CR-SAM: Curvature Regularized Sharpness-Aware Minimization

The capacity to generalize to future unseen data stands as one of the utmost crucial attributes of deep neural networks. Sharpness-Aware Minimization (SAM) aims to enhance the generalizability by minimizing worst-case loss using one-step gradient ascent as an approximation. However, as training progresses, the non-linearity of the loss landscape increases, rendering one-step gradient ascent less effective. On the other hand, multi-step gradient ascent will incur higher training cost. In this paper, we introduce a normalized Hessian trace to accurately measure the curvature of loss landscape on {\em both} training and test sets. In particular, to counter excessive non-linearity of loss landscape, we propose Curvature Regularized SAM (CR-SAM), integrating the normalized Hessian trace as a SAM regularizer. Additionally, we present an efficient way to compute the trace via finite differences with parallelism. Our theoretical analysis based on PAC-Bayes bounds establishes the regularizer's efficacy in reducing generalization error. Empirical evaluation on CIFAR and ImageNet datasets shows that CR-SAM consistently enhances classification performance for ResNet and Vision Transformer (ViT) models across various datasets. Our code is available at https://github.com/TrustAIoT/CR-SAM.