The Tamed Unadjusted Langevin Algorithm

TL;DR

This paper introduces the Tamed Unadjusted Langevin Algorithm (TULA) to address instability issues in Langevin-based sampling methods for superlinear potentials, providing theoretical bounds and numerical validation.

Contribution

The paper proposes TULA, a novel algorithm that stabilizes Langevin sampling for superlinear potentials, with rigorous non-asymptotic error bounds and practical numerical experiments.

Findings

TULA achieves stable sampling from complex distributions with superlinear potentials.

Non-asymptotic bounds in total variation and Wasserstein distances are established for TULA.

Numerical experiments confirm the theoretical stability and accuracy of TULA.

Abstract

In this article, we consider the problem of sampling from a probability measure $\pi$ having a density on $\mathbb{R}^d$ known up to a normalizing constant, $x\mapsto \mathrm{e}^{-U(x)} / \int_{\mathbb{R}^d} \mathrm{e}^{-U(y)} \mathrm{d} y$. The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable in a precise sense, when the potential $U$ is superlinear, i.e. $\liminf_{\Vert x \Vert\to+\infty} \Vert \nabla U(x) \Vert / \Vert x \Vert = +\infty$. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in $V$-total variation norm and Wasserstein distance of order $2$ between the iterates of TULA and $\pi$, as well as weak error bounds. Numerical experiments are presented which support our findings.