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arXiv · 1710.05559

The Tamed Unadjusted Langevin Algorithm

Nicolas Brosse·Alain Durmus·Éric Moulines·Sotirios Sabanis
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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.

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Nicolas Brosse, Alain Durmus, Éric Moulines, Sotirios Sabanis. 2017-10-16. The Tamed Unadjusted Langevin Algorithm. https://arxiv.org/abs/1710.05559

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