RSAM: Learning on Manifolds with Riemannian Sharpness-Aware Minimization

Tuan Truong, Hoang-Phi Nguyen, Tung Pham, Minh-Tuan Tran, Mehrtash Harandi, Dinh Phung, Trung Le

Keywords: manifold, constrained optimization, sharpness-aware minimization, optimization

TL;DR: We introduce a Riemannian sharpness-aware optimizer that improves generalization ability.

Abstract: Nowadays, understanding the geometry of the loss landscape shows promise in enhancing a model's generalization ability. In this work, we draw upon prior works that apply geometric principles to optimization and present a novel approach to improve robustness and generalization ability for constrained optimization problems. Indeed, this paper aims to generalize the Sharpness-Aware Minimization (SAM) optimizer to Riemannian manifolds. In doing so, we first extend the concept of sharpness and introduce a novel notion of sharpness on manifolds. To support this notion of sharpness, we present a theoretical analysis characterizing generalization capabilities with respect to manifold sharpness, which demonstrates a tighter bound on the generalization gap, a result not known before. Motivated by this analysis, we introduce our algorithm, Riemannian Sharpness-Aware Minimization (RSAM). To demonstrate RSAM's ability to enhance generalization ability, we evaluate and contrast our algorithm on a broad set of problems, such as image classification and contrastive learning across different datasets, including CIFAR100, CIFAR10, and FGVCAircraft. Our code is publicly available at \url{https://t.ly/RiemannianSAM}.

Primary Area: metric learning, kernel learning, and sparse coding

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Submission Number: 4869