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Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm

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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Contributor : Alain Durmus <>
Submitted on : Monday, December 19, 2016 - 9:51:09 AM
Last modification on : Tuesday, December 8, 2020 - 10:29:24 AM


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  • HAL Id : hal-01176132, version 3
  • ARXIV : 1507.05021


Alain Durmus, Éric Moulines. Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm. 2016. ⟨hal-01176132v3⟩



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