Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Concentration Inequalities and UQ Bounds for Hypocoercive MCMC Samplers

Jeremiah Birrell, Luc Rey-Bellet

TL;DR

The article develops non-asymptotic performance guarantees for hypocoercive, non-reversible MCMC samplers, including Langevin, HMC, Bouncy Particle, and Zig-Zag, by deriving a Bernstein-type concentration inequality for ergodic averages and studying uncertainty due to model-form perturbations through relative entropy. The core methodology combines Wu’s Feynman-Kac semigroups with the Dolbeault–Mouhot–Schmeiser hypocoercivity framework, using a modified Poincaré inequality in a parameterized inner product to obtain explicit constants. This yields non-asymptotic confidence intervals for $\mu_*[f]$ from finite-time data and a robust UQ bound on the bias under perturbed dynamics, quantified by path-space entropy rates. The results extend coercive (reversible) concentration theory to hypocoercive dynamics and provide practical, explicit constants for rigorous error control in advanced MCMC methods used in Bayesian inference and molecular dynamics.

Abstract

In this work we provide performance guarantees for hypocoercive non-reversible MCMC samplers $X_t$ with invariant measure $μ_*$; our results apply in particular to the Langevin equation, Hamiltonian Monte-Carlo, and the bouncy particle and zig-zag samplers. Specifically, we establish a concentration inequality of Bernstein type for ergodic averages $\frac{1}{T} \int_0^T f(X_t)\, dt$. As a consequence we provide two types of performance guarantees: (a) explicit non-asymptotic confidence intervals for $\int f dμ_*$ when using a finite time ergodic average with given initial condition $μ$ and (b) uncertainty quantification (UQ) bounds, expressed in terms of relative entropy rate, on the bias of $\int f dμ_*$ when using an alternative or approximate processes $\widetilde{X}_t$. (Results in (b) generalize results (arXiv:1812.05174) from the authors for coercive dynamics.) The concentration inequality is proved by combining the approach via Feynman-Kac semigroups first noted by Wu with the hypocoercive estimates of Dolbeault, Mouhot and Schmeiser (arXiv:1005.1495) developed for the Langevin equation and generalized to partially deterministic Markov processes by Andrieu et al. (arXiv:1808.08592).

Concentration Inequalities and UQ Bounds for Hypocoercive MCMC Samplers

TL;DR

The article develops non-asymptotic performance guarantees for hypocoercive, non-reversible MCMC samplers, including Langevin, HMC, Bouncy Particle, and Zig-Zag, by deriving a Bernstein-type concentration inequality for ergodic averages and studying uncertainty due to model-form perturbations through relative entropy. The core methodology combines Wu’s Feynman-Kac semigroups with the Dolbeault–Mouhot–Schmeiser hypocoercivity framework, using a modified Poincaré inequality in a parameterized inner product to obtain explicit constants. This yields non-asymptotic confidence intervals for from finite-time data and a robust UQ bound on the bias under perturbed dynamics, quantified by path-space entropy rates. The results extend coercive (reversible) concentration theory to hypocoercive dynamics and provide practical, explicit constants for rigorous error control in advanced MCMC methods used in Bayesian inference and molecular dynamics.

Abstract

In this work we provide performance guarantees for hypocoercive non-reversible MCMC samplers with invariant measure ; our results apply in particular to the Langevin equation, Hamiltonian Monte-Carlo, and the bouncy particle and zig-zag samplers. Specifically, we establish a concentration inequality of Bernstein type for ergodic averages . As a consequence we provide two types of performance guarantees: (a) explicit non-asymptotic confidence intervals for when using a finite time ergodic average with given initial condition and (b) uncertainty quantification (UQ) bounds, expressed in terms of relative entropy rate, on the bias of when using an alternative or approximate processes . (Results in (b) generalize results (arXiv:1812.05174) from the authors for coercive dynamics.) The concentration inequality is proved by combining the approach via Feynman-Kac semigroups first noted by Wu with the hypocoercive estimates of Dolbeault, Mouhot and Schmeiser (arXiv:1005.1495) developed for the Langevin equation and generalized to partially deterministic Markov processes by Andrieu et al. (arXiv:1808.08592).

Paper Structure

This paper contains 8 sections, 7 theorems, 66 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Hypocoercive MCMC samplers
  3. Concentration inequalities and UQ bounds via Feynman-Kac semigroups
  4. Background
  5. Concentration inequalities
  6. Robustness bounds on steady state bias due to model-form uncertainty
  7. Application to hypocoercive samplers
  8. Additional Proofs

Key Result

Theorem 1.1

(Non-asymptotic confidence intervals) Suppose that the Markov process $(X_t,P^\mu)$ satisfies the hypocoercive estimate eq:hypocoercive-epsilon. Then for any bounded observable $f$, any time $T>0$, and tolerance level $0< 1-\delta < 1$ we have where with $\Lambda(\epsilon)$ is defined in Eq. (Lambda_m_def) and $\epsilon\in(0,1)$ must satisfy Eq. (eq:Lambda_pos).

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Corollary 1
  • proof
  • Remark 3.3
  • Theorem 3.4
  • ...and 3 more