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# Mean-field Underdamped Langevin Dynamics and its Spacetime Discretization

*Proceedings of the 41st International Conference on Machine Learning*, PMLR 235:14175-14206, 2024.

#### Abstract

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.