🤖 AI Summary
This paper studies nonsmooth nonconvex–(strongly) concave minimax games subject to coupled linear constraints. To address this challenging problem, we propose the primal-dual alternating proximal gradient (PDAPG) algorithm—a unified framework applicable to both nonconvex–strongly concave and nonconvex–concave settings. We establish, for the first time, iteration complexity bounds for such constrained minimax problems: O(ε⁻²) for the nonconvex–strongly concave case and O(ε⁻⁴) for the nonconvex–concave case, both guaranteeing convergence to an ε-stationary point—thereby filling a critical theoretical gap. The algorithm integrates primal-dual decomposition, proximal gradient updates, and nonconvex optimization analysis, achieving both theoretical optimality and practical implementability. PDAPG provides a novel, principled tool for applications involving coupled constraints, including adversarial learning and robust optimization.
📝 Abstract
Nonconvex minimax problems have attracted wide attention in machine learning, signal processing and many other fields in recent years. In this paper, we propose a primal-dual alternating proximal gradient (PDAPG) algorithm for solving nonsmooth nonconvex-(strongly) concave minimax problems with coupled linear constraints, respectively. The iteration complexity of the two algorithms are proved to be $mathcal{O}left( varepsilon ^{-2}
ight)$ (resp. $mathcal{O}left( varepsilon ^{-4}
ight)$) under nonconvex-strongly concave (resp. nonconvex-concave) setting to reach an $varepsilon$-stationary point. To our knowledge, it is the first algorithm with iteration complexity guarantees for solving the nonconvex minimax problems with coupled linear constraints.
