Alternating and Parallel Proximal Gradient Methods for Nonsmooth, Nonconvex Minimax: A Unified Convergence Analysis
本文针对非光滑、非凸-强凹的极小极大问题，提出了交替和并行两种近端梯度方法，并通过统一框架证明其收敛到临界点，适用于更广泛的优化问题。
There is growing interest in nonconvex minimax problems that is driven by an abundance of applications. Our focus is on nonsmooth, nonconvex-strongly concave minimax, thus departing from the more common weakly convex and smooth models assumed in the recent literature. We present proximal gradient schemes with either parallel or alternating steps. We show that both methods can be analyzed through a single scheme within a unified analysis that relies on expanding a general convergence mechanism used for analyzing nonconvex, nonsmooth optimization problems. In contrast to the current literature, which focuses on the complexity of obtaining nearly approximate stationary solutions, we prove subsequence convergence to a critical point of the primal objective and global convergence when the latter is semialgebraic. Furthermore, the complexity results we provide are with respect to approximate stationary solutions. Lastly, we expand the scope of problems that can be addressed by generalizing one of the steps with a Bregman proximal gradient update, and together with a few adjustments to the analysis, this allows us to extend the convergence and complexity results to this broader setting. Funding: The research of E. Cohen was partially supported by a doctoral fellowship from the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240]. The research of M. Teboulle was partially supported by the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240].