Stochastic Particle-Optimization Sampling and the Non-Asymptotic Convergence Theory

Particle-optimization-based sampling (POS) is a recently developed technique to generate high-quality samples from a target distribution by iteratively updating a set of interactive particles. A representative algorithm is the Stein variational gradient descent (SVGD). Though obtaining significant empirical success, {\em non-asymptotic} convergence behaviors of SVGD remain unknown. We show in this paper that under certain conditions, SVGD experiences a theoretical pitfall where particles tend to collapse, rendering development of general non-asymptotic convergence theory challenging. To overcome this issue, we generalize POS to a stochasticity setting by injecting random noise into particle updates, thus termed stochastic particle-optimization sampling (SPOS). Notably, for the first time, we develop {\em non-asymptotic convergence theory} for the SPOS framework, characterizing algorithm convergence in terms of the 1-Wasserstein distance w.r.t.\! the number of particles and iterations, under both convex- and noncovex-energy-function settings. Our theory also shreds light on the non-asymptotic convergence of standard SVGD. Our development is based on analysis of nonlinear stochastic differential equations, serving as an extension and a complementary development to the asymptotic convergence theory for SVGD such as \citep{liu2017stein_flow}. Extensive experimental results verify our theory and demonstrate the effectiveness of our proposed framework.