We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

翻译：摘要：针对概率测度空间上定义的一类特殊非线性泛函优化问题，我们提出了一种名为N粒子欠阻尼朗之万算法的新方法。此类问题的典型实例包括平均场神经网络训练、最大均值差异最小化以及核斯坦因差异最小化。该算法基于平均场欠阻尼朗之万动力学的新型时空离散化方案，并为此提供了新的快速混合性保证。进一步地，我们证明了该算法在总变差距离意义下具有全局收敛性，从而弥合了动力学理论与其实际实现之间的理论鸿沟。