## The Annals of Applied Probability

### Nonasymptotic convergence analysis for the unadjusted Langevin algorithm

#### Abstract

In this paper, we study a method to sample from a target distribution $\pi$ over $\mathbb{R}^{d}$ having a positive density with respect to the Lebesgue measure, known up to a normalisation factor. This method is based on the Euler discretization of the overdamped Langevin stochastic differential equation associated with $\pi$. For both constant and decreasing step sizes in the Euler discretization, we obtain nonasymptotic bounds for the convergence to the target distribution $\pi$ in total variation distance. A particular attention is paid to the dependency on the dimension $d$, to demonstrate the applicability of this method in the high-dimensional setting. These bounds improve and extend the results of Dalalyan [J. R. Stat. Soc. Ser. B. Stat. Methodol. (2017) 79 651–676].

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 3 (2017), 1551-1587.

Dates
Revised: August 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1500451235

Digital Object Identifier
doi:10.1214/16-AAP1238

Mathematical Reviews number (MathSciNet)
MR3678479

Zentralblatt MATH identifier
1377.65007

#### Citation

Durmus, Alain; Moulines, Éric. Nonasymptotic convergence analysis for the unadjusted Langevin algorithm. Ann. Appl. Probab. 27 (2017), no. 3, 1551--1587. doi:10.1214/16-AAP1238. https://projecteuclid.org/euclid.aoap/1500451235

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