diff options
| -rw-r--r-- | THEORY.md | 35 | ||||
| -rw-r--r-- | experiments/verify_theory.py | 26 |
2 files changed, 61 insertions, 0 deletions
@@ -470,6 +470,41 @@ The relevant timescale is therefore the product of learning rate and number of neutral updates. Too few neutral updates leave traffic; task-period updates converge to the wrong coefficient regardless of speed. +### Intermittent feedback tracking can hide behind a final cosine + +An idealized mirror event every `k` task updates uses + +```text +Q_m^+ = Q_m + eta_M (W_m - Q_m). +``` + +Let `E_m=Q_m-W_m` immediately before that event and let +`U_m=W_(m+1)-W_m` be the sum of the following `k` forward-weight changes. +The tracking error obeys the exact recurrence + +```text +E_(m+1) = (1 - eta_M) E_m - U_m. +``` + +If every task update has norm at most `d`, then `||U_m|| <= k d` and + +```text +limsup_m ||E_m|| <= k d / eta_M. +``` + +The bound is attained in the scalar or constant-collinear case. Thus reducing +the mirror cadence or increasing its learning rate only controls lag relative +to how fast the task weights move; a high feedback cosine is not automatic. +For the frozen RRM setting `k=16, eta_M=0.1`, the ideal worst-case lag scale is +`160` single-task-update norms before stochastic finite-probe error is added. + +Conversely, after the task updates stop (`U_m=0`), the endpoint error decays as +`(1-eta_M)^m`. A learning-rate drop or quiet tail can therefore produce an +excellent final cosine even when the feedback was badly out of date during +the loss-producing part of training. Endpoint alignment must be reported with +the alignment trajectory and task loss. This is a standard tracking fact used +to audit the mirror baseline, not an SDIL novelty claim. + ## 4. Hardware-independent complexity Let `H` be the number of hidden layers, `K` the perturbation directions per diff --git a/experiments/verify_theory.py b/experiments/verify_theory.py index ba07487..8054b0d 100644 --- a/experiments/verify_theory.py +++ b/experiments/verify_theory.py @@ -196,6 +196,31 @@ def check_innovation_identification(): assert abs(retained_fraction - (1.0 - predictable_fraction)) < 0.01 +def check_intermittent_feedback_tracking(): + eta_m = 0.1 + cadence = 16 + per_step_change = 0.002 + block_change = cadence * per_step_change + + # Constant collinear forward-weight motion attains the norm bound. + error = 0.0 + for _ in range(1000): + error = (1.0 - eta_m) * error - block_change + steady_state = -block_change / eta_m + assert abs(error - steady_state) < 1e-14 + + # Once forward motion stops, endpoint tracking can look excellent despite + # the large error accumulated during the task-active trajectory. + error_before_quiet_tail = abs(error) + for _ in range(80): + error *= 1.0 - eta_m + expected_endpoint = error_before_quiet_tail * (1.0 - eta_m) ** 80 + print("\nINTERMITTENT FEEDBACK TRACKING") + print(f"steady_error={abs(steady_state):.6f} bound={cadence * per_step_change / eta_m:.6f} " + f"endpoint_after_quiet_tail={abs(error):.9e}") + assert abs(abs(error) - expected_endpoint) < 1e-15 + + def main(): check_simultaneous_variance() check_sigma_bias() @@ -203,6 +228,7 @@ def main(): check_conditional_projection() check_predictor_timescale() check_innovation_identification() + check_intermittent_feedback_tracking() print("\nALL THEORY CHECKS PASSED") |