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.

翻译：判定非凸-非凹问题的鞍点是否存在或是否可逼近通常是棘手的。本文旨在理解一类保持可处理性的广泛非凸-非凹极小极大问题。具体而言，本文研究了定义在测地度量空间上的极小极大问题，这一框架极大推广了经典的凸-凹鞍点问题。论文的第一个主要结果是Sion极小极大定理在测地度量空间版本；我们相信该证明具有新颖性且易于理解，因为它仅依赖于有限交性质。第二个主要结果是该定理在测地完备黎曼流形上的特化：在此背景下，我们设计并分析了用于光滑极小极大问题的一阶方法的复杂度。