From Continuous Dynamics to Practical Gradient-Based Samplers

James Chok

REVIEW 3 major objections 5 minor 66 references

From Continuous Dynamics to Practical Gradient-Based Samplers

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Gradient-based samplers are one family—continuous dynamics, discretization, then Metropolis correction—and the practical bottleneck is geometry, which a randomized step size can address for funnel-like targets.

desk verdict Solid tutorial worth reading, but the central RS-MAKLA recipe has a sign error in Eq. (14) that must be fixed before the practical claims can be trusted. read the letter →

arxiv 2608.05425 v1 pith:IWQETABZ submitted 2026-08-05 stat.CO stat.ME

James Chok This is my paper · ORCID

classification stat.COstat.ME MSC 65C0560J2262F15

keywords HamiltonianMonteCarloLangevindynamicsMetropolisadjustmentNeal'sfunnelrandomizedstepsizepreconditioninghierarchicalmodelsMCMC

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper builds a unified picture of gradient-based MCMC: every standard sampler begins as an exact continuous-time dynamics, becomes practical through numerical discretization, and recovers exactness through a Metropolis accept/reject step. On that scaffolding it identifies the real cause of slow sampling as geometry, not asymptotic bias: global anisotropy that a fixed mass matrix can whiten, and local multiscale structure such as Neal's funnel—a hierarchical scale model where one variable sets the width of many others—that a fixed step cannot resolve. The central practical proposal is RS-MAKLA, a Metropolis-adjusted underdamped Langevin algorithm whose step size is drawn per position from a distribution that shrinks where the gradient is large, corrected by a density-ratio factor in the acceptance probability. The paper's message is that a state-dependent time rescaling can make a simple, vectorizable, fixed-work kernel robust to funnel-like targets, and it tests this against NUTS on three hierarchical posteriors.

What carries the argument

The load-bearing object is the Metropolis acceptance ratio for a randomized step size, Eq. (13): $\alpha(z,z') = \min\big(1, \frac{e^{\pi(z')} q_h(z',z)}{e^{\pi(z)} q_h(z,z')} \cdot \frac{r_\ell(\ell \mid z')}{r_\ell(\ell \mid z)}\big)$. Here $\ell = \log h$ is drawn from a position-dependent proposal density $r_\ell(\cdot \mid x)$, Eq. (14), whose center scales like $h_*\big(1 + (\|\nabla U(x)\|/\sqrt{d})^2\big)^{1/2}$, so small steps are proposed where the force is large. The extra ratio of step-size densities is what preserves detailed balance when the proposal kernel itself depends on the random tuning parameter. Around this sits the paper's organizational machinery: splitting underdamped Langevin into exact O (momentum refreshment) and BAB (Hamiltonian) flows, with the Metropolis correction applied to the Hamiltonian error of the BAB step, followed by a final O refresh.

What would settle it

Run RS-MAKLA, with the Eq. (13) acceptance ratio, on Neal's funnel with $d=10$ and $\sigma=3$, and compare the empirical distribution of the neck variable $v$ against the known marginal; if the mean or a tail quantile of $v$ is biased beyond Monte Carlo error after long runs, the exact-invariance claim falls. A quicker check is to verify detailed balance on a small discrete state space with the same randomized-step acceptance rule and an asymmetric proposal, since any stationary bias there falsifies the formula.

Watch

Extended reading notes

Core claim

The central claim is that the four canonical gradient-based samplers—HMC, MALA, NUTS, and MAKLA—are not separate algorithms but stages of the same construction: exact continuous dynamics, followed by discretization, followed by Metropolis correction of the resulting bias, followed by geometric adaptation. On top of this, the paper claims that replacing the fixed step size with a random step size whose log is drawn from a truncated Normal centered at a force-scale value (Eq. 14), and multiplying the usual Metropolis ratio by the conditional density ratio of that log step size (Eq. 13), yields a sampler that remains exactly invariant while adapting its numerical resolution to local curvature. In RS-MAKLA this randomized step is embedded in an OBABABO discretization of underdamped Langevin dynamics, with the momentum-refreshment parameter tied to the realized step and rejection handled by momentum negation. The paper argues that this kernel reaches deeper into the neck of Neal's funnel than fixed-step alternatives while keeping per-iteration work fixed and GPU-vectorizable.

Load-bearing premise

The load-bearing premise is that the randomized-step acceptance ratio in Eq. (13), including the step-density ratio $r_\ell(\ell \mid x')/r_\ell(\ell \mid x)$, preserves the exact invariant distribution of the RS-MAKLA chain; the paper takes this as given from the literature rather than proving it.

Editorial extensions

If this is right

Fixed-work sampling: each RS-MAKLA iteration uses a constant number of gradient evaluations (one for OBABO, two for OBABABO), so it can be vectorized across many chains on GPUs in a way that NUTS's tree-building cannot.
The four-layer error separation gives a diagnostic recipe: if a sampler is slow, decide whether the problem is transient mixing, Monte Carlo variance, discretization bias, or numerical instability, and choose the adaptation accordingly.
For hierarchical posteriors, the paper predicts that non-centered reparameterization, if available, should be tried first; otherwise MAP-Hessian preconditioning, then randomized step sizes, in that order.
Funnel-like geometry is identified as a failure mode that no fixed mass matrix can fix, so the randomized-step mechanism is the appropriate intervention when reparameterization is impractical.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the exactness of Eq. (13) survives scrutiny, the same randomized-parameter Metropolization is a general template: any sampler whose proposal has a state-dependent tuning parameter could be made locally adaptive by multiplying the acceptance ratio by the density ratio of that parameter, at the cost of one extra density evaluation per proposal.
The force-magnitude proxy in Eq. (14) is a heuristic; a natural testable extension is to replace it with a local curvature estimate, such as the norm of the Hessian or a diagonal Fisher information, and compare on targets where force and curvature disagree.
The paper's comparison of RS-MAKLA with NUTS on Gaussian and funnel-like targets suggests a sharper separating experiment: on a hierarchical target whose likelihood truncates the neck, like the radon example, the advantage of local time rescaling should shrink as data volume grows, and that prediction is directly testable.
A proof that the rejection momentum-negation step in RS-MAKLA preserves detailed balance would close the main theoretical gap; until then, a small finite-state counterexample to the acceptance rule would decide the matter.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague — the short version: this is a genuinely useful tutorial and deserves a serious referee, but the paper's main new recipe, RS-MAKLA, has a load-bearing internal inconsistency in Eq. (14). The text says the conditional mean of log step size should decrease as the force magnitude grows, which is what allows the sampler to slow down in the funnel neck. The formula for µ(x), however, increases with ||∇U(x)||. As written, the sampler would propose larger steps precisely where the curvature is highest, which is the opposite of Figure 9 and of the intended behavior. The likely fix is a missing reciprocal in µ(x), but until that is corrected, the recipe cannot deliver the claimed funnel performance.

What the paper does well: the unified continuous-dynamics view is clear and pedagogically valuable, the Gaussian laboratory calculations are neat illustrations, and the decomposition of errors (transient, Monte Carlo, discretization, numerical instability) is a genuinely helpful framing. The standard math that is actually proved is correct, and the practical guidance on preconditioning and non-centering is sensible and well-cited. Shipping code is a plus.

Now the soft spots, roughly in order of importance. First, the paper is a tutorial rather than a primary research contribution: the randomized step-size idea is drawn from [23] and [52–55], and no new theorem or algorithm is proved. Second, Eq. (13) — the randomized-step-size Metropolis ratio — is asserted with a citation but not proved here; it is standard and justifiable by an augmented-state detailed-balance argument, but the paper should either provide that argument or cite it precisely. Third, the benchmarks are small illustrations, not a systematic comparison; they favor NUTS on the real-data examples, and the RS-MAKLA advantage appears mainly in the funnel example. Tables 1 and 2 report no uncertainty on ESS or R-hat, and several step sizes are hand-tuned or chosen post hoc. Fourth, the performance claim for RS-MAKLA leans heavily on the author's own companion paper [23]; that is not damning, but it is not independently established here either.

The audience is applied statisticians who want a mental model of gradient-based samplers and practical recipes. The tutorial half is solid; the central recipe, as written, is not reliable until Eq. (14) is fixed and the funnel experiment is re-run with the corrected formula.

My recommendation: send it to peer review, but require the author to correct the sign error, prove or properly cite the exactness in Eq. (13), and add uncertainty estimates to the benchmark tables. With those changes, this could be a widely used reference.

Referee Report

3 major / 5 minor

Summary. The paper develops a unified exposition of gradient-based MCMC samplers, organizing the field around four stages: exact continuous-time dynamics, numerical discretization, Metropolis adjustment, and geometric preconditioning. It reviews exact HMC, overdamped and underdamped Langevin dynamics, ULA, MALA, HMC, NUTS, and MAKLA, then proposes practical design choices: Hessian-at-MAP or adaptively estimated mass matrices for global anisotropy, position-dependent metrics, and a randomized step-size MAKLA (RS-MAKLA) for funnel-like geometries. The second half reports experiments on Neal's funnel, German credit, and Minnesota radon data comparing RS-MAKLA with NUTS, and concludes with a practitioner guideline. The central research claim is that RS-MAKLA, with the acceptance ratio in Eq. (13) and the conditional log-step-size distribution in Eq. (14), is an exact, fixed-work, vectorizable sampler that handles state-dependent curvature better than a fixed-step kernel.

Significance. If the RS-MAKLA construction is correct, the paper would make a useful practical contribution: it offers a fixed-work gradient-based sampler with local step-size adaptation, which is attractive for GPU-vectorized inference on multiscale posteriors. The tutorial component is well structured, and the explicit separation of transient error, Monte Carlo error, discretization bias, and numerical instability is pedagogically valuable. The authors also responsibly report settings in which NUTS outperforms RS-MAKLA, which strengthens the credibility of the experimental narrative. However, the central formula behind the randomized step size currently contradicts its own stated goal, the exact-invariance claim for the randomized acceptance step is not proved in the manuscript, and the quantitative comparisons lack uncertainty quantification. These issues need to be resolved before the practical recipe can be adopted.

major comments (3)

[5.5] The text states that the conditional mean of the log step size should decrease as ||∇U(x)|| increases, but the displayed formula does the opposite: μ(x) = h*(1 + (||∇U(x)||/√d)^2)^{1/2} is an increasing function of the force norm. In Neal's funnel, the neck region has exponentially large force magnitude, so Eq. (14) would assign the largest typical step sizes exactly where the local curvature demands the smallest ones. This contradicts the mechanism claimed in the surrounding paragraph and Figure 9, which shows realized step sizes decreasing by roughly two orders of magnitude from mouth to neck. As written, RS-MAKLA would propose very large jumps in the high-curvature region and cannot deliver the claimed funnel behavior. The most plausible fix is a reciprocal construction, μ(x) = h* / (1 + (||∇U(x)||/√d)^2)^{1/2}, but the correction must be stated explicitly and used consistently in the algorithm and experiments.
[5.5] The exact π-invariance of the randomized-step-size Metropolis correction is asserted with a citation to [55] but is not proved in this manuscript. In particular, the paper does not demonstrate the required proposal/reversal pairing for the full OBABO transition, the treatment of the momentum flip upon rejection, or the role of the ratio r_ℓ(ℓ|x')/r_ℓ(ℓ|x) in maintaining detailed balance. Since the central practical claim is that RS-MAKLA is an exact sampler, this is a load-bearing point. The manuscript should either include a self-contained detailed-balance or augmented-state argument, or state precisely which existing theorem covers exactly the algorithm as written.
[Appendix A] The quantitative comparisons report ESS/Grad., ESS/Sec., and rank-normalized split R-hat as point estimates without any Monte Carlo uncertainty, standard errors, or repeated-run variability. Figures 5 and 6 state that step sizes were tuned to achieve an acceptance rate of 0.9, but the actual values are not reported, and the caption in Appendix A.2 concedes that the funnel improvement is 'in this run'. These choices weaken the experimental support for the general claim that RS-MAKLA is the method of choice for funnel-like posteriors. I am not asking for a full benchmark suite, but the manuscript should either provide uncertainty-aware diagnostics or explicitly label the comparisons as illustrative rather than decisive.

minor comments (5)

[4.2] The phrase 'at the exception quarter-period HMC sampler' should read 'at the exceptional quarter-period HMC sampler.'
[5.5] The notation 'increases, and vice versa' is ambiguous: the sentence should specify that the conditional mean of ℓ decreases as ||∇U(x)|| increases, and increases as ||∇U(x)|| decreases, to avoid any confusion with the formula in Eq. (14).
[7] The paper uses 'bR' in the text and tables, but the standard notation is 'R-hat' or 'R'; please define the symbol at first use and keep it consistent.
[7] The footnotes 'I am guilty of this' and 'feel free to email me' are informal for a journal manuscript; they should be removed or moved to an acknowledgments section.
[7] The claim that RS-MAKLA 'outperforms NUTS on a wide range of posterior distributions' rests on the companion paper [23], which is not available to the reader. Please either summarize the relevant experimental evidence or clearly mark this statement as dependent on unpublished work.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the randomized-step-size correction is a standard augmented-MH result, and the only author self-citation ([23]) is not load-bearing for the algorithmic correctness claim.

full rationale

The claimed derivation chain is not circular. The continuous-time samplers (Section 2), the discretizations (Section 3), and the Metropolis corrections (Section 4) are standard material re-derived from the detailed-balance condition (6), and the randomized-step-size adjustment in Eq. (13) is the ordinary augmented-state Metropolis ratio for the target pi(x) r_l(ell|x) with the step size refreshed independently; the paper credits this to external prior work [52-55] rather than to its own conclusions. The concrete step-size model (14) is a proposed functional form, not a quantity fitted to data and then relabeled as a prediction, and the experimental hyperparameters (h*, sigma=0.5, gamma=0.1, truncation bounds) are fixed tuning choices, not predictions. The only self-citation with an empirical claim is [23] for the statement that RS-MAKLA outperforms NUTS on a wide range of posterior distributions; that is a supplementary performance pointer, and the paper's own Appendix A independently compares the samplers, so the citation is not the load-bearing justification for the algorithm's correctness or for the funnel mechanism. Two non-circular concerns should be noted for the correctness pass rather than the circularity pass: the exactness of Eq. (13) is cited to [55] rather than proved, and the displayed formula (14) contradicts its own prose and Figure 9 by making the conditional step-size mean increase with the force norm when the text and figure require a decrease. Neither concern is a self-referential derivation, so the circularity score is 0.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The ledger shows a tutorial that leans on standard math for its derivations and on hand-tuned algorithmic constants for its practical recipes. The randomized log step size is an auxiliary algorithmic variable from cited work, not a new entity. The main circularity burden is evidential: the performance claim for RS-MAKLA is supported by the author's own [23] and by hand-set parameters.

free parameters (6)

Reference step size h* in RS-MAKLA = tuned to a fixed acceptance rate, exact value not reported
Controls the mean log step size in Eq. (14) and is adapted during warmup; the efficiency comparison depends on this tuning.
Log step-size standard deviation sigma = 0.5
Hand-set in Section A.1 for Eq. (14); no sensitivity analysis is provided.
Momentum persistence gamma = 0.1
Hand-set in Section A.1; couples momentum refreshment to the realized step size through eta_n = exp(-gamma h_n/2).
Step-size truncation bounds h_min, h_max = 1e-4, 1
Hand-set truncation bounds for the truncated Normal in Eq. (14).
Regularization epsilon in adaptive covariance = not specified
Used in Eq. (8) and in the FA-MALA diagonal metric to ensure positive definiteness; the value is left unspecified.
Step sizes in Figures 5-6 = 0.25 for preconditioned runs, 0.1 for unpreconditioned Student-t
Selected post hoc; the footnote admits that h=0.25 for the unpreconditioned Student-t sampler produced a one-order-of-magnitude bias, so a smaller h was used.

assumptions (8)

standard math Under mild conditions, overdamped and underdamped Langevin diffusions have pi as invariant distribution.
Stated in Sections 2.2-2.3 with citation [5]; this is standard invariant-measure theory for elliptic diffusions.
standard math The leapfrog integrator is symplectic, volume-preserving, and reversible, so the Metropolis correction yields exact pi.
Used in Sections 3.2 and 4.1; standard geometric integration theory.
standard math The Metropolis-Hastings accept/reject step removes discretization bias and preserves pi.
Section 4, Eq. (6); standard Markov chain theory.
domain assumption The randomized step-size Metropolis ratio in Eq. (13) preserves pi exactly.
Section 5.5, Eq. (13); asserted with citation to [55] but not proved in the paper. The RS-MAKLA recipe depends on this.
domain assumption No fixed global metric can whiten both the neck and mouth of Neal's funnel, so identity metric is a fair baseline there.
Sections 5.3 and A.2; geometric intuition about exponential variation in the x-block Hessian, not a formal proof.
standard math The Bernstein-von Mises theorem makes centered posteriors approximately Gaussian in large-data regimes, so global preconditioning may suffice.
Section 5.3; standard asymptotic Bayesian result invoked to justify why centered parameterizations can be harmless with enough data.
standard math At stationarity of the identity Gaussian target, X0 and P0 are independent N(0,I), and the expected No-U-Turn criteria are d sin t and d sin(2t).
Sections 2.4 and 4.2; direct Gaussian calculations used to motivate the quarter-period interpretation of NUTS.
ad hoc to paper The No-U-Turn condition mimics the quarter-period exact HMC sampler.
Section 4.2; a heuristic intuition, with a footnote acknowledging that the realized NUTS stopping time is random and need not equal the quarter-period state.

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Cite this review

Pith. "Pith review of From Continuous Dynamics to Practical Gradient-Based Samplers." pith.science (2026). https://pith.science/paper/IWQETABZ

@misc{pith2026260805425,
  author       = {Pith},
  title        = {Pith review of: From Continuous Dynamics to Practical Gradient-Based Samplers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWQETABZ}},
  note         = {Machine review of arXiv:2608.05425}
}

read the original abstract

Gradient-based Markov chain Monte Carlo methods are often introduced as a catalog of algorithms: Hamiltonian Monte Carlo (HMC), the Metropolis-adjusted Langevin algorithm (MALA), the No-U-Turn Sampler (NUTS), and several underdamped variants. This presentation obscures the common structure of the methods and, more importantly, the reasons why a sampler that is correct in principle may be ineffective in practice. We develop a unified account, beginning with exact continuous-time dynamics that represent idealized sampling methods and for which Metropolis adjustments are not required. Numerical discretization makes the dynamics computationally feasible but introduces bias. Metropolis adjustment removes the asymptotic bias by converting numerical errors into rejection, leading to HMC, MALA, NUTS, and the Metropolis-adjusted kinetic Langevin algorithm (MAKLA). The second half of the paper presents geometric design choices that determine practical performance, namely, although MAKLA and NUTS have nice theoretical properties, their sampling efficiency may be slow in practice. Importantly, a fixed mass matrix can whiten globally anisotropic targets, often fixing sampling inefficiency in Bayesian posteriors with large data. Whereas hierarchical posteriors introduce their own problem, causing state-dependent variation in the Hessian (e.g., Neal's funnel). We explain how a randomized step size can be used effectively to sample from such a distribution. The resulting paper is both a tutorial on the mechanics of gradient-based sampling and a set of practical recipes to improve sampler performance.

Figures

Figures reproduced from arXiv: 2608.05425 by the authors.

**Figure 2.** Discretization bias for different step sizes of ULA and LM-ULA to compute [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Discretization bias for different h for unadjusted HMC using the leapfrog iteration to compute E[∥x∥] for x ∼ N (0, I) and x ∈ R 25 . With corresponding discretization maps [15, 16, 17] A(x, p) = x + hM−1 p, p B(x, p) = x, p − h 2 ∇U(x) O(x, p) = x, e−γh/2 p + p 1 − e−γhM1/2 ξ , ξ ∼ N (0, I). (5) Classic discretizations of underdamped Langevin dynamics consider palindromic compositions such as BAOAB or OB… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Discretization bias for different step sizes of OBABO and OBABABO to compute [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Evolution of the empirical estimate of E[∥Σ −1/2x∥], where x ∼ N (0, Σ), λmin(Σ) = 10−2 and λmax(Σ) = 102 , and x ∈ R 25. The left panel compares unadjusted OBABABO with no preconditioning, M = I, and inverseHessian preconditioning, M = Σ, using h = 0.25. The right pa…

**Figure 6.** Figure 6: Evolution of the empirical estimate of E[∥Σ −1/2x∥], where x ∼ t4(0, Σ), λmin(Σ) = 10−2 and λmax(Σ) = 102 , and x ∈ R 25. The left panel compares unadjusted OBABABO with no preconditioning, M = I, using h = 0.1, and inverse-Hessian preconditioning, M = (ν − 2)(ν + d) −…

**Figure 7.** Figure 7: Samples from Neal’s funnel projected onto the [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: Retained samples from Neal’s funnel projected onto a representative pair [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

**Figure 9.** Figure 9: Realized RS-MAKLA step size h against the funnel variable v for Neal’s funnel. The logarithmic horizontal axis makes the local time rescaling explicit: small steps are selected in the high-curvature neck and much larger steps in the low-curvature mouth. The quantitativ…

**Figure 10.** Figure 10: Centered German credit posterior for four representative local scale–coefficient pairs. The black solid curve [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]

**Figure 11.** Figure 11: Minnesota radon posterior for four representative counties. The first two columns show centered NUTS and [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

66 extracted references · 59 canonical work pages

[23]

J. Chok, M. W. Lee, D. Paulin, and G. M. Vasil. Divide, interact, sample: The two-system paradigm, 2026

work page 2026
[55]

Grazzi, S

S. Grazzi, S. Livingstone, and L. Riou-Durand. On randomized step sizes in Metropolis-Hastings algorithms, 2026

work page 2026
[1]

Duane, A

S. Duane, A. Kennedy, B. J. Pendleton, and D. Roweth. Hybrid Monte Carlo.Physics Letters B, 1987

work page 1987
[2]

R. M. Neal. MCMC using Hamiltonian dynamics. In A. Gelman, G. Jones, X.-L. Meng, and S. Brooks, editors, Handbook of Markov Chain Monte Carlo, chapter 6, pages 113–162. Chapman and Hall/CRC, 2011

work page 2011
[3]

Bou-Rabee and J

N. Bou-Rabee and J. M. Sanz-Serna. Randomized Hamiltonian Monte Carlo.The Annals of Applied Probability, 2017

work page 2017
[4]

M. D. Hoffman and A. Gelman. The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo.Journal of Machine Learning Research, 2014

work page 2014
[5]

G. A. Pavliotis.Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Springer New York, 2014

2014
[6]

A. M. Horowitz. A generalized guided Monte Carlo algorithm.Physics Letters B, 1991

work page 1991

Show all 66 references

[7]

Girolami and B

M. Girolami and B. Calderhead. Riemann manifold Langevin and Hamiltonian Monte Carlo methods.Journal of the Royal Statistical Society Series B: Statistical Methodology, 2011

2011
[8]

Riou-Durand and J

L. Riou-Durand and J. V ogrinc. Metropolis adjusted Langevin trajectories: a robust alternative to Hamiltonian Monte Carlo.arXiv preprint arXiv:2202.13230, 2023

2023 arXiv
[9]

Leimkuhler and C

B. Leimkuhler and C. Matthews. Rational construction of stochastic numerical methods for molecular sampling. Applied Mathematics Research eXpress, 2012

2012
[10]

experiments

L. Verlet. Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Physical Review, 1967

1967
[11]

G. Strang. On the construction and comparison of difference schemes.SIAM Journal on Numerical Analysis, 1968

1968
[12]

Bou-Rabee and J

N. Bou-Rabee and J. M. Sanz-Serna. Geometric integrators and the Hamiltonian Monte Carlo method.Acta Numerica, 2018

2018
[13]

Leimkuhler and S

B. Leimkuhler and S. Reich.Simulating Hamiltonian Dynamics. Cambridge University Press, February 2005

2005
[14]

Bilbao, M

S. Bilbao, M. Ducceschi, and F. Zama. Explicit exactly energy-conserving methods for Hamiltonian systems. Journal of Computational Physics, 2023

2023
[15]

Bussi and M

G. Bussi and M. Parrinello. Accurate sampling using Langevin dynamics.Physical Review E, 2007

2007
[16]

R. D. Skeel and J. A. Izaguirre. An impulse integrator for Langevin dynamics.Molecular Physics, 2002

2002
[17]

Melchionna

S. Melchionna. Design of quasisymplectic propagators for Langevin dynamics.The Journal of Chemical Physics, 2007

2007
[18]

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, 1953

1953
[19]

W. K. Hastings. Monte Carlo sampling methods using Markov chains and their applications.Biometrika, 1970

1970
[20]

Kennedy, R

A. Kennedy, R. Edwards, H. Mino, and B. Pendleton. Tuning the generalized hybrid Monte Carlo algorithm. Nuclear Physics B - Proceedings Supplements, 1996

1996
[21]

Riou-Durand, P

L. Riou-Durand, P. Sountsov, J. V ogrinc, C. Margossian, and S. Power. Adaptive tuning for Metropolis adjusted Langevin trajectories. In F. Ruiz, J. Dy, and J.-W. van de Meent, editors,Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, ...

2023
[22]

Bou-Rabee and S

N. Bou-Rabee and S. Oberdörster. Mixing of Metropolis-adjusted Markov chains via couplings: The high acceptance regime.Electronic Journal of Probability, 2024

2024
[24]

R. H. Swendsen and J.-S. Wang. Replica Monte Carlo simulation of spin-glasses.Physical Review Letters, 1986. 19 Continuous Dynamics to Practical Gradient-Based SamplersA PREPRINT

1986
[25]

Hukushima and K

K. Hukushima and K. Nemoto. Exchange Monte Carlo method and application to spin glass simulations.Journal of the Physical Society of Japan, 1996

1996
[26]

D. J. Earl and M. W. Deem. Parallel tempering: Theory, applications, and new perspectives.Physical Chemistry Chemical Physics, 2005

2005
[27]

Miasojedow, E

B. Miasojedow, E. Moulines, and M. Vihola. An adaptive parallel tempering algorithm.Journal of Computational and Graphical Statistics, 2013

2013
[28]

W. D. V ousden, W. M. Farr, and I. Mandel. Dynamic temperature selection for parallel tempering in Markov chain Monte Carlo simulations.Monthly Notices of the Royal Astronomical Society, 2015

2015
[29]

S. Syed, A. Bouchard-Côté, G. Deligiannidis, and A. Doucet. Non-reversible parallel tempering: A scalable highly parallel MCMC scheme.Journal of the Royal Statistical Society Series B: Statistical Methodology, 2021

2021
[30]

B. P. Welford. Note on a method for calculating corrected sums of squares and products.Technometrics, 1962

1962
[31]

T. F. Chan, G. H. Golub, and R. J. Leveque. Algorithms for computing the sample variance: Analysis and recommendations.The American Statistician, 1983

1983
[32]

Andrieu and J

C. Andrieu and J. Thoms. A tutorial on adaptive MCMC.Statistics and Computing, 2008

2008
[33]

Haario, E

H. Haario, E. Saksman, and J. Tamminen. An adaptive Metropolis algorithm.Bernoulli, 2001

2001
[34]

G. O. Roberts and J. S. Rosenthal. Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. Journal of Applied Probability, 2007

2007
[35]

Clarté, A

G. Clarté, A. Diez, and J. Feydy. Collective proposal distributions for nonlinear MCMC samplers: Mean-field theory and fast implementation.Electronic Journal of Statistics, 2022

2022
[36]

Sprungk, S

B. Sprungk, S. Weissmann, and J. Zech. Metropolis-adjusted interacting particle sampling.Statistics and Computing, 2025

2025
[37]

Liang, C

F. Liang, C. Liu, and R. J. Carroll.Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples. Wiley, July 2010

2010
[38]

Laitinen and M

P. Laitinen and M. Vihola. An invitation to adaptive Markov chain Monte Carlo convergence theory, 2024

2024
[39]

R. M. Neal. Slice sampling.The Annals of Statistics, 2003

2003
[40]

M. I. Gorinova, D. Moore, and M. D. Hoffman. Automatic reparameterisation of probabilistic programs. In Proceedings of the 37th International Conference on Machine Learning, ICML’20. JMLR.org, 2020

2020
[41]

Xifara, C

T. Xifara, C. Sherlock, S. Livingstone, S. Byrne, and M. Girolami. Langevin diffusions and the Metropolis-adjusted Langevin algorithm.Statistics & Probability Letters, 2014

2014
[42]

Roy and L

V . Roy and L. Zhang. Convergence of position-dependent MALA with application to conditional simulation in GLMMs.Journal of Computational and Graphical Statistics, 2022

2022
[43]

Hsieh, A

Y .-P. Hsieh, A. Kavis, P. Rolland, and V . Cevher. Mirrored Langevin dynamics.Advances in Neural Information Processing Systems, 2018

2018
[44]

Chok and D

J. Chok and D. Petzinna. Constrained Dikin–Langevin diffusion for polyhedra.IMA Journal of Applied Mathematics, 2026

2026
[45]

Y . F. Atchadé. An adaptive version for the Metropolis adjusted Langevin algorithm with a truncated drift. Methodology and Computing in Applied Probability, 2006

2006
[46]

Brosse, A

N. Brosse, A. Durmus, É. Moulines, and S. Sabanis. The tamed unadjusted Langevin algorithm.Stochastic Processes and their Applications, 2019

2019
[47]

Hutzenthaler, A

M. Hutzenthaler, A. Jentzen, and P. E. Kloeden. Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients.The Annals of Applied Probability, 2012

2012
[48]

S. Sabanis. A note on tamed Euler approximations.Electronic Communications in Probability, 2013

2013
[49]

Johnston, N

T. Johnston, N. Makras, and S. Sabanis. Taming the interacting particle Langevin algorithm: The superlinear case. Applied Mathematics & Optimization, 2025

2025
[50]

Leroy, B

A. Leroy, B. Leimkuhler, J. Latz, and D. J. Higham. Adaptive stepsize algorithms for Langevin dynamics.SIAM Journal on Scientific Computing, 2024. 20 Continuous Dynamics to Practical Gradient-Based SamplersA PREPRINT

2024
[51]

Leimkuhler, R

B. Leimkuhler, R. Lohmann, and P. Whalley. A Langevin sampling algorithm inspired by the Adam optimizer, 2025

2025
[52]

Bou-Rabee, B

N. Bou-Rabee, B. Carpenter, T. S. Kleppe, and M. Marsden. Incorporating local step-size adaptivity into the No-U-Turn sampler using Gibbs self-tuning.The Journal of Chemical Physics, 2025

2025
[53]

Bou-Rabee, B

N. Bou-Rabee, B. Carpenter, and M. Marsden. GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo.Statistics Surveys, 2026

2026
[54]

Biron-Lattes, N

M. Biron-Lattes, N. Surjanovic, S. Syed, T. Campbell, and A. Bouchard-Côté. autoMALA: Locally adaptive Metropolis-adjusted Langevin algorithm, 2023

2023
[56]

Tierney and A

L. Tierney and A. Mira. Some adaptive Monte Carlo methods for Bayesian inference.Statistics in Medicine, 1999

1999
[57]

A. Mira. On Metropolis-Hastings algorithms with delayed rejection.Metron - International Journal of Statistics, 2001

2001
[58]

C. Modi, A. Barnett, and B. Carpenter. Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions.Bayesian Analysis, 2024

2024
[59]

Turok, C

G. Turok, C. Modi, and B. Carpenter. Sampling from multiscale densities with delayed rejection Generalized Hamiltonian Monte Carlo, 2024

2024
[60]

Paulin, P

D. Paulin, P. A. Whalley, N. K. Chada, and B. J. Leimkuhler. Sampling from Bayesian neural network posteriors with symmetric minibatch splitting Langevin dynamics. InProceedings of The 28th International Conference on Artificial Intelligence and Statistics. PMLR, 2025

2025
[61]

X. Zuo, S. Osher, and W. Li. Gradient-adjusted underdamped Langevin dynamics for sampling.SIAM/ASA Journal on Uncertainty Quantification, 2025

2025
[62]

H. Hofmann. Statlog (German credit data). https://archive.ics.uci.edu/dataset/144, 1994. Accessed: 2026-08-01

1994
[63]

Gelman and J

A. Gelman and J. Hill.Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press, December 2006. 21 Continuous Dynamics to Practical Gradient-Based SamplersA PREPRINT A Applications This appendix compares NUTS with the randomized step-size MA...

2006
[64]

Reparameterize to a non-centered form when available and when it improves the posterior geometry
[65]

Otherwise, use NUTS as the default for moderate-dimensional CPU-based inference, especially when warmup yields a reliable Euclidean metric, and the posterior does not exhibit an extreme neck
[66]

draws_matrix

RS-MAKLA is most compelling when the posterior remains strongly multiscale after global preconditioning, if reparameterization is computationally impractical, when predictable one-step kernels are needed, or when many chains can be vectorized on accelerators. Indeed, [23] expl...

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [23]

J. Chok, M. W. Lee, D. Paulin, and G. M. Vasil. Divide, interact, sample: The two-system paradigm, 2026

work page 2026

[2] [55]

Grazzi, S

S. Grazzi, S. Livingstone, and L. Riou-Durand. On randomized step sizes in Metropolis-Hastings algorithms, 2026

work page 2026

[3] [1]

Duane, A

S. Duane, A. Kennedy, B. J. Pendleton, and D. Roweth. Hybrid Monte Carlo.Physics Letters B, 1987

work page 1987

[4] [2]

R. M. Neal. MCMC using Hamiltonian dynamics. In A. Gelman, G. Jones, X.-L. Meng, and S. Brooks, editors, Handbook of Markov Chain Monte Carlo, chapter 6, pages 113–162. Chapman and Hall/CRC, 2011

work page 2011

[5] [3]

Bou-Rabee and J

N. Bou-Rabee and J. M. Sanz-Serna. Randomized Hamiltonian Monte Carlo.The Annals of Applied Probability, 2017

work page 2017

[6] [4]

M. D. Hoffman and A. Gelman. The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo.Journal of Machine Learning Research, 2014

work page 2014

[7] [5]

G. A. Pavliotis.Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Springer New York, 2014

2014

[8] [6]

A. M. Horowitz. A generalized guided Monte Carlo algorithm.Physics Letters B, 1991

work page 1991