This paper studies the convergence of the Optimistic Multiplicative Weights Update algorithm (OMWU) in two player zero-sum games. Recent works have identified instances on which the last-iterate of OMWU can converge arbitrarily slowly, but understanding when and why this slow convergence occurs has remained open. In this work, we develop a new analysis framework that gives sharp, quantitative explanations for this behavior. Our analysis is based on viewing the algorithm's dual iterates as an optimistic skew-gradient descent with respect to an energy function. We prove over the dual iterates that energy is dissipative, and by establishing tight bounds on the magnitude of dissipation, our analysis quantifies the geometric bottlenecks that arise when the corresponding primal iterates are close to the simplex boundary. This further translates into a new linear last-iterate convergence rate in KL divergence on games with a unique and interior Nash equilibrium. Compared to prior work, this new rate contains a much sharper dependence on game-specific constants, and we prove this dependence is optimal. Moreover, these geometric insights further translate into new separations on uniform convergence rates for OMWU. On the one hand, we prove constant lower bounds on the uniform best-iterate convergence rate in KL divergence and total variation distance from Nash. On the other hand, we establish for the $2\times 2$ setting a new ${\widetilde O}(T^{-1/2})$ best-iterate rate in duality gap, improving substantially over prior work. Together, this shows in general that uniform convergence rate guarantees do not transfer across different measures of distance to Nash.

翻译：本文研究乐观乘性权重更新算法在二人零和博弈中的收敛性。近期研究已识别出该算法末次迭代可能任意慢收敛的实例，但对其发生条件与内在机制的理解仍属空白。本文建立了一个新的分析框架，为该行为提供精确的定量解释。我们的分析基于将算法的对偶迭代视为关于能量函数的乐观偏斜梯度下降。我们证明了对偶迭代中能量耗散性成立，并通过建立耗散幅度的紧致界，定量刻画了原始迭代接近单纯形边界时产生的几何瓶颈。这进一步转化为具有唯一内点纳什均衡的博弈中KL散度的线性末次迭代收敛速率。相比于既有研究，该新速率包含对博弈特定常数更为敏锐的依赖关系，且我们证明这种依赖是最优的。此外，这些几何洞见进一步转化为OMWU均匀收敛速率的新分离性质。一方面，我们证明了KL散度与总变差距离度量下纳什均衡的均匀最优迭代收敛速率具有常数下界。另一方面，针对$2\times 2$情形，我们在对偶间隙上建立了新的${\widetilde O}(T^{-1/2})$最优迭代速率，较既有工作有显著提升。这些结果共同表明：一般而言，均匀收敛速率保证在不同纳什距离度量之间不可传递。