Bregman Golden Ratio Algorithms for Variational Inequalities

TL;DR

This paper introduces Bregman modifications to the GRAAL algorithm for variational inequalities, improving convergence and efficiency in solving saddle-point and related problems with applications in communication and game theory.

Contribution

It proposes two Bregman-based variants of GRAAL, one with fixed step-size and one with adaptive step-size, enhancing convergence under different Lipschitz conditions.

Findings

The fixed step-size Bregman GRAAL converges under global Lipschitz assumptions.

The adaptive step-size Bregman GRAAL performs well on communication and game problems.

Numerical experiments demonstrate improved efficiency with Bregman distances.

Abstract

Variational inequalities provide a framework through which many optimisation problems can be solved, in particular, saddle-point problems. In this paper, we study modifications to the so-called Golden RAtio ALgorithm (GRAAL) for variational inequalities -- a method which uses a fully explicit adaptive step-size, and provides convergence results under local Lipschitz assumptions without requiring backtracking. We present and analyse two Bregman modifications to GRAAL: the first uses a fixed step-size and converges under global Lipschitz assumptions, and the second uses an adaptive step-size rule. Numerical performance of the former method is demonstrated on a bimatrix game arising in network communication, and of the latter on two problems, namely, power allocation in Gaussian communication channels and $N$-person Cournot completion games. In all of these applications, an appropriately chosen Bregman distance simplifies the projection steps computed as part of the algorithm.