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arXiv · 1808.00728

Higher Order Langevin Monte Carlo Algorithm

Sotirios Sabanis·Ying Zhang
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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 $π$ that has a density $\hatπ$ on $\mathbb{R}^d$ known up to a normalizing constant. Moreover, $-\log \hatπ$ is assumed to have a locally Lipschitz gradient and its third derivative is locally Hölder continuous with exponent $β\in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+ β/2$ 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 $-\log \hatπ$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.

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Sotirios Sabanis, Ying Zhang. 2019-09-25. Higher Order Langevin Monte Carlo Algorithm. https://doi.org/10.1214/19-ejs1615

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