Module 2 · Data structures · ~25 min

2.3 Lists of lists = matrices

microgpt’s knowledge lives in tables of numbers: 27 rows of 16 for the letters, 16 rows of 16 for positions (one row for each position in a name, up to 16), and more. In Python a table is just a list of lists: a matrix.

1Watch

[[1, 2], [3, 4]] is a matrix with 2 rows of 2. m[1][0] is row 1, column 0: the number 3.

To build one, you repeat things a set number of times. range(3) counts 0, 1, 2, so for r in range(3): runs three times with r = 0, then 1, then 2. A matrix needs two loops, one inside the other: the outer loop makes each row, and for every row the inner loop fills it with numbers. Start every row as a fresh empty list; if all the rows share one list, they all end up the same.

microgpt’s matrix(nout, nin) (line 80) builds a table of nout rows with nin numbers in each, and fills it with random.gauss(0, std): a random number that is usually close to 0, where std (0.08 here) sets how far it typically strays. microgpt writes the two loops in one line, as a list comprehension; the loop version below does the same job.

The state_dict (line 81) names every table, and params (line 89) flattens them all into one long list. The dict’s .values() gives each table without its name. microgpt has nine tables, 4,192 numbers in all; in the lab you build the first three, which hold 1,120.

2Explore

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

3Build

One blank: matrix(nout, nin). nout is how many rows, nin how many numbers in each row. A plan:

Start with an empty list m (already there).
Loop nout times. Each time, make a fresh empty row.
Inside, loop nin times, appending random.gauss(0, std) to the row.
After the inner loop, append the row to m.

import random
random.seed(42)  # fix the dice (lesson 1.2): the same random numbers every run

# A matrix is a list of rows; each row is a list of numbers.
# microgpt line 80 makes its knobs this way (Value comes in lesson 7.4).
# nout = how many rows, nin = how many numbers in each row.

# std=0.08 is a default: matrix(3, 4) uses std 0.08, matrix(3, 4, std=0.5) overrides it.
def matrix(nout, nin, std=0.08):
    m = []
    # TODO: repeat nout times (range(nout) counts 0, 1, ..., nout - 1):
    #   make a FRESH empty row,
    #   append nin numbers to it, each one random.gauss(0, std),
    #   then append the row to m
    return m


wte = matrix(27, 16)            # one row of 16 numbers per symbol
# isinstance(x, list) asks "is x a list?" and gives True or False.
# a if cond else b is a one-line if/else: a when cond is True, otherwise b.
if wte and not isinstance(wte[0], list):
    print('wte[0] should be a row (a list of numbers), got', wte[0])
else:
    print('rows:', len(wte), ' numbers per row:', len(wte[0]) if wte else 0)
    if wte:
        print('the row for symbol 0:', [round(v, 3) for v in wte[0][:4]], '...')

# Name three of microgpt's tables, then flatten every number into one list.
state_dict = {'wte': wte, 'wpe': matrix(16, 16), 'lm_head': matrix(27, 16)}
params = []
for mat in state_dict.values():     # .values() gives each table (not its name)
    for row in mat:
        if isinstance(row, list):
            for p in row:
                params.append(p)
print('numbers in these three tables:', len(params))

4Check yourself

1. A matrix with 27 rows of 16 holds how many numbers?

2. m = [[1, 2], [3, 4]]. What does print(m[1][0]) show?

3. What does print(list(range(3))) show?

4. What is params in microgpt?

5Unlocked in microgpt

Lines 74 to 90: the sizes, matrix(), the state_dict of tables and the flat params list.

Line 80 writes matrix with lambda, a one-line def: it means exactly what your function means. Lines 75 to 79 are the sizes (how many layers, how wide each letter’s row is, the longest name it reads, and so on); what each one controls is explained in Modules 6 to 9. // divides and drops the remainder, and f'layer{i}.attn_wq' builds a table name with the number i slotted in.

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microgpt.py33 / 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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