IDEAS home Printed from https://ideas.repec.org/p/tse/wpaper/129118.html
   My bibliography  Save this paper

Discretisation of Langevin diffusion in the weak log-concave case

Author

Listed:
  • Crespo, Marelys

Abstract

The Euler discretisation of Langevin diffusion, also known as Unadjusted Langevin Algorithm, is commonly used in machine learning for sampling from a given distribution µ ∝ e−U . In this paper we investigate a potential U : Rd −→ R which is a weakly convex function and has Lipschitz gradient. We parameterize the weak convexity with the help of the Kurdyka-Lojasiewicz (KL) inequality, that permits to handle a vanishing curvature settings, which is far less restrictive when compared to the simple strongly convex case. We prove that the final horizon of simulation to obtain an ε approximation (in terms of entropy) is of the order ε−1d1+2(1+r)2Poly(log(d), log(ε−1)), where the parameter r is involved in the KL inequality and varies between 0 (strongly convex case) and 1 (limiting Laplace situation).

Suggested Citation

  • Crespo, Marelys, 2024. "Discretisation of Langevin diffusion in the weak log-concave case," TSE Working Papers 24-1506, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:129118
    as

    Download full text from publisher

    File URL: https://www.tse-fr.eu/sites/default/files/TSE/documents/doc/wp/2024/wp_tse_1506.pdf
    File Function: Full Text
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Gareth O. Roberts & Jeffrey S. Rosenthal, 1998. "Optimal scaling of discrete approximations to Langevin diffusions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 255-268.
    Full references (including those not matched with items on IDEAS)

    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. Jorge I. Figueroa-Zúñiga & Cristian L. Bayes & Víctor Leiva & Shuangzhe Liu, 2022. "Robust beta regression modeling with errors-in-variables: a Bayesian approach and numerical applications," Statistical Papers, Springer, vol. 63(3), pages 919-942, June.
    2. Mamatzakis, Emmanuel C. & Tsionas, Mike G., 2021. "Making inference of British household's happiness efficiency: A Bayesian latent model," European Journal of Operational Research, Elsevier, vol. 294(1), pages 312-326.
    3. Gael M. Martin & David T. Frazier & Christian P. Robert, 2020. "Computing Bayes: Bayesian Computation from 1763 to the 21st Century," Monash Econometrics and Business Statistics Working Papers 14/20, Monash University, Department of Econometrics and Business Statistics.
    4. Delis, Manthos D. & Tsionas, Mike G., 2018. "Measuring management practices," International Journal of Production Economics, Elsevier, vol. 199(C), pages 65-77.
    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.
    6. Aknouche, Abdelhakim & Dimitrakopoulos, Stefanos, 2020. "On an integer-valued stochastic intensity model for time series of counts," MPRA Paper 105406, University Library of Munich, Germany.
    7. 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.
    8. O. F. Christensen & J. Møller & R. P. Waagepetersen, 2001. "Geometric Ergodicity of Metropolis-Hastings Algorithms for Conditional Simulation in Generalized Linear Mixed Models," Methodology and Computing in Applied Probability, Springer, vol. 3(3), pages 309-327, September.
    9. M Ludkin & C Sherlock, 2023. "Hug and hop: a discrete-time, nonreversible Markov chain Monte Carlo algorithm," Biometrika, Biometrika Trust, vol. 110(2), pages 301-318.
    10. Tsionas, Mike G. & Michaelides, Panayotis G., 2017. "Neglected chaos in international stock markets: Bayesian analysis of the joint return–volatility dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 482(C), pages 95-107.
    11. G. O. Roberts & O. Stramer, 2002. "Langevin Diffusions and Metropolis-Hastings Algorithms," Methodology and Computing in Applied Probability, Springer, vol. 4(4), pages 337-357, December.
    12. Shao, Wei & Guo, Guangbao & Meng, Fanyu & Jia, Shuqin, 2013. "An efficient proposal distribution for Metropolis–Hastings using a B-splines technique," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 465-478.
    13. Anandamayee Majumdar & Corinna Gries & Jason Walker, 2011. "A non-stationary spatial generalized linear mixed model approach for studying plant diversity," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(9), pages 1935-1950, October.
    14. Reihaneh Entezari & Patrick E. Brown & Jeffrey S. Rosenthal, 2020. "Bayesian spatial analysis of hardwood tree counts in forests via MCMC," Environmetrics, John Wiley & Sons, Ltd., vol. 31(4), June.
    15. N. Englezos & X. Kartala & P. Koundouri & M. Tsionas & A. Alamanos, 2023. "A Novel HydroEconomic - Econometric Approach for Integrated Transboundary Water Management Under Uncertainty," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 84(4), pages 975-1030, April.
    16. Moffa, Giusi & Kuipers, Jack, 2014. "Sequential Monte Carlo EM for multivariate probit models," Computational Statistics & Data Analysis, Elsevier, vol. 72(C), pages 252-272.
    17. Lambert, Philippe & Eilers, Paul H.C., 2009. "Bayesian density estimation from grouped continuous data," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1388-1399, February.
    18. Peter Neal & Gareth Roberts, 2008. "Optimal Scaling for Random Walk Metropolis on Spherically Constrained Target Densities," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 277-297, June.
    19. Palczewski, Andrzej & Palczewski, Jan, 2019. "Black–Litterman model for continuous distributions," European Journal of Operational Research, Elsevier, vol. 273(2), pages 708-720.
    20. Burda Martin & Maheu John M., 2013. "Bayesian adaptively updated Hamiltonian Monte Carlo with an application to high-dimensional BEKK GARCH models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 17(4), pages 345-372, September.

    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:tse:wpaper:129118. 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: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/tsetofr.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.