  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.
