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Deciding whether saddle points exist or are approximable fornonconvex-nonconcave problems is usually intractable. This paper takes a steptowards understanding a broad class of nonconvex-nonconcave minimax problemsthat do remain tractable. Specifically, it studies minimax problems overgeodesic metric spaces, which provide a vast generalization of the usualconvex-concave saddle point problems. The first main result of the paper is ageodesic metric space version of Sion's minimax theorem; we believe our proofis novel and broadly accessible as it relies on the finite intersectionproperty alone. The second main result is a specialization to geodesicallycomplete Riemannian manifolds: here, we devise and analyze the complexity offirst-order methods for smooth minimax problems.

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