Multilevel Stein variational gradient descent with applications to Bayesian inverse problems

TL;DR

This paper introduces a multilevel Stein variational gradient descent method that leverages a hierarchy of discretizations to efficiently sample from complex distributions, significantly reducing computational costs in Bayesian inverse problems.

Contribution

It proposes a novel multilevel approach to Stein variational gradient descent, improving efficiency by utilizing multiple discretization levels and reducing the number of expensive high-fidelity iterations.

Findings

Error decay rate improves by a log factor over single-level methods.

Numerical experiments demonstrate over tenfold speedup in Bayesian inverse problems.

Multilevel method maintains accuracy while reducing computational costs.

Abstract

This work presents a multilevel variant of Stein variational gradient descent to more efficiently sample from target distributions. The key ingredient is a sequence of distributions with growing fidelity and costs that converges to the target distribution of interest. For example, such a sequence of distributions is given by a hierarchy of ever finer discretization levels of the forward model in Bayesian inverse problems. The proposed multilevel Stein variational gradient descent moves most of the iterations to lower, cheaper levels with the aim of requiring only a few iterations on the higher, more expensive levels when compared to the traditional, single-level Stein variational gradient descent variant that uses the highest-level distribution only. Under certain assumptions, in the mean-field limit, the error of the proposed multilevel Stein method decays by a log factor faster than the error of the single-level counterpart with respect to computational costs. Numerical experiments with Bayesian inverse problems show speedups of more than one order of magnitude of the proposed multilevel Stein method compared to the single-level variant that uses the highest level only.