Module 3 · Algorithms · ~25 min
3.5 How much work?
Why do AI companies buy so many GPUs? Count the work. Even tiny microgpt does thousands of multiplications per letter; real models do billions.
1Watch
A model’s knowledge lives in tables of numbers called weights(like the lists of lists from lesson 2.3). When a letter passes through a table, each weight is used in exactly one multiplication.
Take a small table: two rows of three weights, and three inputs, 10, 20 and 30. Each row multiplies its weights by the inputs, one pair at a time, and adds them up:
- row [1, 2, 3]: 1×10 + 2×20 + 3×30 = 140 (3 multiplications)
- row [4, 5, 6]: 4×10 + 5×20 + 6×30 = 320 (3 multiplications)
Two rows × three columns = 6 multiplications. The rule is rows × columns, so a 16 × 16 table costs 256.
For each letter (models call the pieces they read tokens; here a token is one letter), microgpt’s seven weight tables need 3,504 multiplications. Make it 1,000 numbers wide instead of 16 and it’s about 12 million per letter, and that’s just the weight tables. A GPU is a chip with thousands of small workers that do multiplications side by side, instead of a few at a time.
2Explore
Change anything and run again. You can’t break it.
3Build
Two blanks. First, multiplications(rows, cols) counts the multiplications in a table, row by row, like the 2 × 3 example. Second, per_token(width) adds up all seven of microgpt’s tables for one letter (their shapes are listed in the starter). Then per_token(16) and per_token(1000) come from your own code.
# How much work? Count the multiplications.
# A table of weights with 2 rows of 3, and 3 inputs. Each row multiplies
# its weights by the inputs, one pair at a time, and adds them up.
table = [[1, 2, 3],
[4, 5, 6]]
inputs = [10, 20, 30]
count = 0
for row in table:
total = 0
for w, x in zip(row, inputs):
total += w * x
count += 1 # one multiplication
print('row total:', total)
print('multiplications:', count)
def multiplications(rows, cols):
count = 0
# TODO: count them row by row, like above: each row does
# one multiplication per column
return count
# A token is one letter here. These are microgpt's tables for one letter
# (you'll meet them by name in Modules 8 and 9). width is 16 in microgpt.
# 4 square tables: width x width
# 1 table 4 times taller: (4 * width) x width
# 1 table 4 times wider: width x (4 * width)
# 1 output table: 27 x width (one row per symbol)
def per_token(width):
return 0 # TODO: add up multiplications(...) for all 7 tables
print('microgpt, 16 wide:', per_token(16))
print('1,000 wide:', per_token(1000))
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5Unlocked in microgpt
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microgpt.py50 / 175 lines learned
1"""
2The most atomic way to train and run inference for a GPT in pure, dependency-free Python.
3This file is the complete algorithm.
4Everything else is just efficiency.
5
6@karpathy
7"""
8
9import os # os.path.exists
10import math # math.log, math.exp
11import random # random.seed, random.choices, random.gauss, random.shuffle
12random.seed(42) # Let there be order among chaos
13
14# Let there be a Dataset `docs`: list[str] of documents (e.g. a list of names)
15if not os.path.exists('input.txt'):
16 import urllib.request
17 names_url = 'https://raw.githubusercontent.com/karpathy/makemore/988aa59/names.txt'
18 urllib.request.urlretrieve(names_url, 'input.txt')
19docs = [line.strip() for line in open('input.txt') if line.strip()]
20random.shuffle(docs)
21print(f"num docs: {len(docs)}")
22
23# Let there be a Tokenizer to translate strings to sequences of integers ("tokens") and back
24uchars = sorted(set(''.join(docs))) # unique characters in the dataset become token ids 0..n-1
25BOS = len(uchars) # token id for a special Beginning of Sequence (BOS) token
26vocab_size = len(uchars) + 1 # total number of unique tokens, +1 is for BOS
27print(f"vocab size: {vocab_size}")
28
29# Let there be Autograd to recursively apply the chain rule through a computation graph
30class Value:
31 __slots__ = ('data', 'grad', '_children', '_local_grads') # Python optimization for memory usage
32
33 def __init__(self, data, children=(), local_grads=()):
34 self.data = data # scalar value of this node calculated during forward pass
35 self.grad = 0 # derivative of the loss w.r.t. this node, calculated in backward pass
36 self._children = children # children of this node in the computation graph
37 self._local_grads = local_grads # local derivative of this node w.r.t. its children
38
39 def __add__(self, other):
40 other = other if isinstance(other, Value) else Value(other)
41 return Value(self.data + other.data, (self, other), (1, 1))
42
43 def __mul__(self, other):
44 other = other if isinstance(other, Value) else Value(other)
45 return Value(self.data * other.data, (self, other), (other.data, self.data))
46
47 def __pow__(self, other): return Value(self.data**other, (self,), (other * self.data**(other-1),))
48 def log(self): return Value(math.log(self.data), (self,), (1/self.data,))
49 def exp(self): return Value(math.exp(self.data), (self,), (math.exp(self.data),))
50 def relu(self): return Value(max(0, self.data), (self,), (float(self.data > 0),))
51 def __neg__(self): return self * -1
52 def __radd__(self, other): return self + other
53 def __sub__(self, other): return self + (-other)
54 def __rsub__(self, other): return other + (-self)
55 def __rmul__(self, other): return self * other
56 def __truediv__(self, other): return self * other**-1
57 def __rtruediv__(self, other): return other * self**-1
58
59 def backward(self):
60 topo = []
61 visited = set()
62 def build_topo(v):
63 if v not in visited:
64 visited.add(v)
65 for child in v._children:
66 build_topo(child)
67 topo.append(v)
68 build_topo(self)
69 self.grad = 1
70 for v in reversed(topo):
71 for child, local_grad in zip(v._children, v._local_grads):
72 child.grad += local_grad * v.grad
73
74# Initialize the parameters, to store the knowledge of the model
75n_layer = 1 # depth of the transformer neural network (number of layers)
76n_embd = 16 # width of the network (embedding dimension)
77block_size = 16 # maximum context length of the attention window (note: the longest name is 15 characters)
78n_head = 4 # number of attention heads
79head_dim = n_embd // n_head # derived dimension of each head
80matrix = lambda nout, nin, std=0.08: [[Value(random.gauss(0, std)) for _ in range(nin)] for _ in range(nout)]
81state_dict = {'wte': matrix(vocab_size, n_embd), 'wpe': matrix(block_size, n_embd), 'lm_head': matrix(vocab_size, n_embd)}
82for i in range(n_layer):
83 state_dict[f'layer{i}.attn_wq'] = matrix(n_embd, n_embd)
84 state_dict[f'layer{i}.attn_wk'] = matrix(n_embd, n_embd)
85 state_dict[f'layer{i}.attn_wv'] = matrix(n_embd, n_embd)
86 state_dict[f'layer{i}.attn_wo'] = matrix(n_embd, n_embd)
87 state_dict[f'layer{i}.mlp_fc1'] = matrix(4 * n_embd, n_embd)
88 state_dict[f'layer{i}.mlp_fc2'] = matrix(n_embd, 4 * n_embd)
89params = [p for mat in state_dict.values() for row in mat for p in row] # flatten params into a single list[Value]
90print(f"num params: {len(params)}")
91
92# Define the model architecture: a function mapping tokens and parameters to logits over what comes next
93# Follow GPT-2, blessed among the GPTs, with minor differences: layernorm -> rmsnorm, no biases, GeLU -> ReLU
94def linear(x, w):
95 return [sum(wi * xi for wi, xi in zip(wo, x)) for wo in w]
96
97def softmax(logits):
98 max_val = max(val.data for val in logits)
99 exps = [(val - max_val).exp() for val in logits]
100 total = sum(exps)
101 return [e / total for e in exps]
102
103def rmsnorm(x):
104 ms = sum(xi * xi for xi in x) / len(x)
105 scale = (ms + 1e-5) ** -0.5
106 return [xi * scale for xi in x]
107
108def gpt(token_id, pos_id, keys, values):
109 tok_emb = state_dict['wte'][token_id] # token embedding
110 pos_emb = state_dict['wpe'][pos_id] # position embedding
111 x = [t + p for t, p in zip(tok_emb, pos_emb)] # joint token and position embedding
112 x = rmsnorm(x) # note: not redundant due to backward pass via the residual connection
113
114 for li in range(n_layer):
115 # 1) Multi-head Attention block
116 x_residual = x
117 x = rmsnorm(x)
118 q = linear(x, state_dict[f'layer{li}.attn_wq'])
119 k = linear(x, state_dict[f'layer{li}.attn_wk'])
120 v = linear(x, state_dict[f'layer{li}.attn_wv'])
121 keys[li].append(k)
122 values[li].append(v)
123 x_attn = []
124 for h in range(n_head):
125 hs = h * head_dim
126 q_h = q[hs:hs+head_dim]
127 k_h = [ki[hs:hs+head_dim] for ki in keys[li]]
128 v_h = [vi[hs:hs+head_dim] for vi in values[li]]
129 attn_logits = [sum(q_h[j] * k_h[t][j] for j in range(head_dim)) / head_dim**0.5 for t in range(len(k_h))]
130 attn_weights = softmax(attn_logits)
131 head_out = [sum(attn_weights[t] * v_h[t][j] for t in range(len(v_h))) for j in range(head_dim)]
132 x_attn.extend(head_out)
133 x = linear(x_attn, state_dict[f'layer{li}.attn_wo'])
134 x = [a + b for a, b in zip(x, x_residual)]
135 # 2) MLP block
136 x_residual = x
137 x = rmsnorm(x)
138 x = linear(x, state_dict[f'layer{li}.mlp_fc1'])
139 x = [xi.relu() for xi in x]
140 x = linear(x, state_dict[f'layer{li}.mlp_fc2'])
141 x = [a + b for a, b in zip(x, x_residual)]
142
143 logits = linear(x, state_dict['lm_head'])
144 return logits
145
146# Let there be Adam, the blessed optimizer and its buffers
147learning_rate, beta1, beta2, eps_adam = 0.01, 0.85, 0.99, 1e-8
148m = [0.0] * len(params) # first moment buffer
149v = [0.0] * len(params) # second moment buffer
150
151# Repeat in sequence
152num_steps = 1000 # number of training steps
153for step in range(num_steps):
154
155 # Take single document, tokenize it, surround it with BOS special token on both sides
156 doc = docs[step % len(docs)]
157 tokens = [BOS] + [uchars.index(ch) for ch in doc] + [BOS]
158 n = min(block_size, len(tokens) - 1)
159
160 # Forward the token sequence through the model, building up the computation graph all the way to the loss
161 keys, values = [[] for _ in range(n_layer)], [[] for _ in range(n_layer)]
162 losses = []
163 for pos_id in range(n):
164 token_id, target_id = tokens[pos_id], tokens[pos_id + 1]
165 logits = gpt(token_id, pos_id, keys, values)
166 probs = softmax(logits)
167 loss_t = -probs[target_id].log()
168 losses.append(loss_t)
169 loss = (1 / n) * sum(losses) # final average loss over the document sequence. May yours be low.
170
171 # Backward the loss, calculating the gradients with respect to all model parameters
172 loss.backward()
173
174 # Adam optimizer update: update the model parameters based on the corresponding gradients
175 lr_t = learning_rate * (1 - step / num_steps) # linear learning rate decay
176 for i, p in enumerate(params):
177 m[i] = beta1 * m[i] + (1 - beta1) * p.grad
178 v[i] = beta2 * v[i] + (1 - beta2) * p.grad ** 2
179 m_hat = m[i] / (1 - beta1 ** (step + 1))
180 v_hat = v[i] / (1 - beta2 ** (step + 1))
181 p.data -= lr_t * m_hat / (v_hat ** 0.5 + eps_adam)
182 p.grad = 0
183
184 print(f"step {step+1:4d} / {num_steps:4d} | loss {loss.data:.4f}", end='\r')
185
186# Inference: may the model babble back to us
187temperature = 0.5 # in (0, 1], control the "creativity" of generated text, low to high
188print("\n--- inference (new, hallucinated names) ---")
189for sample_idx in range(20):
190 keys, values = [[] for _ in range(n_layer)], [[] for _ in range(n_layer)]
191 token_id = BOS
192 sample = []
193 for pos_id in range(block_size):
194 logits = gpt(token_id, pos_id, keys, values)
195 probs = softmax([l / temperature for l in logits])
196 token_id = random.choices(range(vocab_size), weights=[p.data for p in probs])[0]
197 if token_id == BOS:
198 break
199 sample.append(uchars[token_id])
200 print(f"sample {sample_idx+1:2d}: {''.join(sample)}")
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