## The Annals of Applied Probability

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

### Nonasymptotic convergence analysis for the unadjusted Langevin algorithm

Alain Durmus and Éric Moulines

#### 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**

Received: March 2016

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

**Subjects**

Primary: 65C05: Monte Carlo methods 60F05: Central limit and other weak theorems 62L10: Sequential analysis

Secondary: 65C40: Computational Markov chains 60J05: Discrete-time Markov processes on general state spaces 93E35: Stochastic learning and adaptive control

**Keywords**

Total variation distance Langevin diffusion Markov Chain Monte Carlo Metropolis adjusted Langevin algorithm rate of convergence

#### 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