Nonadiabatic Training Controller

1. Integration point: add a controller around Adam or SGD rather than modifying backpropagation. Maintain B short replicas sharing the same architecture and initialization but receiving independently sampled mini-batches, or use B stochastic perturbations around the current weights. Let z be a low-dimensional observable such as per-example loss, logits projected by a fixed random matrix, or the first 16 principal components of recent gradients. At each controller update, collect z from the current finite-rate schedule p_t and from a slow reference run q_t using a learning rate 10 to 50 times smaller. 2. Pseudocode: run m optimizer steps at learning rate eta; collect z_p and z_q; fit Gaussian means and covariances (mu_p,Sigma_p) and (mu_q,Sigma_q); compute D_hat=0.5*(log(det(Sigma_q)/det(Sigma_p))-d+tr(Sigma_q^{-1}Sigma_p)+(mu_q-mu_p)^T Sigma_q^{-1}(mu_q-mu_p)); update eta <- eta*exp(-kappa*max(0,D_hat-D_star)); repeat. Optionally estimate block work from w_i=loss_i,end-loss_i,start plus any explicit schedule-driving cost and monitor mean(exp(-beta*w_i)). 3. Taken from the paper: relative-entropy nonnegativity, excess-work interpretation, and exponential fluctuation relation. Estimated empirically: p_t, q_t, beta, work, and D_star; beta can initially be set from the inverse variance scale of block losses. 4. First cheap experiment: train a two-layer MLP on MNIST with SGD, comparing fixed learning rates and cosine decay against this controller. Use B=16 replicas, m=20 steps, 16 projected observables, and a reference rate eta/20. Sweep eta geometrically. The falsifiable prediction is that loss variance and catastrophic spikes increase sharply when D_hat crosses D_star; the controller should keep D_hat near the threshold. The exponential-work statistic should be closest to one for small eta and deviate monotonically as eta increases.