On the Convergence of Proximal Algorithms for Weakly-convex Min-max Optimization

TL;DR

This paper proves convergence of simplified alternating first-order algorithms for weakly-convex-strongly-concave min-max problems, enabling broader step-size choices and applications in imaging inverse problems.

Contribution

It provides a simplified convergence proof for alternating gradient methods and extends results to doubly proximal algorithms in weakly convex-strongly concave settings.

Findings

Convergence of alternating gradient descent-ascent algorithm with enlarged step-size range.

Proof of convergence for a doubly proximal algorithm in weakly convex-strongly concave problems.

Application of these algorithms to regularized imaging inverse problems with neural networks.

Abstract

We study alternating first-order algorithms with no inner loops for solving nonconvex-strongly-concave min-max problems. We show the convergence of the alternating gradient descent--ascent algorithm method by proposing a substantially simplified proof compared to previous ones. It allows us to enlarge the set of admissible step-sizes. Building on this general reformulation, we also prove the convergence of a doubly proximal algorithm in the weakly convex-strongly concave setting. Finally, we show how this new result opens the way to new applications of min-max optimization algorithms for solving regularized imaging inverse problems with neural networks in a plug-and-play manner.