The tamed unadjusted Langevin algorithm

Nicolas Brosse; Alain Durmus; Éric Moulines; Sotirios Sabanis

doi:10.1016/j.spa.2018.10.002

The tamed unadjusted Langevin algorithm

Nicolas Brosse
, Alain Durmus
, Éric Moulines
, Sotirios Sabanis

Ecole polytechnique
Université Paris-Saclay
University of Edinburgh
The Alan Turing Institute

Research output: Contribution to journal › Article › peer-review

Abstract

In this article, we consider the problem of sampling from a probability measure π having a density on R^d proportional to x↦e^−U(x). The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential U is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.

Original language	English
Pages (from-to)	3638-3663
Number of pages	26
Journal	Stochastic Processes and their Applications
Volume	129
Issue number	10
DOIs	https://doi.org/10.1016/j.spa.2018.10.002
Publication status	Published - 1 Oct 2019

Keywords

Markov chain Monte Carlo
Tamed unadjusted Langevin algorithm
Total variation distance
Wasserstein distance

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10.1016/j.spa.2018.10.002

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title = "The tamed unadjusted Langevin algorithm",

abstract = "In this article, we consider the problem of sampling from a probability measure π having a density on Rd proportional to x↦e−U(x). The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential U is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.",

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AU - Brosse, Nicolas

AU - Durmus, Alain

AU - Moulines, Éric

AU - Sabanis, Sotirios

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Y1 - 2019/10/1

N2 - In this article, we consider the problem of sampling from a probability measure π having a density on Rd proportional to x↦e−U(x). The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential U is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.

AB - In this article, we consider the problem of sampling from a probability measure π having a density on Rd proportional to x↦e−U(x). The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential U is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings.

KW - Markov chain Monte Carlo

KW - Tamed unadjusted Langevin algorithm

KW - Total variation distance

KW - Wasserstein distance

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SP - 3638

EP - 3663

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 10

ER -

The tamed unadjusted Langevin algorithm

Abstract

Keywords

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