IDEAS home Printed from https://ideas.repec.org/p/crs/wpaper/2017-21.html
   My bibliography  Save this paper

Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent

Author

Listed:
  • Arnak Dalalyan

    (ENSAE;CREST)

Abstract

In this paper, we revisit the recently established theoretical guarantees for the convergence of the Langevin Monte Carlo algorithm of sampling from a smooth and (strongly) log-concave density. We improve the existing results when the convergence is measured in the Wasserstein distance and provide further insights on the very tight relations between, on the one hand, the Langevin Monte Carlo for sampling and, on the other hand, the gradient descent for optimization. Finally, we also establish guarantees for the convergence of a version of the Langevin Monte Carlo algorithm that is based on noisy evaluations of the gradient

Suggested Citation

  • Arnak Dalalyan, 2017. "Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent," Working Papers 2017-21, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2017-21
    as

    Download full text from publisher

    File URL: http://crest.science/RePEc/wpstorage/2017-21.pdf
    File Function: CREST working paper version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ghaderi, Susan & Ahookhosh, Masoud & Arany, Adam & Skupin, Alexander & Patrinos, Panagiotis & Moreau, Yves, 2024. "Smoothing unadjusted Langevin algorithms for nonsmooth composite potential functions," Applied Mathematics and Computation, Elsevier, vol. 464(C).
    2. Florian Maire & Nial Friel & Pierre ALQUIER, 2017. "Informed Sub-Sampling MCMC: Approximate Bayesian Inference for Large Datasets," Working Papers 2017-40, Center for Research in Economics and Statistics.
    3. Vincent Lemaire & Gilles Pag`es & Christian Yeo, 2023. "Swing contract pricing: with and without Neural Networks," Papers 2306.03822, arXiv.org, revised Mar 2024.
    4. Chau, Huy N. & Rásonyi, Miklós, 2022. "Stochastic Gradient Hamiltonian Monte Carlo for non-convex learning," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 341-368.
    5. Dalalyan, Arnak S. & Karagulyan, Avetik, 2019. "User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5278-5311.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Tengyuan Liang & Weijie J. Su, 2019. "Statistical inference for the population landscape via moment‐adjusted stochastic gradients," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 431-456, April.
    2. Dalalyan, Arnak S. & Karagulyan, Avetik, 2019. "User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5278-5311.
    3. Loaiza-Maya, Rubén & Nibbering, Didier & Zhu, Dan, 2024. "Hybrid unadjusted Langevin methods for high-dimensional latent variable models," Journal of Econometrics, Elsevier, vol. 241(2).
    4. Menz, Georg & Schlichting, André & Tang, Wenpin & Wu, Tianqi, 2022. "Ergodicity of the infinite swapping algorithm at low temperature," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 519-552.
    5. Ruben Loaiza-Maya & Didier Nibbering & Dan Zhu, 2023. "Hybrid unadjusted Langevin methods for high-dimensional latent variable models," Papers 2306.14445, arXiv.org.
    6. Denis Belomestny & Leonid Iosipoi, 2019. "Fourier transform MCMC, heavy tailed distributions and geometric ergodicity," Papers 1909.00698, arXiv.org, revised Dec 2019.
    7. Villeneuve, Stéphane & Bolte, Jérôme & Miclo, Laurent, 2022. "Swarm gradient dynamics for global optimization: the mean-field limit case," TSE Working Papers 22-1302, Toulouse School of Economics (TSE).
    8. Belomestny, Denis & Iosipoi, Leonid, 2021. "Fourier transform MCMC, heavy-tailed distributions, and geometric ergodicity," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 351-363.
    9. Yang, Jun & Roberts, Gareth O. & Rosenthal, Jeffrey S., 2020. "Optimal scaling of random-walk metropolis algorithms on general target distributions," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6094-6132.
    10. Samuel Livingstone & Giacomo Zanella, 2022. "The Barker proposal: Combining robustness and efficiency in gradient‐based MCMC," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 496-523, April.
    11. Crespo, Marelys & Gadat, Sébastien & Gendre, Xavier, 2023. "Stochastic Langevin Monte Carlo for (weakly) log-concave posterior distributions," TSE Working Papers 23-1398, Toulouse School of Economics (TSE).
    12. M. Barkhagen & S. García & J. Gondzio & J. Kalcsics & J. Kroeske & S. Sabanis & A. Staal, 2023. "Optimising portfolio diversification and dimensionality," Journal of Global Optimization, Springer, vol. 85(1), pages 185-234, January.
    13. 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.
    14. Chau, Huy N. & Rásonyi, Miklós, 2022. "Stochastic Gradient Hamiltonian Monte Carlo for non-convex learning," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 341-368.
    15. Vincent Lemaire & Gilles Pag`es & Christian Yeo, 2023. "Swing contract pricing: with and without Neural Networks," Papers 2306.03822, arXiv.org, revised Mar 2024.
    16. Tung Duy Luu & Jalal Fadili & Christophe Chesneau, 2021. "Sampling from Non-smooth Distributions Through Langevin Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1173-1201, December.
    17. Gadat, Sébastien & Panloup, Fabien & Pellegrini, C., 2020. "On the cost of Bayesian posterior mean strategy for log-concave models," TSE Working Papers 20-1155, Toulouse School of Economics (TSE), revised Feb 2022.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:crs:wpaper:2017-21. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Secretariat General (email available below). General contact details of provider: https://edirc.repec.org/data/crestfr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.