Certificate-Aware Gradient-Noise Probing

Integrate the controller into SGD or AdamW between ordinary minibatch updates. Maintain an EMA mean gradient \(\bar g_t\) and estimate gradient-noise variance from two independently augmented minibatches: compute \(g_t^{(1)}\) and \(g_t^{(2)}\), set \(r_t=(g_t^{(1)}-g_t^{(2)})/\sqrt{2}\), and update \(\hat\theta_t=(1-\beta)\hat\theta_{t-1}+\beta\|r_t\|^2/d\), where \(d\) is the number of sampled gradient coordinates. Approximate the posterior over \(\theta\) with a Gamma distribution updated from the squared residuals. Estimate \(L_\eta\) by occasional finite differences of the short-window loss slope under learning-rate perturbations, and estimate \(\kappa\) from the local slope of update-norm or loss-floor versus measured noise. At each controller interval, calculate the displayed posterior VoNI. If it is below \(\varepsilon_C\), use one minibatch and the normal learning rate. If it exceeds the threshold, collect two to four diagnostic minibatches, update the posterior, and reduce the learning rate or increase batch size until the estimate falls below the budget. The paper supplies the certificate-aware objective and model/radius decomposition; the neural implementation estimates the posterior and sensitivities empirically. First test a two-layer MLP on MNIST with SGD under an abrupt fourfold gradient-noise increase halfway through training. Compare fixed control, posterior-entropy probing, and VoNI probing. Record diagnostic minibatches, loss divergence, and recovery time. The prediction is a crossover at \(\widehat{\operatorname{VoNI}}_t=\varepsilon_C\): intervention should occur near this boundary, and post-intervention loss-slope degradation should stay within the chosen certificate budget, while entropy probing should react later or spend more diagnostics before the boundary.