The extragradient method has recently gained increasing attention, due to its convergence behavior on smooth games. In $n$-player differentiable games, the eigenvalues of the Jacobian of the vector field are distributed on the complex plane. Thus, compared to classical (i.e., single player) minimization, games exhibit more convoluted dynamics, where the extragradient method succeeds while simple gradient method could fail. Yet, in this work, instead of focusing on a specific problem class, we follow a reverse path: starting from the momentum extragradient method as the selected optimizer, and using polynomial-based analyses, we identify problem subclasses where the use of momentum in extragradient motions lead to optimal performance. Based on the hyperparameter setup, we show that the extragradient with momentum exhibits three different modes of convergence: when the eigenvalues are distributed $i)$ on the real line, $ii)$ both on the real line along with complex conjugates, and $iii)$ only as complex conjugates. We then derive the optimal hyperparameters for each case, and show that it achieves an accelerated convergence rate.

翻译：外推梯度法因其在平滑博弈中的收敛行为，近期受到日益关注。在$n$玩家可微博弈中，向量场雅可比矩阵的特征值分布在复平面上。因此，与经典（即单玩家）极小化相比，博弈展现出更复杂的动力学特性，其中外推梯度法能够成功收敛，而简单梯度法则可能失效。然而，在本工作中，我们并未聚焦于特定问题类别，而是采取逆向路径：以动量外推梯度法作为所选优化器，并基于多项式分析，识别出使用动量外推梯度运动能够实现最优性能的问题子类。基于超参数设置，我们展示了带动量的外推梯度呈现出三种不同的收敛模式：特征值分布为$i)$实线上，$ii)$实线上与共轭复数并存，以及$iii)$仅作为共轭复数。我们随后推导出每种情况下的最优超参数，并证明其能达到加速收敛速率。