Calm local optimality for nonconvex-nonconcave minimax problems

TL;DR

This paper introduces the concept of calm local minimax points for nonconvex-nonconcave problems, providing new optimality conditions and establishing their equivalence with local minimax points under certain conditions.

Contribution

It defines calm local minimax points and derives first and second-order optimality conditions for nonsmooth problems, extending the theory of local minimax optimality.

Findings

Calm local minimax points have a calm radius function.

Optimality conditions are established for a broad class of nonsmooth problems.

Calm local minimax optimality coincides with local minimax optimality under weak conditions.

Abstract

Nonconvex-nonconcave minimax problems have found numerous applications in various fields including machine learning. However, questions remain about what is a good surrogate for local minimax optimum and how to characterize the minimax optimality. Recently Jin, Netrapalli, and Jordan (ICML 2020) introduced a concept of local minimax point and derived optimality conditions for the smooth and unconstrained case. In this paper, we introduce the concept of calm local minimax point, which is a local minimax point with a calm radius function. With the extra calmness property we obtain first and second-order sufficient and necessary optimality conditions for a very general class of nonsmooth nonconvex-nonconcave minimax problem. Moreover we show that the calm local minimax optimality and the local minimax optimality coincide under a weak sufficient optimality condition for the maximization problem. This equivalence allows us to derive stronger optimality conditions under weaker assumptions for local minimax optimality.