A first-order method for nonconvex-strongly-concave constrained minimax optimization

2025-12-28

AI Summary

This work studies constrained minimax optimization problems where the objective is nonconvex in the primal variable and strongly concave in the dual variable. We propose a first-order augmented Lagrangian framework whose subproblems are unconstrained nonconvex–strongly concave minimax problems, and design a specialized first-order inner solver that exploits strong concavity to accelerate convergence. Under standard assumptions, our method achieves— for the first time— a gradient complexity of $O(varepsilon^{-3.5}logvarepsilon^{-1})$, improving upon the previous best rate by a factor of $varepsilon^{-0.5}$. Moreover, it efficiently computes an $varepsilon$-KKT point. This work provides a new algorithmic tool for applications such as nonconvex adversarial training and constrained games, offering both theoretical guarantees and computational tractability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Adversarial Learning & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper we study a nonconvex-strongly-concave constrained minimax problem. Specifically, we propose a first-order augmented Lagrangian method for solving it, whose subproblems are nonconvex-strongly-concave unconstrained minimax problems and suitably solved by a first-order method developed in this paper that leverages the strong concavity structure. Under suitable assumptions, the proposed method achieves an emph{operation complexity} of $O(varepsilon^{-3.5}logvarepsilon^{-1})$, measured in terms of its fundamental operations, for finding an $varepsilon$-KKT solution of the constrained minimax problem, which improves the previous best-known operation complexity by a factor of $varepsilon^{-0.5}$.

Problem

• Proposes a first-order method for nonconvex-strongly-concave constrained minimax optimization
• Develops an augmented Lagrangian approach to solve constrained minimax problems efficiently
• Improves operation complexity for finding approximate KKT solutions in minimax optimization

Innovation

• First-order augmented Lagrangian method for constrained minimax
• Subproblems solved by first-order method leveraging strong concavity
• Achieves improved operation complexity for epsilon-KKT solution