Scaling limit of the Stein variational gradient descent: the mean field regime
This work provides theoretical foundations for SVGD, a popular sampling algorithm, by analyzing its mean-field behavior, which is incremental but important for understanding its convergence properties.
The authors studied the large-particle limit of Stein variational gradient descent (SVGD), proving that the empirical measure of the particle system converges to a solution of a non-local nonlinear PDE, with global existence, uniqueness, and regularity established, and showing long-time convergence to the invariant solution.
We study an interacting particle system in $\mathbf{R}^d$ motivated by Stein variational gradient descent [Q. Liu and D. Wang, NIPS 2016], a deterministic algorithm for sampling from a given probability density with unknown normalization. We prove that in the large particle limit the empirical measure of the particle system converges to a solution of a non-local and nonlinear PDE. We also prove 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 long time limit.