pith. sign in

arxiv: 2605.24070 · v1 · pith:SZHGBPYCnew · submitted 2026-05-22 · 📊 stat.CO · cs.NA· math.NA· math.PR

Convergence and non-asymptotic error analysis for kinetic Langevin samplers using the exact harmonic Langevin integrator

Katharina Schuh This is my paper

Pith reviewed 2026-06-30 14:59 UTC · model grok-4.3

classification 📊 stat.CO cs.NAmath.NAmath.PR
keywords kinetic Langevinsplitting integratorWasserstein convergencenon-asymptotic boundsstrongly log-concaveharmonic oscillator
0
0 comments X

The pith

A splitting scheme for kinetic Langevin sampling matches the contraction rate of the continuous dynamics.

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

This paper proposes a kinetic Langevin sampler that uses a splitting scheme based on the exact harmonic Langevin integrator. For strongly log-concave targets, the potential is split into a quadratic term and a convex perturbation with Lipschitz gradient. The first- and second-order versions of this scheme are shown to converge in L2-Wasserstein distance with rates of the same order as the continuous process. Non-asymptotic error bounds are derived, and the second-order scheme needs step sizes similar to those of standard methods like OBABO and UBU to reach a given accuracy.

Core claim

For strongly log-concave target measures, the first- and second-order splitting schemes associated with the exact harmonic Langevin integrator establish convergence rates in L²-Wasserstein distance as well as non-asymptotic error bounds, with the contraction rate being of the same order as that of the underlying continuous dynamics.

What carries the argument

The splitting scheme that incorporates the exact harmonic Langevin integrator and exploits the quadratic-convex decomposition of the strongly convex potential.

If this is right

  • The required step size for the second-order scheme to achieve ε-accuracy is comparable to that of OBABO or UBU.
  • The contraction rate matches the order of the continuous kinetic Langevin dynamics.
  • The approach applies to sampling in machine learning and molecular dynamics contexts.
  • Explicit non-asymptotic error bounds are available for the discrete schemes.

Where Pith is reading between the lines

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

  • If the continuous contraction rate is known for a given potential, this method provides a discrete sampler with matching performance without additional degradation.
  • The technique might extend to other Langevin-type dynamics or integrators beyond the harmonic case.
  • In high-dimensional settings where the Lipschitz constant of the perturbation is controlled, the method could offer practical efficiency gains over generic integrators.

Load-bearing premise

The target measures must be strongly log-concave to allow decomposition of the potential into a quadratic part plus a convex perturbation whose gradient is Lipschitz continuous.

What would settle it

A counterexample where the L2-Wasserstein contraction rate of the discrete scheme is strictly slower than that of the continuous dynamics by more than a multiplicative constant independent of dimension would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.24070 by Katharina Schuh.

Figure 1
Figure 1. Figure 1: Potential with oscillations: Blue lines on the plot are the contraction of the av [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Logistic type potential: Blue lines on the plot are the contraction of the averaged dis [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
read the original abstract

We propose a novel kinetic Langevin sampler based on a specific splitting scheme using the exact harmonic Langevin integrator. For strongly log-concave target measures, the sampler exploits a decomposition of the strongly convex potential into a quadratic part and a convex perturbation with Lipschitz continuous gradient. For the resulting first- and second-order schemes associated with this splitting we establish convergence rates in $L^2$-Wasserstein distance as well as non-asymptotic error bounds. In particular, the contraction rate is of the same order as that of the underlying continuous dynamics. To achieve $\varepsilon$-accuracy, the required step size for the second-order scheme is comparable to that of established splitting schemes such as OBABO or UBU, which are widely used in machine learning and molecular dynamics.

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 / 2 minor

Summary. The manuscript proposes a novel kinetic Langevin sampler based on a splitting scheme that uses the exact harmonic Langevin integrator. For strongly log-concave targets, the potential is decomposed into a quadratic part plus a convex perturbation with Lipschitz gradient. The authors establish L²-Wasserstein convergence rates and non-asymptotic error bounds for the resulting first- and second-order schemes, claiming that the contraction rate matches the order of the underlying continuous dynamics. They further state that the step size needed for ε-accuracy with the second-order scheme is comparable to that of OBABO and UBU.

Significance. If the stated rates and bounds hold, the work adds a rigorously analyzed splitting integrator to the kinetic Langevin literature. The matching contraction order and comparable step-size requirement position the method as competitive with existing schemes used in machine learning and molecular dynamics. The analysis relies on the standard quadratic-plus-Lipschitz-perturbation decomposition, allowing direct comparison to prior OBABO/UBU results.

minor comments (2)
  1. [Abstract] Abstract: the splitting scheme and the precise definition of the first- and second-order integrators are not described, making it difficult to assess novelty from the abstract alone.
  2. The manuscript should include a short comparison table of step-size restrictions and contraction constants versus OBABO and UBU to substantiate the 'comparable' claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the recognition of its novelty in the kinetic Langevin literature, and the recommendation for minor revision. The report correctly identifies the key contributions regarding the splitting scheme, L2-Wasserstein rates matching the continuous dynamics, and comparable step-size requirements to OBABO/UBU.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central results establish L²-Wasserstein convergence rates and non-asymptotic bounds for first- and second-order splitting schemes that employ the exact harmonic Langevin integrator. These rates are shown to match the order of the underlying continuous kinetic Langevin dynamics under the standard structural assumption that the target is strongly log-concave, allowing decomposition of the potential into a quadratic term plus a convex perturbation whose gradient is Lipschitz. The analysis proceeds by controlling the perturbation term via its Lipschitz constant, exactly as in prior analyses of comparable splittings (OBABO, UBU). No load-bearing step reduces by definition to its own inputs, renames a fitted quantity as a prediction, or rests on a self-citation chain whose validity is internal to the paper. The derivation is therefore self-contained and draws on externally verifiable techniques from the sampling literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; full paper may have more parameters or assumptions in the proofs.

axioms (1)
  • domain assumption Target distribution is strongly log-concave
    Stated in abstract as assumption for the analysis.

pith-pipeline@v0.9.1-grok · 5663 in / 1169 out tokens · 51864 ms · 2026-06-30T14:59:58.726929+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

47 extracted references · 24 canonical work pages

  1. [1]

    Ableidinger, E

    M. Ableidinger, E. Buckwar, and H. Hinterleitner , A stochastic version of the jansen and rit neural mass model: analysis and numerics , The Journal of Mathematical Neuroscience, 7 (2017), p. 8

    2017

  2. [2]

    Andrieu, N

    C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan , An introduction to mcmc for machine learning , Machine learning, 50 (2003), pp. 5--43

    2003

  3. [3]

    Bakry, P

    D. Bakry, P. Cattiaux, and A. Guillin , Rate of convergence for ergodic continuous markov processes: Lyapunov versus poincar \'e , Journal of Functional Analysis, 254 (2008), pp. 727--759

    2008

  4. [4]

    Bakry, I

    D. Bakry, I. Gentil, and M. Ledoux , Analysis and geometry of M arkov diffusion operators , vol. 348 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Cham, 2014, https://doi.org/10.1007/978-3-319-00227-9, https://doi.org/10.1007/978-3-319-00227-9

    work page doi:10.1007/978-3-319-00227-9 2014

  5. [5]

    Bou-Rabee, A

    N. Bou-Rabee, A. Eberle, and R. Zimmer , Coupling and convergence for H amiltonian M onte C arlo , Ann. Appl. Probab., 30 (2020), pp. 1209--1250, https://doi.org/10.1214/19-AAP1528, https://doi.org/10.1214/19-AAP1528

    work page doi:10.1214/19-aap1528 2020

  6. [6]

    Bou-Rabee and K

    N. Bou-Rabee and K. Schuh , Convergence of unadjusted hamiltonian monte carlo for mean-field models , Electronic Journal of Probability, 28 (2023), pp. 1--40

    2023

  7. [7]

    H. J. Brascamp and E. H. Lieb , On extensions of the B runn- M inkowski and P r\'ekopa- L eindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation , J. Functional Analysis, 22 (1976), pp. 366--389, https://doi.org/10.1016/0022-1236(76)90004-5, https://doi.org/10.1016/0022-1236(76)90004-5

    work page doi:10.1016/0022-1236(76)90004-5 1976

  8. [8]

    Bussi and M

    G. Bussi and M. Parrinello , Accurate sampling using langevin dynamics , Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 75 (2007), p. 056707

    2007

  9. [9]

    Camrud, A

    E. Camrud, A. Durmus, P. Monmarch \'e , and G. Stoltz , Second order quantitative bounds for unadjusted generalized hamiltonian monte carlo , arXiv preprint arXiv:2306.09513, (2023)

    work page arXiv 2023

  10. [10]

    Y. Cao, J. Lu, and L. Wang , On explicit l 2-convergence rate estimate for underdamped langevin dynamics , Archive for Rational Mechanics and Analysis, 247 (2023), p. 90

    2023

  11. [11]

    Cattiaux and A

    P. Cattiaux and A. Guillin , Trends to equilibrium in total variation distance , Ann. Inst. Henri Poincar\'e Probab. Stat., 45 (2009), pp. 117--145, https://doi.org/10.1214/07-AIHP152, https://doi.org/10.1214/07-AIHP152

    work page doi:10.1214/07-aihp152 2009

  12. [12]

    Chak and P

    M. Chak and P. Monmarch \'e , Reflection coupling for unadjusted generalized hamiltonian monte carlo in the nonconvex stochastic gradient case , IMA Journal of Numerical Analysis, (2025), p. draf045

    2025

  13. [13]

    Y. Chen, K. Gatmiry, and M. Jiang , When does metropolized hamiltonian monte carlo provably outperform metropolis-adjusted langevin algorithm? , arXiv preprint arXiv:2304.04724, (2023)

    work page arXiv 2023

  14. [14]

    Chen and S

    Z. Chen and S. S. Vempala , Optimal convergence rate of hamiltonian monte carlo for strongly logconcave distributions , Theory of Computing, 18 (2022), pp. 1--18

    2022

  15. [15]

    Cheng, N

    X. Cheng, N. S. Chatterji, P. L. Bartlett, and M. I. Jordan , Underdamped langevin mcmc: A non-asymptotic analysis , in Conference on learning theory, PMLR, 2018, pp. 300--323

    2018

  16. [16]

    A. S. Dalalyan , Theoretical guarantees for approximate sampling from smooth and log-concave densities , J. R. Stat. Soc. Ser. B. Stat. Methodol., 79 (2017), pp. 651--676, https://doi.org/10.1111/rssb.12183, https://doi.org/10.1111/rssb.12183

    work page doi:10.1111/rssb.12183 2017

  17. [17]

    A. S. Dalalyan and L. Riou-Durand , On sampling from a log-concave density using kinetic L angevin diffusions , Bernoulli, 26 (2020), pp. 1956--1988, https://doi.org/10.3150/19-BEJ1178

    work page doi:10.3150/19-bej1178 2020

  18. [18]

    Deligiannidis, D

    G. Deligiannidis, D. Paulin, A. Bouchard-C\^ o t\' e , and A. Doucet , Randomized H amiltonian M onte C arlo as scaling limit of the bouncy particle sampler and dimension-free convergence rates , Ann. Appl. Probab., 31 (2021), pp. 2612--2662, https://doi.org/10.1214/20-aap1659, https://doi.org/10.1214/20-aap1659

    work page doi:10.1214/20-aap1659 2021

  19. [19]

    Dolbeault, C

    J. Dolbeault, C. Mouhot, and C. Schmeiser , Hypocoercivity for kinetic equations with linear relaxation terms , Comptes Rendus. Math \'e matique, 347 (2009), pp. 511--516

    2009

  20. [20]

    Dolbeault, C

    J. Dolbeault, C. Mouhot, and C. Schmeiser , Hypocoercivity for linear kinetic equations conserving mass , Transactions of the American Mathematical Society, 367 (2015), pp. 3807--3828

    2015

  21. [21]

    Durmus and E

    A. Durmus and E. Moulines , Nonasymptotic convergence analysis for the unadjusted L angevin algorithm , Ann. Appl. Probab., 27 (2017), pp. 1551--1587, https://doi.org/10.1214/16-AAP1238, https://doi.org/10.1214/16-AAP1238

    work page doi:10.1214/16-aap1238 2017

  22. [22]

    Durmus and E

    A. Durmus and E. Moulines , High-dimensional B ayesian inference via the unadjusted L angevin algorithm , Bernoulli, 25 (2019), pp. 2854--2882, https://doi.org/10.3150/18-BEJ1073, https://doi.org/10.3150/18-BEJ1073

    work page doi:10.3150/18-bej1073 2019

  23. [23]

    Eberle , Reflection couplings and contraction rates for diffusions , Probab

    A. Eberle , Reflection couplings and contraction rates for diffusions , Probab. Theory Related Fields, 166 (2016), pp. 851--886, https://doi.org/10.1007/s00440-015-0673-1, https://doi.org/10.1007/s00440-015-0673-1

    work page doi:10.1007/s00440-015-0673-1 2016

  24. [24]

    ‘Couplings and quantitative contraction rates for Langevin dy- namics’

    A. Eberle, A. Guillin, and R. Zimmer , Couplings and quantitative contraction rates for L angevin dynamics , Ann. Probab., 47 (2019), pp. 1982--2010, https://doi.org/10.1214/18-AOP1299, https://doi.org/10.1214/18-AOP1299

    work page doi:10.1214/18-aop1299 2019

  25. [25]

    Gelman, J

    A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin , Bayesian data analysis , Chapman and Hall/CRC, 1995

    1995

  26. [26]

    Gouraud, P

    N. Gouraud, P. L. Bris, A. Majka, and P. Monmarch\' e , Hmc and underdamped langevin united in the unadjusted convex smooth case , SIAM/ASA Journal on Uncertainty Quantification, 13 (2025), pp. 278--303, https://doi.org/10.1137/23M1608963, https://doi.org/10.1137/23M1608963

    work page doi:10.1137/23m1608963 2025

  27. [27]

    W. K. Hastings , Monte carlo sampling methods using markov chains and their applications , (1970)

    1970

  28. [28]

    P. E. Kloeden and E. Platen , Numerical solution of stochastic differential equations , vol. 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992, https://doi.org/10.1007/978-3-662-12616-5, https://doi.org/10.1007/978-3-662-12616-5

    work page doi:10.1007/978-3-662-12616-5 1992

  29. [29]

    Leimkuhler and C

    B. Leimkuhler and C. Matthews , Rational construction of stochastic numerical methods for molecular sampling , Appl. Math. Res. Express. AMRX, (2013), pp. 34--56, https://doi.org/10.1093/amrx/abs010, https://doi.org/10.1093/amrx/abs010

    work page doi:10.1093/amrx/abs010 2013

  30. [30]

    Leimkuhler, C

    B. Leimkuhler, C. Matthews, and G. Stoltz , The computation of averages from equilibrium and nonequilibrium L angevin molecular dynamics , IMA J. Numer. Anal., 36 (2016), pp. 13--79, https://doi.org/10.1093/imanum/dru056, https://doi.org/10.1093/imanum/dru056

    work page doi:10.1093/imanum/dru056 2016

  31. [31]

    Leimkuhler, D

    B. Leimkuhler, D. Paulin, and P. A. Whalley , Contraction rate estimates of stochastic gradient kinetic L angevin integrators , ESAIM Math. Model. Numer. Anal., 58 (2024), pp. 2255--2286, https://doi.org/10.1051/m2an/2024038, https://doi.org/10.1051/m2an/2024038

    work page doi:10.1051/m2an/2024038 2024

  32. [32]

    B. J. Leimkuhler, D. Paulin, and P. A. Whalley , Contraction and convergence rates for discretized kinetic L angevin dynamics , SIAM J. Numer. Anal., 62 (2024), pp. 1226--1258, https://doi.org/10.1137/23M1556289, https://doi.org/10.1137/23M1556289

    work page doi:10.1137/23m1556289 2024

  33. [33]

    Lelievre and G

    T. Lelievre and G. Stoltz , Partial differential equations and stochastic methods in moleculardynamics , Acta Numerica, 25 (2016), pp. 681--880

    2016

  34. [34]

    Mangoubi and A

    O. Mangoubi and A. Smith , Mixing of H amiltonian M onte C arlo on strongly log-concave distributions: continuous dynamics , Ann. Appl. Probab., 31 (2021), pp. 2019--2045, https://doi.org/10.1214/20-aap1640, https://doi.org/10.1214/20-aap1640

    work page doi:10.1214/20-aap1640 2021

  35. [35]

    R. I. McLachlan and G. R. W. Quispel , Splitting methods , Acta Numerica, 11 (2002), pp. 341--434

    2002

  36. [36]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller , Equation of state calculations by fast computing machines , The journal of chemical physics, 21 (1953), pp. 1087--1092

    1953

  37. [37]

    Monmarch\'e , High-dimensional MCMC with a standard splitting scheme for the underdamped L angevin diffusion

    P. Monmarch\'e , High-dimensional MCMC with a standard splitting scheme for the underdamped L angevin diffusion. , Electron. J. Stat., 15 (2021), pp. 4117--4166, https://doi.org/10.1214/21-ejs1888, https://doi.org/10.1214/21-ejs1888

    work page doi:10.1214/21-ejs1888 2021

  38. [38]

    R. M. Neal , Bayesian learning for neural networks , vol. 118, Springer Science & Business Media, 2012

    2012

  39. [39]

    Nesterov,Lectures on Convex Optimization, vol

    Y. Nesterov , Lectures on convex optimization , vol. 137 of Springer Optimization and Its Applications, Springer, Cham, second ed., 2018, https://doi.org/10.1007/978-3-319-91578-4, https://doi.org/10.1007/978-3-319-91578-4

    work page doi:10.1007/978-3-319-91578-4 2018

  40. [40]

    wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations

    D. Paulin and P. A. Whalley , Correction to" wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations" , Journal of Machine Learning Research, 25 (2024), pp. 1--9

    2024

  41. [41]

    G. A. Pavliotis , Stochastic processes and applications , Texts in applied mathematics, 60 (2014), pp. 41--43

    2014

  42. [42]

    J. M. Sanz-Serna and K. C. Zygalakis , Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations , J. Mach. Learn. Res., 22 (2021), pp. Paper No. 242, 37

    2021

  43. [43]

    Schuh , Global contractivity for L angevin dynamics with distribution-dependent forces and uniform in time propagation of chaos , Ann

    K. Schuh , Global contractivity for L angevin dynamics with distribution-dependent forces and uniform in time propagation of chaos , Ann. Inst. Henri Poincar\'e Probab. Stat., 60 (2024), pp. 753--789, https://doi.org/10.1214/22-aihp1337, https://doi.org/10.1214/22-aihp1337

    work page doi:10.1214/22-aihp1337 2024

  44. [44]

    Schuh and P

    K. Schuh and P. A. Whalley , Convergence of kinetic langevin samplers for non-convex potentials , arXiv preprint arXiv:2405.09992, (2024)

    work page arXiv 2024

  45. [45]

    Villani , Hypocoercivity , vol

    C. Villani , Hypocoercivity , vol. 202, American Mathematical Society, 2009

    2009

  46. [46]

    Welling and Y

    M. Welling and Y. W. Teh , Bayesian learning via stochastic gradient L angevin dynamics , in Proceedings of the 28th international conference on machine learning (ICML-11), 2011, pp. 681--688

    2011

  47. [47]

    A. A. Zapatero , Word series for the numerical integration of stochastic differential equations , PhD thesis, Universidad de Valladolid, 2017

    2017