Controlling Moments with Kernel Stein Discrepancies

TL;DR

This paper investigates the ability of Kernel Stein Discrepancies to control moments and convergence types, proposing new variants that can precisely characterize Wasserstein convergence.

Contribution

It introduces diffusion KSDs that control both moments and weak convergence, including the first KSDs to characterize q-Wasserstein convergence.

Findings

Standard KSDs fail to control moment convergence.

Diffusion KSDs can control both moment and weak convergence.

New KSDs characterize q-Wasserstein convergence for all q > 0.

Abstract

Kernel Stein discrepancies (KSDs) measure the quality of a distributional approximation and can be computed even when the target density has an intractable normalizing constant. Notable applications include the diagnosis of approximate MCMC samplers and goodness-of-fit tests for unnormalized statistical models. The present work analyzes the convergence control properties of KSDs. We first show that standard KSDs used for weak convergence control fail to control moment convergence. To address this limitation, we next provide sufficient conditions under which alternative diffusion KSDs control both moment and weak convergence. As an immediate consequence we develop, for each $q > 0$, the first KSDs known to exactly characterize $q$-Wasserstein convergence.