A Note on the Convergence of Mirrored Stein Variational Gradient Descent under $(L_0,L_1)-$Smoothness Condition

TL;DR

This paper proves a convergence property for the Mirrored Stein Variational Gradient Method under a broad class of non-smooth conditions, extending its applicability to constrained sampling problems.

Contribution

It establishes a descent lemma for MSVGD that does not depend on path information and applies to non-smooth potentials, broadening its theoretical foundation.

Findings

MSVGD can be applied to non-smooth $V$ in constrained sampling.

A descent lemma for MSVGD is established without relying on path information.

Analysis of MSVGD complexity in high dimensions.

Abstract

In this note, we establish a descent lemma for the population limit Mirrored Stein Variational Gradient Method~(MSVGD). This descent lemma does not rely on the path information of MSVGD but rather on a simple assumption for the mirrored distribution $\nabla\Psi_{\#}\pi\propto\exp(-V)$. Our analysis demonstrates that MSVGD can be applied to a broader class of constrained sampling problems with non-smooth $V$. We also investigate the complexity of the population limit MSVGD in terms of dimension $d$.