pith. sign in

arxiv: 2606.26086 · v1 · pith:5MJTSNG5new · submitted 2026-06-24 · 🧮 math.PR · math.AP· stat.CO

On the entropic convergence for piecewise deterministic samplers: speedup and obstruction

Pierre Monmarch\'e , Lihan Wang This is my paper

Pith reviewed 2026-06-25 19:37 UTC · model grok-4.3

classification 🧮 math.PR math.APstat.CO
keywords piecewise deterministic Markov processesrelative entropy convergencehypocoercivitydiffusive-to-ballistic speedupRHMCBouncy Particle SamplerZig-Zag Samplerlog-concave targets
0
0 comments X

The pith

RHMC achieves the diffusive-to-ballistic speedup in relative entropy while BPS and ZZS lose all exponential entropy convergence even on Gaussians.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the quadratic improvement in convergence rates known from L2 analysis extends to relative entropy for Randomized Hamiltonian Monte Carlo on log-concave targets. For the Bouncy Particle Sampler and Zig-Zag Sampler the same improvement and even exponential convergence in relative entropy fail, as shown by explicit obstruction on the standard Gaussian. A reader would care because relative entropy controls information loss and uncertainty in sampling applications where L2 bounds are insufficient. The distinction arises from how each sampler's dynamics interact with the entropy functional under the ballistic regime.

Core claim

The diffusive-to-ballistic speedup holds in relative entropy for RHMC. For BPS and ZZS, exponential convergence in relative entropy fails even for the standard Gaussian target.

What carries the argument

The diffusive-to-ballistic speedup, which quadratically improves convergence rates with suitable parameters relative to overdamped Langevin, carries the positive result for RHMC while the failure of entropy dissipation under the jump mechanisms obstructs it for BPS and ZZS.

If this is right

  • RHMC admits parameter choices that yield quadratically faster entropy convergence than overdamped Langevin.
  • BPS and ZZS require convergence analyses that do not rely on exponential entropy decay.
  • The choice among piecewise deterministic samplers affects whether entropy-based performance guarantees are available.
  • Log-concavity is not sufficient by itself to guarantee entropy speedup across all such samplers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The jump mechanisms in BPS and ZZS may prevent the entropy dissipation that continuous Hamiltonian flow preserves.
  • When entropy bounds matter for downstream tasks, RHMC may be preferable to the other two samplers.
  • Hybrid constructions that retain ballistic motion while altering jump rules could be tested for restored entropy convergence.

Load-bearing premise

The log-concavity of the target together with the standard generator and jump mechanisms of each process are required for both the speedup and the obstruction.

What would settle it

A direct computation or simulation of the relative entropy along the BPS trajectory on the standard Gaussian that exhibits exponential decay at a dimension-independent positive rate would refute the obstruction result.

read the original abstract

For piecewise deterministic samplers such as Randomized Hamiltonian Monte Carlo (RHMC), Bouncy Particle Sampler (BPS) or Zig-Zag Process (ZZP), long-time exponential convergence rates have been established in previous works using Harris or $L^2$ hypocoercivity approaches. In particular, in the $L^2$ framework, a so-called \emph{diffusive-to-ballistic} speedup was known for log-concave targets, according to which the convergence rates of these samplers, with suitable parameters, are quadratically improved with respect to the standard overdamped Langevin diffusion process. A recent work by Jianfeng Lu showed that this speedup also holds for the kinetic Langevin diffusion process when the convergence is stated in terms of relative entropy, raising the question whether this also holds for piecewise deterministic samplers. The present work provides a positive and a negative answer to this: first, we show that the speedup holds in entropy for RHMC; second, we show that for BPS or ZZS, even for a standard Gaussian target, a similar result cannot hold, and even that exponential convergence (at any rate) in entropy fails.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes a positive result that the diffusive-to-ballistic speedup (quadratic improvement over overdamped Langevin) previously known in L² for log-concave targets extends to relative entropy for Randomized Hamiltonian Monte Carlo (RHMC). It also establishes a negative result: for the Bouncy Particle Sampler (BPS) and Zig-Zag Sampler (ZZS), exponential convergence in relative entropy fails at any rate, even when the target is the standard Gaussian.

Significance. If the derivations hold, the work supplies a precise distinction among PDMP samplers regarding entropy dissipation, extending the L²/Harris hypocoercivity literature and the recent entropy result for kinetic Langevin. The combination of a positive speedup for RHMC and an explicit obstruction for BPS/ZZS is a substantive clarification for the design of entropy-based analyses of piecewise deterministic processes.

minor comments (3)
  1. [§2] The abstract and introduction reference the standard generators and jump mechanisms of RHMC, BPS, and ZZS; the manuscript should include a short self-contained paragraph (perhaps §2) recalling the precise infinitesimal generators and the parameter choices that realize the diffusive-to-ballistic regime.
  2. [negative-result section] In the negative result for the Gaussian target, the entropy functional and the precise notion of “exponential convergence at any rate” should be stated explicitly before the obstruction argument (e.g., whether it is the relative entropy H(μ_t | π) or a weighted variant).
  3. A few typographical inconsistencies appear in the notation for the target density and the jump rates; these do not affect the logic but should be unified.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results on RHMC entropy speedup versus the BPS/ZZS obstruction, and recommendation of minor revision. No major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central claims establish a diffusive-to-ballistic speedup in relative entropy for RHMC under log-concave targets (extending prior L2/Harris hypocoercivity results) and demonstrate failure of exponential entropy convergence for BPS/ZZS even on the standard Gaussian. The stated assumptions (standard generators, jump mechanisms, and log-concavity) align with referenced external literature without reducing any prediction or rate to a fitted parameter, self-defined quantity, or self-citation chain. No load-bearing self-citations, ansatzes smuggled via prior work, or self-definitional steps are present; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on the log-concavity assumption inherited from the L2 speedup literature and on the standard definitions of the three piecewise deterministic processes; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Target distribution is log-concave
    Referenced when stating the diffusive-to-ballistic speedup context from previous works.

pith-pipeline@v0.9.1-grok · 5736 in / 1303 out tokens · 41679 ms · 2026-06-25T19:37:38.678149+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 3 linked inside Pith

  1. [1]

    Variational methods for the kinetic Fokker–Planck equation.Analysis & PDE, 17(6):1953–2010, 2024

    Dallas Albritton, Scott Armstrong, Jean-Christophe Mourrat, and Matthew Novack. Variational methods for the kinetic Fokker–Planck equation.Analysis & PDE, 17(6):1953–2010, 2024

    1953

  2. [2]

    Shifted composition IV: toward ballistic acceleration for log-concave sampling.arXiv preprint arXiv:2506.23062, 2025

    Jason M Altschuler, Sinho Chewi, and Matthew S Zhang. Shifted composition IV: toward ballistic acceleration for log-concave sampling.arXiv preprint arXiv:2506.23062, 2025

    arXiv 2025

  3. [3]

    Hypocoercivity of piecewise determin- istic Markov process-Monte Carlo.The Annals of Applied Probability, 31(5):2478–2517, 2021

    Christophe Andrieu, Alain Durmus, Nikolas Nüsken, and Julien Roussel. Hypocoercivity of piecewise determin- istic Markov process-Monte Carlo.The Annals of Applied Probability, 31(5):2478–2517, 2021

    2021

  4. [4]

    ENTROPIC CONVERGENCE FOR PDMP 13

    JorisBierkens, PaulFearnhead, andGarethRoberts.Thezig-zagprocessandsuper-efficientsamplingforBayesian analysis of big data.The Annals of Statistics, 47(3):1288–1320, 2019. ENTROPIC CONVERGENCE FOR PDMP 13

    2019

  5. [5]

    A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model.The Annals of Applied Probability, 27(2):846 – 882, 2017

    Joris Bierkens and Gareth Roberts. A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model.The Annals of Applied Probability, 27(2):846 – 882, 2017

    2017

  6. [6]

    Ergodicity of the zigzag process.The Annals of Applied Probability, 29(4):2266–2301, 2019

    Joris Bierkens, Gareth O Roberts, and Pierre-André Zitt. Ergodicity of the zigzag process.The Annals of Applied Probability, 29(4):2266–2301, 2019

    2019

  7. [7]

    Tail-Sensitive KL and Rényi Convergence of Unad- justed Hamiltonian Monte Carlo via One-Shot Couplings.arXiv e-prints, page arXiv:2601.09019, January 2026

    Nawaf Bou-Rabee, Siddharth Mitra, and Andre Wibisono. Tail-Sensitive KL and Rényi Convergence of Unad- justed Hamiltonian Monte Carlo via One-Shot Couplings.arXiv e-prints, page arXiv:2601.09019, January 2026

    arXiv 2026

  8. [8]

    Randomized Hamiltonian Monte Carlo.The Annals of Applied Probability, 27(4):2159–2194, 2017

    Nawaf Bou-Rabee and Jesus Maria Sanz-Serna. Randomized Hamiltonian Monte Carlo.The Annals of Applied Probability, 27(4):2159–2194, 2017

    2017

  9. [9]

    The bouncy particle sampler: A non- reversible rejection-free Markov chain Monte Carlo method.Journal of the American Statistical Association, 113(522):855–867, 2018

    Alexandre Bouchard-Côté, Sebastian J Vollmer, and Arnaud Doucet. The bouncy particle sampler: A non- reversible rejection-free Markov chain Monte Carlo method.Journal of the American Statistical Association, 113(522):855–867, 2018

    2018

  10. [10]

    Hypocoercivity meets lifts.Kinetic and Related Models, 20:34– 55, 2025

    Giovanni Brigati, Francis Lörler, and Lihan Wang. Hypocoercivity meets lifts.Kinetic and Related Models, 20:34– 55, 2025

    2025

  11. [11]

    Second order quantitative bounds for unadjusted generalized Hamiltonian Monte Carlo.arXiv e-prints, page arXiv:2306.09513, June 2023

    Evan Camrud, Alain Oliviero Durmus, Pierre Monmarché, and Gabriel Stoltz. Second order quantitative bounds for unadjusted generalized Hamiltonian Monte Carlo.arXiv e-prints, page arXiv:2306.09513, June 2023

    arXiv 2023

  12. [12]

    On explicitL2-convergence rate estimate for underdamped Langevin dynamics.Archive for Rational Mechanics and Analysis, 247(5):90, 2023

    Yu Cao, Jianfeng Lu, and Lihan Wang. On explicitL2-convergence rate estimate for underdamped Langevin dynamics.Archive for Rational Mechanics and Analysis, 247(5):90, 2023

    2023

  13. [13]

    On cutoff via rigidity for high dimensional curved diffusions.arXiv preprint arXiv:2412.15969, 2024

    Djalil Chafaï and Max Fathi. On cutoff via rigidity for high dimensional curved diffusions.arXiv preprint arXiv:2412.15969, 2024

    arXiv 2024

  14. [14]

    Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics.Electronic Journal of Probability, 29(none):1 – 43, 2024

    Fan Chen, Yiqing Lin, Zhenjie Ren, and Songbo Wang. Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics.Electronic Journal of Probability, 29(none):1 – 43, 2024

    2024

  15. [15]

    Lifting Markov chains to speed up mixing

    Fang Chen, László Lovász, and Igor Pak. Lifting Markov chains to speed up mixing. InProceedings of the thirty- first annual ACM symposium on Theory of computing, pages 275–281, 1999

    1999

  16. [16]

    Randomized Hamiltonian Monte Carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates.The Annals of Applied Probability, 31(6):2612 – 2662, 2021

    George Deligiannidis, Daniel Paulin, Alexandre Bouchard-Côté, and Arnaud Doucet. Randomized Hamiltonian Monte Carlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates.The Annals of Applied Probability, 31(6):2612 – 2662, 2021

    2021

  17. [17]

    Analysis of a nonreversible Markov chain sampler.Annals of Applied Probability, pages 726–752, 2000

    Persi Diaconis, Susan Holmes, and Radford M Neal. Analysis of a nonreversible Markov chain sampler.Annals of Applied Probability, pages 726–752, 2000

    2000

  18. [18]

    JeanDolbeault, Clément Mouhot, andChristian Schmeiser.Hypocoercivity forlinear kineticequationsconserving mass.Transactions of the American Mathematical Society, 367(6):3807–3828, 2015

    2015

  19. [19]

    Geometric ergodicity of the bouncy particle sampler

    Alain Durmus, Arnaud Guillin, and Pierre Monmarché. Geometric ergodicity of the bouncy particle sampler. The Annals of Applied Probability, 30(5):2069–2098, 2020

    2069

  20. [20]

    Piecewise deterministic Markov processes and their invariant measures

    Alain Durmus, Arnaud Guillin, and Pierre Monmarché. Piecewise deterministic Markov processes and their invariant measures. InAnnales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, volume 57, pages 1442–1475. Institut Henri Poincaré, 2021

    2021

  21. [21]

    Couplings and quantitative contraction rates for Langevin dynamics.The Annals of Probability, 47(4):1982–2010, 2019

    Andreas Eberle, Arnaud Guillin, and Raphael Zimmer. Couplings and quantitative contraction rates for Langevin dynamics.The Annals of Probability, 47(4):1982–2010, 2019

    1982

  22. [22]

    Non-reversible lifts of reversible diffusion processes and relaxation times

    Andreas Eberle and Francis Lörler. Non-reversible lifts of reversible diffusion processes and relaxation times. Probability Theory and Related Fields, 194:173–203, 2026

    2026

  23. [23]

    Sharp hypocoercive convergence estimates for underdamped Langevin dynamics via the modifiedL2 method.arXiv preprint arXiv:2604.10068, 2026

    Zexi Fan, Bowen Li, and Jianfeng Lu. Sharp hypocoercive convergence estimates for underdamped Langevin dynamics via the modifiedL2 method.arXiv preprint arXiv:2604.10068, 2026

    Pith/arXiv arXiv 2026

  24. [24]

    Long time behavior of telegraph processes under convex potentials.Stochastic Processes and their Applications, 126(10):3077–3101, 2016

    Joaquin Fontbona, Hélène Guérin, and Florent Malrieu. Long time behavior of telegraph processes under convex potentials.Stochastic Processes and their Applications, 126(10):3077–3101, 2016

    2016

  25. [25]

    Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation.Asymptotic Analysis, 46(3-4):349–359, 2006

    Frédéric Hérau. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation.Asymptotic Analysis, 46(3-4):349–359, 2006

    2006

  26. [26]

    Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential.Archive for Rational Mechanics and Analysis, 171(2):151–218, 2004

    Frédéric Hérau and Francis Nier. Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential.Archive for Rational Mechanics and Analysis, 171(2):151–218, 2004

    2004

  27. [27]

    Partial differential equations and stochastic methods in molecular dynamics

    Tony Lelievre and Gabriel Stoltz. Partial differential equations and stochastic methods in molecular dynamics. Acta Numerica, 25:681–880, 2016

    2016

  28. [28]

    A sharp hypocoercive entropy decay estimate for underdamped Langevin dynamics.arXiv preprint arXiv:2605.01933, 2026

    Jianfeng Lu. A sharp hypocoercive entropy decay estimate for underdamped Langevin dynamics.arXiv preprint arXiv:2605.01933, 2026

    Pith/arXiv arXiv 2026

  29. [29]

    On explicitL 2-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms.The Annals of Applied Probability, 32(2):1333–1361, 2022

    Jianfeng Lu and Lihan Wang. On explicitL 2-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms.The Annals of Applied Probability, 32(2):1333–1361, 2022

    2022

  30. [30]

    Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L

    Yi-An Ma, Niladri S. Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L. Bartlett, and Michael I. Jordan. Is there an analog of Nesterov acceleration for gradient-based MCMC?Bernoulli, 27(3):1942 – 1992, 2021

    1942

  31. [31]

    Mattingly, Andrew M

    Jonathan C. Mattingly, Andrew M. Stuart, and Desmond J. Higham. Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise.Stochastic Processes and their Applications, 101(2):185–232, October 2002. 14 PIERRE MONMARCHÉ AND LIHAN W ANG

    2002

  32. [32]

    Forward event-chain Monte Carlo: Fast sampling by randomness control in irreversible Markov chains.Journal of Computational and Graphical Statistics, 29(4):689– 702, 2020

    Manon Michel, Alain Durmus, and Stéphane Sénécal. Forward event-chain Monte Carlo: Fast sampling by randomness control in irreversible Markov chains.Journal of Computational and Graphical Statistics, 29(4):689– 702, 2020

    2020

  33. [33]

    Piecewise deterministic simulated annealing.ALEA Lat

    Pierre Monmarché. Piecewise deterministic simulated annealing.ALEA Lat. Am. J. Probab. Math. Stat., 13(1):357–398, 2016

    2016

  34. [34]

    Long-time behaviour and propagation of chaos for mean field kinetic particles.Stochastic Process

    Pierre Monmarché. Long-time behaviour and propagation of chaos for mean field kinetic particles.Stochastic Process. Appl., 127(6):1721–1737, 2017

    2017

  35. [35]

    Nesterov acceleration for the Wasserstein minimization of displacement-convex free energies

    Pierre Monmarché. Nesterov acceleration for the Wasserstein minimization of displacement-convex free energies. arXiv e-prints, page arXiv:2605.13186, May 2026

    Pith/arXiv arXiv 2026

  36. [36]

    Exact targeting of Gibbs distributions using velocity- jump processes.Stochastics and Partial Differential Equations: Analysis and Computations, pages 1–40, 2022

    Pierre Monmarché, Mathias Rousset, and Pierre-André Zitt. Exact targeting of Gibbs distributions using velocity- jump processes.Stochastics and Partial Differential Equations: Analysis and Computations, pages 1–40, 2022

    2022

  37. [37]

    Generalizationof an inequality byTalagrand and links withthe logarithmic Sobolev inequality.Journal of Functional Analysis, 173(2):361–400, 2000

    Felix Ottoand Cédric Villani. Generalizationof an inequality byTalagrand and links withthe logarithmic Sobolev inequality.Journal of Functional Analysis, 173(2):361–400, 2000

    2000

  38. [38]

    E. A. J. F. Peters and G. de With. Rejection-free Monte Carlo sampling for general potentials.Phys. Rev. E 85, 026703, 2012

    2012

  39. [39]

    Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretiza- tion by the implicit Euler scheme.Markov Process

    Denis Talay. Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretiza- tion by the implicit Euler scheme.Markov Process. Related Fields, 8(2):163–198, 2002

    2002

  40. [40]

    Irreversible Monte Carlo algorithms for efficient sampling.Physica D: Nonlinear Phenomena, 240(4-5):410–414, 2011

    Konstantin S Turitsyn, Michael Chertkov, and Marija Vucelja. Irreversible Monte Carlo algorithms for efficient sampling.Physica D: Nonlinear Phenomena, 240(4-5):410–414, 2011

    2011

  41. [41]

    Rapid convergence of the unadjusted langevin algorithm: Isoperimetry suffices.Advances in neural information processing systems, 32, 2019

    Santosh Vempala and Andre Wibisono. Rapid convergence of the unadjusted langevin algorithm: Isoperimetry suffices.Advances in neural information processing systems, 32, 2019

    2019

  42. [42]

    Hypocoercivity.Memoirs of the American Mathematical Society, 202(950), 2009

    Cédric Villani. Hypocoercivity.Memoirs of the American Mathematical Society, 202(950), 2009

    2009

  43. [43]

    Lifting-a nonreversible Markov chain Monte Carlo algorithm.American Journal of Physics, 84(12):958–968, 2016

    Marija Vucelja. Lifting-a nonreversible Markov chain Monte Carlo algorithm.American Journal of Physics, 84(12):958–968, 2016

    2016

  44. [44]

    Zhang, Jason M

    Matthew S. Zhang, Jason M. Altschuler, and Sinho Chewi. Algorithmic warm starts for Hamiltonian Monte Carlo.arXiv preprint 2603.22741, 2026. LAMA, Université Gusta ve Eiffel, 77420 Champs-sur-Marne, France. Email address:pierre.monmarche@univ-eiffel.fr Department of Mathematics, National University of Singapore, 119076 Singapore Email address:lihanw@nus.edu.sg

    arXiv 2026