Last-iterate Convergence Separation between Extra-gradient and Optimism in Constrained Periodic Games
This addresses a theoretical gap for researchers in game theory and optimization by extending separation results to more common constrained games, though it is incremental.
The paper tackles the problem of last-iterate convergence in constrained periodic games, showing that extra-gradient method converges to equilibrium while optimistic method diverges, similar to prior results in unconstrained settings.
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.