Variance Reduction in Stochastic Particle-Optimization Sampling

Stochastic particle-optimization sampling (SPOS) is a recently-developed scalable Bayesian sampling framework unifying stochastic gradient MCMC (SG-MCMC) and Stein variational gradient descent (SVGD) algorithms based on Wasserstein gradient flows. With a rigorous non-asymptotic convergence theory developed, SPOS can avoid the particle-collapsing pitfall of SVGD. However, the variance-reduction effect in SPOS has not been clear. In this paper, we address this gap by presenting several variance-reduction techniques for SPOS. Specifically, we propose three variants of variance-reduced SPOS, called SAGA particle-optimization sampling (SAGA-POS), SVRG particle-optimization sampling (SVRG-POS) and a variant of SVRG-POS which avoids full gradient computations, denoted as SVRG-POS$^+$. Importantly, we provide non-asymptotic convergence guarantees for these algorithms in terms of the 2-Wasserstein metric and analyze their complexities. The results show our algorithms yield better convergence rates than existing variance-reduced variants of stochastic Langevin dynamics, though more space is required to store the particles in training. Our theory aligns well with experimental results on both synthetic and real datasets.