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June 2017 Nonasymptotic convergence analysis for the unadjusted Langevin algorithm
Alain Durmus, Éric Moulines
Ann. Appl. Probab. 27(3): 1551-1587 (June 2017). DOI: 10.1214/16-AAP1238


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].


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Alain Durmus. Éric Moulines. "Nonasymptotic convergence analysis for the unadjusted Langevin algorithm." Ann. Appl. Probab. 27 (3) 1551 - 1587, June 2017.


Received: 1 March 2016; Revised: 1 August 2016; Published: June 2017
First available in Project Euclid: 19 July 2017

zbMATH: 1377.65007
MathSciNet: MR3678479
Digital Object Identifier: 10.1214/16-AAP1238

Primary: 60F05 , 62L10 , 65C05
Secondary: 60J05 , 65C40 , 93E35

Keywords: Langevin diffusion , Markov chain Monte Carlo , Metropolis adjusted Langevin algorithm , rate of convergence , total variation distance

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.27 • No. 3 • June 2017
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