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

Published: 07 May 2023, Last Modified: 07 May 2023Accepted by TMLREveryoneRevisionsBibTeX
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 or comparable convergence speed of our stochastic Cubic-LocalMinimax than the state-of-the-art algorithms such as GDA and Minimax Cubic-Newton. In particular, our stochastic Cubic-LocalMinimax was also faster as compared to several other algorithms for minimax optimization on a particular adversarial loss for training a convolutional neural network on MNIST.
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Length: Long submission (more than 12 pages of main content)
Changes Since Last Submission: Dear Action Editor, We have edited the abstract per your suggestion. In addition, since (Luo et al., 2022) proposed inexact cubic-regularization based approach earlier than our work, we removed ''the concurrent work'' throughout the paper and changed the sentences right after Algorithm 4 to below. "A similar algorithm named as Inexact Minimax Cubic-Newton (IMCN) with inexact cubic solver was first proposed by (Luo et al., 2022). Our Algorithm 4 differs from IMCN in the following aspects." Thanks for your suggestion. Authors
Code: https://github.com/datou30/Cubic-Localminimax-experiment
Assigned Action Editor: ~Robert_M._Gower1
Submission Number: 618