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

TL;DR

This paper introduces a first-order augmented Lagrangian method for nonconvex-strongly-concave constrained minimax problems, achieving improved complexity for finding approximate solutions.

Contribution

It develops a novel first-order method leveraging strong concavity, improving operation complexity bounds for constrained minimax optimization.

Findings

Achieves $O( ext{epsilon}^{-3.5} ext{log} ext{epsilon}^{-1})$ complexity for $ ext{epsilon}$-KKT solutions.

Improves previous complexity bounds by a factor of $ ext{epsilon}^{-0.5}$.

Provides a practical approach for constrained minimax problems with strong theoretical guarantees.

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 operation complexity of $O(\varepsilon^{-3.5}\log\varepsilon^{-1})$, 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 $\varepsilon^{-0.5}$.