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"""Analyze softmax attention Jacobian: decompose into diagonal (local) vs off-diagonal (lateral).
The softmax Jacobian J = diag(A) - AA^T acts on gradient g as:
g_S = A ⊙ g - A * (A^T g) (full, has lateral sum)
g_S_diag = A ⊙ (1-A) ⊙ g (diagonal-only, element-wise, same formula as sigmoid)
g_S_ste = g (identity STE)
This script measures:
1. How much energy is in diagonal vs off-diagonal components
2. Cosine between full vs diagonal-only vs STE on real FA training data
3. Per-head, per-layer breakdown
4. Whether removing the lateral sum is catastrophic or tolerable
"""
import pickle
from pathlib import Path
import torch
import torch.nn as nn
import torch.nn.functional as F
from model_local import LocalGPT, LocalGPTConfig
import numpy as np
def get_batch(data_dir, block_size, batch_size, device):
data = np.memmap(data_dir / "train.bin", dtype=np.uint16, mode="r")
ix = torch.randint(len(data) - block_size - 1, (batch_size,))
x = torch.stack([torch.from_numpy(data[i:i+block_size].astype(np.int64)) for i in ix])
y = torch.stack([torch.from_numpy(data[i+1:i+1+block_size].astype(np.int64)) for i in ix])
return x.to(device), y.to(device)
def main():
device = "cuda" if torch.cuda.is_available() else "cpu"
data_dir = Path("data/shakespeare_char")
torch.manual_seed(1337)
with open(data_dir / "meta.pkl", "rb") as f:
meta = pickle.load(f)
# Train a softmax FA model for 500 steps to get meaningful attention patterns
cfg = LocalGPTConfig(
block_size=64, vocab_size=meta["vocab_size"],
n_layer=4, n_head=4, n_embd=128, dropout=0.0,
attn_mode="softmax", method="fa",
)
model = LocalGPT(cfg).to(device)
optimizer = torch.optim.AdamW(model.parameters(), lr=1e-3)
model.train()
for step in range(500):
X, Y = get_batch(data_dir, cfg.block_size, 32, device)
_, loss = model(X, Y)
optimizer.zero_grad()
loss.backward()
optimizer.step()
print(f"Trained 500 steps, final loss: {loss.item():.3f}")
# Hook into attention forward to capture scores and attention weights
attn_data = {}
def make_attn_hook(name, module):
original_forward = module.forward
def hooked_forward(x):
B, T, C = x.shape
q = module.q_proj(x).view(B, T, module.n_head, module.head_dim).transpose(1, 2)
k = module.k_proj(x).view(B, T, module.n_head, module.head_dim).transpose(1, 2)
v = module.v_proj(x).view(B, T, module.n_head, module.head_dim).transpose(1, 2)
scores = (q @ k.transpose(-2, -1)) * (module.head_dim ** -0.5)
mask = module.causal_mask[:T, :T]
scores = scores.masked_fill(~mask, float("-inf"))
attn = F.softmax(scores, dim=-1)
attn_data[name] = {
"scores": scores.detach(),
"attn": attn.detach(),
}
# Need grad wrt attention output for Jacobian analysis
attn_for_grad = attn.clone().requires_grad_(True)
out = (attn_for_grad @ v).transpose(1, 2).contiguous().view(B, T, C)
out = module.resid_drop(module.o_proj(out))
attn_data[name]["attn_for_grad"] = attn_for_grad
return out
module.forward = hooked_forward
return module
# Install hooks
for name, module in model.named_modules():
if hasattr(module, "q_proj") and hasattr(module, "k_proj"):
make_attn_hook(name, module)
# Forward + backward on diagnostic batch
model.eval()
X, Y = get_batch(data_dir, cfg.block_size, 32, device)
logits, loss = model(X, Y)
loss.backward()
# Analyze each attention layer
print(f"\n{'layer':30s} {'A_mean':>8s} {'A_entropy':>10s} {'r_diag':>8s} {'r_offdiag':>10s} "
f"{'cos_diag':>9s} {'cos_ste':>8s}")
print("-" * 100)
for name, d in sorted(attn_data.items()):
A = d["attn"] # (B, n_head, T, T)
attn_ref = d.get("attn_for_grad")
if attn_ref is None or attn_ref.grad is None:
print(f"{name:30s} (no grad captured)")
continue
g = attn_ref.grad.detach() # (B, n_head, T, T) = dL/dA
B_size, n_head, T, _ = A.shape
# Per-head analysis
for h in range(n_head):
A_h = A[:, h, :, :] # (B, T, T)
g_h = g[:, h, :, :] # (B, T, T)
# Full softmax backward: g_S = A * (g - A @ g sum along last dim)
Ag_sum = (A_h * g_h).sum(dim=-1, keepdim=True) # (B, T, 1) = sum_j A_j g_j per query
g_full = A_h * (g_h - Ag_sum) # (B, T, T)
# Diagonal-only (element-wise, sigmoid-like): g_diag = A*(1-A)*g
g_diag = A_h * (1 - A_h) * g_h # (B, T, T)
# STE: g_ste = g
g_ste = g_h
# Energy fractions
g_full_norm = (g_full * g_full).sum((-2, -1)).mean()
g_diag_norm = (g_diag * g_diag).sum((-2, -1)).mean()
diff_norm = ((g_full - g_diag) * (g_full - g_diag)).sum((-2, -1)).mean()
# Cosines (flatten per-sample)
def cos(a, b):
af = a.reshape(B_size, -1)
bf = b.reshape(B_size, -1)
return F.cosine_similarity(af, bf, dim=-1).mean().item()
cos_diag = cos(g_diag, g_full)
cos_ste = cos(g_ste, g_full)
# Attention statistics
# Mask out -inf positions for stats
valid_mask = A_h > 0
A_valid = A_h[valid_mask]
A_mean = A_valid.mean().item()
# Entropy per query row
entropy = -(A_h * (A_h + 1e-10).log()).sum(-1).mean().item()
r_diag = g_diag_norm / (g_full_norm + 1e-12)
print(f"{name}.head{h:d} "
f" {A_mean:8.4f} {entropy:10.3f} {r_diag.item():8.3f} "
f"{(1-r_diag).item():10.3f} {cos_diag:9.4f} {cos_ste:8.4f}")
# Summary
print(f"\nKey: r_diag = ||g_diag||^2 / ||g_full||^2 (energy in diagonal/element-wise part)")
print(f" cos_diag = cosine(diagonal-only, full softmax backward)")
print(f" cos_ste = cosine(identity STE, full softmax backward)")
print(f"\nIf cos_diag ≈ 1: diagonal-only (sigmoid-like) approximation is good → lateral sum not needed")
print(f"If cos_diag << 1: off-diagonal (lateral sum) is critical → need to keep or find local surrogate")
if __name__ == "__main__":
main()
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