Module 5 · Language as probability · train0.py · ~30 min

5.1 Counting letter pairs

Quick: in an English name, what letter comes after q? You said u without thinking. Your brain is carrying statistics around. In this lesson you put those statistics into a table, and that table is already a working (if silly) language model.

1Watch

Take all 32,033 names in our dataset. Wrap each one in a start/end marker, so emma becomes .emma., then slide a two-letter window along it: .e, em, mm, ma, a.. Count every pair you see. That’s 228,146 pairs, landing in a 27 × 27 grid (26 letters plus the marker).

One row of that grid is a weighted die. The row for q says: 206 times the next character was u, 28 times the name ended, 13 times a… Roll that die and you get a plausible next letter.

To generate a name: start at the marker, roll its die, move to the letter you rolled, roll that letter’s die, and stop when you roll the marker again. That’s it. That’s the model. In AI language this is called a bigram model (“bi” = two characters at a time).

Why do the names come out weird? The model has a memory of exactly one letter. It has no idea it already wrote three vowels in a row. The whole rest of this course is about giving the model a better memory, until it’s a GPT.

2Explore

Loading 32,033 names…

3Build

Two blanks to fill, both marked TODO:

In count_pairs, add 1 to the count for the pair (a, b). The first time you meet a pair it isn’t in the dictionary yet, so you need a way to read “its count so far, or 0”.
Show me the line
counts[(a, b)] = counts.get((a, b), 0) + 1
In sample_name, roll the weighted die over options, using weights as the weights, and take the one letter it lands on. You met the die in lesson 3.4; rng is just the random module (the checks swap in a seeded one). It’s the exact function microgpt uses on line 196.
Show me the line
nxt = rng.choices(options, weights=weights)[0]
choices returns a list of picks, so [0] takes the single letter out of it.

Press Run to see your names, then Check my code.

# `docs` is already loaded for you: a list of 32,033 names.
docs = [line.strip() for line in open('names.txt') if line.strip()]

# 1) Count every pair of neighbouring characters.
#    Wrap each name in '.' markers first, so "emma" becomes ".emma."
#    and the pairs are ('.','e'), ('e','m'), ('m','m'), ('m','a'), ('a','.')
def count_pairs(docs):
    counts = {}
    for name in docs:
        chars = '.' + name + '.'
        for a, b in zip(chars, chars[1:]):
            pass  # TODO: add 1 to counts[(a, b)]
            # (hint: counts.get((a, b), 0) is 0 the first time you see a pair)
    return counts


# 2) Generate a name: start at '.', roll a weighted die over the letters
#    that followed it, move to the letter you rolled, stop at '.'.
import random

# rng is the random module, unless the checks pass in a seeded one (lesson 3.4)
def sample_name(counts, rng=random):
    name = ''
    current = '.'
    while True:
        options = [b for (a, b) in counts if a == current]
        weights = [counts[(current, b)] for b in options]
        nxt = '.'  # TODO: roll the weighted die (rng.choices) and take the one letter it picks
        if nxt == '.':
            return name
        name += nxt
        current = nxt


counts = count_pairs(docs)
print('pairs counted:', sum(counts.values()))
after_q = {}
for (a, b) in counts:
    if a == 'q':
        after_q[b] = counts[(a, b)]
print('after q:', after_q)
print([sample_name(counts) for _ in range(10)])

4Check yourself

1. Why wrap every name in "." markers before counting?

2. The row for "q" has u: 206, end: 28, a: 13, i: 13, and 12 more sightings spread over 7 other letters. Roughly how likely is "u" next?

3. Predict: what happens if you always pick the MOST common next letter instead of rolling the die?

4. What is the biggest weakness of this model?

5Unlocked in microgpt

Line 157 wraps each name in the BOS marker, exactly like your '.' + name + '.'. Line 196 is your weighted die: random.choices(…, weights=…).

The difference: microgpt doesn’t count to get the weights. It learns them, and it looks at the whole name so far, not just the last letter. Counting scores 2.45 on our “surprise” scale (lower is better). The finished GPT reaches about 2.28.

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microgpt.py58 / 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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