Controlling Moments with Kernel Stein Discrepancies
This work addresses a theoretical gap for researchers in statistics and machine learning dealing with distributional approximations, though it appears incremental as it builds on existing KSD methods.
The paper tackled the limitation of standard kernel Stein discrepancies (KSDs) in controlling moment convergence, and showed that alternative diffusion KSDs can control both moment and weak convergence, leading to the first KSDs that exactly characterize q-Wasserstein convergence for each q > 0.
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.