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Workshop: OPT 2023: Optimization for Machine Learning

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

Peiyuan Zhang · Jingzhao Zhang · Suvrit Sra


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.

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