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

TL;DR

This paper introduces a cubic regularization algorithm that guarantees convergence to local minimax points in nonconvex minimax problems, improving upon gradient descent-ascent methods by providing theoretical convergence guarantees and practical efficiency.

Contribution

The paper develops a novel cubic regularization-based algorithm for nonconvex minimax optimization, with convergence analysis, complexity bounds, and a stochastic variant for large-scale problems.

Findings

Global convergence to local minimax points established

Sublinear convergence rate proven for the proposed algorithm

Experimental results show faster convergence than existing methods

Abstract

Gradient descent-ascent (GDA) is a widely used algorithm for minimax optimization. However, GDA has been proved to converge to stationary points for nonconvex minimax optimization, which are suboptimal compared with local minimax points. In this work, we develop cubic regularization (CR) type algorithms that globally converge to local minimax points in nonconvex-strongly-concave minimax optimization. We first show that local minimax points are equivalent to second-order stationary points of a certain envelope function. Then, inspired by the classic cubic regularization algorithm, we propose an algorithm named Cubic-LocalMinimax for finding local minimax points, and provide a comprehensive convergence analysis by leveraging its intrinsic potential function. Specifically, we establish the global convergence of Cubic-LocalMinimax to a local minimax point at a sublinear convergence rate and characterize its iteration complexity. Also, we propose a GDA-based solver for solving the cubic subproblem involved in Cubic-LocalMinimax up to certain pre-defined accuracy, and analyze the overall gradient and Hessian-vector product computation complexities of such an inexact Cubic-LocalMinimax algorithm. Moreover, we propose a stochastic variant of Cubic-LocalMinimax for large-scale minimax optimization, and characterize its sample complexity under stochastic sub-sampling. Experimental results demonstrate faster convergence of our stochastic Cubic-LocalMinimax than some existing algorithms.