Scaling Limit of the Stein Variational Gradient Descent: The Mean Field Regime

We study an interacting particle system in Rᵈ motivated by Stein variational gradient descent [Q. Liu and D. Wang, Proceedings of NIPS, 2016], a deterministic algorithm for approximating a given probability density with unknown normalization based on particles. We prove that in the large particle limit the empirical measure of the particle system converges to a solution of a nonlocal and nonlinear PDE. We also prove the global existence, uniqueness, and regularity of the solution to the limiting PDE. Finally, we prove that the solution to the PDE converges to the unique invariant solution in a long time limit.