Min-Max Optimization under Delays

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 ($\mathtt{\text{EG}}$) to diverge on simple instances for which $\mathtt{\text{EG}}$ 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 ($\mathtt{\text{GDA}}$) and $\mathtt{\text{EG}}$ 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.