arxiv: 2605.07939 · v1 · submitted 2026-05-08 · 🧮 math.ST · cs.NA· math.NA· stat.TH

Recognition: no theorem link

Accelerating Langevin Monte Carlo via Efficient Stochastic Runge--Kutta Methods beyond Log-Concavity

Bin Yang, Xiaojie Wang

Pith reviewed 2026-05-11 03:01 UTC · model grok-4.3

classification 🧮 math.ST cs.NAmath.NAstat.TH

keywords Langevin Monte Carlostochastic Runge-Kuttanon-log-concave samplingWasserstein convergenceoverdamped Langevinuniform-in-time boundslog-smooth potentials

0 comments

The pith

A stochastic Runge-Kutta scheme for overdamped Langevin dynamics achieves uniform W2 convergence of order O(d^{3/2} h^{3/2}) under only log-smoothness.

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

The paper introduces a Hessian-free stochastic Runge-Kutta integrator of strong order 1.5 for the overdamped Langevin equation that requires only two gradient evaluations per iteration. It derives non-asymptotic error bounds in Wasserstein-2 distance that hold uniformly in time for targets whose potentials are log-smooth. The resulting rate O(d^{3/2} h^{3/2}) matches the best previously known rate yet applies without the log-concavity assumption used in earlier work.

Core claim

The proposed efficient stochastic Runge-Kutta discretization of the overdamped Langevin dynamics produces a sampling algorithm whose law converges to the target at a uniform-in-time rate of order O(d^{3/2} h^{3/2}) in the 2-Wasserstein metric, provided only that the potential satisfies a log-smoothness condition; the same rate had been established earlier only under the stronger assumption of log-concavity.

What carries the argument

The efficient stochastic Runge-Kutta integrator of strong order 1.5, which approximates the overdamped Langevin SDE using two gradient evaluations per step and supplies the higher-order local error terms required for the non-log-concave analysis.

Load-bearing premise

The target density has a gradient whose Lipschitz constant is bounded uniformly over the whole space.

What would settle it

A numerical computation of the W2 distance after a fixed number of steps on a family of non-log-concave Gaussian-mixture targets with growing dimension d would refute the claimed scaling if the observed exponent on d deviates from 3/2.

Figures

Figures reproduced from arXiv: 2605.07939 by Bin Yang, Xiaojie Wang.

**Figure 1.** Figure 1: Convergence Rates of LMC, SRK-LD and RKLMC-2G. 8 9 10 11 12 13 14 15 16 Dimensions 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Mean-square errors1/2 10-3 Dimension Dependence for GMM SRK-LD RKLMC-2G order 1.5 6 7 8 9 10 11 12 13 14 Dimensions 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Mean-square errors1/2 10-3 Dimension Dependence for BLR SRK-LD RKLMC-2G order 1.5 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Figure 2: Dimension Dependence of LMC, SRK-LD and RKLMC2G. 6, 8, 10, 12, 14 for BLR. The reference stepsize is chosen as href = 2−9 for GMM and href = 2−11 for BLR and the coarse approximations are computed with the fixed stepsizes h = 2−4 and h = 2−6 , respectively. Root mean-square errors are then plotted against the dimension on a log-log scale. As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 4.** Figure 4: Scatter Plots for LMC, SRK-LD and RKLMC-2G on 8-Mode GMM. quality of this paper. This work was supported by Natural Science Foundation of China (Nos. 12471394, 12371417). Impact Statement This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here. References Altschuler,… view at source ↗

**Figure 3.** Figure 3: Histogram of the First Component for RKLMC-2G on Two-mode GMM. 5. Conclusion and Future Work In the present work, we propose a class of Runge–Kutta LMC algorithms, including a particular one with only two gradient evaluations per every iteration. Moreover, under certain non-log-concavity condition, we obtain the non-asymptotic error bound in the W2-distance of order O(d 3/2h 3/2 ). In future work, (i) we … view at source ↗

read the original abstract

Sampling from a high-dimensional probability distribution is a fundamental algorithmic task arising in wide-ranging applications across multiple disciplines, including scientific computing, computational statistics and machine learning. Langevin Monte Carlo (LMC) algorithms are among the most widely used sampling methods in high-dimensional settings. This paper introduces a novel higher-order and Hessian-free LMC sampling algorithm based on an efficient stochastic Runge--Kutta method of strong order $1.5$ for the overdamped Langevin dynamics. In contrast to the existing Runge--Kutta type LMC (Li et al., 2019) involved with three gradient evaluations, the newly proposed algorithm is computationally cheaper and requires only two gradient evaluations for one iteration. Under certain log-smooth conditions, non-asymptotic error bounds of the proposed algorithms are analyzed in $\mathcal{W}_2$-distance. In particular, a uniform-in-time convergence rate of order $O(d ^{\frac32} h^{\frac32})$ is derived in a non-log-concave setting, matching the convergence rate proved in the aforementioned work but under the log-concavity condition. Numerical experiments are finally presented to demonstrate the effectiveness of the new sampling algorithm.

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.

Desk Editor's Note private letter to a colleague

This paper cuts gradient evaluations in a Runge-Kutta LMC scheme and claims the same uniform W2 rate without log-concavity, but the non-log-concave analysis may rest on unstated growth conditions.

read the letter

The punchline is that this work gives a cheaper two-gradient stochastic Runge-Kutta integrator for overdamped Langevin sampling and claims to keep the O(d^{3/2} h^{3/2}) uniform-in-time W2 bound even when the target is not log-concave. It improves on the Li et al. 2019 method by halving the gradient calls per step while matching the convergence order. The numerical experiments are a plus for showing it works in practice on some examples. The main soft spot is the theoretical claim for the non-log-concave case. Log-concavity usually supplies the contraction needed for uniform bounds, and log-smoothness by itself does not guarantee dissipativity or moment control. The abstract refers to certain log-smooth conditions but does not list them or sketch how the proof avoids the usual requirements. Without explicit growth conditions or a new coupling argument, it is unclear whether the uniform rate actually holds or if the analysis has a gap. This paper is for people working on high-order discretizations of Langevin dynamics in statistics and machine learning. Readers who care about reducing computational cost in sampling would find the algorithmic change useful, and the experiments give some evidence of effectiveness. I would send it for peer review. The idea is specific and the potential improvement is clear enough that referees can check the details and see if the extension works.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces an efficient stochastic Runge-Kutta discretization of the overdamped Langevin dynamics requiring only two gradient evaluations per iteration while achieving strong order 1.5. Under log-smooth conditions on the potential, it derives non-asymptotic W_2 error bounds of order O(d^{3/2} h^{3/2}) that are uniform in time, extending the rate previously obtained by Li et al. (2019) to non-log-concave targets.

Significance. If the uniform-in-time W_2 bounds hold rigorously under the stated log-smoothness without hidden dissipativity assumptions, the result would be significant: it supplies a computationally cheaper higher-order LMC method whose non-asymptotic analysis applies to a broader class of targets than existing log-concave analyses, while preserving the same dimension-and-step-size dependence.

major comments (1)

The central uniform-in-time W_2 bound of order O(d^{3/2} h^{3/2}) (stated in the abstract and presumably proved in the main theorem) is claimed under only 'certain log-smooth conditions' in a non-log-concave setting. Log-smoothness controls local Lipschitz constants but does not by itself yield global contraction or moment bounds for the continuous overdamped Langevin flow; standard coupling or Gronwall arguments for uniform-in-time discretization error therefore require an explicit dissipativity condition such as <∇V(x), x> ≥ a|x|^2 − b. The manuscript must either add this assumption explicitly or provide a new contraction argument that closes without it; otherwise the claimed rate cannot be verified from the given hypotheses.

minor comments (2)

The abstract refers to 'certain log-smooth conditions' without listing them; the introduction or assumption section should state the precise regularity and growth hypotheses on V (e.g., Lipschitz gradient constant L, any moment or dissipativity requirements) so that the scope of the theorem is immediately clear.
Numerical experiments should report wall-clock time or gradient-evaluation counts alongside W_2 or ESS metrics to quantify the claimed computational saving relative to the three-evaluation Runge-Kutta scheme of Li et al. (2019).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below and have revised the manuscript to strengthen the presentation of the assumptions.

read point-by-point responses

Referee: The central uniform-in-time W_2 bound of order O(d^{3/2} h^{3/2}) (stated in the abstract and presumably proved in the main theorem) is claimed under only 'certain log-smooth conditions' in a non-log-concave setting. Log-smoothness controls local Lipschitz constants but does not by itself yield global contraction or moment bounds for the continuous overdamped Langevin flow; standard coupling or Gronwall arguments for uniform-in-time discretization error therefore require an explicit dissipativity condition such as <∇V(x), x> ≥ a|x|^2 − b. The manuscript must either add this assumption explicitly or provide a new contraction argument that closes without it; otherwise the claimed rate cannot be verified from the given hypotheses.

Authors: We thank the referee for highlighting this point. Our proof of the uniform-in-time W_2 bound proceeds via standard coupling and Gronwall estimates on the continuous overdamped Langevin flow. While log-smoothness supplies the local Lipschitz control for the discretization error, the global moment bounds and contraction indeed rely on a dissipativity condition of the form ⟨∇V(x), x⟩ ≥ a|x|^2 − b (a > 0). This condition was used implicitly in our derivations to close the estimates in the non-log-concave regime, but we agree it was not stated with sufficient clarity among the “certain log-smooth conditions.” We will revise the manuscript by (i) explicitly listing the dissipativity assumption in the main theorem and assumptions section, (ii) updating the abstract and introduction to reflect the precise hypotheses, and (iii) adding a brief remark that the condition is standard for non-log-concave targets and compatible with the claimed rate. No new contraction argument is required; the existing proof carries through once the assumption is stated. This clarification does not alter the algorithmic contribution or the O(d^{3/2} h^{3/2}) rate. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence rate derived from discretization analysis under explicit assumptions

full rationale

The paper presents a new stochastic Runge-Kutta discretization for overdamped Langevin dynamics and derives non-asymptotic W2 bounds under log-smoothness conditions. The uniform-in-time O(d^{3/2} h^{3/2}) rate is obtained by extending the analysis of Li et al. (2019) to the non-log-concave case; the extension relies on the paper's own error estimates and moment bounds rather than re-using a fitted quantity or self-referential definition. No step reduces the claimed rate to an input by construction, and the cited prior work is external. The derivation chain is self-contained once the log-smoothness and growth conditions are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard overdamped Langevin SDE and the domain assumption of log-smoothness; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption The target distribution satisfies log-smooth conditions
Invoked to obtain the non-asymptotic W2 error bounds and uniform-in-time rate.

pith-pipeline@v0.9.0 · 5516 in / 1294 out tokens · 48739 ms · 2026-05-11T03:01:04.527353+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

137 extracted references · 137 canonical work pages · 1 internal anchor

[1]

Langley , title =

P. Langley , title =. Proceedings of the 17th International Conference on Machine Learning (ICML 2000) , address =. 2000 , pages =

work page 2000
[2]

T. M. Mitchell. The Need for Biases in Learning Generalizations. 1980

work page 1980
[3]

M. J. Kearns , title =

work page
[4]

Machine Learning: An Artificial Intelligence Approach, Vol. I. 1983

work page 1983
[5]

R. O. Duda and P. E. Hart and D. G. Stork. Pattern Classification. 2000

work page 2000
[6]

Suppressed for Anonymity , author=

work page
[7]

Newell and P

A. Newell and P. S. Rosenbloom. Mechanisms of Skill Acquisition and the Law of Practice. Cognitive Skills and Their Acquisition. 1981

work page 1981
[8]

A. L. Samuel. Some Studies in Machine Learning Using the Game of Checkers. IBM Journal of Research and Development. 1959

work page 1959
[9]

International Conference on Machine Learning , pages=

Grenioux, Louis and Noble, Maxence and Gabri. International Conference on Machine Learning , pages=. 2024 , organization=

work page 2024
[10]

Neufeld, Ariel and Zhang, Ying , journal=

work page
[11]

2024 , publisher=

Altschuler, Jason M and Chewi, Sinho , journal=. 2024 , publisher=

work page 2024
[12]

1993 , publisher=

Hairer, Ernst and Wanner, Gerhard and N. 1993 , publisher=

work page 1993
[13]

2000 , publisher=

Burrage, Kevin and Burrage, Pamela M , journal=. 2000 , publisher=

work page 2000
[14]

Li, Lei and Wang, Chen and Wang, Mengchao , journal=

work page
[15]

Altschuler and Sinho Chewi , booktitle =

Jason M. Altschuler and Sinho Chewi , booktitle =. ArXiv , title =

work page
[16]

Wang, Xiaojie and Yang, Bin , journal=

work page
[17]

2021 , organization=

Erdogdu, Murat A and Hosseinzadeh, Rasa , booktitle=. 2021 , organization=

work page 2021
[18]

2016 , publisher=

Lelievre, Tony and Stoltz, Gabriel , journal=. 2016 , publisher=

work page 2016
[19]

2003 , publisher=

Andrieu, Christophe and De Freitas, Nando and Doucet, Arnaud and Jordan, Michael I , journal=. 2003 , publisher=

work page 2003
[20]

2013 , publisher=

Cotter, Simon L and Roberts, Gareth O and Stuart, Andrew M and White, David , journal=. 2013 , publisher=

work page 2013
[21]

When Langevin Monte Carlo Meets Randomization: New Sampling Algorithms with Non-asymptotic Error Bounds beyond Log-Concavity and Gradient Lipschitzness

Xiaojie Wang and Bin Yang , year=. 2509.25630 , archivePrefix=

work page internal anchor Pith review Pith/arXiv arXiv
[22]

2008 , publisher=

Villani, C. 2008 , publisher=

work page 2008
[23]

SIAM Journal on Numerical Analysis , volume=

R. SIAM Journal on Numerical Analysis , volume=. 2010 , publisher=

work page 2010
[24]

Song, Yang and Sohl-Dickstein, Jascha and Kingma, Diederik P and Kumar, Abhishek and Ermon, Stefano and Poole, Ben , booktitle=

work page
[25]

HASTINGS, WK , journal=

work page
[26]

1953 , publisher=

Metropolis, Nicholas and Rosenbluth, Arianna W and Rosenbluth, Marshall N and Teller, Augusta H and Teller, Edward , journal=. 1953 , publisher=

work page 1953
[27]

1995 , publisher=

Chib, Siddhartha and Greenberg, Edward , journal=. 1995 , publisher=

work page 1995
[28]

The Annals of Probability , number =

Feng-Yu Wang , title =. The Annals of Probability , number =. 2011 , doi =

work page 2011
[29]

Journal of Functional Analysis , volume=

Otto, Felix and Villani, C. Journal of Functional Analysis , volume=. 2000 , publisher=

work page 2000
[30]

2025 , publisher=

Li, Lei and Wang, Yuliang , journal=. 2025 , publisher=

work page 2025
[31]

2025 , publisher=

Lytras, Iosif and Sabanis, Sotirios , journal=. 2025 , publisher=

work page 2025
[32]

2023 , organization=

Mousavi-Hosseini, Alireza and Farghly, Tyler K and He, Ye and Balasubramanian, Krishna and Erdogdu, Murat A , booktitle=. 2023 , organization=

work page 2023
[33]

Yang, Bin and Wang, Xiaojie , journal=

work page
[34]

2017 , publisher=

Kruse, Raphael and Wu, Yue , journal=. 2017 , publisher=

work page 2017
[35]

2019 , publisher=

Kruse, Raphael and Wu, Yue , journal=. 2019 , publisher=

work page 2019
[36]

Applied Numerical Mathematics , volume=

Przyby. Applied Numerical Mathematics , volume=. 2014 , publisher=

work page 2014
[37]

2009 , publisher=

Jentzen, Arnulf and Neuenkirch, Andreas , journal=. 2009 , publisher=

work page 2009
[38]

2008 , publisher=

Heinrich, Stefan and Milla, Bernhard , journal=. 2008 , publisher=

work page 2008
[39]

Journal of Complexity , volume =

Thomas Daun , keywords =. Journal of Complexity , volume =. 2011 , note =. doi:https://doi.org/10.1016/j.jco.2010.07.002 , url =

work page doi:10.1016/j.jco.2010.07.002 2011
[40]

Stengle, Gilbert , TITLE =. Numer. Math. , FJOURNAL =. 1995 , NUMBER =. doi:10.1007/s002110050113 , URL =

work page doi:10.1007/s002110050113 1995
[41]

Stengle, Gilbert , TITLE =. Appl. Math. Lett. , FJOURNAL =. 1990 , NUMBER =. doi:10.1016/0893-9659(90)90040-I , URL =

work page doi:10.1016/0893-9659(90)90040-i 1990
[42]

2014 , publisher=

Wang, Feng-Yu , volume=. 2014 , publisher=

work page 2014
[43]

Infinite Dimensional Analysis, Quantum Probability and Related Topics , volume=

R. Infinite Dimensional Analysis, Quantum Probability and Related Topics , volume=. 2010 , publisher=

work page 2010
[44]

2024 , publisher=

Pang, Chenxu and Wang, Xiaojie , journal=. 2024 , publisher=

work page 2024
[45]

2010 , publisher=

Wang, Feng-Yu , journal=. 2010 , publisher=

work page 2010
[46]

2024 , publisher=

Chewi, Sinho and Erdogdu, Murat A and Li, Mufan and Shen, Ruoqi and Zhang, Matthew S , journal=. 2024 , publisher=

work page 2024
[47]

Shen, Ruoqi and Lee, Yin Tat , journal=

work page
[48]

Yu, Lu and Karagulyan, Avetik and Dalalyan, Arnak , booktitle=

work page
[49]

Roberts, Gareth O and Tweedie, Richard L , journal=

work page
[50]

1999 , publisher=

Robert, Christian P and Casella, George and Casella, George , volume=. 1999 , publisher=

work page 1999
[51]

2001 , publisher=

Liu, Jun S and Liu, Jun S , volume=. 2001 , publisher=

work page 2001
[52]

Xu, Pan and Chen, Jinghui and Zou, Difan and Gu, Quanquan , journal=

work page
[53]

2011 , organization=

Welling, Max and Teh, Yee W , booktitle=. 2011 , organization=

work page 2011
[54]

2014 , publisher=

Pavliotis, Grigorios A , volume=. 2014 , publisher=

work page 2014
[55]

2405.05679 , archivePrefix=

Ariel Neufeld and Ying Zhang , year=. 2405.05679 , archivePrefix=

work page arXiv
[56]

2025 , publisher=

Neufeld, Ariel and Zhang, Ying and others , journal=. 2025 , publisher=

work page 2025
[57]

Annals of Applied Probability , volume=

Majka, Mateusz B and Mijatovi. Annals of Applied Probability , volume=. 2020 , publisher=

work page 2020
[58]

Li, Xiang and Wang, Feng-Yu and Xu, Lihu , journal=

work page
[59]

The Annals of Applied Probability , volume=

Pag. The Annals of Applied Probability , volume=. 2023 , publisher=

work page 2023
[60]

Wainwright and Peter L

Wenlong Mou and Nicolas Flammarion and Martin J. Wainwright and Peter L. Bartlett , title =. Bernoulli , number =

work page
[61]

Li, Ruilin and Zha, Hongyuan and Tao, Molei , booktitle=

work page
[62]

arXiv preprint arXiv:1805.01648 , year=

Xiang Cheng and Niladri S. Chatterji and Yasin Abbasi-Yadkori and Peter L. Bartlett and Michael I. Jordan , year=. 1805.01648 , archivePrefix=

work page arXiv
[63]

Neal, Radford , booktitle =

work page
[64]

1988 , publisher=

Milstein, GN , journal=. 1988 , publisher=

work page 1988
[65]

2021 , organization=

Chewi, Sinho and Lu, Chen and Ahn, Kwangjun and Cheng, Xiang and Le Gouic, Thibaut and Rigollet, Philippe , booktitle=. 2021 , organization=

work page 2021
[66]

Durmus, Alain and Moulines, \'Eric , TITLE =. Ann. Appl. Probab. , FJOURNAL =. 2017 , NUMBER =

work page 2017
[67]

Kakade, Sham Machandranath , year=

work page
[68]

2019 , publisher=

Liang, Tengyuan and Su, Weijie J , journal=. 2019 , publisher=

work page 2019
[69]

2018 , organization=

Wibisono, Andre , booktitle=. 2018 , organization=

work page 2018
[70]

Song, Yang and Ermon, Stefano , journal=

work page
[71]

Sabanis, Sotirios and Zhang, Ying , journal=

work page
[72]

Li, Xuechen and Wu, Yi and Mackey, Lester and Erdogdu, Murat A , journal=

work page
[73]

2019 , publisher=

Dalalyan, Arnak S and Karagulyan, Avetik , journal=. 2019 , publisher=

work page 2019
[74]

2017 , publisher=

Dalalyan, Arnak S , journal=. 2017 , publisher=

work page 2017
[75]

Sqrt (d) Dimension Dependence of Langevin Monte Carlo , author=

work page
[76]

Durmus, Alain and Moulines, \'Eric , journal=

work page
[77]

Journal of Machine Learning Research , volume=

Durmus, Alain and Majewski, Szymon and Miasojedow, B. Journal of Machine Learning Research , volume=

work page
[78]

2018 , organization=

Cheng, Xiang and Bartlett, Peter , booktitle=. 2018 , organization=

work page 2018
[79]

2017 , organization=

Dalalyan, Arnak , booktitle=. 2017 , organization=

work page 2017
[80]

2025 , author =

Journal of Computational Physics , volume =. 2025 , author =

work page 2025

Showing first 80 references.