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Theoretical guarantees for neural control variates in MCMC

Denis Belomestny, Artur Goldman, Alexey Naumov, Sergey Samsonov

TL;DR

A variance reduction approach for Markov chains based on additive control variates and the minimization of an appropriate estimate for the asymptotic variance is proposed.

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.

Theoretical guarantees for neural control variates in MCMC

TL;DR

A variance reduction approach for Markov chains based on additive control variates and the minimization of an appropriate estimate for the asymptotic variance is proposed.

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.
Paper Structure (21 sections, 19 theorems, 166 equations, 3 figures, 11 tables, 2 algorithms)

This paper contains 21 sections, 19 theorems, 166 equations, 3 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. ESVM algorithm and control variates
  4. Assumptions
  5. Markov Chain assumptions
  6. VR rates with ESVM procedure
  7. Numerical study
  8. Implementation specifics
  9. Funnel distribution
  10. Banana-shaped distribution
  11. Logistic regression
  12. Conclusion
  13. Acknowledgements
  14. Proofs
  15. Proof of Theorem \ref{['th:bound_approx']}
  16. ...and 6 more sections

Key Result

Theorem 1

Assume assu:AUF, assu:ge, and assu:br. Then for any $x_0 \in \mathsf{S}$ and $\delta \in (0,1)$ there exists $n_0=n_0(\delta, R_0, d, \beta, \mathcal{D})>0$ such that for all $n\geqslant n_0$, $n\in\mathbb{N}$ by setting $R=\log n$, $K = n^{\frac{1}{2\beta + d}}$, $b_n=2(\log(1/\rho))^{-1}\log(n)$, where $C_{th:bound_approx, 1}=C_{th:bound_approx, 1}(\beta, \mathcal{D})$ is independent of the pro

Figures (3)

  • Figure 1: Funnel distribution
  • Figure 2: Banana-shaped
  • Figure 3: Logistic regression, Pima dataset

Theorems & Definitions (34)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4: belomestny_variance_2020_esvm
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 24 more