论文标题
最小值平衡的同时运输演化
Simultaneous Transport Evolution for Minimax Equilibria on Measures
论文作者
论文摘要
在几个关键的机器学习设置中出现了Min-Max优化问题，包括对抗性学习和生成建模。在其一般形式中，在没有凸度/凹度假设的情况下，找到基本的两人零和游戏的纯平衡在计算上很难[Daskalakis等，2021]。在这项工作中，我们集中于寻找混合平衡，并考虑概率措施空间中相关的提升问题。通过添加熵正则化，我们的主要结果通过使用同时使用与沃斯坦斯坦度量的同时梯度上升，建立了全局趋于朝着全局均衡的融合，这种动力学是一种动力学，该动力学与熵镜下降相反，该动力学可以在高维度中有效地粒子离散化。我们与相关的熵调查损失相互补充，该损失不是双线性的，但仍在Wasserstein几何形状中凸出，并且同时动力学尚未融合时间尺度分离。综上所述，这些结果展示了双线性游戏在措施空间中的良性几何形状，从而使粒子动态具有全球定性收敛的保证。
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.
