Module 1 · Computational thinking & Python · ~25 min

1.1 Thinking like a computer

A computer is fast, obedient and completely literal. It will do exactly what you say, nothing more. Computational thinking is the skill of breaking a job into steps so small and exact that a machine can follow them.

1Watch

Four habits do most of the work. Here they are on one question: “Which letter do the most names start with?”

Decomposition: split a big job into smaller ones. Collect the first letters, count each one, pick the biggest count.
Patterns: notice when steps repeat, so you write them once. “Take the first letter” is the same step for every name.
Abstraction: ignore details that don’t matter for the question. How long a name is, or what its other letters are, doesn’t matter here.
Algorithms: write the steps as a recipe, in order. The three numbered steps in the lab below are an algorithm.

Each step is small enough to write in a line or two of Python. That’s how all of microgpt is built: 200 lines, each doing one small, exact thing.

Three bits of Python you’ll meet today. You don’t need to write them yet, just recognise them.

print(...) shows things on screen.
name[0] is the first letter of name (Python counts from 0), and name[-1] is the last letter. So if name is 'emma', name[0] is 'e' and name[-1] is 'a'.
A line starting with # is a note for humans. Python skips it. Deleting the # turns the line back into code.

2Explore

Change anything and run again. You can’t break it.

Don’t worry about every symbol yet: press Run, then change a name and run again. n[0] means “the first letter of n”, and [n[0] for n in names] means “the first letter of each name in names”. You’ll learn to write lines like it in lesson 1.5.

3Build

The three steps are already written, using eight names. Press Run first: the winner is “a”, twice. Then make two small edits (they’re also listed at the bottom of the file):

Use all 32,033 names: delete the # and the space at the start of the names = open('names.txt')… line, so the line starts at the left edge.
Ask a new question, “which letter do the most names end with?”: in step 1, change name[0] to name[-1].

Then press Check my code. You don’t need to understand steps 2 and 3 yet; you’ll write code like that in Module 2. Notice that you only changed one step and the rest of the recipe still worked.

# Computational thinking: break a big job into small, exact steps.
# Job: which letter do the most names start with?
# Lines starting with # are notes for humans. Python skips them.

# Step 0: the names. Eight to start with.
names = ['emma', 'olivia', 'ava', 'isabella', 'sophia', 'mia', 'amelia', 'harper']
# names = open('names.txt').read().split()   # all 32,033 names in the course's file

# Step 1: take one letter from every name.  name[0] is the first letter of name.
letters = [name[0] for name in names]

# Steps 2 and 3 are given. You'll learn to write lines like these in Module 2.
# Step 2: count each letter (look, add one, repeat).
counts = {}
for letter in letters:
    counts[letter] = counts.get(letter, 0) + 1

# Step 3: pick the letter with the biggest count.
winner = max(counts, key=counts.get)

print('names:', len(names))
print('letters:', letters[:8], '...')
print('most common letter:', winner, '-', counts[winner], 'times')

# YOUR TURN (two small edits, then press Check my code):
#  1. Use all the names: delete the "# " at the start of the names = open(...) line.
#  2. Ask a new question, "which letter do the most names END with?":
#     in step 1, change name[0] to name[-1]  (name[-1] is the last letter).

4Check yourself

1. Breaking “find the most common first letter” into collect, count, pick is called…

2. To find the most common first letter, you ignore how long each name is. Which habit is that?

3. name = 'olivia'. What does print(name[0], name[-1]) show?

4. Why does a computer need such exact steps?

5Unlocked in microgpt

No lines yet. Every one of microgpt’s 200 lines is one of these small, exact steps.

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microgpt.py0 / 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.

6@karpathy

7"""

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

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)}")

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}")

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

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

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))

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))

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

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

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)}")

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]

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)}")

this lessonlearnedKarpathy’s original

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