A Finite-Particle Convergence Rate for Stein Variational Gradient Descent

We provide the first finite-particle convergence rate for Stein variational gradient descent (SVGD), a popular algorithm for approximating a probability distribution with a collection of particles. Specifically, whenever the target distribution is sub-Gaussian with a Lipschitz score, SVGD with n particles and an appropriate step size sequence drives the kernel Stein discrepancy to zero at an order {1/}{\sqrt{\log\log n}} rate. We suspect that the dependence on n can be improved, and we hope that our explicit, non-asymptotic proof strategy will serve as a template for future refinements.