An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function

TL;DR

This paper introduces OGAProx, a novel accelerated algorithm for convex-concave saddle point problems with nonsmooth coupling, achieving improved convergence rates and validated on practical machine learning tasks.

Contribution

The paper proposes the OGAProx algorithm, combining optimistic gradient ascent and proximal steps, with theoretical convergence guarantees for nonsmooth convex-concave saddle point problems.

Findings

Achieves $ ext{O}(1/K)$ convergence rate for iterates.

Attains linear convergence $ ext{O}( heta^K)$ under certain conditions.

Validates effectiveness on SVM training and fairness classification tasks.

Abstract

In this work we aim to solve a convex-concave saddle point problem, where the convex-concave coupling function is smooth in one variable and nonsmooth in the other and not assumed to be linear in either. The problem is augmented by a nonsmooth regulariser in the smooth component. We propose and investigate a novel algorithm under the name of OGAProx, consisting of an optimistic gradient ascent step in the smooth variable coupled with a proximal step of the regulariser, and which is alternated with a {proximal step} in the nonsmooth component of the coupling function. We consider the situations convex-concave, convex-strongly concave and strongly convex-strongly concave related to the saddle point problem under investigation. Regarding iterates we obtain (weak) convergence, a convergence rate of order $ \mathcal{O}(\frac{1}{K}) $ and linear convergence like $\mathcal{O}(\theta^{K})$ with $ \theta < 1 $, respectively. In terms of function values we obtain ergodic convergence rates of order $ \mathcal{O}(\frac{1}{K}) $, $ \mathcal{O}(\frac{1}{K^{2}}) $ and $ \mathcal{O}(\theta^{K}) $ with $ \theta < 1 $, respectively. We validate our theoretical considerations on a nonsmooth-linear saddle point problem, the training of multi kernel support vector machines and a classification problem incorporating minimax group fairness.