Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm

Alain Durmus; Eric Moulines

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arXiv · 1507.05021

Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm

Alain Durmus·Eric Moulines

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Abstract

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 non-asymptotic 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 2014).

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BibTeX RIS

Alain Durmus, Eric Moulines. 2015-07-17. Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm. https://arxiv.org/abs/1507.05021

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