Polynomial-Lyapunov Training Controller

Integrate this controller into SGD or Adam after computing a minibatch gradient but before permanently applying the update. Maintain a nonnegative monitored error E_k: on supervised learning use the current loss minus a running lower bound, clipped below by epsilon_E. Pseudocode: compute gradient g_k; propose theta_trial = theta_k - eta_k g_k; evaluate a small fixed monitoring minibatch and obtain E_trial; compute c_hat = (E_k - E_trial)/(E_k^(1+m) delta_t); reject the proposal and halve eta if E_trial exceeds E_k; otherwise accept it and update eta using the displayed multiplicative controller toward c_star. Smooth c_hat with an exponential moving average to reduce minibatch noise. The Lyapunov comparison law and degree interpretation come from the paper; c_star, the loss floor, smoothing coefficient, and rejection thresholds are empirical. First run a two-layer MLP on MNIST and a small ResNet on CIFAR-10, comparing SGD, Adam, and controllers with m equal to 0, 1, and 2. Plot loss on log-linear and log-log axes. The falsifiable prediction is exponential decay for m equal to zero and a final log-log slope near -1/m for m greater than zero; estimate the slope over the final third of training and require agreement within 20 percent. Also test learning rates around the acceptance boundary: the controller should reject steps when c_hat becomes nonpositive, and the measured transition from stable to divergent updates should be reproducible across random seeds.