Floor-Aware SAM Radius Scheduling

Modify only the radius-selection line of an existing SAM implementation. Maintain the previous parameter vector $x_{t-1}$ and ordinary minibatch gradient $g_{t-1}$, not the perturbed gradient. At step $t$, compute the current ordinary gradient $g_t$ and estimate local smoothness as Lhat_raw=norm(g_t-g_prev)/(norm(x_t-x_prev)+eps). To avoid noisy or artificially small estimates, update Lhat=beta*Lhat+(1-beta)*Lhat_raw and clamp it to a running maximum or a conservative lower bound. Choose a target gradient tolerance $\varepsilon_t$: keep it fixed for a simple test, or set it to the running median gradient norm multiplied by a decay factor. Set rho=min(rho_max, epsilon_t**alpha/(c*Lhat)), where $c=2$ or $4$ is a safety margin, then perform ordinary SAM with this $\rho_t$: perturb by rho_t*g_t/(norm(g_t)+eps)**alpha, recompute the gradient, restore parameters, and update with AdamW or SGD. The paper supplies the inversion $\rho\le\varepsilon^\alpha/L$; the curvature estimate and target schedule are empirical engineering choices. First run a CIFAR-10 ResNet-18 experiment with fixed SAM radius, cosine-decayed radius, and this floor-aware schedule, using the same optimizer and two-pass budget. Evaluate gradient norm at the end of training, validation accuracy, and loss descent per step. The predicted signal is fewer late-training oscillations and a lower final gradient norm than fixed SAM, while preserving its early-training validation improvement. Measure whether reducing perturbation also prevents unstable runs; per-step FLOPs remain approximately unchanged.