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Nonasymptotic analysis of Stochastic Gradient Hamiltonian Monte Carlo under local conditions for nonconvex optimization

Ömer Deniz Akyildiz, Sotirios Sabanis

TL;DR

It is proved that the Wasserstein-2 distance between the target and the law of the SGHMC is uniformly controlled by the step-size of the algorithm, therefore it is demonstrated that the S GHMC can provide high-precision results uniformly in the number of iterations.

Abstract

We provide a nonasymptotic analysis of the convergence of the stochastic gradient Hamiltonian Monte Carlo (SGHMC) to a target measure in Wasserstein-2 distance without assuming log-concavity. Our analysis quantifies key theoretical properties of the SGHMC as a sampler under local conditions which significantly improves the findings of previous results. In particular, we prove that the Wasserstein-2 distance between the target and the law of the SGHMC is uniformly controlled by the step-size of the algorithm, therefore demonstrate that the SGHMC can provide high-precision results uniformly in the number of iterations. The analysis also allows us to obtain nonasymptotic bounds for nonconvex optimization problems under local conditions and implies that the SGHMC, when viewed as a nonconvex optimizer, converges to a global minimum with the best known rates. We apply our results to obtain nonasymptotic bounds for scalable Bayesian inference and nonasymptotic generalization bounds.

Nonasymptotic analysis of Stochastic Gradient Hamiltonian Monte Carlo under local conditions for nonconvex optimization

TL;DR

It is proved that the Wasserstein-2 distance between the target and the law of the SGHMC is uniformly controlled by the step-size of the algorithm, therefore it is demonstrated that the S GHMC can provide high-precision results uniformly in the number of iterations.

Abstract

We provide a nonasymptotic analysis of the convergence of the stochastic gradient Hamiltonian Monte Carlo (SGHMC) to a target measure in Wasserstein-2 distance without assuming log-concavity. Our analysis quantifies key theoretical properties of the SGHMC as a sampler under local conditions which significantly improves the findings of previous results. In particular, we prove that the Wasserstein-2 distance between the target and the law of the SGHMC is uniformly controlled by the step-size of the algorithm, therefore demonstrate that the SGHMC can provide high-precision results uniformly in the number of iterations. The analysis also allows us to obtain nonasymptotic bounds for nonconvex optimization problems under local conditions and implies that the SGHMC, when viewed as a nonconvex optimizer, converges to a global minimum with the best known rates. We apply our results to obtain nonasymptotic bounds for scalable Bayesian inference and nonasymptotic generalization bounds.

Paper Structure

This paper contains 24 sections, 21 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Main results and overview
  3. Related work and contributions
  4. Preliminaries
  5. Proof of Theorem \ref{['thm:ConvRate']}
  6. Proof of Theorem \ref{['thm:NonconvexBound']}
  7. Applications
  8. Convergence rates for scalable Bayesian inference
  9. A generalization bound for machine learning
  10. Conclusions
  11. Constants in Theorem \ref{['thm:Eberle']}
  12. Additional lemmata
  13. Proofs of the preliminary results
  14. Proof of Lemma \ref{['lem:DiscreteTimeLemma']}
  15. Proof of Lemma \ref{['lem:zetaprocessmoment']}
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let Assumptions ass:nonnegativity--assum:dissipativity hold. Then, there exist constants $C_1^\star, C_2^\star, C_3^\star, C_4^\star > 0$ such that, for every $0 < \eta \leq \eta_{\textnormal{max}}$, where the constants $C_1^\star,C_2^\star,C_3^\star,C_4^\star$ are explicitly provided in the Appendix.

Theorems & Definitions (27)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.1
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.2
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • ...and 17 more