feat: add model comparisons and sanitize session files
- Rename gamma to glm5 and model to minimax-m2.7 - Add model_comparison/ directory with head-to-head analyses - Sanitize all session.jsonl files: remove absolute paths and usernames - Remove __pycache__ artifacts - Add .gitignore
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"""
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Benchmark and numerical stability comparison for layer_norm_backward.py
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"""
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import numpy as np
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import time
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from layer_norm_backward import layer_norm_forward, layer_norm_backward
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# ---------------------------------------------------------------------------
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# 1. Numerical stability: two-pass vs naive variance
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# ---------------------------------------------------------------------------
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def naive_variance(x, axis=-1):
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"""Naive one-pass variance: E[x²] - E[x]² — prone to cancellation."""
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return np.mean(x ** 2, axis=axis) - np.mean(x, axis=axis) ** 2
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def two_pass_variance(x, axis=-1):
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"""Two-pass variance: center first, then compute — numerically stable."""
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mu = np.mean(x, axis=axis, keepdims=True)
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return np.mean((x - mu) ** 2, axis=axis)
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def demo_variance_stability():
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print("=" * 70)
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print("NUMERICAL STABILITY: TWO-PASS vs NAIVE VARIANCE")
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print("=" * 70)
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print()
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print("When mean² ≫ var, the naive formula E[x²] - E[x]² suffers from")
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print("catastrophic cancellation. The two-pass algorithm avoids this.")
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print()
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# Construct a pathological case: large offset, tiny variance
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offset = 1e8
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true_values = np.array([0.0, 1.0, 2.0, 3.0, 4.0], dtype=np.float64)
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true_var = np.var(true_values) # 2.0
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x_shifted = true_values + offset
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naive_var = naive_variance(x_shifted[np.newaxis, np.newaxis, :])
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stable_var = two_pass_variance(x_shifted[np.newaxis, np.newaxis, :])
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print(f" True values: {true_values}")
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print(f" True variance: {true_var:.15f}")
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print(f" Offset: {offset:.0e}")
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print(f" Shifted values: {x_shifted}")
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print()
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print(f" Naive (E[x²]-E[x]²): {naive_var[0,0]:.15f} (error: {abs(naive_var[0,0] - true_var):.2e})")
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print(f" Two-pass (centered): {stable_var[0,0]:.15f} (error: {abs(stable_var[0,0] - true_var):.2e})")
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print()
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# Show how it gets worse with larger offsets
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print(" Worsening with larger offsets:")
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for exp in range(4, 16, 2):
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offset = 10 ** exp
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x = true_values + offset
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nv = naive_variance(x[np.newaxis, np.newaxis, :])[0, 0]
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sv = two_pass_variance(x[np.newaxis, np.newaxis, :])[0, 0]
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print(f" offset=1e{exp:2d}: naive={nv:15.6f} stable={sv:15.6f} true=2.000000")
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print()
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# ---------------------------------------------------------------------------
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# 2. Performance benchmark
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# ---------------------------------------------------------------------------
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def benchmark(B, T, D, n_warmup=5, n_iter=50):
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"""Benchmark forward + backward throughput."""
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x = np.random.randn(B, T, D).astype(np.float32)
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gamma = np.random.randn(D).astype(np.float32)
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beta = np.random.randn(D).astype(np.float32)
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dy = np.random.randn(B, T, D).astype(np.float32)
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# Warmup
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for _ in range(n_warmup):
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y, cache = layer_norm_forward(x, gamma, beta)
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dx, dgamma, dbeta = layer_norm_backward(dy, cache)
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# Benchmark forward
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times_fwd = []
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for _ in range(n_iter):
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t0 = time.perf_counter()
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y, cache = layer_norm_forward(x, gamma, beta)
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times_fwd.append(time.perf_counter() - t0)
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# Benchmark backward
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times_bwd = []
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for _ in range(n_iter):
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t0 = time.perf_counter()
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dx, dgamma, dbeta = layer_norm_backward(dy, cache)
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times_bwd.append(time.perf_counter() - t0)
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N = B * T * D
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fwd_ms = np.median(times_fwd) * 1000
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bwd_ms = np.median(times_bwd) * 1000
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fwd_tflops = (6 * N) / (fwd_ms * 1e-3) / 1e12
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bwd_tflops = (9 * N) / (bwd_ms * 1e-3) / 1e12
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return {
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"shape": f"({B}, {T}, {D})",
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"N": N,
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"fwd_ms": fwd_ms,
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"bwd_ms": bwd_ms,
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"fwd_tflops": fwd_tflops,
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"bwd_tflops": bwd_tflops,
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}
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def run_benchmarks():
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print("=" * 70)
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print("PERFORMANCE BENCHMARK (NumPy, single CPU core)")
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print("=" * 70)
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print()
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print(f"{'Shape':<20} {'Elements':>10} {'Fwd (ms)':>10} {'Bwd (ms)':>10} {'Fwd TF/s':>10} {'Bwd TF/s':>10}")
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print("-" * 72)
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configs = [
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(1, 1, 64),
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(1, 1, 1024),
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(1, 1, 4096),
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(2, 128, 64),
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(2, 128, 1024),
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(2, 128, 4096),
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(4, 512, 1024),
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(4, 512, 4096),
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]
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for B, T, D in configs:
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result = benchmark(B, T, D)
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print(
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f"{result['shape']:<20} {result['N']:>10,} "
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f"{result['fwd_ms']:>10.4f} {result['bwd_ms']:>10.4f} "
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f"{result['fwd_tflops']:>10.4f} {result['bwd_tflops']:>10.4f}"
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)
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print()
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print(" Note: NumPy is multithreaded for large arrays (BLAS).")
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print(" These numbers are memory-bandwidth bound, not compute bound.")
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# ---------------------------------------------------------------------------
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# 3. Backward formula verification: alternative derivation
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# ---------------------------------------------------------------------------
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def verify_backward_alternative():
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"""
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Verify the backward formula using an alternative derivation path.
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Alternative: compute dx by explicitly differentiating through each step
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(mean → centered → normalized → affine) rather than using the compact
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projection formula. This serves as a cross-check.
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"""
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print("=" * 70)
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print("BACKWARD CROSS-CHECK: ALTERNATIVE DERIVATION")
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print("=" * 70)
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print()
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B, T, D = 3, 5, 8
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x = np.random.randn(B, T, D).astype(np.float64)
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gamma = np.random.randn(D).astype(np.float64)
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beta = np.random.randn(D).astype(np.float64)
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dy = np.random.randn(B, T, D).astype(np.float64)
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# Forward
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mu = x.mean(axis=-1, keepdims=True) # (B, T, 1)
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x_c = x - mu # (B, T, D)
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var = np.mean(x_c ** 2, axis=-1, keepdims=True) # (B, T, 1)
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std = np.sqrt(var + 1e-5) # (B, T, 1)
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x_hat = x_c / std # (B, T, D)
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y = gamma * x_hat + beta
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# --- Alternative backward: step-by-step chain rule ---
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# Step 4: y = γ·x_hat + β → ∂L/∂x_hat = γ·dy
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dx_hat = gamma[np.newaxis, np.newaxis, :] * dy # (B, T, D)
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# Step 3: x_hat = x_c / std
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# ∂x_hat_i/∂x_c_j = δ_ij/std - x_c_i·(Σ_k x_c_k·∂x_c_k/∂x_c_j)/(D·std³)
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# But since std depends on x_c, we need the full derivative.
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# ∂x_hat_i/∂x_c_j = (δ_ij·std - x_hat_i·x_hat_j/std) / std
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# = (δ_ij - x_hat_i·x_hat_j) / std
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# Wait, that's not quite right. Let me be more careful.
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#
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# x_hat_i = x_c_i / σ where σ = sqrt(mean(x_c²) + ε)
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# ∂σ/∂x_c_j = x_c_j / (D·σ)
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# ∂x_hat_i/∂x_c_j = (δ_ij·σ - x_c_i·∂σ/∂x_c_j) / σ²
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# = (δ_ij·σ - x_c_i·x_c_j/(D·σ)) / σ²
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# = δ_ij/σ - x_hat_i·x_hat_j/(D·σ)
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# = (1/σ) · (δ_ij - x_hat_i·x_hat_j/D)
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#
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# So: ∂L/∂x_c_j = Σ_i dx_hat_i · (1/σ) · (δ_ij - x_hat_i·x_hat_j/D)
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# = (1/σ) · [dx_hat_j - (1/D)·Σ_i(dx_hat_i·x_hat_i)·x_hat_j]
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std_inv = 1.0 / std[..., 0] # (B, T)
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dx_hat_sum_xhat = np.sum(dx_hat * x_hat, axis=-1, keepdims=True) # (B, T, 1)
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dx_c = std_inv[..., np.newaxis] * (dx_hat - dx_hat_sum_xhat * x_hat / D)
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# Step 2: x_c = x - μ
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# ∂x_c_i/∂x_j = δ_ij - 1/D
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# ∂L/∂x_j = Σ_i ∂L/∂x_c_i · (δ_ij - 1/D)
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# = dx_c_j - (1/D)·Σ_i dx_c_i
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dx_c_sum = np.sum(dx_c, axis=-1, keepdims=True) # (B, T, 1)
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dx_alt = dx_c - dx_c_sum / D
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# --- Our compact backward ---
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y2, cache = layer_norm_forward(x, gamma, beta)
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dx_ours, _, _ = layer_norm_backward(dy, cache)
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# Compare
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rel_err = np.max(np.abs(dx_alt - dx_ours)) / np.max(np.abs(dx_alt) + np.abs(dx_ours) + 1e-30)
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print(f" Alternative derivation (step-by-step chain rule)")
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print(f" Compact derivation (projection formula)")
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print(f" Relative error: {rel_err:.2e}")
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print(f" {'✓ MATCH' if rel_err < 1e-10 else '✗ MISMATCH'}")
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print()
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# ---------------------------------------------------------------------------
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# Main
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# ---------------------------------------------------------------------------
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if __name__ == "__main__":
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np.random.seed(42)
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demo_variance_stability()
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run_benchmarks()
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verify_backward_alternative()
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