A Finite-Particle Convergence Rate for Stein Variational Gradient Descent

TL;DR

This paper establishes the first finite-particle convergence rate for Stein variational gradient descent (SVGD), showing that it effectively approximates sub-Gaussian distributions with a specific convergence rate, and provides a new proof strategy for future improvements.

Contribution

It provides the first explicit finite-particle convergence rate for SVGD with non-asymptotic analysis under sub-Gaussian assumptions.

Findings

SVGD with n particles converges at a rate of 1/sqrt(log log n)

The convergence is measured via the kernel Stein discrepancy

The proof strategy is explicit and non-asymptotic

Abstract

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.