What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?
Chi Jin,u00a0Praneeth Netrapalli,u00a0Michael Jordan
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 arisesu2014u201cwhat is a proper definition of local optima?u201d 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 settingu2014local minimax, as well as to present its properties and existence results. Finally, we establish a strong connection to a basic local search algorithmu2014gradient descent ascent (GDA): under mild conditions, all stable limit points of GDA are exactly local minimax points up to some degenerate points.
