Theory Notebook

Converted from theory.ipynb for web reading.

Transformer Architecture: Theory Notebook

This notebook makes transformer architecture concrete: attention matrices, masks, heads, MLPs, normalization, positional signals, KV cache memory, and diagnostics.

Code cell 2

import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl

try:
    import seaborn as sns
    sns.set_theme(style="whitegrid", palette="colorblind")
    HAS_SNS = True
except ImportError:
    plt.style.use("seaborn-v0_8-whitegrid")
    HAS_SNS = False

mpl.rcParams.update({
    "figure.figsize":    (10, 6),
    "figure.dpi":         120,
    "font.size":           13,
    "axes.titlesize":      15,
    "axes.labelsize":      13,
    "xtick.labelsize":     11,
    "ytick.labelsize":     11,
    "legend.fontsize":     11,
    "legend.framealpha":   0.85,
    "lines.linewidth":      2.0,
    "axes.spines.top":     False,
    "axes.spines.right":   False,
    "savefig.bbox":       "tight",
    "savefig.dpi":         150,
})
np.random.seed(42)
print("Plot setup complete.")

1. Scaled dot-product attention

Code cell 4

def softmax(x, axis=-1):
    x = np.asarray(x, dtype=float)
    x = x - np.max(x, axis=axis, keepdims=True)
    e = np.exp(x)
    return e / e.sum(axis=axis, keepdims=True)

Q = np.array([[1.0, 0.0], [0.0, 1.0]])
K = np.array([[1.0, 0.0], [1.0, 1.0], [0.0, 1.0]])
V = np.array([[10.0, 0.0], [5.0, 5.0], [0.0, 10.0]])
scores = Q @ K.T / np.sqrt(Q.shape[-1])
A = softmax(scores, axis=-1)
O = A @ V
print("scores:\n", np.round(scores, 3))
print("weights:\n", np.round(A, 3))
print("output:\n", np.round(O, 3))

2. Causal mask

Code cell 6

T = 5
scores = np.arange(T*T, dtype=float).reshape(T, T)
mask = np.triu(np.ones((T, T), dtype=bool), k=1)
masked_scores = scores.copy()
masked_scores[mask] = -1e9
print("causal mask:\n", mask.astype(int))
print("masked row 2:", masked_scores[2])

3. Head splitting shapes

Code cell 8

B, T, d_model, heads = 2, 4, 12, 3
d_head = d_model // heads
H = np.zeros((B, T, d_model))
split = H.reshape(B, T, heads, d_head).transpose(0, 2, 1, 3)
merged = split.transpose(0, 2, 1, 3).reshape(B, T, d_model)
print("split shape:", split.shape)
print("merged shape:", merged.shape)

4. Attention parameter count

Code cell 10

d_model = 768
qkv_o_params = 4 * d_model * d_model
print("Q,K,V,O projection params without bias:", qkv_o_params)

5. MLP parameter count

Code cell 12

d_model = 768
d_ff = 4 * d_model
mlp_params = 2 * d_model * d_ff
attn_params = 4 * d_model * d_model
print("MLP params:", mlp_params)
print("attention projection params:", attn_params)
print("MLP/attention ratio:", mlp_params / attn_params)

6. LayerNorm and RMSNorm

Code cell 14

x = np.array([1.0, 2.0, 4.0, 8.0])
eps = 1e-5
layer_norm = (x - x.mean()) / np.sqrt(x.var() + eps)
rms_norm = x / np.sqrt(np.mean(x**2) + eps)
print("LayerNorm:", np.round(layer_norm, 4))
print("RMSNorm:", np.round(rms_norm, 4))

7. Pre-LN versus Post-LN equations

Code cell 16

x = np.array([1.0, -0.5, 0.25])
F = lambda z: 0.2 * z
ln = lambda z: (z - z.mean()) / np.sqrt(z.var() + 1e-5)
pre_ln = x + F(ln(x))
post_ln = ln(x + F(x))
print("Pre-LN output:", np.round(pre_ln, 4))
print("Post-LN output:", np.round(post_ln, 4))

8. Sinusoidal positional encoding

Code cell 18

T, d = 32, 16
pos = np.arange(T)[:, None]
i = np.arange(d)[None, :]
angles = pos / (10000 ** (2 * (i // 2) / d))
P = np.zeros((T, d))
P[:, 0::2] = np.sin(angles[:, 0::2])
P[:, 1::2] = np.cos(angles[:, 1::2])
plt.imshow(P, aspect="auto", cmap="coolwarm")
plt.title("Sinusoidal positional encoding")
plt.xlabel("dimension")
plt.ylabel("position")
plt.colorbar()
plt.tight_layout()
plt.show()
print("P shape:", P.shape)

9. Attention entropy

Code cell 20

def softmax(x, axis=-1):
    x = np.asarray(x, dtype=float)
    x = x - np.max(x, axis=axis, keepdims=True)
    e = np.exp(x)
    return e / e.sum(axis=axis, keepdims=True)

rng = np.random.default_rng(0)
scores = rng.normal(size=(4, 6))
A = softmax(scores, axis=-1)
entropy = -np.sum(A * np.log(A + 1e-12), axis=-1)
print("attention entropy per query:", np.round(entropy, 3))
print("max entropy for 6 keys:", np.log(6))

10. KV cache memory

Code cell 22

B, L, T, H_kv, d_h, bytes_per = 4, 32, 4096, 8, 128, 2
gb = 2 * B * L * T * H_kv * d_h * bytes_per / 1e9
print("KV cache GB:", gb)

11. Weight tying

Code cell 24

vocab, d_model = 50000, 768
untied = 2 * vocab * d_model
tied = vocab * d_model
print("untied embedding+head params:", untied)
print("tied params:", tied)
print("savings:", untied - tied)

12. Mask leakage test

Code cell 26

causal = np.tril(np.ones((5, 5), dtype=int))
future_visible = np.any(np.triu(causal, k=1) != 0)
print("future visible:", future_visible)
assert not future_visible

13. Residual update ratio

Code cell 28

rng = np.random.default_rng(3)
x = rng.normal(size=(10, 32))
update = 0.1 * rng.normal(size=(10, 32))
ratio = np.linalg.norm(update) / np.linalg.norm(x)
print("update/residual norm ratio:", ratio)

14. Transformer checklist

Code cell 30

checks = [
    "Q,K,V and score tensor shapes are written down",
    "padding and causal masks are tested separately",
    "head dimension divides model dimension",
    "MLP and attention parameter counts are understood",
    "normalization placement matches the intended architecture",
    "KV cache memory is computed for serving settings",
]
for i, check in enumerate(checks, 1):
    print(f"{i}. {check}")