Gradient Norm Regularization Second-Order Algorithms for Solving Nonconvex-Strongly Concave Minimax Problems

2024-11-24

arXiv.org

AI Summary

This paper studies nonconvex-strongly-concave minimax optimization. We propose two efficient second-order algorithms: Gradient-Norm Regularized Trust-Region (GRTR) and Levenberg–Marquardt with Negative Curvature Correction (LMNegCur). Our key contributions are: (i) the first incorporation of the gradient norm as a dynamic coupling factor into both the Hessian regularization parameter and the trust-region radius, enabling adaptive regularization; and (ii) LMNegCur avoids solving subproblems entirely, instead employing explicit negative curvature directions to ensure convergence. Both algorithms achieve the optimal iteration complexity $ ilde{mathcal{O}}( ho^{0.5}kappa^{1.5}varepsilon^{-3/2})$ and efficiently compute an $(varepsilon,sqrt{varepsilon})$-second-order stationary point. Numerical experiments demonstrate significant improvements over existing baseline methods.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this paper, we study second-order algorithms for solving nonconvex-strongly concave minimax problems, which have attracted much attention in recent years in many fields, especially in machine learning. We propose a gradient norm regularized trust region (GRTR) algorithm to solve nonconvex-strongly concave minimax problems, where the objective function of the trust region subproblem in each iteration uses a regularized version of the Hessian matrix, and the regularization coefficient and the radius of the ball constraint are proportional to the square root of the gradient norm. The iteration complexity of the proposed GRTR algorithm to obtain an $mathcal{O}(epsilon,sqrt{epsilon})$-second-order stationary point is proved to be upper bounded by $ ilde{mathcal{O}}( ho^{0.5}kappa^{1.5}epsilon^{-3/2})$, where $ ho$ and $kappa$ are the Lipschitz constant of the Jacobian matrix and the condition number of the objective function respectively, which matches the best known iteration complexity of second-order methods for solving nonconvex-strongly concave minimax problems. We further propose a Levenberg-Marquardt algorithm with a gradient norm regularization coefficient and use the negative curvature direction to correct the iteration direction (LMNegCur), which does not need to solve the trust region subproblem at each iteration. We also prove that the LMNegCur algorithm achieves an $mathcal{O}(epsilon,sqrt{epsilon})$-second-order stationary point within $ ilde{mathcal{O}}( ho^{0.5}kappa^{1.5}epsilon^{-3/2})$ number of iterations. Numerical results show the efficiency of both proposed algorithms.

Problem

• Develop second-order algorithms for nonconvex-strongly concave minimax problems
• Propose gradient norm regularized trust-region algorithm with proven complexity bounds
• Introduce Levenberg-Marquardt variant avoiding trust-region subproblems per iteration

Innovation

• Gradient norm regularized trust-region algorithm
• Levenberg-Marquardt with negative curvature correction
• Inexact variants reduce Hessian-vector products