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:由于在平滑的游戏中出现了趋同行为,超升级方法最近受到越来越多的关注。在以美元计价的不同游戏中,矢量场Jacobian的叶基值分布在复杂的平面上。因此,与古典(即单玩家)最小化相比,游戏呈现出更复杂的动态,即超升级方法成功,而简单的梯度方法可能失败。然而,在这项工作中,我们没有把重点放在特定的问题类别上,而是走相反的道路:从动力超升级方法开始,作为选定的优化者,并使用多元基分析,我们找出在异常动作中使用动力导致最佳性能的问题子类。在超参数设置的基础上,我们显示,与动力相比,超成熟的动态呈现出三种不同的趋同模式:在实际线上分配eigen值时,在实际线上分配$(i)美元),在与复杂的组合线上两美元(美元),以及仅作为复杂的组合值的美元(美元),我们随后为每个案例计算出最佳的超模度趋同率,并显示其达到加速速度。