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{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# This exercise wasn't exactly smooth sailing for me, but I did try to understand most of it. Will try to come back to this whenever I can"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# there no change change in the first several cells from last lecture\n",
"\n",
"import torch\n",
"import torch.nn.functional as F\n",
"import matplotlib.pyplot as plt # for making figures\n",
"%matplotlib inline"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# download the names.txt file from github\n",
"!wget https://raw.githubusercontent.com/karpathy/makemore/master/names.txt"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"# read in all the words\n",
"words = open('names.txt', 'r').read().splitlines()\n",
"# print(len(words))\n",
"# print(max(len(w) for w in words))\n",
"# print(words[:8])\n",
"\n",
"# build the vocabulary of characters and mappings to/from integers\n",
"chars = sorted(list(set(''.join(words))))\n",
"stoi = {s:i+1 for i,s in enumerate(chars)}\n",
"stoi['.'] = 0\n",
"itos = {i:s for s,i in stoi.items()}\n",
"vocab_size = len(itos)\n",
"# print(itos)\n",
"# print(vocab_size)\n",
"\n",
"# build the dataset\n",
"block_size = 3 # context length: how many characters do we take to predict the next one?\n",
"\n",
"def build_dataset(words):\n",
" X, Y = [], []\n",
"\n",
" for w in words:\n",
" context = [0] * block_size\n",
" for ch in w + '.':\n",
" ix = stoi[ch]\n",
" X.append(context)\n",
" Y.append(ix)\n",
" context = context[1:] + [ix] # crop and append\n",
"\n",
" X = torch.tensor(X)\n",
" Y = torch.tensor(Y)\n",
" # print(X.shape, Y.shape)\n",
" return X, Y\n",
"\n",
"import random\n",
"random.seed(42)\n",
"random.shuffle(words)\n",
"n1 = int(0.8*len(words))\n",
"n2 = int(0.9*len(words))\n",
"\n",
"Xtr, Ytr = build_dataset(words[:n1]) # 80%\n",
"Xdev, Ydev = build_dataset(words[n1:n2]) # 10%\n",
"Xte, Yte = build_dataset(words[n2:]) # 10%"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"# utility function we will use later when comparing manual gradients to PyTorch gradients\n",
"def cmp(s, dt, t):\n",
" ex = torch.all(dt == t.grad).item()\n",
" app = torch.allclose(dt, t.grad)\n",
" maxdiff = (dt - t.grad).abs().max().item()\n",
" print(f'{s:15s} | exact: {str(ex):5s} | approximate: {str(app):5s} | maxdiff: {maxdiff}')"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"4137\n"
]
}
],
"source": [
"n_embd = 10 # the dimensionality of the character embedding vectors\n",
"n_hidden = 64 # the number of neurons in the hidden layer of the MLP\n",
"\n",
"g = torch.Generator().manual_seed(2147483647) # for reproducibility\n",
"C = torch.randn((vocab_size, n_embd), generator=g)\n",
"# Layer 1\n",
"W1 = torch.randn((n_embd * block_size, n_hidden), generator=g) * (5/3)/((n_embd * block_size)**0.5)\n",
"b1 = torch.randn(n_hidden, generator=g) * 0.1 # using b1 just for fun, it's useless because of BN\n",
"# Layer 2\n",
"W2 = torch.randn((n_hidden, vocab_size), generator=g) * 0.1\n",
"b2 = torch.randn(vocab_size, generator=g) * 0.1\n",
"# BatchNorm parameters\n",
"bngain = torch.randn((1, n_hidden))*0.1 + 1.0\n",
"bnbias = torch.randn((1, n_hidden))*0.1\n",
"\n",
"# Note: I am initializating many of these parameters in non-standard ways\n",
"# because sometimes initializating with e.g. all zeros could mask an incorrect\n",
"# implementation of the backward pass.\n",
"\n",
"parameters = [C, W1, b1, W2, b2, bngain, bnbias]\n",
"print(sum(p.nelement() for p in parameters)) # number of parameters in total\n",
"for p in parameters:\n",
" p.requires_grad = True"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
"batch_size = 32\n",
"n = batch_size # a shorter variable also, for convenience\n",
"# construct a minibatch\n",
"ix = torch.randint(0, Xtr.shape[0], (batch_size,), generator=g)\n",
"Xb, Yb = Xtr[ix], Ytr[ix] # batch X,Y"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor(3.3479, grad_fn=<NegBackward0>)"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# forward pass, \"chunkated\" into smaller steps that are possible to backward one at a time\n",
"\n",
"emb = C[Xb] # embed the characters into vectors\n",
"embcat = emb.view(emb.shape[0], -1) # concatenate the vectors\n",
"# Linear layer 1\n",
"hprebn = embcat @ W1 + b1 # hidden layer pre-activation\n",
"# BatchNorm layer\n",
"bnmeani = 1/n*hprebn.sum(0, keepdim=True)\n",
"bndiff = hprebn - bnmeani\n",
"bndiff2 = bndiff**2\n",
"bnvar = 1/(n-1)*(bndiff2).sum(0, keepdim=True) # note: Bessel's correction (dividing by n-1, not n)\n",
"bnvar_inv = (bnvar + 1e-5)**-0.5\n",
"bnraw = bndiff * bnvar_inv\n",
"hpreact = bngain * bnraw + bnbias\n",
"# Non-linearity\n",
"h = torch.tanh(hpreact) # hidden layer\n",
"# Linear layer 2\n",
"logits = h @ W2 + b2 # output layer\n",
"# cross entropy loss (same as F.cross_entropy(logits, Yb))\n",
"logit_maxes = logits.max(1, keepdim=True).values\n",
"norm_logits = logits - logit_maxes # subtract max for numerical stability\n",
"counts = norm_logits.exp()\n",
"counts_sum = counts.sum(1, keepdims=True)\n",
"counts_sum_inv = counts_sum**-1 # if I use (1.0 / counts_sum) instead then I can't get backprop to be bit exact...\n",
"probs = counts * counts_sum_inv\n",
"logprobs = probs.log()\n",
"loss = -logprobs[range(n), Yb].mean()\n",
"\n",
"# PyTorch backward pass\n",
"for p in parameters:\n",
" p.grad = None\n",
"for t in [logprobs, probs, counts, counts_sum, counts_sum_inv, # afaik there is no cleaner way\n",
" norm_logits, logit_maxes, logits, h, hpreact, bnraw,\n",
" bnvar_inv, bnvar, bndiff2, bndiff, hprebn, bnmeani,\n",
" embcat, emb]:\n",
" t.retain_grad()\n",
"loss.backward()\n",
"loss"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"#The entire Exercise 1 implementation combined\n",
"\n",
"dlogprobs = torch.zeros_like(logprobs)\n",
"dlogprobs[range(n), Yb] = -1.0/n\n",
"dprobs = (1.0 / probs) * dlogprobs\n",
"dcounts_sum_inv = (counts * dprobs).sum(1, keepdim=True)\n",
"dcounts = counts_sum_inv * dprobs\n",
"dcounts_sum = (-counts_sum**-2) * dcounts_sum_inv\n",
"dcounts += torch.ones_like(counts) * dcounts_sum\n",
"dnorm_logits = counts * dcounts\n",
"dlogits = dnorm_logits.clone()\n",
"dlogit_maxes = (-dnorm_logits).sum(1, keepdim=True)\n",
"dlogits += F.one_hot(logits.max(1).indices, num_classes=logits.shape[1]) * dlogit_maxes\n",
"dh = dlogits @ W2.T\n",
"dW2 = h.T @ dlogits\n",
"db2 = dlogits.sum(0)\n",
"dhpreact = (1.0 - h**2) * dh\n",
"dbngain = (bnraw * dhpreact).sum(0, keepdim=True)\n",
"dbnraw = bngain * dhpreact\n",
"dbnbias = dhpreact.sum(0, keepdim=True)\n",
"dbndiff = bnvar_inv * dbnraw\n",
"dbnvar_inv = (bndiff * dbnraw).sum(0, keepdim=True)\n",
"dbnvar = (-0.5*(bnvar + 1e-5)**-1.5) * dbnvar_inv\n",
"dbndiff2 = (1.0/(n-1))*torch.ones_like(bndiff2) * dbnvar\n",
"dbndiff += (2*bndiff) * dbndiff2\n",
"dhprebn = dbndiff.clone()\n",
"dbnmeani = (-dbndiff).sum(0)\n",
"dhprebn += 1.0/n * (torch.ones_like(hprebn) * dbnmeani)\n",
"dembcat = dhprebn @ W1.T\n",
"dW1 = embcat.T @ dhprebn\n",
"db1 = dhprebn.sum(0)\n",
"demb = dembcat.view(emb.shape)\n",
"dC = torch.zeros_like(C)\n",
"for k in range(Xb.shape[0]):\n",
" for j in range(Xb.shape[1]):\n",
" ix = Xb[k,j]\n",
" dC[ix] += demb[k,j]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Similar boiler plate codes as done in the prev exercise and provided in the starter code^\n",
"\n",
"------------"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"[1:36:38](https://youtu.be/q8SA3rM6ckI?si=Lo5Ly5jApvwIBfy9&t=6516) to 1:48:35 - Pen and Paper derivation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"[1:48:36](https://youtu.be/q8SA3rM6ckI?si=Lo5Ly5jApvwIBfy9&t=6516) to - Implementation of the derivation in code"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max diff: tensor(7.1526e-07, grad_fn=<MaxBackward1>)\n"
]
}
],
"source": [
"# Exercise 3: backprop through batchnorm but all in one go\n",
"# to complete this challenge look at the mathematical expression of the output of batchnorm,\n",
"# take the derivative w.r.t. its input, simplify the expression, and just write it out\n",
"# BatchNorm paper: https://arxiv.org/abs/1502.03167\n",
"\n",
"# forward pass\n",
"\n",
"# before:\n",
"# bnmeani = 1/n*hprebn.sum(0, keepdim=True)\n",
"# bndiff = hprebn - bnmeani\n",
"# bndiff2 = bndiff**2\n",
"# bnvar = 1/(n-1)*(bndiff2).sum(0, keepdim=True) # note: Bessel's correction (dividing by n-1, not n)\n",
"# bnvar_inv = (bnvar + 1e-5)**-0.5\n",
"# bnraw = bndiff * bnvar_inv\n",
"# hpreact = bngain * bnraw + bnbias\n",
"\n",
"# now:\n",
"hpreact_fast = bngain * (hprebn - hprebn.mean(0, keepdim=True)) / torch.sqrt(hprebn.var(0, keepdim=True, unbiased=True) + 1e-5) + bnbias\n",
"print('max diff:', (hpreact_fast - hpreact).abs().max())"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"hprebn | exact: False | approximate: True | maxdiff: 9.313225746154785e-10\n"
]
}
],
"source": [
"# backward pass\n",
"\n",
"# before we had:\n",
"# dbnraw = bngain * dhpreact\n",
"# dbndiff = bnvar_inv * dbnraw\n",
"# dbnvar_inv = (bndiff * dbnraw).sum(0, keepdim=True)\n",
"# dbnvar = (-0.5*(bnvar + 1e-5)**-1.5) * dbnvar_inv\n",
"# dbndiff2 = (1.0/(n-1))*torch.ones_like(bndiff2) * dbnvar\n",
"# dbndiff += (2*bndiff) * dbndiff2\n",
"# dhprebn = dbndiff.clone()\n",
"# dbnmeani = (-dbndiff).sum(0)\n",
"# dhprebn += 1.0/n * (torch.ones_like(hprebn) * dbnmeani)\n",
"\n",
"# calculate dhprebn given dhpreact (i.e. backprop through the batchnorm)\n",
"# (you'll also need to use some of the variables from the forward pass up above)\n",
"\n",
"#This is a direct implementation of what sensei did, as he said in the video this equation itself has a lot of breakdown steps to be considered\n",
"#But this is what we come up with at the end\n",
"dhprebn = bngain*bnvar_inv/n * (n*dhpreact - dhpreact.sum(0) - n/(n-1)*bnraw*(dhpreact*bnraw).sum(0))\n",
"\n",
"cmp('hprebn', dhprebn, hprebn) # I can only get approximate to be true, my maxdiff is 9e-10"
]
}
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