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

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  • Dalalyan, Arnak S.
  • Karagulyan, Avetik

Abstract

In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide non asymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:12:p:5278-5311
    DOI: 10.1016/j.spa.2019.02.016
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    1. Jarner, Søren Fiig & Hansen, Ernst, 2000. "Geometric ergodicity of Metropolis algorithms," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 341-361, February.
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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. Peiyi Zhang & Tianning Dong & Faming Liang, 2024. "An extended Langevinized ensemble Kalman filter for non-Gaussian dynamic systems," Computational Statistics, Springer, vol. 39(6), pages 3347-3372, September.
    3. 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.
    4. 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).
    5. 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.
    6. 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.
    7. 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.

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