论文标题
Sion的Minimax定理在测量公制空间和Riemannian外部算法中
Sion's Minimax Theorem in Geodesic Metric Spaces and a Riemannian Extragradient Algorithm
论文作者
论文摘要
确定鞍点是否存在或与非convex-nonconcave问题相似，通常是棘手的。本文迈出了一步，朝着理解一系列的非convex-nonconcave minimax问题，这些问题仍然可以解决。具体而言，它研究了在测量公制空间上的最小值问题，这些问题对通常的凸连接鞍点问题提供了广泛的概括。该论文的第一个主要结果是Sion的Minimax定理的大地测量空间版本。我们认为，我们的证据是新颖的，并且仅依赖有限的交叉属性，因此可以广泛使用。第二个主要结果是针对地理上完成riemannian歧管的专业化：在这里，我们设计和分析了对于平滑的最小问题的一阶方法的复杂性。
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.
