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Journal Articles Annals of Applied Probability Year : 2017

Nonasymptotic convergence analysis for the unadjusted Langevin algorithm

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Abstract

In this paper, we study a method to sample from a target distribution π over Rd 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 π. For both constant and decreasing step sizes in the Euler discretization, we obtain nonasymptotic bounds for the convergence to the target distribution π 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 [J. R. Stat. Soc. Ser. B. Stat. Methodol. (2017) 79 651–676].

Dates and versions

hal-01668245 , version 1 (19-12-2017)

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Cite

Alain Durmus, Éric Moulines. Nonasymptotic convergence analysis for the unadjusted Langevin algorithm. Annals of Applied Probability, 2017, 27 (3), pp.1551 - 1587. ⟨10.1214/16-AAP1238⟩. ⟨hal-01668245⟩
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