论时变零和博弈中的最后迭代收敛性：Extra Gradient在Optimism失效的场景下取得成功
On the Last-iterate Convergence in Time-varying Zero-sum Games: Extra Gradient Succeeds where Optimism Fails
AI总结：
该文研究时变双线性零和博弈的算法最后迭代行为，发现周期博弈中EG收敛而OGDA、动量法发散，收敛扰动博弈中若博弈以快于1/t的速率稳定则所有算法均收敛。
摘要：
从双线性博弈、凸-凹博弈到满足MVI条件的设定，两人零和博弈中的最后迭代收敛性已得到广泛研究。在上述博弈中表现出最后迭代收敛性的典型方法包括extra-gradient（EG，外梯度法）和optimistic gradient descent ascent（OGDA，乐观梯度下降上升法）。然而，所有已确立的最后迭代收敛性结果都仅适用于底层重复博弈不随时间变化的限制性设定。近期有一系列研究聚焦于OGDA在时变博弈（即收益随时间演化的博弈）中的悔值分析，但OGDA和EG在时变环境下的最后迭代行为仍不明确。本文研究了多种算法在两类无约束时变双线性零和博弈中的最后迭代行为：周期扰动博弈和收敛扰动博弈。这些模型拓展了常规的重复博弈公式，纳入了外部环境因素，例如物种竞争的季节效应和逐渐消失的外部噪声。在周期博弈中，我们证明EG会收敛，而OGDA和动量法会发散。这一结果十分出人意料，因为据我们所知，这是首个表明EG和OGDA的最后迭代行为存在本质差异、且表现并不相似的结论。在收敛扰动博弈中，我们证明只要博弈本身以快于$1/t$的速率稳定，所有这些算法都会收敛。
英文摘要：
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 $1/t$.
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