Stein Variational Gradient Descent (SVGD) is a method that uses interacting particles to sample from a target distribution when you have access to its score (gradient of log-density). In the continuous-time, infinite-particle limit, the method is known to converge, but prior work lacked quantitative rates for the final distribution in strong norms. This paper proves explicit polynomial convergence rates in L²-norm for the mean-field dynamics on a torus, assuming the initial and target distributions are smooth and start close together. The rates depend on dimension, kernel regularity, and smoothness of the densities. They also show these rates are tight in some regimes and recover prior exponential convergence results for Coulomb-type kernels as a special case.
Main takeaways:
- Establishes the first quantitative local convergence rates for SVGD's continuous-time mean-field limit in strong (L²) norms
- Rates are polynomial and depend explicitly on dimension, kernel type, and smoothness of initial/target densities
- Assumes initialization is close to the target and both are smooth
- Shows the rates are sharp (cannot be improved) in certain parameter regimes
- Recovers prior global exponential convergence for Coulomb kernels as a special case