{ "metadata": { "timestamp_utc": "2026-09-09T17:58:25.052691+00:00", "python": "3.14.4 (main, Apr 7 2026, 13:13:20) [Clang 21.0.0 (clang-2100.0.123.102)]", "platform": "macOS-26.6.2-arm64-arm-64bit-Mach-O", "machine": "arm64", "cpu": "Apple M5 Max", "logical_cpus": 18, "torch": "2.14.0", "numpy": "2.5.3", "torch_threads": 1, "torch_interop_threads": 1, "device": "cpu", "precision": "float32 training/timing; float64 exact/equivalence checks", "source_paper": "https://arxiv.org/abs/2609.05275", "source_version": "v1, 2026-09-04" }, "exact": { "protocol": "Exact mask enumeration, float64 arithmetic; scalar inputs x=1,a=0.5,b=0.4,target=0.7; half-MSE loss.", "claims": [ "Inverted dropout preserves an isolated residual branch's conditional mean.", "The Eq.6 shared mask biases even an affine two-sublayer block's mean by ab*p/(1-p)*x.", "Independent masks remove that affine covariance, but nonlinear composition still creates mean bias.", "Even one branch with an unbiased mean adds a dropout-dependent MSE penalty and changes its expected gradient." ], "rows": [ { "dropout_p": 0.0, "density": 1.0, "single_residual_mean": 1.5, "single_residual_dense": 1.5, "shared_mask_dense_prediction": 2.0999999999999996, "shared_mask_expected_prediction": 2.1, "shared_mask_prediction_bias": 4.440892098500626e-16, "shared_mask_expected_half_mse": 0.9800000000000002, "shared_mask_dense_half_mse": 0.9799999999999995, "shared_mask_expected_gradient_da": 1.96, "shared_mask_dense_gradient_da": 1.9599999999999995, "independent_linear_mean": 2.1, "independent_nonlinear_dense": 2.4, "independent_nonlinear_mean": 2.4000000000000004, "single_residual_loss_penalty": 0.0, "single_residual_expected_gradient_da": 0.8, "single_residual_dense_gradient_da": 0.8 }, { "dropout_p": 0.1, "density": 0.9, "single_residual_mean": 1.5000000000000002, "single_residual_dense": 1.5, "shared_mask_dense_prediction": 2.0999999999999996, "shared_mask_expected_prediction": 2.1222222222222227, "shared_mask_prediction_bias": 0.02222222222222303, "shared_mask_expected_half_mse": 1.0813237311385462, "shared_mask_dense_half_mse": 0.9799999999999995, "shared_mask_expected_gradient_da": 2.23443072702332, "shared_mask_dense_gradient_da": 1.9599999999999995, "independent_linear_mean": 2.1000000000000005, "independent_nonlinear_dense": 2.4, "independent_nonlinear_mean": 2.4111111111111114, "single_residual_loss_penalty": 0.01388888888888889, "single_residual_expected_gradient_da": 0.8555555555555556, "single_residual_dense_gradient_da": 0.8 }, { "dropout_p": 0.2, "density": 0.8, "single_residual_mean": 1.5, "single_residual_dense": 1.5, "shared_mask_dense_prediction": 2.0999999999999996, "shared_mask_expected_prediction": 2.1500000000000004, "shared_mask_prediction_bias": 0.05000000000000071, "shared_mask_expected_half_mse": 1.2165625, "shared_mask_dense_half_mse": 0.9799999999999995, "shared_mask_expected_gradient_da": 2.60625, "shared_mask_dense_gradient_da": 1.9599999999999995, "independent_linear_mean": 2.1000000000000005, "independent_nonlinear_dense": 2.4, "independent_nonlinear_mean": 2.4250000000000007, "single_residual_loss_penalty": 0.03125, "single_residual_expected_gradient_da": 0.925, "single_residual_dense_gradient_da": 0.8 }, { "dropout_p": 0.4, "density": 0.6, "single_residual_mean": 1.5, "single_residual_dense": 1.5, "shared_mask_dense_prediction": 2.0999999999999996, "shared_mask_expected_prediction": 2.2333333333333334, "shared_mask_prediction_bias": 0.13333333333333375, "shared_mask_expected_half_mse": 1.6825925925925933, "shared_mask_dense_half_mse": 0.9799999999999995, "shared_mask_expected_gradient_da": 3.925925925925927, "shared_mask_dense_gradient_da": 1.9599999999999995, "independent_linear_mean": 2.1, "independent_nonlinear_dense": 2.4, "independent_nonlinear_mean": 2.466666666666667, "single_residual_loss_penalty": 0.08333333333333334, "single_residual_expected_gradient_da": 1.1333333333333333, "single_residual_dense_gradient_da": 0.8 }, { "dropout_p": 0.6, "density": 0.4, "single_residual_mean": 1.5, "single_residual_dense": 1.5, "shared_mask_dense_prediction": 2.0999999999999996, "shared_mask_expected_prediction": 2.4, "shared_mask_prediction_bias": 0.30000000000000027, "shared_mask_expected_half_mse": 2.915, "shared_mask_dense_half_mse": 0.9799999999999995, "shared_mask_expected_gradient_da": 7.6, "shared_mask_dense_gradient_da": 1.9599999999999995, "independent_linear_mean": 2.1, "independent_nonlinear_dense": 2.4, "independent_nonlinear_mean": 2.55, "single_residual_loss_penalty": 0.18749999999999997, "single_residual_expected_gradient_da": 1.5499999999999998, "single_residual_dense_gradient_da": 0.8 }, { "dropout_p": 0.8, "density": 0.19999999999999996, "single_residual_mean": 1.5, "single_residual_dense": 1.5, "shared_mask_dense_prediction": 2.0999999999999996, "shared_mask_expected_prediction": 2.9000000000000004, "shared_mask_prediction_bias": 0.8000000000000007, "shared_mask_expected_half_mse": 9.640000000000006, "shared_mask_dense_half_mse": 0.9799999999999995, "shared_mask_expected_gradient_da": 29.400000000000016, "shared_mask_dense_gradient_da": 1.9599999999999995, "independent_linear_mean": 2.1, "independent_nonlinear_dense": 2.4, "independent_nonlinear_mean": 2.8000000000000003, "single_residual_loss_penalty": 0.5000000000000001, "single_residual_expected_gradient_da": 2.8000000000000007, "single_residual_dense_gradient_da": 0.8 } ], "schedule_check": { "layers": 6, "steps": 240, "pmax": 0.8, "measured_mean_dropout": 0.20000000000000004, "expected_pmax_over_4": 0.2 }, "optimizer_skip_check": { "protocol": "Scalar float64 AdamW, parameter initialized1, lr.1, weight_decay.1, default betas. First gradient1 initializes identical momentum. Second dropped step passes either zero or absent gradient; no optimizer synchronization policy is imposed.", "outcomes": { "zero_gradient": { "before_skipped_step": 0.890000001, "after_skipped_step": 0.8140941765241887 }, "absent_gradient": { "before_skipped_step": 0.890000001, "after_skipped_step": 0.890000001 } }, "interpretation": "Compute-then-mask and true skips need an explicit optimizer update policy to give the same training trajectory, even when forward outputs and mathematical gradients agree." }, "all_assertions_passed": true }, "timing": { "protocol": "CPU, float32, 1 PyTorch thread, GELU MLP residual blocks; 4 warmups then 9 randomized-order repetitions of 16 executions. Fixed balanced masks retain exactly half of sequence-block or batch-block work; they isolate execution overhead, not Bernoulli variance. Masks are precomputed. Backward excludes optimizer update. Dense uses no dropout scaling. No compilation.", "seed": 713, "density": 0.5, "equivalence_checks": [ { "mask_granularity": "sequence", "comparison": "full_mask vs gather", "max_forward_error": 0.0, "max_gradient_error": 0.0 }, { "mask_granularity": "batch", "comparison": "full_mask vs gather", "max_forward_error": 0.0, "max_gradient_error": 0.0 }, { "mask_granularity": "batch", "comparison": "full_mask vs batch_skip", "max_forward_error": 0.0, "max_gradient_error": 0.0 } ], "summaries": [ { "shape": "small", "phase": "forward", "mode": "dense", "mask_granularity": "none", "median_ms": 0.119453125, "min_ms": 0.109145875, "max_ms": 0.1876796875, "speedup_vs_dense": 1.0, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "small", "phase": "forward", "mode": "full_mask", "mask_granularity": "sequence", "median_ms": 0.148572875, "min_ms": 0.1311198125, "max_ms": 0.1820494375, "speedup_vs_dense": 0.8040035908304258, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "small", "phase": "forward", "mode": "gather", "mask_granularity": "sequence", "median_ms": 0.1278984375, "min_ms": 0.1217135625, "max_ms": 0.16416925, "speedup_vs_dense": 0.933968603017531, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "small", "phase": "forward", "mode": "full_mask", "mask_granularity": "batch", "median_ms": 0.1452734375, "min_ms": 0.1321640625, "max_ms": 0.2131015625, "speedup_vs_dense": 0.8222640494756656, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "small", "phase": "forward", "mode": "gather", "mask_granularity": "batch", "median_ms": 0.0875755, "min_ms": 0.0795390625, "max_ms": 0.1434244375, "speedup_vs_dense": 1.3640016328767748, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "small", "phase": "forward", "mode": "batch_skip", "mask_granularity": "batch", "median_ms": 0.0744895625, "min_ms": 0.0634765625, "max_ms": 0.096690125, "speedup_vs_dense": 1.6036223195699397, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "small", "phase": "forward_backward", "mode": "dense", "mask_granularity": "none", "median_ms": 0.366341125, "min_ms": 0.3559375, "max_ms": 0.380184875, "speedup_vs_dense": 1.0, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "small", "phase": "forward_backward", "mode": "full_mask", "mask_granularity": "sequence", "median_ms": 0.42796875, "min_ms": 0.424033875, "max_ms": 0.44028125, "speedup_vs_dense": 0.8559997079225995, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "small", "phase": "forward_backward", "mode": "gather", "mask_granularity": "sequence", "median_ms": 0.3809921875, "min_ms": 0.3747760625, "max_ms": 0.38710675, "speedup_vs_dense": 0.9615449791867451, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "small", "phase": "forward_backward", "mode": "full_mask", "mask_granularity": "batch", "median_ms": 0.4215234375, "min_ms": 0.412627625, "max_ms": 0.4284635625, "speedup_vs_dense": 0.8690883884718748, "repetitions": 9, 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"repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward", "mode": "gather", "mask_granularity": "sequence", "median_ms": 0.583877625, "min_ms": 0.568591125, "max_ms": 0.5920625, "speedup_vs_dense": 1.5254873613284978, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward", "mode": "full_mask", "mask_granularity": "batch", "median_ms": 0.9651614375, "min_ms": 0.9556614375, "max_ms": 0.9931796875, "speedup_vs_dense": 0.9228486581551804, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward", "mode": "gather", "mask_granularity": "batch", "median_ms": 0.5025755625, "min_ms": 0.4974296875, "max_ms": 0.5152265625, "speedup_vs_dense": 1.7722667076555279, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward", "mode": "batch_skip", "mask_granularity": "batch", "median_ms": 0.4705546875, "min_ms": 0.4600885, "max_ms": 0.4968150625, "speedup_vs_dense": 1.8928680579767894, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward_backward", "mode": "dense", "mask_granularity": "none", "median_ms": 2.543099, "min_ms": 2.51893225, "max_ms": 2.6889323125, "speedup_vs_dense": 1.0, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward_backward", "mode": "full_mask", "mask_granularity": "sequence", "median_ms": 2.7203828125, "min_ms": 2.6778489375, "max_ms": 2.765510375, "speedup_vs_dense": 0.9348312995930605, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward_backward", "mode": "gather", "mask_granularity": "sequence", "median_ms": 1.6455390625, "min_ms": 1.62659375, "max_ms": 1.6903046875, "speedup_vs_dense": 1.5454503985680985, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward_backward", "mode": "full_mask", "mask_granularity": "batch", "median_ms": 2.7343880625, "min_ms": 2.6784921875, "max_ms": 2.88890625, "speedup_vs_dense": 0.9300431913365261, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward_backward", "mode": "gather", "mask_granularity": "batch", "median_ms": 1.3985416875, "min_ms": 1.38154425, "max_ms": 1.438776, "speedup_vs_dense": 1.8183934184657617, "repetitions": 9, "executions_per_repetition": 16 }, { "shape": "medium", "phase": "forward_backward", "mode": "batch_skip", "mask_granularity": "batch", "median_ms": 1.3219270625, "min_ms": 1.31053125, "max_ms": 1.35028125, "speedup_vs_dense": 1.9237816307282085, "repetitions": 9, "executions_per_repetition": 16 } ], "limitations": "This is a local CPU MLP measurement, not an attention, GPU, CS-3, energy, or distributed benchmark. Same semantic output/gradient is checked only between implementations using the same masks; sequence and batch masks define different stochastic models. Zero and absent gradients are mathematically equivalent for this check but can differ in optimizer state updates." }, "training": { "protocol": { "task": "Online fresh-sample noiseless 8D regression; target sin(x0+.5*x1)+.5*cos(x2-x3)+.25*x4*x5+.1*x6^2-.2*x7, inputs iid N(0,1).", "architecture": "Trainable 8\u219224 tanh embedding, 6 or 5 residual tanh-linear 24\u219224 blocks scaled by 1/sqrt(depth), linear scalar head; no attention or normalization.", "optimizer": "AdamW(weight_decay=.01), cosine LR to 10% of initial LR, batch=64, base_steps=240; no gradient clipping or early stopping.", "mask": "Independent Bernoulli per sample and residual block; inverted 1/(1-p) scaling in training, scale=1 at evaluation. Compute-then-mask implementation.", "tuning": "Equal 5 learning rates \u00d7 2 tuning seeds independently for every config/regime; select by mean MSE on fixed 1,024-point validation set. Evaluation uses 6 disjoint seeds and a fixed separate 2,048-point test set. Initial 3-point pilot hit the upper boundary for every treatment; grid expanded equally to include .03 and .1 before interpreting final comparisons.", "learning_rates": [ 0.001, 0.003, 0.01, 0.03, 0.1 ], "tuning_seeds": [ 10, 11 ], "evaluation_seeds": [ 100, 101, 102, 103, 104, 105 ], "budget": "Matched steps = 240 for every architecture. Matched expected active sequence-block budget = 240*6*64; dropout configurations get 300 steps; dense5 gets 288. Excludes embedding, head, optimizer, and mask overhead. Actual execution computes every row; budget is conceptual, NOT measured FLOPs or elapsed time.", "confidence": "Two-sided Student-t 95% intervals over 6 evaluation seeds (df=5); paired intervals use per-seed MSE differences. No multiple-comparison correction; these characterize this toy protocol only." }, "configs": { "dense6": { "depth": 6, "schedule": "none", "pmax": 0.0, "mean_dropout": 0.0 }, "constant_ild": { "depth": 6, "schedule": "constant", "pmax": 0.4, "mean_dropout": 0.2 }, "decreasing_ild": { "depth": 6, "schedule": "decreasing", "pmax": 0.8, "mean_dropout": 0.2 }, "increasing_ild": { "depth": 6, "schedule": "increasing", "pmax": 0.8, "mean_dropout": 0.2 }, "dense5": { "depth": 5, "schedule": "none", "pmax": 0.0, "mean_dropout": 0.0 } }, "selected_lrs": [ { "config": "dense6", "regime": "matched_steps", "selected_lr": 0.03 }, { "config": "constant_ild", "regime": "matched_steps", "selected_lr": 0.03 }, { "config": "decreasing_ild", "regime": "matched_steps", "selected_lr": 0.03 }, { "config": "increasing_ild", "regime": "matched_steps", "selected_lr": 0.03 }, { "config": "dense5", "regime": "matched_steps", "selected_lr": 0.03 }, { "config": "dense6", "regime": "matched_expected_active_budget", "selected_lr": 0.03 }, { "config": "constant_ild", "regime": "matched_expected_active_budget", "selected_lr": 0.03 }, { "config": "decreasing_ild", "regime": "matched_expected_active_budget", "selected_lr": 0.03 }, { "config": "increasing_ild", "regime": "matched_expected_active_budget", "selected_lr": 0.03 }, { "config": "dense5", "regime": "matched_expected_active_budget", "selected_lr": 0.03 } ], "summaries": [ { "config": "dense6", "regime": "matched_steps", "test_mse": { "n": 6, "mean": 0.019246889588733513, "ci95_low": 0.014997986324559572, "ci95_high": 0.023495792852907454, "std": 0.004048750683419803 }, "early_exit_depth4_test_mse": { "n": 6, "mean": 0.19203738123178482, "ci95_low": 0.06685966038797841, "ci95_high": 0.3172151020755912, "std": 0.1192809888350865 }, "expected_budget_relative_to_dense6": 1.0, "steps": 240, "selected_lr": 0.03, "median_train_seconds": 0.08756937505677342 }, { "config": "constant_ild", "regime": "matched_steps", "test_mse": { "n": 6, "mean": 0.023820065893232822, "ci95_low": 0.0197013478721617, "ci95_high": 0.027938783914303943, "std": 0.003924698061081226 }, "early_exit_depth4_test_mse": { "n": 6, "mean": 0.03163820287833611, "ci95_low": 0.025301004581064185, "ci95_high": 0.037975401175608034, "std": 0.006038672650749278 }, "expected_budget_relative_to_dense6": 0.7999999999999964, "steps": 240, "selected_lr": 0.03, "median_train_seconds": 0.09946945903357118 }, { "config": "decreasing_ild", "regime": "matched_steps", "test_mse": { "n": 6, "mean": 0.026742215889195602, "ci95_low": 0.02009670595645054, "ci95_high": 0.03338772582194067, "std": 0.006332460686676907 }, "early_exit_depth4_test_mse": { "n": 6, "mean": 0.03571749913195769, "ci95_low": 0.026648578165703818, "ci95_high": 0.04478642009821156, "std": 0.008641712384840348 }, "expected_budget_relative_to_dense6": 0.7999999999999994, "steps": 240, "selected_lr": 0.03, "median_train_seconds": 0.09688974998425692 }, { "config": "increasing_ild", "regime": "matched_steps", "test_mse": { "n": 6, "mean": 0.027081420334676903, "ci95_low": 0.02244259947967283, "ci95_high": 0.031720241189680976, "std": 0.004420300472670612 }, "early_exit_depth4_test_mse": { "n": 6, "mean": 0.03479581419378519, "ci95_low": 0.029082039571742054, "ci95_high": 0.04050958881582833, "std": 0.005444616520447294 }, "expected_budget_relative_to_dense6": 0.8000000000000002, "steps": 240, "selected_lr": 0.03, "median_train_seconds": 0.09989185445010662 }, { "config": "dense5", "regime": "matched_steps", "test_mse": { "n": 6, "mean": 0.021738227146367233, "ci95_low": 0.010363445866578563, "ci95_high": 0.0331130084261559, "std": 0.010838950810815593 }, "early_exit_depth4_test_mse": { "n": 6, "mean": 0.07576803428431352, "ci95_low": 0.006903761218256907, "ci95_high": 0.14463230735037014, "std": 0.06562029194458746 }, "expected_budget_relative_to_dense6": 0.8333333333333334, "steps": 240, "selected_lr": 0.03, "median_train_seconds": 0.07534756243694574 }, { "config": "dense6", "regime": 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LR grid is small and selection can land on its boundary. This does not test CompleteP transfer, language loss, token scaling, CS-3 speed, speculative decoding or universal optimal schedules. Sparse toy training uses compute-then-mask, so the budget is theoretical active work." } }