Last-iterate Convergence Separation between Extra-gradient and Optimism in Constrained Periodic Games

TL;DR

This paper investigates the last-iterate convergence behaviors of optimistic and extra-gradient methods in constrained periodic games, revealing a separation similar to unconstrained cases, which challenges previous assumptions about their equivalence.

Contribution

It extends the analysis of last-iterate convergence separation between optimistic and extra-gradient methods to constrained periodic games, a more practical setting.

Findings

Extra-gradient converges to equilibrium in constrained periodic games.

Optimistic method diverges in constrained periodic games.

Separation results mirror those in unconstrained games.

Abstract

Last-iterate behaviors of learning algorithms in repeated two-player zero-sum games have been extensively studied due to their wide applications in machine learning and related tasks. Typical algorithms that exhibit the last-iterate convergence property include optimistic and extra-gradient methods. However, most existing results establish these properties under the assumption that the game is time-independent. Recently, (Feng et al, 2023) studied the last-iterate behaviors of optimistic and extra-gradient methods in games with a time-varying payoff matrix, and proved that in an unconstrained periodic game, extra-gradient method converges to the equilibrium while optimistic method diverges. This finding challenges the conventional wisdom that these two methods are expected to behave similarly as they do in time-independent games. However, compared to unconstrained games, games with constrains are more common both in practical and theoretical studies. In this paper, we investigate the last-iterate behaviors of optimistic and extra-gradient methods in the constrained periodic games, demonstrating that similar separation results for last-iterate convergence also hold in this setting.