Concentration inequalities and UQ bounds for hypocoercive MCMC samplers

In this work we provide performance guarantees for hypocoercive nonreversible Monte-Carlo Markov chain (MCMC) samplers X t X_t with invariant measure μ ∗ 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 1 T ∫ 0 T f ( X t ) d t 1/T ∫ _0^T f(X_t)\, dt . As a consequence we provide two types of performance guarantees: (a) explicit nonasymptotic confidence intervals for ∫ f d μ ∗ ∫ f dμ _* when using a finite time ergodic average with given initial condition μ μ and (b) uncertainty quantification (UQ) bounds, expressed in terms of relative entropy rate, on the bias of ∫ f d μ ∗ ∫ f dμ _* when using an alternative or approximate processes X ~ t {X}_t . (Results in (b) generalize results of the authors [ Uncertainty quantification for Markov processes via variational principles and functional inequalities , SIAM/ASA J. Uncertain. Quantif. 8 (2020), no. 2, 539–572] for coercive dynamics.) The concentration inequality is proved by combining the approach via Feynman-Kac semigroups first noted by Wu [ A deviation inequality for non-reversible Markov processes , Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 435–445] with the hypocoercive estimates of Dolbeault, Mouhot, and Schmeiser [ Hypocoercivity for kinetic equations with linear relaxation terms , C. R. Math. Acad. Sci. Paris 347 (2009), no. 9–10, 511–516] and [ Hypocoercivity for linear kinetic equations conserving mass , Trans. Amer. Math. Soc. 367 (2015), no. 6, 3807–3828] developed for the Langevin equation and generalized to partially deterministic Markov processes by Andrieu, Durmus, Nüsken, and Roussel [ Hypocoercivity of piecewise deterministic Markov process-Monte Carlo , Ann. Appl. Probab. 31 (2021), no. 5, 2478–2517].