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 nonasymptotic 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.
Original language  English 

Pages (fromto)  36383663 
Number of pages  26 
Journal  Stochastic processes and their applications 
Volume  129 
Issue number  10 
Early online date  16 Oct 2018 
DOIs  
Publication status  Published  31 Oct 2019 
Keywords
 stat.ME
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Sotirios Sabanis
 School of Mathematics  Personal Chair of Stochastic Analysis and Algorithms
Person: Academic: Research Active