Convergence of Stein Variational Gradient Descent under a Weaker Smoothness Condition

Abstract

Stein Variational Gradient Descent (SVGD) is an important alternative to the Langevin-type algorithms for sampling from probability distributions of the form π(x) ∝ exp(−V (x)). In the existing theory of Langevin-type algorithms and SVGD, the potential function V is often assumed to be L-smooth. However, this restrictive condition excludes a large class of potential functions such as polynomials of degree greater than 2. Our paper studies the convergence of the SVGD algorithm for distributions with (L0, L1)-smooth potentials. This relaxed smoothness assumption was introduced by Zhang et al. [2019a] for the analysis of gradient clipping algorithms. With the help of trajectory-independent auxiliary conditions, we provide a descent lemma establishing that the algorithm decreases the KL divergence at each iteration and prove a complexity bound for SVGD in the population limit in terms of the Stein Fisher information.