Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm

Published: 26 Oct 2023, Last Modified: 13 Dec 2023NeurIPS 2023 Workshop PosterEveryoneRevisionsBibTeX
Keywords: minimax, geodesic convexity, extragradient
Abstract: Deciding whether saddle points exist or are approximable for nonconvex-nonconcave problems is usually intractable. We take a step toward understanding a broad class of nonconvex-nonconcave minimax problems that do remain tractable. Specifically, we study minimax problems in geodesic metric spaces. The first main result of the paper is a geodesic metric space version of Sion's minimax theorem; we believe our proof is novel and broadly accessible as it relies on the finite intersection property alone. The second main result is a specialization to geodesically complete Riemannian manifolds, for which we analyze first-order methods for smooth minimax problems.
Submission Number: 82