Min-max optimization problems arise in several key machine learning setups, including adversarial learning and generative modeling. In their general form, in absence of convexity/concavity assumptions, finding pure equilibria of the underlying two-player zero-sum game is computationally hard [Daskalakis et al., 2021]. In this work we focus instead in finding mixed equilibria, and consider the associated lifted problem in the space of probability measures. By adding entropic regularization, our main result establishes global convergence towards the global equilibrium by using simultaneous gradient ascent-descent with respect to the Wasserstein metric -- a dynamics that admits efficient particle discretization in high-dimensions, as opposed to entropic mirror descent. We complement this positive result with a related entropy-regularized loss which is not bilinear but still convex-concave in the Wasserstein geometry, and for which simultaneous dynamics do not converge yet timescale separation does. Taken together, these results showcase the benign geometry of bilinear games in the space of measures, enabling particle dynamics with global qualitative convergence guarantees.

本文利用熵正则化方法，通过针对概率测度空间上的混合平衡点问题，采用对称梯度升降法求解Wasserstein距离来解决最小化最大问题，并取得全局收敛性。同时，提出Wasserstein几何下的凸凹可用于解决相关熵正则化的损失函数。

测度上极小最大值均衡的同时传输演化