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 $\mu_*$; 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 $\int f d\mu_*$ 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 $\int f d\mu_*$ 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).