## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 4 (2017), 2195-2237.

### Fast Langevin based algorithm for MCMC in high dimensions

Alain Durmus, Gareth O. Roberts, Gilles Vilmart, and Konstantinos C. Zygalakis

#### Abstract

We introduce new Gaussian proposals to improve the efficiency of the standard Hastings–Metropolis algorithm in Markov chain Monte Carlo (MCMC) methods, used for the sampling from a target distribution in large dimension $d$. The improved complexity is $\mathcal{O}(d^{1/5})$ compared to the complexity $\mathcal{O}(d^{1/3})$ of the standard approach. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterised by its overall acceptance rate (with asymptotical value 0.704), independently of the target distribution. Numerical experiments confirm our theoretical findings.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 4 (2017), 2195-2237.

**Dates**

Received: July 2015

Revised: October 2016

First available in Project Euclid: 30 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1504080030

**Digital Object Identifier**

doi:10.1214/16-AAP1257

**Mathematical Reviews number (MathSciNet)**

MR3693524

**Zentralblatt MATH identifier**

1373.60053

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 65C05: Monte Carlo methods

**Keywords**

Weak convergence Markov chain Monte Carlo diffusion limit exponential ergodicity

#### Citation

Durmus, Alain; Roberts, Gareth O.; Vilmart, Gilles; Zygalakis, Konstantinos C. Fast Langevin based algorithm for MCMC in high dimensions. Ann. Appl. Probab. 27 (2017), no. 4, 2195--2237. doi:10.1214/16-AAP1257. https://projecteuclid.org/euclid.aoap/1504080030

#### Supplemental materials

- Supplement to “Fast Langevin based algorithm for MCMC in high dimensions”. Mathematica notebooks.Digital Object Identifier: doi:10.1214/16-AAP1257SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.