HTML DOM through filters to remove noise (e.g., headers, footers, sidebars, etc.). We are left with the main page content which is then transformed into Markdown, preserving the document structure (e.g., lists, tables, links, image references, articles, and sections). From this structure a table of contents is derived and based on it we break the Markdown page into passages. 2.2 Generation In the generation step, we prompt a large language model to generate a synthetic example. For that, we use the in-context capabilities of the model. We prompt the model with four content-example pairs followed by the extracted content from the corpus with no example (see the prompt in appendix §D). The LLM generates a new example to match the extracted content. Implementation details. We decode greedily, which encourages the models to produce more grounded responses (Honovich et al., 2022b). In addition, we create two variants of the data, one by generating examples using Falcon-40B (Penedo et al., 2023) and another by generating with Llama-2-70B (Touvron et al., 2023). In general, the results using the different models have similar tendencies, with Llama being slightly better (see replications with Llama in an appendix §B). As Falcon is purely pre-trained, without additional steps we mainly report results relying on Falcon, to showcase that our method is not dependent on further alignment and instruction steps. 2.3 Filtering In the filtering step, we score each content-example pair for its format, faithfulness (i.e., grounded), and quality. For each such aspect, we implement a scoring function and filter low-scoring pairs. Format. We filter out examples where parts of the template are missing (e.g., in QA, when the prefixes signifying the start of the question or the answer are absent). Furthermore, we filter examples that are too short (less than ten words) or too long (surpassing 1.5 times the length of the grounding content for LFQA, and 0.25 for Summarization). For QA, we use [document], [question], and [answer] as prefixes before each corresponding element. For summarization [document], [summarize], and [summary] with [summarize] representing the specific summarization instruction. It is important to note that we did not fine-tune these prompts. Faithfulness. To validate that the model-generated examples are grounded in the content, we adopt an off-the-shelf faithfulness metric and filter low-scoring examples. When deployed with trustworthy data, this can serve as a measure of correctness. We test faithfulness by mapping the problem into a Textual Entailment (Dagan et al., 2005) or Natural Language Inference (Bowman et al., 2015) (NLI) problem. NLI involves two input sentences: a hypothesis and a premise. The objective is to determine whether the hypothesis can be inferred from the premise, contradicts it, or is neutral with respect to it. NLI models were widely utilized for faithfulness consistency evaluation (Honovich et al., 2021; Dziri et al., 2022), and most simply by taking the grounding text as the premise and the generated example as the hypothesis (Maynez et al., 2020). Here, we use the fine-tuning T5-11B NLI model presented in (Honovich et al., 2022a) for assessing the generated example faithfulness. Quality. An important aspect of our methodology involves evaluating the quality of the generated examples, specifically quantifying their relevance to the corresponding task. Note that the task may be constant throughout a dataset (as is often the case of summarization) or be dependent upon an instruction (such as the question in question answering). To judge the quality automatically we use a reward model. Reward models are trained on human preference data to give a high reward for answers that human annotators prefer. Such models can quantify quality in a human-like way, considering dimensions that are hard to isolate and measure independently by dedicated metrics. Reward models are used as quality scores for Reinforcement Learning optimization (Ouyang et al., 2022), and also serve as reference-less evaluation metrics for text generation tasks (Touvron et al., 2023). Here, we use the reward model for both purposes and rely on the Open-Assistant model (Köpf et al., 2023), using the DeBERTa-v3 architecture (He et al., 2021). We filter generated examples whose score is below 0.5 by the reward model reward-model-deberta-v3-large-v2.² We chose 0.5 ²https://huggingface.co/OpenAssistant/reward-model-deberta-v3-large-v2 as a threshold based on experimentation. Similarly, we use t5_xxl_true_nli_mixture$^3$ model to filter examples deemed unfaithful by it. ### 3 EXPERIMENTAL SETUP Here we describe the datasets we utilized in our content-grounded generation tasks: LFQA and summarization (§3.1). Subsequently, we outline the various synthetic datasets we generated (§3.2), and finally, we discuss the models employed for training and the evaluation metrics (§3.4). #### 3.1 DATASETS ELI5. (Explain Like I’m Five) \cite{fan2019eli5} comprises open-ended questions and extensive responses authored by users within the Reddit forum of the same name. To these questions and answers, retrieved documents were added as grounding content. In their manual analysis, they have found that the content is sufficient to answer 65% of the questions and have information relevant to 92% of the questions. In this work, we use the KILT version of the dataset \cite{petroni2020kilt}. ASQA. (Answer Summaries for Questions which are Ambiguous) \cite{stelmakh2022asqa} is a dataset that pairs ambiguous questions from the AmbigQA dataset \cite{min2020ambigqa} with meticulously crafted long-form answers generated through crowdsourcing. To add grounding they have used the same method presented in ELI5, but specifically retrieved documents from Wikipedia. NQ. (Natural Questions) \cite{kwiatkowski2019natural} is a dataset of real user questions sourced from the Google search engine. It includes questions and their corresponding passages (named long answers) from Wikipedia which provide potential answers and contain extractive short answers. This dataset does not have long-form answers, and here we will use only its documents for our synthetic data generation process §3.2 and will compare our synthetic questions with the questions from NQ. CNN-DailyMail. \cite{kwiatkowski2019natural} is a dataset commonly used for text summarization. It consists of news articles from CNN and the DailyMail along with their human-written summaries. #### 3.2 GENERATING SYNTHETIC DATASETS The datasets described above were used to create datasets of synthetic data; Wish-QA-NQ. To create this dataset, we draw upon NQ passages \cite{kwiatkowski2019natural}, for our synthetic data generation process. These passages are well-suited for our process because they were originally extracted from Wikipedia pages by annotators and typically consist of well-structured paragraphs, each centered around a specific topic. Wish-QA ELI5/ASQA. For the creation of a dataset that mimics the conditions of ELI5 and ASQA, where answers can be derived from multiple documents, we rely on the top three retrieved passages from either of the corresponding corpus. These passages are used as the grounding documents for constructing this synthetic dataset. In addition, we make a new wish dataset entirely from crawled data: Wish-QA. stands for Wikipedia from Scratch$^4$ is a novel data we constructed following the general approach for crawling and processing as detailed in Section §2.1. It represents a realistic data generation use case from unprocessed content. We note that the extracted passages may exhibit noise and lack of coherence and conciseness. --- $^3$https://huggingface.co/google/t5_xxl_true_nli_mixture $^4$Wish-QA is also the general name for all our synthetic QA datasets 3.3 Models for Extrinsic Evaluation In the Extrinsic evaluation, our goal is to compare the performance of models trained on our synthetic content-grounded data with those trained on data generated by humans. To ensure a fair comparison, we maintain an equal number of examples from each dataset (10,000) and employ identical models for training, using the same set of hyperparameters. The models we use for training are Flan-xl (Wei et al., 2021) and llama-2-13b-Chat (Touvron et al., 2023). These models serve as the foundation for facilitating comparisons across architectural variations, including Encoder-Decoder and Decoder-only models, as well as different variations of instruction fine-tuning and alignment training. 3.4 Evaluation Metrics We evaluate the performance with ROUGE as a lexical similarity metric (Lin, 2004), BERT-Score as a model-based reference-based metric (Zhang et al., 2019b), and Reward model as a model-based reference-less metric. We reuse the ANLI faithfulness metric and reward mentioned in the filtering for evaluation. For faithfulness evaluation, we also calculate the K-Precision lexical similarity metric (Adlakha et al., 2023). Different performance metrics (Post, 2018; Zhang et al., 2019a; and more) showed similar results in initial trials, showing reliability of different forms (Perlitz et al., 2023). ROUGE. Following the conventional approach of assessing generated text quality, including long-form answers (Fan et al., 2019), we report the ROUGE-L score (Lin, 2004). BERT Score. (Zhang et al., 2019b) is a semantic similarity-based metric that leverages pre-trained language models to predict if the model response is semantically equivalent to the gold answer. Kasai et al. (2022) have shown BERT Score F1 is effective in evaluating many generation tasks. K-Precision. Following Adlakha et al. (2023) we report K-Precision, as it showed the highest correlation with human judgments from all lexical metrics. The metric follows the intuition, that in faithful response most words need to come from the content. 4 Intrinsic Evaluation In this section, we perform intrinsic evaluation and validation of Wish-QA. We conduct a micro Turing Test, presenting synthetic and human questions side by side. We show that the questions generated synthetically are more natural than most of those found in available datasets. We also test the whole workflow and show that the filters contribute to the generated data quality and that Genie is cost and time-efficient and creates diverse data. Naturalness Evaluation. To assess the naturalness of our questions, we conducted a human-evaluation experiment. In the experiment, an expert annotator$^5$ was provided with two questions: one human-created and the other synthetic. Both questions were based on the same content. The annotator’s task was to identify the question they believed was human-written. For this experiment, we sampled 100 questions from ELI5, ASQA, and NQ, along with their 100 synthetic counterparts. The results in Table 1 (and App. §B) indicate that for ELI5, the synthetic question was selected as the human-written one in 72% of the cases, for NQ it was 63%, and for ASQA it was 49%. These results suggest that our synthetic questions are more natural and human-like than questions collected from sources like Reddit and Google Search engine. Additionally, they are indistinguishable from questions written by experts, such as those in the ASQA dataset. As a side finding, we also find that the ASQA dataset is of higher quality than the others, which experiments below replicate. Multi-Dimensional Quality Assessment. In this assessment, we aimed to investigate the qualities of the generated data and the impact of the filtration processes. We focused on the following dimensions: relevance and clarity of the questions, and faithfulness and overall quality of the answers. To accomplish that, we randomly selected 100 questions from the unfiltered and filtered Wish-QA. For each content-question-answer triplet, we asked annotators to answer a list of questions as shown in --- $^5$A non-author, native English speaker with an MA degree. The first two assessment questions aim to assess the relevance and clarity of the question. The clarity question is inspired by the findings of Min et al. (2020), which revealed that more than half of naturally occurring factoid questions are ambiguous. Following that, we include three questions related to the answer quality. These questions are designed to ascertain whether the answer adequately addresses the question while remaining faithful to the underlying content. Lastly, we ask for an overall quality rating on a 5-level Likert scale. Human assessment results in Table 1 demonstrate that the filtration process had a significant impact on the relevance of the questions. Although our filtration setup does not directly assess the questions, we find that our faithfulness filter together with the reward filter provides an indirect signal about the relevance of the question. We also observed an improvement in the percentage of answers that were found to address the question. Faithfulness results show decent improvement, but there is still room for enhancement. Annotators’ interviews reveal that despite the presence of unfaithful cases in the dataset, their granularity was often more subtle. In some instances, the model added missing pieces of information that were subsequently found to be factually correct. We observe a slight improvement in the clarity of questions, coupled with almost all answers addressing the questions. This highlights that our answer is a single relevant response from a wide space of plausible answers, a well-documented phenomenon in LFQA (Krishna et al., 2021). Lastly, we identify an improvement in the overall score, which leads us to the conclusion that the filtering process substantially contributes to the quality and faithfulness of our dataset. Table 1: Multi-Dimensional Quality assessment for synthetic data generated from scratch. Results show a large improvement in question relevance and the percentage of answers that address the question, answers that are faithful, and overall answer scores. \| Quality Review Question \| Wish-QA w/o filters \| Wish-QA w/ filters \| \|-------------------------\|---------------------\|--------------------\| \| Is the question relevant to the content? \| 67% \| 92% \| \| Is the question clear? (not ambiguous) \| 63% \| 67% \| \| Does the answer address the question? \| 80% \| 98% \| \| Is the answer faithful to the content? \| 53% \| 76% \| \| Grade the overall quality of the answer \| 3.48 \| 4.58 \| Diversity. Our synthetic data is built on top of large-scale content that covers many different distinct topics. As a result, our data contain diverse lexicons. We compute vocd-D (McCarthy & Jarvis, 2010) to measure the lexical diversity of our data. We found that the lexical diversity of all synthetic data is higher than their human-generated counterparts (see Table 6). We also can see that most response lengths are similar to the ones in the human writing datasets. Scale. With 300K samples overall (full statistics in App. A), our dataset collection balances scale and quality. ELI5 is of a similar size but noisy, and ASQA is carefully annotated but much smaller. Monetary and Time Cost. Genie is more cost-efficient and time-efficient than the traditional approach of crowd-sourced dataset curation. The cost of API’s calls of models like the ones used typically ranges from $0.02 to $0.04, while the cost of an expert annotator to create a question is approximately $4.45 (Stelmakh et al., 2022). According to this rate, the 300K examples in our synthetic dataset would have cost over $1M. The time it takes to generate 10 examples is less than a minute, i.e. much faster than the time that it would take a human to read the context. ## 5 EXTRINSIC EVALUATION Finding the synthetic data to be of high quality, we test its usefulness for improving training. We present quantitative results from our extrinsic experiments, evaluating models trained on synthetic and human-generated data on the ASQA and ELI5 test sets. In Table 2 (and App. § B), we present Flan-xl results trained on human and synthetic data. We note that here by synthetic in-domain we refer to the case where the train and test come from the same dataset, either ELI5 or ASQA. Results indicate that synthetic data is a competitive alternative even when human-generated data already exists. In all cases, we see substantial gains from training on the synthetic data. For example, Rouge-L almost triples from 10.5 to 28.2 for Synthetic NQ. This gain is over the already strong multitask baseline (Flan) that trained on thousands of tasks, many of which are forms of question answering. Moreover, the synthetic data provides better or comparable results in all metrics even for cases where train and test data come from the same dataset. While – for ASQA – Rouge-L and Bert-Score are slightly lower than the in-domain training data, the synthetic data is even better than the human data on the rest of the scores on ELI5. We conclude that, if no human-generated data exists, automatically generating it has the potential to be as good. ASQA performs better on both ASQA and ELI5 test sets. This observation implies that ASQA is, on the whole, a superior dataset compared to ELI5. This aligns with the substantial annotation efforts invested in the creation of ASQA, in contrast to the noisy and automatically scraped ELI5 data. However, it is important to note that this meticulous curation has led to a considerably smaller dataset for ASQA, totaling approximately 6k examples including the development and test sets (compared to 272k examples in ELI5). This emphasizes the contribution of our approach which allows large-scale high-quality data generation. Another strong support for the effectiveness of our data generation approach is exemplified by the model’s outputs being favored by the preference reward model, achieving comparable or higher results than the gold standard of both datasets. Wish-QA seems to work well even with the noisy content. Wish-QA-NQ data outperformed the synthetic in-domain data across all metrics. This can be due to the quality of the Wish-QA-NQ being favorable or that a signal document generation setup is slightly preferable. The performance on CNN-DailyMail, presented in Table 4, shows that Wish-summarization data improves upon the strong Flan-xl baseline, in Bert-Score and Reward score but not on ROUGE-L. Overall, the dataset seems comparable, attesting to the flexibility of the method. ### 5.1 Faithfulness Results Overall, Table 3 suggests training on our synthetic data leads to more content-faithful models. Models trained on Wish-QA-NQ and Synthetic data, and Wish-QA were more faithful than those trained on ASQA and ELI5 data by k-Precision and ANLI metrics. This aligns with Krishna et al. (2021), indicating LFQA models generated answers that are not grounded in the retrieved documents, and assert that this is one of the hurdles for filed progress. Flan-xl achieves the highest Faithfulness scores followed by the synthetic datasets. Flan’s achievement can be the result of its shorter and almost extractive answers. Taking into account that it is also substantially underperforming, we deduce that the synthetic datasets achieve the best trade-off across performance and faithfulness. The faithfulness results for CNN-DailyMail are consistently high. As we observed, Flan-xl tends to produce more extractive responses. Since CNN-DailyMail primarily contains extractive summarization, it’s no surprise that it exhibits high faithfulness scores. However, the model trained on our data, which doesn’t emphasize extractiveness as a hard requirement, outperforms Flan-xl in terms of k-Precision, matches it in terms of NLI, and achieves the highest average level of faithfulness. In summary, our quantitative analysis affirms that the utilization of synthetic data substantially enhances answer quality in both ASQA and ELI5 datasets. Our approach not only matches human-generated responses but also quantitatively surpasses them in terms of reward, highlighting its potential for generating higher-quality answers. Additionally, our method ensures high faithfulness and grounding in the generated responses, setting it apart from existing datasets. ### 6 Domain Adaptation We have demonstrated that our method can generate synthetic data as good as human-generated data. Next, we hypothesize generating data directly in the target domain is more effective than from another domain for a given task. To investigate this hypothesis, we define our test set as Table 2: Performance comparison of Flan-xl models trained on human-generated and synthetic data. The results reveal that our synthetic data consistently outperforms or achieves comparable performance to human-generated data, as indicated by ROUGE-L and Bert-Score metrics. Additionally, by reward score, models trained on our synthetic data exhibit superior or comparable performance to the gold standard responses. \| Test-set \| ASQA \| ELI5 \| \|----------------\|---------------\|---------------\| \| Train Set \| ROUGE-L \| Bert-Score \| Reward \| ROUGE-L \| Bert-Score \| F1 \| Reward \| \| Flan-xl \| 10.5 \| 49.7 \| 28.8 \| 6.2 \| 46.7 \| 9.2 \| \| ASQA \| 31.4 \| 66.0 \| 68.6 \| 13.5 \| 52.2 \| 24.4 \| \| ELI5 \| 18.7 \| 58.7 \| 37.2 \| 13.1 \| 51.3 \| 11.3 \| \| Wish-QA \| 28.0 \| 67.5 \| 85.1 \| 13.8 \| 55.2 \| 26.7 \| \| Wish-QA-NQ \| 28.2 \| 64.8 \| 80.3 \| 13.2 \| 54.0 \| 30.3 \| \| Wish-QA in-domain \| 27.0 \| 63.4 \| 73.3 \| 13.1 \| 52.8 \| 22.7 \| \| Gold \| - \| - \| 72.1 \| - \| - \| 30.3 \| Table 3: Fiathfullnes performance Comparison of Flan-xl Models Trained on Human-Created and Synthetic Data. The results demonstrate that our synthetic data consistently outperforms both human-generated data and gold responses, as indicated by the k-Precision, and ANLI metrics. Flan-xl stands out with the highest scores, which can be attributed to the extractive nature of its responses. \| Test-set \| ASQA \| ELI5 \| \|----------------\|---------------\|---------------\| \| Train Set \| k-Precision \| ANLI \| k-Precision \| ANLI \| \| Flan-xl \| 98.2 \| 88.7 \| 89.2 \| 84.9 \| \| ASQA \| 67.5 \| 55.7 \| 52.2 \| 34.3 \| \| ELI5 \| 52.9 \| 33.5 \| 29.0 \| 5.6 \| \| Wish-QA \| 77.9 \| 74.9 \| 58.5 \| 37.9 \| \| Wish-QA-NQ \| 79.3 \| 75.5 \| 60.4 \| 43.3 \| \| Wish in-domain \| 81.9 \| 79.1 \| 68.3 \| 52.8 \| \| Gold \| 46.3 \| 25.3 \| 20.6 \| 2.7 \| PubMed-QA, which focuses on LFQA in the medical domain. Accordingly, we create synthetic question-answering data on PubMed papers (Wish-QA-MED) as task data in the target domain. We then compare the performance of models trained on Wish-QA-MED dataset with those trained on Wish-QA-NQ data, as well as with models trained on the human-created ELI5 and ASQA datasets. The results in Table 5 demonstrate that the synthetic dataset outperforms ELI5 and is comparable to or slightly better than ASQA in ROUGE-L and Bert-Score. Additionally, there is a more substantial gap in terms of reward and faithfulness. Interestingly, Wish-QA-NQ and Wish-QA-MED achieve similar results, echoing the finding that Wish-QA outperforms other datasets. This suggests that out-of-domain data holds little disadvantage over in-domain data and can often surpass it. One explanation may be that providing the content with the task (e.g., QA) makes the model rely less on the training domain. Supportive evidence is the finding of Onoe et al. (2023), who found that, in their task, update strategies lag behind the performance of simply concatenating the content to the prompt. This may mean that the model relies on the content more than was previously thought (Neeman et al., 2023). Table 4: Performance Comparison of Flan-xl Models Trained on Human-Created and Wish-Summarization Data. The results reveal that our synthetic data achieves comparable performance to human-generated data. \| Test-set \| CNN-DailyMail \| \|----------------\|---------------\| \| Train-set \| ROUGE-L \| Bert-Score \| Reward \| k-Precision \| ANLI \| \| Flan-xl \| 30.2 \| 70.9 \| 96.3 \| 97.6 \| 98.7 \| \| CNN-DailyMail \| 33.3 \| 72.7 \| 96.5 \| 97.0 \| 99.1 \| \| Wish-Summarization \| 28.6 \| 71.3 \| 97.5 \| 98.2 \| 98.7 \| Table 5: Performance of Flan-xl Models on PubMed test data. The results reveal that our synthetic data consistently outperforms or achieves comparable performance to human-generated data in general and faithfulness metrics. Results suggest that in-domain data don’t provide additional improvement for content-grounded generation, but may help the faithfulness of the model. \| Train-set \| ROUGE-L \| Bert-Score \| Reward \| K-Precision \| ANLI \| \|-----------------\|---------\|------------\|--------\|-------------\|------\| \| Flan-xl \| 12.8 \| 53.8 \| 10.7 \| 60.6 \| 38.2 \| \| ASQA \| 20.5 \| 61.4 \| 37.3 \| 77.2 \| 60.8 \| \| ELI5 \| 15.0 \| 56.3 \| 16.8 \| 32.2 \| 2.2 \| \| Wish-QA-MED \| 22.1 \| 61.6 \| 39.4 \| 78.2 \| 81.8 \| \| Wish-QA-NQ \| 22.0 \| 62.9 \| 44.5 \| 84.2 \| 73.1 \| The faithfulness scores are inconclusive, while ANLI indicates that in-domain synthetic improves faithfulness, the k-Precision says otherwise, suggesting at least parity. We conclude that Genie can be beneficial in creating human-level data for many tasks and domains, however, it can be that LFQA is flexible in terms of its training data domain. We leave for future research to check this finding and to show tasks or dimensions that exhibit improvement due to target domain data and can benefit from our method. 7 RELATED WORK Our work is far from the first to propose synthetic data for training or experimentation (Choshen & Abend, 2019; Agarwal et al., 2020). Recently, generating data from a large language model to train a smaller one was suggested as a weak form of distillation to improve smaller models (West et al., 2022). Our method does not focus on distillation. Apart from using a stronger model for the synthetic data, those methods differ from ours as the learned model mimics a diverse set of skills, rather than becoming an expert on a task. Still, there are a few synthetic methods for specific tasks. Most notably, methods that rely on a 2-step process, generation, and filtering. West et al. (2022) presented a 2-step pipeline for Symbolic Knowledge Distillation, rather than for creating content-grounded data. Kim et al. (2022) apply this method to create a social dialogue dataset. In Unnatural Instructions and Self-Instruct (Honovich et al., 2022b; Wang et al., 2022), they applied this method for the creation of an instruction dataset. Their method relies on model knowledge for content-grounded tasks. Similarly, Bitton et al. (2023) q2d approach uses a 2-step process for creating information-seeking dialogs. Those works share similar mechanisms with our method but differ in the content-grounded aspect of our work. The dialog inpainting approach (Dai et al., 2022), shares a common objective with ours, to generate content-grounded question answering. They add questions between the document sentences to create a dialogue. This approach ensures the groundedness of the dialogue but it comes at the cost of less fluent and neutral conversation. In our approach, we generate the question and answer using the LLM and verify its groundedness and quality to allow both faithfulness and naturalness. 8 DISCUSSION Our work introduces Genie, an efficient and cost-effective automated approach for curating content-grounded datasets. Our method incorporates a novel filtering mechanism to ensure data quality. We demonstrate that our synthetic wish-QA and wish-summarization data achieves parity with expert human datasets in both intrinsic and extrinsic evaluations. Furthermore, we illustrate that our data surpasses human-written datasets in terms of lexical diversity and faithfulness. We have also proven the applicability of our method to noisy crawled data. We want to emphasize the immense potential this approach holds for facilitating the development of content-focused datasets and, consequently, generative models, minimizing the need for costly human annotation. Therefore, our method democratizes the creation of such datasets and models, making them more accessible to the entire community. REFERENCES Vaibhav Adlakha, Parishad BehnamGhader, Xing Han Lu, Nicholas Meade, and Siva Reddy. Evaluating correctness and faithfulness of instruction-following models for question answering. arXiv preprint arXiv:2307.16877, 2023. Oshin Agarwal, Heming Ge, Siamak Shakeri, and Rami Al-Rfou. Knowledge graph based synthetic corpus generation for knowledge-enhanced language model pre-training. arXiv preprint arXiv:2010.12688, 2020. Yonatan Bitton, Shlomi Cohen-Ganor, Ido Hakimi, Yoad Lewenberg, Roei Aharoni, and Enav Weinreb. q2d: Turning questions into dialogs to teach models how to search. arXiv preprint arXiv:2304.14318, 2023. Samuel R Bowman, Gabor Angeli, Christopher Potts, and Christopher D Manning. A large annotated corpus for learning natural language inference. arXiv preprint arXiv:1508.05326, 2015. Leshem Choshen and Omri Abend. Automatically extracting challenge sets for non-local phenomena in neural machine translation. In Conference on Computational Natural Language Learning (CoNLL), pp. 291–303, November 2019. doi: 10.18653/v1/K19-1028. URL [https://aclanthology.org/K19-1028](https://aclanthology.org/K19-1028). Ido Dagan, Oren Glickman, and Bernardo Magnini. The pascal recognising textual entailment challenge. In Machine learning challenges workshop, pp. 177–190. Springer, 2005. Zhuyun Dai, Arun Tejasvi Chaganty, Vincent Y Zhao, Aida Amini, Qazi Mamunur Rashid, Mike Green, and Kelvin Guu. Dialog inpainting: Turning documents into dialogs. In International Conference on Machine Learning, pp. 4558–4586. PMLR, 2022. Nouha Dziri, Hannah Rashkin, Tal Linzen, and David Reitter. Evaluating attribution in dialogue systems: The begin benchmark. Transactions of the Association for Computational Linguistics, 10:1066–1083, 2022. Angela Fan, Yacine Jernite, Ethan Perez, David Grangier, Jason Weston, and Michael Auli. 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URL [https://aclanthology.org/2021.emnlp-main.619](https://aclanthology.org/2021.emnlp-main.619). Or Honovich, Roei Aharoni, Jonathan Herzig, Hagai Taitelbaum, Doron Kukliansy, Vered Cohen, Thomas Scialom, Idan Szpektor, Avinatan Hassidim, and Yossi Matias. True: Re-evaluating factual consistency evaluation. arXiv preprint arXiv:2204.04991, 2022a. Or Honovich, Thomas Scialom, Omer Levy, and Timo Schick. Unnatural instructions: Tuning language models with (almost) no human labor. arXiv preprint arXiv:2212.09689, 2022b. Jungo Kasai, Keisuke Sakaguchi, Ronan Le Bras, Lavinia Dunagan, Jacob Morrison, Alexander Fabbri, Yejin Choi, and Noah A. Smith. Bidimensional leaderboards: Generate and evaluate language hand in hand. In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 3540–3557, Seattle, United States, July 2022. 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