Module 6 · Learning = walking downhill · train1.py · ~25 min
6.3 Many knobs
One knob was easy. Now give the model 27: one for every symbol a name could start with. Same rule, same valley, just in 27 directions at once. Then look at what it costs.
1Watch
Each symbol gets a knob. A knob’s share of the total is that symbol’s chance of starting a name. All 27 start equal, which is the know-nothing model at 3.30.
With many knobs, the slope becomes a list of 27 slopes: the gradient. Entry i says how the loss changes if you nudge knob i alone. Step every knob against its own slope, all at once, and repeat. After 60 steps the loss reaches 2.944, almost exactly what counting first letters gives: the same loss to 3 decimals, with chances that are close but not identical (“a” lands on 0.1335 against counting’s 0.1377).
One change from 6.2: there you nudged both ways, (f(x + h) − f(x − h)) / (2h). Here we nudge one way only, (f(nudged) − base) / h. The base loss is the same for all 27 knobs, so you work it out once and share it: a step costs 1 + 27 = 28 runs instead of 2 × 27 = 54. The answer is a touch less precise, but plenty good for walking downhill. Because the nudge is one way, the distance is h, not 2h.
Now the catch. To measure the gradient by nudging, you run the model once for the starting loss and once more per knob: 28 runs per step. microgpt has 4,192 knobs, so that would be 4,193 runs every step, each through the whole network. Far too slow. Module 7 gets all 4,192 slopes from a single backward pass.
2Explore
Loading 32,033 names…
3Build
Fill the blank in numerical_gradient: after nudging knob i by h, its slope is how much the loss moved away from base, per unit of nudge. At the start, the slope for “a” should come out at about −0.1006. Run it and read how many loss evaluations 60 steps cost.
The update line uses a learning rate of 2.0 (this valley is much flatter than 6.2’s) and stops any knob going below 0, because a share can’t be negative. Lesson 6.4 fixes that problem properly.
import math
docs = [line.strip() for line in open('names.txt') if line.strip()]
N = len(docs)
LETTERS = '.abcdefghijklmnopqrstuvwxyz'
first = {ch: 0 for ch in LETTERS}
for d in docs:
first[d[0]] += 1
# 27 knobs, one per symbol. A knob's share of the total is its chance.
def loss(knobs):
total = sum(knobs)
# enumerate(LETTERS) gives (position, item) pairs: (0, '.'), (1, 'a'), ...
# so i is the knob's number and ch is its letter.
return -sum(first[ch] * math.log(knobs[i] / total)
for i, ch in enumerate(LETTERS) if first[ch]) / N
# The gradient: nudge EACH knob on its own and record how the loss moves.
# One-sided nudge: every knob shares the same base run, so a step costs
# 1 + 27 = 28 runs instead of 2 * 27 = 54.
def numerical_gradient(f, knobs, h=1e-5):
base = f(knobs)
grad = []
for i in range(len(knobs)):
nudged = list(knobs)
nudged[i] += h
grad.append(0) # TODO: how much did the loss move, per unit of nudge?
return grad
knobs = [1.0] * 27
print('start:', round(loss(knobs), 4))
evaluations = 0
for step in range(60):
grad = numerical_gradient(loss, knobs)
evaluations += len(knobs) + 1
# lr 2.0: this valley is much flatter than 6.2's, so the stride is longer.
# max(1e-6, ...) stops a knob going negative, because a share can't be
# negative (lesson 6.4 fixes this properly).
knobs = [max(1e-6, k - 2.0 * g) for k, g in zip(knobs, grad)]
print('after 60 steps:', round(loss(knobs), 4), f'({evaluations} loss evaluations)')
print("learned chance of 'a':", round(knobs[1] / sum(knobs), 4))
4Check yourself
5Unlocked in microgpt
No new lines: microgpt never nudges knobs one by one. That’s precisely why it has lines 59 to 72, backward(). Keep the number 4,193 in mind when you get there.
Progress is saved in this browser. Sign in to keep it across devices.
microgpt.py63 / 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)}")
Was this lesson clear?