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User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

Author

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  • Arnak Dalalyan

    (ENSAE;CREST)

  • Avetik Karagulyan

    (CREST;ENSAE)

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, in terms of constants, the existing results when the accuracy of sampling is measured in the Wasserstein distance and provide further insights on relations between, on the one hand, the Langevin Monte Carlo for sampling and, on the other hand, the gradient descent for optimization. More importantly, we establish non-asymptotic guarantees for the accuracy of a version of the Langevin Monte Carlo algorithm that is based on inaccurate evaluations of the gradient. Finally, we propose a variable-step version of the Langevin Monte Carlo algorithm that has two advantages. First, its step-sizes are independent of the target accuracy and, second, its rate provides a logarithmic improvement over the constant-step Langevin Monte Carlo algorithm ;Classification-JEL: Primary 62J05; secondary 62H12

Suggested Citation

  • Arnak Dalalyan & Avetik Karagulyan, 2017. "User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient," Working Papers 2017-20, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2017-20
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    References listed on IDEAS

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    1. O. Stramer & R. L. Tweedie, 1999. "Langevin-Type Models II: Self-Targeting Candidates for MCMC Algorithms," Methodology and Computing in Applied Probability, Springer, vol. 1(3), pages 307-328, October.
    2. Jarner, Søren Fiig & Hansen, Ernst, 2000. "Geometric ergodicity of Metropolis algorithms," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 341-361, February.
    3. 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.
    4. 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.
    5. 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.
    6. Aurélien Alfonsi & Benjamin Jourdain & Arturo Kohatsu-Higa, 2014. "Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme," Post-Print hal-00727430, HAL.
    7. O. Stramer & R. L. Tweedie, 1999. "Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations," Methodology and Computing in Applied Probability, Springer, vol. 1(3), pages 283-306, October.
    8. 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.
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    Cited by:

    1. 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).
    2. Murray Pollock & Paul Fearnhead & Adam M. Johansen & Gareth O. Roberts, 2020. "Quasi‐stationary Monte Carlo and the ScaLE algorithm," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(5), pages 1167-1221, December.
    3. Sotirios Sabanis & Ying Zhang, 2020. "A fully data-driven approach to minimizing CVaR for portfolio of assets via SGLD with discontinuous updating," Papers 2007.01672, arXiv.org.
    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. 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.

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