Concentration Inequalities and UQ Bounds for Hypocoercive MCMC Samplers

Abstract

In this work we provide performance guarantees for hypocoercive non-reversible MCMC samplers $X_t$ with invariant measure ${\mu }_{\ast }$; 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 $\int fd{\mu }_{\ast }$ when using a finite time ergodic average with given initial condition $\mu$ and (b) uncertainty quantification (UQ) bounds, expressed in terms of relative entropy rate, on the bias of $\int fd{\mu }_{\ast }$ when using an alternative or approximate processes ${\tilde{X}}_t$. (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).

16 pages

References in corpus (9)

• Rejection-free Monte-Carlo sampling for general potentials
• Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics
• Variance Reduction using Nonreversible Langevin Samplers
• Randomized Hamiltonian Monte Carlo
• Ergodicity of the zigzag process
• Improving the convergence of reversible samplers
• Scalable Information Inequalities for Uncertainty Quantification
• How biased is your model? Concentration Inequalities, Information and Model Bias
• Generalized Bouncy Particle Sampler

Cited by in corpus (1)

• Large Deviations of Irreversible Processes