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

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  • 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
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    References listed on IDEAS

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    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.
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    Cited by:

    1. 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.
    2. Vincent Lemaire & Gilles Pag`es & Christian Yeo, 2023. "Swing contract pricing: with and without Neural Networks," Papers 2306.03822, arXiv.org, revised Mar 2024.
    3. 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.
    4. 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.

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