Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile

TL;DR

This paper demonstrates that optimistic mirror descent (OMD) reliably converges in a broad class of saddle-point problems relevant to GAN training, surpassing the limitations of vanilla mirror descent.

Contribution

It introduces the concept of coherence in non-monotone problems and proves that optimism in mirror descent ensures convergence beyond convex-concave settings.

Findings

OMD converges in all coherent problems.

Vanilla mirror descent may fail to converge even in simple bilinear models.

Numerical experiments validate theoretical results across various GAN models.

Abstract

Owing to their connection with generative adversarial networks (GANs), saddle-point problems have recently attracted considerable interest in machine learning and beyond. By necessity, most theoretical guarantees revolve around convex-concave (or even linear) problems; however, making theoretical inroads towards efficient GAN training depends crucially on moving beyond this classic framework. To make piecemeal progress along these lines, we analyze the behavior of mirror descent (MD) in a class of non-monotone problems whose solutions coincide with those of a naturally associated variational inequality - a property which we call coherence. We first show that ordinary, "vanilla" MD converges under a strict version of this condition, but not otherwise; in particular, it may fail to converge even in bilinear models with a unique solution. We then show that this deficiency is mitigated by optimism: by taking an "extra-gradient" step, optimistic mirror descent (OMD) converges in all coherent problems. Our analysis generalizes and extends the results of Daskalakis et al. (2018) for optimistic gradient descent (OGD) in bilinear problems, and makes concrete headway for establishing convergence beyond convex-concave games. We also provide stochastic analogues of these results, and we validate our analysis by numerical experiments in a wide array of GAN models (including Gaussian mixture models, as well as the CelebA and CIFAR-10 datasets).