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Gradient descent-ascent (GDA) is a widely used algorithm for minimaxoptimization. However, GDA has been proved to converge to stationary points fornonconvex minimax optimization, which are suboptimal compared with localminimax points. In this work, we develop cubic regularization (CR) typealgorithms that globally converge to local minimax points innonconvex-strongly-concave minimax optimization. We first show that localminimax points are equivalent to second-order stationary points of a certainenvelope function. Then, inspired by the classic cubic regularizationalgorithm, we propose an algorithm named Cubic-LocalMinimax for finding localminimax points, and provide a comprehensive convergence analysis by leveragingits intrinsic potential function. Specifically, we establish the globalconvergence of Cubic-LocalMinimax to a local minimax point at a sublinearconvergence rate and characterize its iteration complexity. Also, we propose aGDA-based solver for solving the cubic subproblem involved inCubic-LocalMinimax up to certain pre-defined accuracy, and analyze the overallgradient and Hessian-vector product computation complexities of such an inexactCubic-LocalMinimax algorithm. Moreover, we propose a stochastic variant ofCubic-LocalMinimax for large-scale minimax optimization, and characterize itssample complexity under stochastic sub-sampling. Experimental resultsdemonstrate faster convergence of our stochastic Cubic-LocalMinimax than someexisting algorithms.

A Cubic Regularization Approach for Finding Local Minimax Points in Nonconvex Minimax Optimization