Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm
This work addresses a foundational problem in optimization theory for researchers dealing with nonconvex-nonconcave minimax settings, though it is incremental as it extends existing theorems and methods to more general spaces.
The paper tackles the intractability of nonconvex-nonconcave minimax problems by studying them over geodesic metric spaces, proving a generalized version of Sion's minimax theorem and developing a Riemannian extragradient algorithm with analyzed complexity for smooth problems.
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.