Sharpness-Aware Minimization Meets Spectral Norm

We propose a projected variant of Sharpness-Aware Minimization (called \textbf{Spectral-SAM}) that projects the weights onto a per-layer spectral-norm ball at two points inside each SAM step, capping layer gains while keeping the SAM objective. The method couples loss-landscape flattening with function‐level smoothness, and can be dropped into standard pipelines with negligible code changes. On CIFAR-10/100 with \texttt{torchvision} ResNet-18 and with PreActResNet-18, it increases AutoAttack and PGD robustness over SAM and Eigen-SAM, improves calibration (ECE), reduces Hessian trace, and constrains the maximal singular value. We also expose a failure mode: clipping \emph{only} after SAM’s ascent step inflates singular values and harms robustness; clipping after both steps prevents this drift.