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：我们提出一种名为N粒子欠阻尼朗之万算法的新方法，用于优化定义在概率测度空间上的一类特殊非线性泛函。该公式所涵盖的问题包括：训练平均场神经网络、最大化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一种新型时空离散化方法，并为此提供了新的快速混合保证。此外，我们证明了该算法在总变差距离上具有全局收敛性，从而弥合了动力学过程与其实际实现之间的理论差距。