Lyapunov Sign-Search Optimizer

1. Integration point: insert the wrapper after a standard optimizer computes its proposed block update and before parameters are modified. Divide the model into four to eight blocks, such as transformer layers, attention heads, convolutional channel groups, or federated-client parameter groups. Use ordinary SGD or AdamW to produce the nominal proposal $v_t$; do not change its norm, momentum, or variance estimates.

2. Pseudocode: maintain a candidate sign vector $s$. Compute $v_t$, form a probe update $u_t=-\eta(s\odot v_t)$, and evaluate a cheap loss probe at $\theta_t+u_t$ using the current minibatch or a small held-out subset. Set $r=(L_t-L(\theta_t+u_t))/\eta$. If $r\geq\rho\|v_t\|^2$ for $K$ consecutive steps, retain $s$; otherwise advance one block sign in a vector scheduler or enumerate sign patterns with a cyclic scalar scheduler. Apply only the selected update and discard the probe state. Reset the scheduler after $J$ consecutive failures.

3. Computed versus estimated: the diagonal signs, candidate schedule, and update rule are exact design quantities. The smoothness constant $L$, descent constant $c$, and noise scale are estimated empirically. Estimate $L$ with power iteration on a minibatch Hessian or finite-difference gradient ratios. Use three repeated loss probes to estimate variance and select $\rho$ above twice the probe standard deviation.

4. First cheap experiment: train a two-layer MLP on MNIST with SGD, split parameters into four blocks, and inject a planted fixed random sign reversal into each block's effective update. Compare ordinary SGD, AdamW, random sign search, and the proposed wrapper over step sizes from $0.1/L$ to $3/L$. The falsifiable signatures are: the scheduler recovers the planted pattern after at most approximately $K2^m$ probes for scalar enumeration; after recovery, its loss curve matches correctly oriented SGD within probe noise; and the stable-step boundary occurs near $\eta_c=2/L$, within 20 percent of the Hessian estimate. If the injected sign pattern changes faster than the persistence window $K$, permanent recovery should fail gracefully rather than produce a false stability claim.