## Electronic Journal of Statistics

### Higher order Langevin Monte Carlo algorithm

#### Abstract

A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $\pi$ that has a density $\hat{\pi}$ on $\mathbb{R}^{d}$ known up to a normalizing constant. Moreover, $-\log\hat{\pi}$ is assumed to have a locally Lipschitz gradient and its third derivative is locally Hölder continuous with exponent $\beta \in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+\beta/2$ in Wasserstein distance, while it is shown that the rate is 1 in total variation even in the absence of convexity. Finally, in the case where $-\log \hat{\pi}$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.

#### Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 3805-3850.

Dates
First available in Project Euclid: 3 October 2019

https://projecteuclid.org/euclid.ejs/1570068044

Digital Object Identifier
doi:10.1214/19-EJS1615

Mathematical Reviews number (MathSciNet)
MR4015336

Zentralblatt MATH identifier
07113731

Subjects
Primary: 62L10: Sequential analysis 65C05: Monte Carlo methods

#### Citation

Sabanis, Sotirios; Zhang, Ying. Higher order Langevin Monte Carlo algorithm. Electron. J. Statist. 13 (2019), no. 2, 3805--3850. doi:10.1214/19-EJS1615. https://projecteuclid.org/euclid.ejs/1570068044

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