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Theoretical guarantees for neural control variates in MCMC

Author

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  • Belomestny, Denis
  • Goldman, Artur
  • Naumov, Alexey
  • Samsonov, Sergey

Abstract

In this paper, we propose a variance reduction approach for Markov chains based on additive control variates and the minimization of an appropriate estimate for the asymptotic variance. We focus on the particular case when control variates are represented as deep neural networks. We derive the optimal convergence rate of the asymptotic variance under various ergodicity assumptions on the underlying Markov chain. The proposed approach relies upon recent results on the stochastic errors of variance reduction algorithms and function approximation theory.

Suggested Citation

  • Belomestny, Denis & Goldman, Artur & Naumov, Alexey & Samsonov, Sergey, 2024. "Theoretical guarantees for neural control variates in MCMC," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 382-405.
    • Handle: RePEc:eee:matcom:v:220:y:2024:i:c:p:382-405
    DOI: 10.1016/j.matcom.2024.01.019
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    References listed on IDEAS

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    1. Chris J. Oates & Mark Girolami & Nicolas Chopin, 2017. "Control functionals for Monte Carlo integration," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 695-718, June.
    2. Petros Dellaportas & Ioannis Kontoyiannis, 2012. "Control variates for estimation based on reversible Markov chain Monte Carlo samplers," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(1), pages 133-161, January.
    3. Langer, Sophie, 2021. "Approximating smooth functions by deep neural networks with sigmoid activation function," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
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