Convergence of kinetic Langevin samplers for non-convex potentials

TL;DR

This paper analyzes the convergence properties of three kinetic Langevin samplers for non-convex potentials, providing contraction results and complexity bounds in Wasserstein distance, applicable to high-dimensional and mean-field systems.

Contribution

It offers the first contraction and complexity guarantees for these samplers in non-convex settings, using tailored distance functions and coupling methods.

Findings

Contraction results in $L^1$-Wasserstein distance for non-convex potentials.

Complexity bounds of $ ilde{O}(rac{ ext{poly}(d)}{ ext{accuracy}})$ for sampling.

Applicability to interacting particle systems and mean-field measures.

Abstract

We study three kinetic Langevin samplers including the Euler discretization, the BU and the UBU splitting scheme. We provide contraction results in $L^1$-Wasserstein distance for non-convex potentials. These results are based on a carefully tailored distance function and an appropriate coupling construction. Additionally, the error in the $L^1$-Wasserstein distance between the true target measure and the invariant measure of the discretization scheme is bounded. To get an $\varepsilon$-accuracy in $L^1$-Wasserstein distance, we show complexity guarantees of order $\mathcal{O}(\sqrt{d}/\varepsilon)$ for the Euler scheme and $\mathcal{O}(d^{1/4}/\sqrt{\varepsilon})$ for the UBU scheme under appropriate assumptions on the target measure. The results are applicable to interacting particle systems and provide bounds for sampling probability measures of mean-field type.