Proximal gradient descent on the smoothed duality gap to solve saddle point problems

TL;DR

This paper introduces an algorithm that minimizes the self-centered smoothed gap to efficiently solve convex-concave saddle point problems, achieving comparable complexity to existing methods and linear convergence under certain conditions.

Contribution

It proposes a novel algorithm that minimizes the self-centered smoothed gap, transforming saddle point problems into a minimization task with favorable convergence properties.

Findings

Algorithm reduces saddle point problems to minimization.

Achieves worst-case complexity similar to primal-dual hybrid gradient methods.

Exhibits linear convergence in favorable scenarios.

Abstract

In this paper, we minimize the self-centered smoothed gap, a recently introduced optimality measure, in order to solve convex-concave saddle point problems. The self-centered smoothed gap can be computed as the sum of a convex, possibly nonsmooth function and a smooth weakly convex function. Although it is not convex, we propose an algorithm that minimizes this quantity, effectively reducing convex-concave saddle point problems to a minimization problem. Its worst case complexity is comparable to the one of the restarted and averaged primal dual hybrid gradient method, and the algorithm enjoys linear convergence in favorable cases.