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 Xₜ 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 1 T ∫ 0 T f (X t) d t 1T ₀T f (Xₜ) \, dt. As a consequence we provide two types of performance guarantees: (a) explicit nonasymptotic confidence intervals for ∫ f d μ ∗ f d _* 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 ∫ f d μ ∗ f d _* when using an alternative or approximate processes X ~ t Xₜ. (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.