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

TL;DR

This paper extends Sion's minimax theorem to geodesic metric spaces and develops a Riemannian extragradient algorithm for smooth minimax problems, broadening the understanding of nonconvex-nonconcave optimization.

Contribution

It introduces a geodesic metric space version of Sion's minimax theorem and proposes a first-order Riemannian extragradient method with complexity analysis for smooth problems.

Findings

Proved a novel geodesic metric space minimax theorem.

Developed and analyzed a Riemannian extragradient algorithm.

Provided complexity bounds for the proposed method.

Abstract

Deciding whether saddle points exist or are approximable for nonconvex-nonconcave problems is usually intractable. This paper takes a step towards understanding a broad class of nonconvex-nonconcave minimax problems that do remain tractable. Specifically, it studies minimax problems over geodesic metric spaces, which provide a vast generalization of the usual convex-concave saddle point problems. 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: here, we devise and analyze the complexity of first-order methods for smooth minimax problems.