The extragradient method has gained popularity due to its robust convergence properties for differentiable games. Unlike single-objective optimization, game dynamics involve complex interactions reflected by the eigenvalues of the game vector field's Jacobian scattered across the complex plane. This complexity can cause the simple gradient method to diverge, even for bilinear games, while the extragradient method achieves convergence. Building on the recently proven accelerated convergence of the momentum extragradient method for bilinear games \citep{azizian2020accelerating}, we use a polynomial-based analysis to identify three distinct scenarios where this method exhibits further accelerated convergence. These scenarios encompass situations where the eigenvalues reside on the (positive) real line, lie on the real line alongside complex conjugates, or exist solely as complex conjugates. Furthermore, we derive the hyperparameters for each scenario that achieve the fastest convergence rate.

翻译：外梯度法因其在可微博弈中具有鲁棒收敛性而广受欢迎。与单目标优化不同，博弈动力学涉及复杂交互，反映在博弈向量场雅可比矩阵的特征值在复平面上的散布。这种复杂性可能导致简单梯度法即使对双线性博弈也会发散，而外梯度法则能实现收敛。基于近期证明的动量外梯度法在双线性博弈中加速收敛的特性\citep{azizian2020accelerating}，我们采用基于多项式的分析方法，识别出该方法展现进一步加速收敛的三种不同场景。这些场景涵盖特征值位于（正）实轴、特征值位于实轴同时伴随复共轭、以及特征值仅以复共轭形式存在三种情形。此外，我们推导出每种场景下实现最快收敛速率的超参数。