SHARD-SAM: Signed Hessian-Amplified Radius and Direction for Sharpness-Aware Minimizatio

Reconciling optimization and generalization remains a fundamental challenge in deep learning. Sharpness-Aware Minimization (SAM) advances this goal, yet its fixed perturbation radius and first-order geometric approximation limit its adaptability to heterogeneous loss landscapes. We propose SHARD-SAM (Signed Hessian-Amplified Radius and Direction), a sharpness-aware framework that jointly adapts perturbation magnitude and orientation through a shared curvature signal. SHARD-SAM constructs a signed Hessian-amplified spectral surrogate using zeroth-order curvature probes. A symmetric asymptotic envelope adjusts the perturbation radius relative to a running curvature baseline, while magnitude-adaptive gating combines the gradient with a spectral correction. A shared curvature-temperature parameter controls both modules, coupling radius and direction adaptation within a unified perturbation rule. Our theoretical analysis characterizes the surrogate's spectral properties, estimation error, and conditions for local directional benefits. Experiments on CIFAR-10, CIFAR-100, and ImageNet across CNN and ViT architectures demonstrate consistent generalization improvements over representative first-order and second-order geometry-adaptive SAM optimizers, with a favorable accuracy–efficiency trade-off among the evaluated curvature-aware methods. Ablations and direction-source controls support the contribution of the shared curvature signal and its two adaptive modules.