The Annals of Applied Probability

Fast Langevin based algorithm for MCMC in high dimensions

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

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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

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

Received: July 2015
Revised: October 2016
First available in Project Euclid: 30 August 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 65C05: Monte Carlo methods

Weak convergence Markov chain Monte Carlo diffusion limit exponential ergodicity


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.

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Supplemental materials

  • Supplement to “Fast Langevin based algorithm for MCMC in high dimensions”. Mathematica notebooks.