Inverse-Square Adaptive Parameter Reset

First, integrate the mechanism as an outer wrapper around SGD or Adam on a small model, resetting the full parameter vector to the best validation checkpoint. A safer first variant applies it only to optimizer momentum or to an RNN/SSM hidden state. Define the scalar excursion z = norm(theta - theta_ref), where theta_ref is the current reference checkpoint. After every optimizer update, compute the reset probability q = 1 - exp(-delta_t times r0 divided by (z squared plus epsilon squared)). Draw a uniform random number U; if U < q, restore theta to theta_ref and optionally set Adam moments to zero. Pseudocode is: compute minibatch gradient; perform baseline optimizer update; compute z; compute q; sample U; restore the checkpoint if U < q; otherwise continue. Keep theta_ref fixed initially, then test a version updated only after validation improvement. The paper supplies the inverse-square hazard, the power-law exponent, and the critical ratio r0/D = 2. Estimate D empirically from the covariance of parameter increments after subtracting their local mean drift: D approximately equals trace of the increment covariance divided by twice the step duration. The first cheap experiment should use a two-layer MLP on MNIST, comparing ordinary SGD, constant-rate checkpoint resets, and inverse-square resets across a grid of r0 values. Record excursion histograms, reset rates, validation loss, maximum excursion, and fitted tail exponent. The quantitative prediction is a transition near r0/D = 2: above it, excursions remain localized; below it, the complementary cumulative excursion distribution follows a power law with exponent approximately beta minus 1. The measured boundary should be within 20 percent of 2. As controls, a hazard proportional to z raised to minus 3 should fail to localize, while a hazard proportional to z raised to minus 1 should produce stretched-exponential rather than power-law tails.