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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript t"> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">X_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with invariant measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi> μ </mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">\mu _*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over upper T EndFraction integral Subscript 0 Superscript upper T Baseline f left-parenthesis upper X Subscript t Baseline right-parenthesis d t"> <mml:semantics> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>T</mml:mi> </mml:mfrac> <mml:msubsup> <mml:mo> ∫ </mml:mo> <mml:mn>0</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\frac {1}{T} \int _0^T f(X_t)\, dt</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As a consequence we provide two types of performance guarantees: (a) explicit nonasymptotic confidence intervals for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral f d mu Subscript asterisk"> <mml:semantics> <mml:mrow> <mml:mo> ∫ </mml:mo> <mml:mi>f</mml:mi> <mml:mi>d</mml:mi> <mml:msub> <mml:mi> μ </mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\int f d\mu _*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when using a finite time ergodic average with given initial condition <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and (b) uncertainty quantification (UQ) bounds, expressed in terms of relative entropy rate, on the bias of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral f d mu Subscript asterisk"> <mml:semantics> <mml:mrow> <mml:mo> ∫ </mml:mo> <mml:mi>f</mml:mi> <mml:mi>d</mml:mi> <mml:msub> <mml:mi> μ </mml:mi> <mml:mo> ∗ </mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\int f d\mu _*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when using an alternative or approximate processes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X overTilde Subscript t"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>X</mml:mi> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:mi>t</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\widetilde {X}_t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . (Results in (b) generalize results of the authors [ <italic>Uncertainty quantification for Markov processes via variational principles and functional inequalities</italic> , 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 [ <italic>A deviation inequality for non-reversible Markov processes</italic> , Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 435–445] with the hypocoercive estimates of Dolbeault, Mouhot, and Schmeiser [ <italic>Hypocoercivity for kinetic equations with linear relaxation terms</italic> , C. R. Math. Acad. Sci. Paris 347 (2009), no. 9–10, 511–516] and [ <italic>Hypocoercivity for linear kinetic equations conserving mass</italic> , 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 [ <italic>Hypocoercivity of piecewise deterministic Markov process-Monte Carlo</italic> , Ann. Appl. Probab. 31 (2021), no. 5, 2478–2517].