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The tamed unadjusted Langevin algorithm

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  • Brosse, Nicolas
  • Durmus, Alain
  • Moulines, Éric
  • Sabanis, Sotirios

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.

Suggested Citation

  • Brosse, Nicolas & Durmus, Alain & Moulines, Éric & Sabanis, Sotirios, 2019. "The tamed unadjusted Langevin algorithm," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3638-3663.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:10:p:3638-3663
    DOI: 10.1016/j.spa.2018.10.002
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    References listed on IDEAS

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    1. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    2. Arnak S. Dalalyan, 2017. "Theoretical guarantees for approximate sampling from smooth and log-concave densities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 651-676, June.
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    2. Crucinio, Francesca R. & De Bortoli, Valentin & Doucet, Arnaud & Johansen, Adam M., 2024. "Solving a class of Fredholm integral equations of the first kind via Wasserstein gradient flows," Stochastic Processes and their Applications, Elsevier, vol. 173(C).
    3. Xinghu Jin & Guodong Pang & Lihu Xu & Xin Xu, 2025. "An Approximation to the Invariant Measure of the Limiting Diffusion of G / Ph / n + GI Queues in the Halfin–Whitt Regime and Related Asymptotics," Mathematics of Operations Research, INFORMS, vol. 50(2), pages 783-812, May.
    4. Lytras, Iosif & Sabanis, Sotirios, 2025. "Taming under isoperimetry," Stochastic Processes and their Applications, Elsevier, vol. 188(C).
    5. Yuyun Hidayat & Titi Purwandari & Subiyanto & Sukono, 2021. "Identifying Unwanted Conditions through Chaotic Area Determination in the Context of Indonesia’s Economic Resilience at the City Level," Sustainability, MDPI, vol. 13(9), pages 1-15, May.

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