Module 3 · Algorithms · ~25 min

3.3 Topological sort

Socks before shoes. Underwear before trousers before belt. Find an order for everything that respects every rule: that’s a topological sort, and it’s the backbone of how neural networks learn.

1Watch

The rules live in a dict called needs: it maps each item to the items that must come before it, so 'shoes': ['socks', 'trousers'] means socks and trousers go on before shoes.

The recipe is depth-first search (lesson 3.2) with one twist: add an item to the order only after everything it needs has been added. The function calls itself on each need first, so the needs land first. Here is the trace from the shoes:

build('shoes') → build('socks'): add socks → build('trousers') → build('underwear'): add underwear → back in trousers: add trousers → back in shoes: add shoes

In the code, build is a helper function written inside topo_order; it can use order and visited from the outer function. microgpt does the same.

microgpt’s build_topo (lines 60 to 68) is exactly this, on its graph of numbers: each Value remembers the values it was made from in _children, and those are its needs. Then backward() walks the order in reverse, like undressing: shoes come off first. Lesson 7.5 shows why that reverse order is precisely what the chain rule needs.

2Explore

start at

visited checkwrite the item

def build(item):
    if item not in visited:
        visited.add(item)
        for before in needs[item]:
            build(before)
        order.append(item)

now: Start: explore 'shoes'.

path: …

visited = set()

order = []

step 1 of 19

Lines go down from each item to what it needs. Orange outline: the path of calls still open. Green: already written into order.

3Build

The starter appends each item before visiting its needs, which is wrong. Move the order.append(item) line so items are added after their needs: below the for loop, but still inside the if. The checks also try a diamond, where two paths meet.

# Topological sort: an order where everything comes AFTER what it needs.
# Socks before shoes. This is exactly microgpt's build_topo (lines 60-68).

# needs maps each item to the items that must come before it.
needs = {
    'shoes': ['socks', 'trousers'],
    'socks': [],
    'trousers': ['underwear'],
    'underwear': [],
    'belt': ['trousers'],
}

def topo_order(goal):
    order = []
    visited = set()
    def build(item):   # a helper inside topo_order: it can use order and visited
        if item not in visited:
            visited.add(item)
            order.append(item)   # TODO: wrong place: move this line so it runs after the for loop
            for before in needs[item]:
                build(before)
    build(goal)
    return order


print('to put on shoes:', topo_order('shoes'))
print('to put on a belt:', topo_order('belt'))

4Check yourself

1. In a topological order, an item comes…

2. Is there only one valid order?

3. Predict: what does topo_order('belt') return?

4. Predict: in the diamond (top needs left and right, both need bottom), what is topo_order('top')?

5Unlocked in microgpt

Lines 60 to 68: topo, visited and build_topo, the first half of backward().

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microgpt.py49 / 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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