Last-iterate convergence has received extensive study in two player zero-sum games starting from bilinear, convex-concave up to settings that satisfy the MVI condition. Typical methods that exhibit last-iterate convergence for the aforementioned games include extra-gradient (EG) and optimistic gradient descent ascent (OGDA). However, all the established last-iterate convergence results hold for the restrictive setting where the underlying repeated game does not change over time. Recently, a line of research has focused on regret analysis of OGDA in time-varying games, i.e., games where payoffs evolve with time; the last-iterate behavior of OGDA and EG in time-varying environments remains unclear though. In this paper, we study the last-iterate behavior of various algorithms in two types of unconstrained, time-varying, bilinear zero-sum games: periodic and convergent perturbed games. These models expand upon the usual repeated game formulation and incorporate external environmental factors, such as the seasonal effects on species competition and vanishing external noise. In periodic games, we prove that EG will converge while OGDA and momentum method will diverge. This is quite surprising, as to the best of our knowledge, it is the first result that indicates EG and OGDA have qualitatively different last-iterate behaviors and do not exhibit similar behavior. In convergent perturbed games, we prove all these algorithms converge as long as the game itself stabilizes with a faster rate than $1/t$.

翻译：最后迭代收敛性在双人零和博弈中已得到广泛研究，从双线性、凸-凹设定到满足MVI条件的场景。典型展现最后迭代收敛性的方法包括额外梯度（EG）和乐观梯度上升下降（OGDA）。然而，所有已建立的最后迭代收敛性结果均局限于底层重复博弈不随时间变化的限制性设定。近年来，一系列研究聚焦于OGDA在时变博弈（即收益随时间演化的博弈）中的后悔分析，但OGDA和EG在时变环境中的最后迭代行为仍不明确。本文研究了两类无约束时变双线性零和博弈中多种算法的最后迭代行为：周期性博弈和收敛性扰动博弈。这些模型扩展了通常的重复博弈形式，并融入了外部环境因素，例如物种竞争的季节性效应和衰减的外部噪声。在周期性博弈中，我们证明EG会收敛，而OGDA和动量法会发散。这相当令人惊讶——据我们所知，这是首次表明EG和OGDA具有定性不同的最后迭代行为，而非表现相似。在收敛性扰动博弈中，只要博弈本身以快于$1/t$的速率稳定，我们证明所有算法都会收敛。