Min-Max Optimization under Delays

TL;DR

This paper investigates how delays affect the convergence of min-max optimization algorithms, providing theoretical guarantees and complexity bounds for delayed Gradient Descent-Ascent and Extra-gradient methods.

Contribution

It offers the first theoretical analysis of delayed min-max optimization algorithms, showing convergence guarantees under certain conditions and quantifying delays' impact.

Findings

Small delays can cause divergence in standard algorithms like Extra-gradient.

GDA and EG with delays still converge under convex-concave assumptions.

Delays slow down convergence, as shown by derived complexity bounds.

Abstract

Delays and asynchrony are inevitable in large-scale machine-learning problems where communication plays a key role. As such, several works have extensively analyzed stochastic optimization with delayed gradients. However, as far as we are aware, no analogous theory is available for min-max optimization, a topic that has gained recent popularity due to applications in adversarial robustness, game theory, and reinforcement learning. Motivated by this gap, we examine the performance of standard min-max optimization algorithms with delayed gradient updates. First, we show (empirically) that even small delays can cause prominent algorithms like Extra-gradient (\texttt{EG}) to diverge on simple instances for which \texttt{EG} guarantees convergence in the absence of delays. Our empirical study thus suggests the need for a careful analysis of delayed versions of min-max optimization algorithms. Accordingly, under suitable technical assumptions, we prove that Gradient Descent-Ascent (\texttt{GDA}) and \texttt{EG} with delayed updates continue to guarantee convergence to saddle points for convex-concave and strongly convex-strongly concave settings. Our complexity bounds reveal, in a transparent manner, the slow-down in convergence caused by delays.