Stein Variational Gradient Descent as Moment Matching

TL;DR

This paper provides a theoretical analysis of Stein Variational Gradient Descent (SVGD), revealing its moment matching properties and how kernel choices affect its inference accuracy, with implications for designing more efficient algorithms.

Contribution

It introduces the Stein matching set concept, analyzes non-asymptotic properties of SVGD, and explores how different kernels influence its inference capabilities.

Findings

SVGD with linear kernels exactly estimates means and variances for Gaussian distributions.

Random Fourier features enable probabilistic bounds for distributional approximation.

Theoretical framework connects SVGD properties to kernel choices and Stein's identity.

Abstract

Stein variational gradient descent (SVGD) is a non-parametric inference algorithm that evolves a set of particles to fit a given distribution of interest. We analyze the non-asymptotic properties of SVGD, showing that there exists a set of functions, which we call the Stein matching set, whose expectations are exactly estimated by any set of particles that satisfies the fixed point equation of SVGD. This set is the image of Stein operator applied on the feature maps of the positive definite kernel used in SVGD. Our results provide a theoretical framework for analyzing the properties of SVGD with different kernels, shedding insight into optimal kernel choice. In particular, we show that SVGD with linear kernels yields exact estimation of means and variances on Gaussian distributions, while random Fourier features enable probabilistic bounds for distributional approximation. Our results offer a refreshing view of the classical inference problem as fitting Stein's identity or solving the Stein equation, which may motivate more efficient algorithms.