Hypocoercivity of piecewise deterministic Markov process-Monte Carlo. (English) Zbl 1476.60124

Andrieu, Christophe; Durmus, Alain; Nüsken, Nikolas; Roussel, Julien

Ann. Appl. Probab. 31, No. 5, 2478-2517 (2021).

Summary: In this work, we establish \(\mathrm{L}^2\)-exponential convergence for a broad class of piecewise deterministic Markov processes recently proposed in the context of Markov process Monte Carlo methods and covering in particular the randomized Hamiltonian Monte Carlo [J. Dolbeault et al., Trans. Am. Math. Soc. 367, No. 6, 3807–3828 (2015; Zbl 1342.82115); N. Bou-Rabee and J. M. Sanz-Serna, Ann. Appl. Probab. 27, No. 4, 2159–2194 (2017; Zbl 1373.60129)], the zig-zag process [J. Bierkens et al., Ann. Stat. 47, No. 3, 1288–1320 (2019; Zbl 1417.65008)] and the bouncy particle Sampler [E. A. J. F. Peters and G. de With, “Rejection-free Monte Carlo sampling for general potentials”, Phys. Rev. E 85, Article ID 026703, 5 p. (2012; doi:10.1080/00031305.2017.1419145); A. Bouchard-Côté et al., J. Am. Stat. Assoc. 113, No. 522, 855–867 (2018; Zbl 1398.60084)]. The kernel of the symmetric part of the generator of such processes is nontrivial, and we follow the ideas recently introduced in [J. Dolbeault et al., C. R., Math., Acad. Sci. Paris 347, No. 9–10, 511–516 (2009; Zbl 1177.35054); Dolbeault 2015, loc. cit.] to develop a rigorous framework for hypocoercivity in a fairly general and unifying set-up, while deriving tractable estimates of the constants involved in terms of the parameters of the dynamics. As a by-product we characterize the scaling properties of these algorithms with respect to the dimension of classes of problems, therefore providing some theoretical evidence to support their practical relevance.

MSC:

60J22	Computational methods in Markov chains
60J25	Continuous-time Markov processes on general state spaces
65C40	Numerical analysis or methods applied to Markov chains

Keywords:

geometric convergence; hypoellipticity; PDMCMC

Citations:

Zbl 1342.82115; Zbl 1373.60129; Zbl 1417.65008; Zbl 1398.60084; Zbl 1177.35054

Software:

BayesDA

References:

[1]	Achleitner, F., Arnold, A. and Carlen, E. A. (2016). On linear hypocoercive BGK models. In From Particle Systems to Partial Differential Equations. III. Springer Proc. Math. Stat. 162 1-37. Springer, Cham. · Zbl 1353.35113 · doi:10.1007/978-3-319-32144-8_1
[2]	Andrieu, C. and Livingstone, S. (2021). Peskun-Tierney ordering for Markovian Monte Carlo: Beyond the reversible scenario. Ann. Statist. 49 1958-1981. · Zbl 1489.65005 · doi:10.1214/20-AOS2008
[3]	Andrieu, C., Durmus, A., Nüsken, N. and Roussel, J. (2021). Supplement to “Hypocoercivity of piecewise deterministic Markov process-Monte Carlo.” https://doi.org/10.1214/20-AAP1653SUPP
[4]	Bakry, D., Barthe, F., Cattiaux, P. and Guillin, A. (2008). A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case. Electron. Commun. Probab. 13 60-66. · Zbl 1186.26011 · doi:10.1214/ECP.v13-1352
[5]	Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Springer, Cham. · Zbl 1376.60002 · doi:10.1007/978-3-319-00227-9
[6]	Bhatnagar, P. L., Gross, E. P. and Krook, M. (1954). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94 511-525. · Zbl 0055.23609
[7]	Bierkens, J., Fearnhead, P. and Roberts, G. (2019). The zig-zag process and super-efficient sampling for Bayesian analysis of big data. Ann. Statist. 47 1288-1320. · Zbl 1417.65008 · doi:10.1214/18-AOS1715
[8]	Bierkens, J., Kamatani, K. and Roberts, G. O. (2018). High-dimensional scaling limits of piecewise deterministic sampling algorithms. ArXiv e-prints.
[9]	Bierkens, J., Roberts, G. O. and Zitt, P.-A. (2019). Ergodicity of the zigzag process. Ann. Appl. Probab. 29 2266-2301. · Zbl 1467.60020 · doi:10.1214/18-AAP1453
[10]	Bobkov, S. G. (2003). Spectral gap and concentration for some spherically symmetric probability measures. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1807 37-43. Springer, Berlin. · Zbl 1052.60003 · doi:10.1007/978-3-540-36428-3_4
[11]	Bonnefont, M., Joulin, A. and Ma, Y. (2016). Spectral gap for spherically symmetric log-concave probability measures, and beyond. J. Funct. Anal. 270 2456-2482. · Zbl 1347.60114 · doi:10.1016/j.jfa.2016.02.007
[12]	Bou-Rabee, N. and Sanz-Serna, J. M. (2017). Randomized Hamiltonian Monte Carlo. Ann. Appl. Probab. 27 2159-2194. · Zbl 1373.60129 · doi:10.1214/16-AAP1255
[13]	Bouchard-Côté, A., Vollmer, S. J. and Doucet, A. (2018). The bouncy particle sampler: A nonreversible rejection-free Markov chain Monte Carlo method. J. Amer. Statist. Assoc. 113 855-867. · Zbl 1398.60084 · doi:10.1080/01621459.2017.1294075
[14]	Bouin, E., Dolbeault, J., Mischler, S., Mouhot, C. and Schmeiser, C. (2020). Hypocoercivity without confinement. Pure Appl. Funct. Anal. 2 203-232. · Zbl 1448.82035 · doi:10.2140/paa.2020.2.203
[15]	Bouin, E., Hoffmann, F. and Mouhot, C. (2017). Exponential decay to equilibrium for a fiber lay-down process on a moving conveyor belt. SIAM J. Math. Anal. 49 3233-3251. · Zbl 1379.35318 · doi:10.1137/16M1077490
[16]	Brosse, N., Durmus, A., Moulines, É. and Sabanis, S. (2019). The tamed unadjusted Langevin algorithm. Stochastic Process. Appl. 129 3638-3663. · Zbl 07107458 · doi:10.1016/j.spa.2018.10.002
[17]	Davies, E. B. (1995). Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics 42. Cambridge Univ. Press, Cambridge. · Zbl 0893.47004 · doi:10.1017/CBO9780511623721
[18]	Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models (with discussion). J. Roy. Statist. Soc. Ser. B 46 353-388. · Zbl 0565.60070
[19]	Davis, M. H. A. (1993). Markov Models and Optimization. Monographs on Statistics and Applied Probability 49. CRC Press, London. · Zbl 0780.60002 · doi:10.1007/978-1-4899-4483-2
[20]	Deligiannidis, G., Bouchard-Côté, A. and Doucet, A. (2019). Exponential ergodicity of the bouncy particle sampler. Ann. Statist. 47 1268-1287. · Zbl 1467.60057 · doi:10.1214/18-AOS1714
[21]	Deligiannidis, G., Paulin, D. and Doucet, A. (2018). Randomized Hamiltonian Monte Carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates. Preprint. Available at arXiv:1808.04299.
[22]	Dolbeault, J., Mouhot, C. and Schmeiser, C. (2009). Hypocoercivity for kinetic equations with linear relaxation terms. C. R. Math. Acad. Sci. Paris 347 511-516. · Zbl 1177.35054 · doi:10.1016/j.crma.2009.02.025
[23]	Dolbeault, J., Mouhot, C. and Schmeiser, C. (2015). Hypocoercivity for linear kinetic equations conserving mass. Trans. Amer. Math. Soc. 367 3807-3828. · Zbl 1342.82115 · doi:10.1090/S0002-9947-2015-06012-7
[24]	Douc, R., Moulines, E., Priouret, P. and Soulier, P. (2018). Markov Chains. Springer Series in Operations Research and Financial Engineering. Springer, Cham. · Zbl 1429.60002 · doi:10.1007/978-3-319-97704-1
[25]	Duane, S., Kennedy, A. D., Pendleton, B. J. and Roweth, D. (1987). Hybrid Monte Carlo. Phys. Lett. B 195 216-222. · doi:10.1016/0370-2693(87)91197-x
[26]	Durmus, A., Guillin, A. and Monmarché, P. (2018). Piecewise deterministic Markov processes and their invariant measure. ArXiv e-prints.
[27]	Durmus, A., Guillin, A. and Monmarché, P. (2020). Geometric ergodicity of the bouncy particle sampler. Ann. Appl. Probab. 30 2069-2098. · Zbl 1472.60118 · doi:10.1214/19-AAP1552
[28]	Eckmann, J.-P. and Hairer, M. (2003). Spectral properties of hypoelliptic operators. Comm. Math. Phys. 235 233-253. · Zbl 1040.35016 · doi:10.1007/s00220-003-0805-9
[29]	Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. · Zbl 0592.60049 · doi:10.1002/9780470316658
[30]	Evans, J. (2021). Hypocoercivity in phi-entropy for the linear relaxation Boltzmann equation on the torus. SIAM J. Math. Anal. 53 1357-1378. · Zbl 1462.35234 · doi:10.1137/19M1277631
[31]	Faggionato, A., Gabrielli, D. and Ribezzi Crivellari, M. (2009). Non-equilibrium thermodynamics of piecewise deterministic Markov processes. J. Stat. Phys. 137 259-304. · Zbl 1179.82108 · doi:10.1007/s10955-009-9850-x
[32]	Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014). Bayesian Data Analysis, 3rd ed. Texts in Statistical Science Series. CRC Press, Boca Raton, FL. · Zbl 1279.62004
[33]	Grothaus, M. and Mertin, M. (2020). Hypocoercivity of langevin-type dynamics on abstract smooth manifolds.
[34]	Grothaus, M. and Stilgenbauer, P. (2014). Hypocoercivity for Kolmogorov backward evolution equations and applications. J. Funct. Anal. 267 3515-3556. · Zbl 1347.37007 · doi:10.1016/j.jfa.2014.08.019
[35]	Grothaus, M. and Stilgenbauer, P. (2015). A hypocoercivity related ergodicity method for singularly distorted non-symmetric diffusions. Integral Equations Operator Theory 83 331-379. · Zbl 1361.37007 · doi:10.1007/s00020-015-2254-1
[36]	Grothaus, M. and Stilgenbauer, P. (2016). Hilbert space hypocoercivity for the Langevin dynamics revisited. Methods Funct. Anal. Topology 22 152-168. · Zbl 1363.47089
[37]	Grothaus, M. and Wang, F.-Y. (2019). Weak Poincaré inequalities for convergence rate of degenerate diffusion processes. Ann. Probab. 47 2930-2952. · Zbl 1472.60096 · doi:10.1214/18-AOP1328
[38]	Han-Kwan, D. and Léautaud, M. (2015). Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium. Ann. PDE 1 Art. 3, 84. · Zbl 1398.35142 · doi:10.1007/s40818-015-0003-z
[39]	Hérau, F. (2006). Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal. 46 349-359. · Zbl 1096.35019
[40]	Hérau, F. and Nier, F. (2004). Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171 151-218. · Zbl 1139.82323 · doi:10.1007/s00205-003-0276-3
[41]	Holley, R. and Stroock, D. (1987). Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys. 46 1159-1194. · Zbl 0682.60109 · doi:10.1007/BF01011161
[42]	Hörmander, L. (1967). Hypoelliptic second order differential equations. Acta Math. 119 147-171. · Zbl 0156.10701 · doi:10.1007/BF02392081
[43]	Liu, J. S. (2008). Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. Springer, New York. · Zbl 1132.65003
[44]	Meyn, S. and Tweedie, R. L. (2012). Markov Chains and Stochastic Stability. Springer, New York.
[45]	Michel, M., Kapfer, S. C. and Krauth, W. (2014). Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps. J. Chem. Phys. 140 054116.
[46]	Michel, M. and Sénécal, S. (2017). Forward event-chain Monte Carlo: A general rejection-free and irreversible Markov chain simulation method. Preprint. Available at arXiv:1702.08397.
[47]	Monmarché, P. (2021). A note on Fisher information hypocoercive decay for the linear Boltzmann equation. Anal. Math. Phys. 11 Paper No. 1, 11. · Zbl 1470.35251 · doi:10.1007/s13324-020-00437-5
[48]	Mouhot, C. and Neumann, L. (2006). Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity 19 969-998. · Zbl 1169.82306 · doi:10.1088/0951-7715/19/4/011
[49]	O’Donnell, R. (2014). Analysis of Boolean Functions. Cambridge Univ. Press, New York. · Zbl 1336.94096 · doi:10.1017/CBO9781139814782
[50]	Pardoux, E. and Veretennikov, A. Y. (2001). On the Poisson equation and diffusion approximation. I. Ann. Probab. 29 1061-1085. · Zbl 1029.60053 · doi:10.1214/aop/1015345596
[51]	Pedersen, G. K. (1989). Analysis Now. Graduate Texts in Mathematics 118. Springer, New York. · Zbl 0668.46002 · doi:10.1007/978-1-4612-1007-8
[52]	Persson, A. (1960). Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scand. 8 143-153. · Zbl 0145.14901 · doi:10.7146/math.scand.a-10602
[53]	Peters, E. A. J. F. and de With, G. (2012). Rejection-free Monte Carlo sampling for general potentials. Phys. Rev. E 85 026703.
[54]	Redon, S., Stoltz, G. and Trstanova, Z. (2016). Error analysis of modified Langevin dynamics. J. Stat. Phys. 164 735-771. · Zbl 1348.82066 · doi:10.1007/s10955-016-1544-6
[55]	Reed, M. and Simon, B. (1972). Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York. · Zbl 0242.46001
[56]	Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer Texts in Statistics. Springer, New York. · Zbl 0935.62005 · doi:10.1007/978-1-4757-3071-5
[57]	Vanetti, P., Bouchard-Côté, A., Deligiannidis, G. and Doucet, A. (2017). Piecewise deterministic Markov chain Monte Carlo. Preprint. Available at arXiv:1707.05296.
[58]	Villani, C. (2006). Hypocoercive diffusion operators. In International Congress of Mathematicians. Vol. III 473-498. Eur. Math. Soc., Zürich. · Zbl 1130.35027
[59]	Villani, C. (2009). Hypocoercivity. Mem. Amer. Math. Soc. 202 iv+141. · Zbl 1197.35004 · doi:10.1090/S0065-9266-09-00567-5
[60]	Wu, C. and Robert, C. P. (2017). Generalized bouncy particle sampler. Preprint. Available at arXiv:1706.04781.
[61]	Yoshida, K. (1980). Functional Analysis. Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen Mit Besonderer Berücksichtigung der Anwendungsgebiete 123, 6th ed. Springer, Berlin