Convergence and stability results for the particle system in the Stein gradient descent method

TL;DR

This paper analyzes the convergence and stability of the Stein gradient descent particle system, showing it remains close to the asymptotic state for algebraic time scales, improving upon previous results.

Contribution

It extends the understanding of the Stein gradient descent method by establishing stability over larger time scales using a novel functional invariant approach.

Findings

Particles stay close to the asymptotic state for time scales t ≈ √N

The method improves stability results from logarithmic to algebraic time scales

A new functional invariant technique is applied to linearized equations

Abstract

There has been recently a lot of interest in the analysis of the Stein gradient descent method, a deterministic sampling algorithm. It is based on a particle system moving along the gradient flow of the Kullback-Leibler divergence towards the asymptotic state corresponding to the desired distribution. Mathematically, the method can be formulated as a joint limit of time $t$ and number of particles $N$ going to infinity. We first observe that the recent work of Lu, Lu and Nolen (2019) implies that if $t \approx \log \log N$, then the joint limit can be rigorously justified in the Wasserstein distance. Not satisfied with this time scale, we explore what happens for larger times by investigating the stability of the method: if the particles are initially close to the asymptotic state (with distance $\approx 1/N$), how long will they remain close? We prove that this happens in algebraic time scales $t \approx \sqrt{N}$ which is significantly better. The exploited method, developed by Caglioti and Rousset for the Vlasov equation, is based on finding a functional invariant for the linearized equation. This allows to eliminate linear terms and arrive at an improved Gronwall-type estimate.