Tung Lam Pham

Tuan Truong, Hoang-Phi Nguyen, Haocheng Luo et al.

Recent studies highlight the effectiveness of flat minima in enhancing generalization, with sharpness-aware minimization (SAM) achieving state-of-the-art performance. Additionally, insights into the intrinsic geometry of the loss landscape have shown promise for improving model generalization. Building on these advancements, we introduce a novel sharpness-aware, geometry-aware teleportation mechanism to further enhance robustness and generalization. The core innovation of our approach is to decompose each iteration into a teleportation step within a local orbit and a sharpness-aware step that transitions between different orbits, leveraging the Riemannian quotient manifold. Our approach is grounded in a theoretical framework that analyzes the generalization gap between population loss and worst-case empirical loss within the context of Riemannian manifolds. To demonstrate the effectiveness of our method, we evaluate and compare our algorithm on diverse vision benchmarks with various datasets and Riemannian manifolds.

Haocheng Luo, Tuan Truong, Tung Pham et al.

Sharpness-Aware Minimization (SAM) has attracted significant attention for its effectiveness in improving generalization across various tasks. However, its underlying principles remain poorly understood. In this work, we analyze SAM's training dynamics using the maximum eigenvalue of the Hessian as a measure of sharpness, and propose a third-order stochastic differential equation (SDE), which reveals that the dynamics are driven by a complex mixture of second- and third-order terms. We show that alignment between the perturbation vector and the top eigenvector is crucial for SAM's effectiveness in regularizing sharpness, but find that this alignment is often inadequate in practice, limiting SAM's efficiency. Building on these insights, we introduce Eigen-SAM, an algorithm that explicitly aims to regularize the top Hessian eigenvalue by aligning the perturbation vector with the leading eigenvector. We validate the effectiveness of our theory and the practical advantages of our proposed approach through comprehensive experiments. Code is available at https://github.com/RitianLuo/EigenSAM.