@inproceedings{
zhang2023sions,
title={Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm},
author={Peiyuan Zhang and Jingzhao Zhang and Suvrit Sra},
booktitle={OPT 2023: Optimization for Machine Learning},
year={2023},
url={https://openreview.net/forum?id=D8WJ7gQEG1}
}
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.
Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm
Peiyuan Zhang Jingzhao Zhang Suvrit Sra