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arXiv · 2306.17443

Calm local optimality for nonconvex-nonconcave minimax problems

Xiaoxiao Ma·Wei Yao·Jane J. Ye·Jin Zhang
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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.

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Xiaoxiao Ma, Wei Yao, Jane J. Ye, Jin Zhang. 2023-06-30. Calm local optimality for nonconvex-nonconcave minimax problems. https://arxiv.org/abs/2306.17443

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