Delay-aware event-triggered optimizer

Integrate the mechanism at the optimizer communication or update layer, initially using SGD on a small quadratic problem. Maintain parameters theta, a last-transmitted reference theta_hat, and a queue of corrections with future execution times. At optimizer step k, compute gradient g_k and the proposed correction u_k=-eta g_k. Trigger an event only when the observable drift satisfies ||theta_k-theta_hat||^2 > epsilon V_k, using V_k=||g_k||^2+lambda||theta_k-theta_hat||^2 for a practical proxy. When triggered, enqueue u_k with execution time k+d and set theta_hat=theta_k. At each step, apply all queued corrections whose execution time has arrived; this explicitly creates stale delayed impulses. Pseudocode is: compute g; perform the local optimizer step; evaluate drift and V; if drift^2 exceeds epsilon V, enqueue the correction and refresh theta_hat; apply due corrections; record V and event times. Estimate mu from the maximum observed local secant growth rate, and estimate q and r by fitting the ratio of post-event to pre-event quadratic energies. The theoretical mechanism is the multiplier exp(2 mu d)(q+r epsilon); these quantities are empirically estimated rather than assumed exact. First test one-dimensional and 20-dimensional quadratic regression, then MNIST logistic regression, comparing ordinary SGD, fixed-delay SGD, and the event-triggered variant. Sweep delay d and threshold epsilon. The quantitative prediction is a divergence boundary near exp(2 mu d)(q+r epsilon)=1: below the boundary, the energy envelope decays geometrically; above it, delayed corrections cause oscillation or divergence. The measured boundary should agree with the estimated condition within 20 percent, and event times should have a positive minimum gap rather than accumulating.