What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?

TL;DR

This paper defines local optimality for sequential nonconvex-nonconcave minimax problems, introduces local minimax, and links it to the stability of gradient descent ascent, addressing a fundamental gap in understanding such problems.

Contribution

It proposes a new mathematical definition of local minimax for sequential games and analyzes its properties and existence, connecting it to GDA stability.

Findings

Local minimax is a proper notion of local optimality for sequential minimax problems.

Stable limit points of GDA correspond to local minimax points under mild conditions.

The paper establishes existence results for local minimax points.

Abstract

Minimax optimization has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs), adversarial training and multi-agent reinforcement learning. As most of these applications involve continuous nonconvex-nonconcave formulations, a very basic question arises---"what is a proper definition of local optima?" Most previous work answers this question using classical notions of equilibria from simultaneous games, where the min-player and the max-player act simultaneously. In contrast, most applications in machine learning, including GANs and adversarial training, correspond to sequential games, where the order of which player acts first is crucial (since minimax is in general not equal to maximin due to the nonconvex-nonconcave nature of the problems). The main contribution of this paper is to propose a proper mathematical definition of local optimality for this sequential setting---local minimax, as well as to present its properties and existence results. Finally, we establish a strong connection to a basic local search algorithm---gradient descent ascent (GDA): under mild conditions, all stable limit points of GDA are exactly local minimax points up to some degenerate points.