In this paper, we study the problem of finding mixed Nash equilibrium for mean-field two-player zero-sum games. Solving this problem requires optimizing over two probability distributions. We consider a quasistatic Wasserstein gradient flow dynamics in which one probability distribution follows the Wasserstein gradient flow, while the other one is always at the equilibrium. Theoretical analysis are conducted on this dynamics, showing its convergence to the mixed Nash equilibrium under mild conditions. Inspired by the continuous dynamics of probability distributions, we derive a quasistatic Langevin gradient descent method with inner-outer iterations, and test the method on different problems, including training mixture of GANs.
翻译:在本文中,我们研究了为平均场两玩零和游戏寻找混合纳什平衡的问题。解决这个问题需要优化两种概率分布。我们考虑了一种准静态瓦森斯坦梯度流动态,其中一种概率分布跟随瓦森斯坦梯度流,而另一种则始终处于平衡状态。我们对这一动态进行了理论分析,在温和条件下显示其与混合纳什平衡的趋同。在概率分布持续动态的启发下,我们得出一种准静态兰格文梯度下降法,并用内外迭代法,并测试不同问题的方法,包括GANs的培训混合。