A Finite-Particle Convergence Rate for Stein Variational Gradient Descent
This provides the first explicit convergence guarantee for SVGD, addressing a theoretical gap for researchers in machine learning and statistics, though it is incremental as it focuses on refining existing analysis.
The paper tackles the problem of establishing a finite-particle convergence rate for Stein Variational Gradient Descent (SVGD), showing that under sub-Gaussian and Lipschitz score conditions, SVGD drives the kernel Stein discrepancy to zero at a rate of 1/sqrt(log log n) with n particles.
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.