Last-iterate convergence rates for min-max optimization

TL;DR

This paper establishes linear last-iterate convergence rates for Hamiltonian Gradient Descent and Consensus Optimization algorithms in certain convex-concave min-max problems, extending guarantees beyond bilinear and strongly concave cases.

Contribution

It proves last-iterate convergence for HGD and CO algorithms in more general convex-concave settings, advancing understanding of their effectiveness.

Findings

HGD achieves linear convergence under a 'sufficiently bilinear' condition.

CO algorithm also attains similar convergence rates for specific parameters.

Results extend last-iterate convergence guarantees beyond classical bilinear and strongly concave cases.

Abstract

While classic work in convex-concave min-max optimization relies on average-iterate convergence results, the emergence of nonconvex applications such as training Generative Adversarial Networks has led to renewed interest in last-iterate convergence guarantees. Proving last-iterate convergence is challenging because many natural algorithms, such as Simultaneous Gradient Descent/Ascent, provably diverge or cycle even in simple convex-concave min-max settings, and previous work on global last-iterate convergence rates has been limited to the bilinear and convex-strongly concave settings. In this work, we show that the Hamiltonian Gradient Descent (HGD) algorithm achieves linear convergence in a variety of more general settings, including convex-concave problems that satisfy a "sufficiently bilinear" condition. We also prove similar convergence rates for the Consensus Optimization (CO) algorithm of [MNG17] for some parameter settings of CO.