diff options
Diffstat (limited to 'THEORY.md')
| -rw-r--r-- | THEORY.md | 35 |
1 files changed, 35 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 A learning-rate drop or quiet tail can therefore produce an +excellent final cosine even when the feedback was badly out of date during +the loss-producing part of training.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 |