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High-Order Langevin Monte Carlo Algorithms

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proposes $P$-th order Langevin Monte Carlo algorithms and claims mixing times of order $O(d^{1/R}/\epsilon^{1/(2R)})$ for $R=4$ ($P=3$) and $R=2P-1$ ($P\ge 4$).

desk verdict The construction is real, but the advertised epsilon exponent is not the one proved; with the corrected 2/R rate and the strong H2 condition disclosed, this is a credible within-subfield extension that needs revision before acceptance. read the letter →

arxiv 2508.17545 v1 pith:LFZAN2Q5 submitted 2025-08-24 stat.ML cs.LGmath.PR

Thanh Dang , Mert Gurbuzbalaban , Mohammad Rafiqul Islam , Nian Yao , Lingjiong Zhu This is my paper · ORCID
classification stat.MLcs.LGmath.PR
keywords LangevinMonteCarlohigh-orderdynamicslog-concavesamplingWassersteinconvergencemixingtimesplittingschemepolynomialgradientapproximationMarkovchain
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The pith

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

The reading

This paper proposes a family of sampling algorithms based on discretizing $P$-th order Langevin dynamics for any $P\ge 3$, and argues that the mixing time improves as $P$ grows. Each step of the algorithm splits the dynamics into $P-1$ stages: the nonlinear coordinate is updated from a high-degree Taylor approximation of the gradient, while the remaining coordinates are integrated exactly as a linear Ornstein-Uhlenbeck process, so the next iterate is conditionally Gaussian. For strongly convex, $L$-smooth potentials satisfying an extra high-order derivative-growth condition, the claimed 2-Wasserstein mixing time is $O(d^{1/R}/\epsilon^{1/(2R)})$ with $R=4$ for $P=3$ and $R=2P-1$ for $P\ge 4$. If correct, this gives an explicit positive answer to the conjecture that higher-order Langevin dynamics can reduce both the dimension and accuracy dependence of log-concave sampling.

What carries the argument

The load-bearing object is the $P-1$-stage splitting discretization of the $P$-th order Langevin dynamics. In each stage, the coordinate that carries the nonlinear potential, $v_1$, is advanced with a polynomial approximation $g^{\mathrm{st}_j}(t)=P_{\alpha-1}(\theta^{\mathrm{st}_{j-1}}(t))$ of the gradient, while the remaining coordinates evolve as a linear multivariate Ornstein-Uhlenbeck process that can be integrated exactly; the whole scheme is built so that $\mathbb{E}[x^{(k+1)}\mid x^{(k)}]$ is multivariate normal. The argument also uses a symmetric positive-definite matrix $M$ from the continuous-time contraction theory, which supplies a dimension-free dissipation rate $\rho$ and lets the discretization error be tracked in the $M$-norm. Condition H2 is the matching condition that makes the Taylor remainder contribute at the same order $\eta^R$ as the splitting error; without it the polynomial approximation would dominate and the rate would not improve with $P$.

What would settle it

Check the exponent by solving the paper's constraint $C d \eta^R \le \epsilon^2/2$ for $\eta$ and substituting into the mixing-time factor $1/(h\eta)$; the $\epsilon$-exponent this returns can be compared with the theorem's stated $\epsilon^{-1/(2R)}$. Numerically, run the fourth-order algorithm on a quadratic potential with known covariance, record the largest stepsize reaching a fixed $W_2$ error, and compare that scaling with the theorem's formula.

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Extended reading notes

Core claim

Under Condition H1 ($U$ is $m$-strongly convex and $L$-smooth) and Condition H2 (the scaled operator norms $L_\alpha/\alpha!$ of the high-order derivatives are controlled relative to $d$ and the stepsize), the paper's $P$-th order Langevin Monte Carlo algorithm is claimed to have 2-Wasserstein mixing time $O(d^{1/R}/\epsilon^{1/(2R)})$ with $R=4\cdot 1_{\{P=3\}}+(2P-1)\cdot 1_{\{P\ge 4\}}$. The algorithm is a $P-1$-stage splitting discretization: at each stage the velocity component that feels the potential is updated using a degree $\alpha-1$ multivariate Taylor polynomial of $\nabla U$, the remaining components are integrated as an exactly solvable Ornstein-Uhlenbeck process, and the next iterate is drawn from the resulting multivariate normal distribution. The proof bounds the one-step discretization error by $C d \eta^R$ plus an exponentially decaying initialization term, using a dimension-free contraction matrix for the continuous dynamics; Condition H2 is exactly the statement that the polynomial approximation error is no larger than the splitting error.

Load-bearing premise

The whole rate rests on Condition H2: the potential's very high derivatives, after factorial scaling, must stay small enough relative to the dimension and the stepsize that a polynomial approximation of the gradient is no worse than the splitting error; for non-polynomial smooth potentials this becomes harder to satisfy as the target accuracy shrinks.

Editorial extensions

If this is right

  • For $P=4$, the paper claims a mixing time of $O(d^{1/7}/\epsilon^{1/14})$, improving on the third-order rate $O(d^{1/4}/\epsilon^{1/2})$ in both dimension and accuracy.
  • Each iteration is a single Gaussian draw with closed-form mean and covariance when the potential is quadratic, so the per-step cost is set by sampling a $Pd$-dimensional normal and forming the polynomial coefficients.
  • For non-polynomial smooth losses such as logistic regression, the required Taylor coefficients are approximated to fixed degree, which is what the numerical experiments use.
  • If the central claim holds, it settles the conjecture that high-order Langevin discretizations reduce the dependence on dimension and accuracy, at least for potentials satisfying Condition H2.
  • As $P$ increases, $R=2P-1$ grows linearly, so the claimed dimension and accuracy exponents improve; the practical bottleneck moves to computing higher-degree Taylor coefficients of $\nabla U$.

Reading between the lines

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

  • The same Gaussian-transition construction should extend to stochastic-gradient settings, with an extra gradient-noise term added to the discretization error; the paper does not analyze this variant.
  • The $P-1$-stage limit suggests a tradeoff: more stages raise the splitting order but require stronger smoothness, so for non-polynomial potentials the optimal order may be finite rather than the $P\to\infty$ improvement suggested by the rate formula.
  • A non-polynomial implementation with higher-degree Taylor or Chebyshev approximations could be tested against the fourth-order algorithm's observed gains, isolating how much of the improvement comes from the splitting order versus the polynomial degree.
  • The dimension-free contraction constants suggest that in practice the Condition H2 constant $c$ and the choice of friction $\gamma$ will matter as much as the asymptotic $d^{1/R}$ term.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes P-th order Langevin Monte Carlo algorithms (P >= 3) based on splitting discretizations of generalized Langevin dynamics, with the gradient inside each stage replaced by a Taylor polynomial. Under an m-strongly convex, L-smooth potential (Condition H1) and an additional high-order smoothness/growth condition (Condition H2), it claims 2-Wasserstein mixing time O(d^{1/R}/epsilon^{1/(2R)}) with R=4 for P=3 and R=2P-1 for P>=4, thereby improving on the third-order result of MMW+21. The proof borrows a continuous-time contraction estimate from Monmarché, bounds the one-step discretization error against the continuous dynamics, and then solves for the step size and number of iterations. Numerical experiments on Bayesian linear and logistic regression compare third- and fourth-order variants.

Significance. If the advertised rate were established, the paper would make a useful incremental contribution: it gives a concrete, implementable family of high-order LMC samplers and a plausible proof route via splitting plus Taylor approximation, with a dimension-independent contraction estimate imported in a transparent way. The authors are explicit that Condition H2 is stronger than the corresponding assumption in MMW+21, and they provide worked formulas for the Gaussian updates in the fourth-order case. However, the central advertised scaling is not the scaling proved in the displayed equations: the proof yields epsilon^{-2/R}, not epsilon^{-1/(2R)}. Since the main quantitative claim of the abstract and Theorem 2.19 is the load-bearing contribution, the paper currently overstates its own result. The contribution remains valuable once the rate is corrected to epsilon^{-2/R} and the missing proofs are supplied.

major comments (4)
  1. [Abstract and Theorem 2.19] The advertised mixing time O(d^{1/R}/epsilon^{1/(2R)}) is not what the proof gives. Proposition 2.26 bounds the discretization error by eC3 d eta^R (up to the contraction term). Solving eC3 d eta^R <= epsilon^2/2 for eta yields eta = (epsilon^2/(2 eC3 d))^{1/R}, and the contraction term e^{-(k+1)h eta} <= epsilon^2/2 requires k of order 1/(h eta), so k* is of order d^{1/R}/epsilon^{2/R}. This is exactly the exponent stated in Theorem 2.9 for P=4, namely epsilon^{2/7}, so the 1/(2R) exponent in the abstract and in Theorem 2.19 is internally inconsistent with the paper's own equations and should be corrected to 2/R.
  2. [Appendix C, 'Proof of Theorem 2.19'] The proof of the general-P theorem is absent: the appendix contains literally 'Proof of Theorem 2.19. □', and the main-text proof only says the argument is the same as for Theorem 2.9. Since the general-P discretization error in Proposition 2.26 is the load-bearing ingredient and its proof depends on the skipped details of Lemma 2.25, the theorem is not proven as printed. A complete derivation of the claimed mixing time from Proposition 2.26 must be supplied.
  3. [Lemma C.4] The proof of Lemma C.4 is omitted with the statement that it is very similar to the proof of Lemma B.4. This lemma provides the moment bound E||x(k+1)||^{2alpha} <= (eC1)^alpha (d+2alpha)^alpha that is used in Lemma 2.25 and hence in Proposition 2.26. Because the stage-wise difference exponents in Lemma 2.25 rely on the combination of this moment bound with Condition H2, the omission concerns a load-bearing ingredient rather than a routine verification. A full proof should be included in a revised version.
  4. [Condition H2 and abstract] Condition H2 is load-bearing for the stated improvement, but the abstract and the contribution bullet in Section 1 advertise the result as holding for convex-smooth potentials without conveying the assumption's strength. The condition contains eta on the right-hand side and, as Remark 2.7 acknowledges, is much stronger than Assumption 2 of MMW+21. The advertised 'convex-smooth' guarantee is therefore misleading; the condition should be stated explicitly wherever the main rate is announced.
minor comments (4)
  1. [Proposition 2.18] The statement displays the discretization bound as C3 d eta^8, while the proof concludes with C3 d eta^7 and Theorem 2.9 solves C3 d eta^7 <= epsilon^2/2. These should be harmonized.
  2. [Proposition 2.26] The phrase 'it holds when and P>=4' is garbled and should read 'for P>=4' (and similarly a separate statement for P=3).
  3. [Theorem 2.19] The step-size definition uses eta0 := (epsilon^2/(2 eC1 d))^{1/R} with eC1 from Lemma C.4, while the discretization error that is solved for uses eC3 from Proposition 2.26; the notation should be aligned so that the displayed eta0 matches the constant in the equation actually solved.
  4. [Lemma 2.25, item (b)] The right-hand side writes v^{stj-1}_n(t) where the index should presumably be P-j; the current notation is confusing and should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central mixing-time derivation is a genuine discretization analysis; the main defects are an exponent inconsistency and an omitted proof, not circular reasoning.

full rationale

The paper's derivation chain is not circular. The continuous-time contraction estimate (Theorem 2.1, with the matrix M and rate ρ) is imported from Monmarché [Mon23], an external result, and is used as a black box; the paper's own contribution is the splitting/Taylor discretization. Lemmas 2.16, 2.17, 2.23–2.26 and C.1–C.4 bound the Taylor remainder, the stage differences, and the one-step discretization error. Proposition 2.26 ends with E‖x((k+1)η)−x^(k+1)‖² ≤ eC3 d η^R + eC4 e^(−(k+1)hη) E|Z−x(0)|², where R = 4 for P = 3 and R = 2P−1 for P ≥ 4. The mixing time is then obtained by equating eC3 d η^R with ε²/2 and solving for k; this is a genuine derivation, not a fitted value. Condition H2 is an explicit sufficient condition: its right-hand side contains η^R, and Remark 2.7 openly acknowledges that it is much stronger than Assumption 2 of MMW+21 and that it is designed to make the polynomial-approximation error be dominated by the splitting error. This is a transparent assumption, not a renamed conclusion, so it does not make the argument circular. The self-citations in the references (e.g., GGZ20, GGZ22, GGHZ21, GIWZ24) are background literature citations and do not carry the derivation. I therefore find no circular step. I do flag, as non-circular correctness risks: (i) the abstract and Theorem 2.19 advertise a mixing-time exponent ε^(−1/(2R)), whereas solving the paper's own Proposition 2.26 inequality for η gives ε^(−2/R), and Theorem 2.9 itself states ε^(−2/7) for P = 4; (ii) the general proof of Theorem 2.19 is missing, being reduced to 'Proof of Theorem 2.19. □' at the end of Appendix C. These concern correctness, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numbers are fitted to data in the derivation. Algorithmic hyperparameters gamma, eta, and Taylor degree alpha are user inputs of the method, not fitted constants; the proof treats them as free but does not optimize them against data. No new physical entities are introduced.

assumptions (4)
  • domain assumption Condition H1: U is m-strongly convex and L-smooth, mI_d <= grad^2 U(x) <= LI_d for all x.
    Standard log-concavity and smoothness condition used for the continuous-time contraction and for L-smoothness bounds.
  • ad hoc to paper Condition H2: there exists c independent of d and integer alpha large enough that (L_alpha/alpha!)^2 (e^{C1})^alpha (d+2alpha)^alpha <= c d (1_{P=3} eta^4 + 1_{P>=4} eta^{2P-1}).
    Tailored to make the Taylor polynomial error O(d eta^R); depends on eta and d; much stronger than Assumption 2 in MMW+21 (Remark 2.7).
  • domain assumption Continuous-time contraction (Monmarche, Theorem 9): for gamma >= gamma0 there is a positive definite M with MJ_b + J_b^T M <= -2 rho M and lambda_min,M, lambda_max,M independent of d.
    Pulled from the prior literature; the paper reproves the dimension-independence claim in Corollary A.5.
  • standard math Taylor's theorem with remainder and Faa di Bruno's formula for derivatives of grad U along polynomial paths.
    Used in Lemma 2.16 and Appendix D for the polynomial approximation error.

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Pith. "Pith review of High-Order Langevin Monte Carlo Algorithms." pith.science (2026). https://pith.science/paper/LFZAN2Q5

@misc{pith2026250817545,
  author       = {Pith},
  title        = {Pith review of: High-Order Langevin Monte Carlo Algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LFZAN2Q5}},
  note         = {Machine review of arXiv:2508.17545}
}
abstract

Langevin algorithms are popular Markov chain Monte Carlo (MCMC) methods for large-scale sampling problems that often arise in data science. We propose Monte Carlo algorithms based on the discretizations of $P$-th order Langevin dynamics for any $P\geq 3$. Our design of $P$-th order Langevin Monte Carlo (LMC) algorithms is by combining splitting and accurate integration methods. We obtain Wasserstein convergence guarantees for sampling from distributions with log-concave and smooth densities. Specifically, the mixing time of the $P$-th order LMC algorithm scales as $O\left(d^{\frac{1}{R}}/\epsilon^{\frac{1}{2R}}\right)$ for $R=4\cdot 1_{\{ P=3\}}+ (2P-1)\cdot 1_{\{ P\geq 4\}}$, which has a better dependence on the dimension $d$ and the accuracy level $\epsilon$ as $P$ grows. Numerical experiments illustrate the efficiency of our proposed algorithms.

Figures

Figures reproduced from arXiv: 2508.17545 by the authors.

Figure 1
Figure 1. Comparative performance of the 3rd- and 4th-order Langevin Monte Carlo algorithms The tuned hyperparameters for the third-order Langevin dynamics γ = 5, η = 0.011, and ξ = 2, and for the fourth-order Langevin dynamics γ = 1 and η = 0.011. For both dynamics, we draw N = 1, 000 samples from the posterior distribution and compute the W2 (2-Wasserstein) distance from the known posterior defined in (45). The shaded regio… view at source ↗
Figure 2
Figure 2. Comparative performance of the 3rd- and 4th-order Langevin Monte Carlo algorithms for the same stepsize η and varying the friction parameter γ in terms of the 2-Wasserstein distance for smaller γ values. However, this is not always the case for the 3rd-order LMC algorithm. 3.2. Bayesian logistic regression. In this section, we provide the implementation of the 4th-order LMC algorithm for sampling in a classification… view at source ↗
Figure 3
Figure 3. Performance of the 4th-order LMC algorithm in sampling from a non-polynomial potential function the model parameters and run a grid search for the hyperparameters η and γ. To avoid overfitting, we use a larger penalty λ = 25. From [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

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Works this paper leans on

50 extracted references · 28 canonical work pages

  1. [1]

    An introduction to MCMC for machine learning

    Christophe Andrieu, Nando De Freitas, Arnaud Doucet, and Michael I Jordan. An introduction to MCMC for machine learning. Machine Learning , 50(1):5--43, 2003

    2003

  2. [2]

    Sharp entropy decay for hypocoercive and non-symmetric F okker- P lanck equations with linear drift

    Anton Arnold and Jan Erb. Sharp entropy decay for hypocoercive and non-symmetric F okker- P lanck equations with linear drift. arXiv preprint arXiv:1409.5425 , 2014

    arXiv 2014

  3. [3]

    Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODE s to linear kinetic equations

    Anton Arnold, Shi Jin, and Tobias W \"o hrer. Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODE s to linear kinetic equations. Journal of Differential Equations , 268(3):1156--1204, 2020

    work page 2020

  4. [4]

    Towards a theory of non-log-concave sampling: First-order stationarity guarantees for L angevin M onte C arlo

    Krishna Balasubramanian, Sinho Chewi, Murat A Erdogdu, Adil Salim, and Shunshi Zhang. Towards a theory of non-log-concave sampling: First-order stationarity guarantees for L angevin M onte C arlo. In Proceedings of Thirty Fifth Conference on Learning Theory , volume 178, pages 2896--2923. PMLR, 2022

    work page 2022

  5. [5]

    On stochastic gradient L angevin dynamics with dependent data streams in the logconcave case

    Mathias Barkhagen, Ngoc Huy Chau, \' E ric Moulines, Mikl\' o s R\' a sonyi, Sotirios Sabanis, and Ying Zhang. On stochastic gradient L angevin dynamics with dependent data streams in the logconcave case. Bernoulli , 27(1):1--33, 2021

    work page 2021

  6. [6]

    Differential Calculus

    Henri Cartan. Differential Calculus . Hermann, 1971

    work page 1971

  7. [7]

    Bartlett

    Xiang Cheng and Peter L. Bartlett. Convergence of L angevin MCMC in KL -divergence. In Proceedings of the 29th International Conference on Algorithmic Learning Theory (ALT) , volume 83, pages 186--211. PMLR, 2018

    2018

  8. [8]

    Chatterji , Yasin Abbasi-Yadkori , Peter L

    Xiang Cheng , Niladri S. Chatterji , Yasin Abbasi-Yadkori , Peter L. Bartlett , and Michael I. Jordan . Sharp Convergence Rates for L angevin Dynamics in the Nonconvex Setting . arXiv:1805.01648 , 2018

    arXiv 2018

Show all 50 references
  1. [9]

    Underdamped L angevin MCMC : A non-asymptotic analysis

    Xiang Cheng, Niladri S Chatterji, Peter L Bartlett, and Michael I Jordan. Underdamped L angevin MCMC : A non-asymptotic analysis. In Conference on learning theory , pages 300--323. PMLR, 2018

    2018

  2. [10]

    Diffusion for global optimization in R ^n

    Tzuu-Shuh Chiang, Chii-Ruey Hwang, and Shuenn Jyi Sheu. Diffusion for global optimization in R ^n . SIAM Journal on Control and Optimization , 25(3):737--753, 1987

    1987

  3. [11]

    Complexity of randomized algorithms for underdamped L angevin dynamics

    Yu Cao, Jianfeng Lu, and Lihan Wang. Complexity of randomized algorithms for underdamped L angevin dynamics. Communications in Mathematical Sciences , 19(7):1827--1853, 2021

    2021

  4. [12]

    On explicit L^ 2 -convergence rate estimate for underdamped L angevin dynamics

    Yu Cao, Jianfeng Lu, and Lihan Wang. On explicit L^ 2 -convergence rate estimate for underdamped L angevin dynamics. Archive for Rational Mechanics and Analysis , 247(90):1--34, 2023

    2023

  5. [13]

    On stochastic gradient L angevin dynamics with dependent data streams: the fully non-convex case

    Ngoc Huy Chau, \' E ric Moulines, Miklos R\' a sonyi, Sotirios Sabanis, and Ying Zhang. On stochastic gradient L angevin dynamics with dependent data streams: the fully non-convex case. SIAM Journal of Mathematics of Data Science , 3(3):959--986, 2021

    2021

  6. [14]

    Theoretical guarantees for approximate sampling from smooth and log-concave densities

    Arnak S Dalalyan. Theoretical guarantees for approximate sampling from smooth and log-concave densities. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 79(3):651--676, 2017

    2017

  7. [15]

    Dalalyan and Avetik G

    Arnak S. Dalalyan and Avetik G. Karagulyan. User-friendly guarantees for the L angevin M onte C arlo with inaccurate gradient. Stochastic Processes and their Applications , 129(12):5278--5311, 2019

    2019

  8. [16]

    Non-asymptotic convergence analysis for the U nadjusted L angevin A lgorithm

    Alain Durmus and Eric Moulines. Non-asymptotic convergence analysis for the U nadjusted L angevin A lgorithm. Annals of Applied Probability , 27(3):1551--1587, 2017

    2017

  9. [17]

    High-dimensional B ayesian inference via the U nadjusted L angevin A lgorithm

    Alain Durmus and Eric Moulines. High-dimensional B ayesian inference via the U nadjusted L angevin A lgorithm. Bernoulli , 25(4A):2854--2882, 2019

    2019

  10. [18]

    Efficient B ayesian computation by proximal M arkov C hain M onte C arlo: When L angevin meets M oreau

    Alain Durmus, Eric Moulines, and Marcelo Pereyra. Efficient B ayesian computation by proximal M arkov C hain M onte C arlo: When L angevin meets M oreau. SIAM Journal on Imaging Sciences , 11(1):473--506, 2018

    2018

  11. [19]

    On sampling from a log-concave density using kinetic L angevin diffusions

    Arnak S Dalalyan and Lionel Riou-Durand. On sampling from a log-concave density using kinetic L angevin diffusions. Bernoulli , 26(3):1956--1988, 2020

    1956

  12. [20]

    Couplings and quantitative contraction rates for L angevin dynamics

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

    1982

  13. [21]

    Erdogdu and Rasa Hosseinzadeh

    Murat A. Erdogdu and Rasa Hosseinzadeh. On the convergence of L angevin M onte C arlo: The interplay between tail growth and smoothness. In Proceedings of Thirty Fourth Conference on Learning Theory , volume 134, pages 1776--1822. PMLR, 2021

    2021

  14. [22]

    Murat A Erdogdu, Rasa Hosseinzadeh, and Matthew S. Zhang. Convergence of L angevin M onte C arlo in chi-squared and R \' e nyi divergence. In Proceedings of the 25th International Conference on Artificial Intelligence and Statistics , volume 151, pages 8151--8175. PMLR, 2022

    2022

  15. [23]

    A note on T aylor's theorem

    Jos \'e A Facenda Aguirre. A note on T aylor's theorem. The American Mathematical Monthly , 96(3):244--247, 1989

    1989

  16. [24]

    Bayesian Data Analysis

    Andrew Gelman, John B Carlin, Hal S Stern, and Donald B Rubin. Bayesian Data Analysis . Chapman & Hall/CRC Press, 1995

    1995

  17. [25]

    Decentralized stochastic gradient L angevin dynamics and H amiltonian M onte C arlo

    Mert G\"urb\"uzbalaban, Xuefeng Gao, Yunhan Hu, and Lingjiong Zhu. Decentralized stochastic gradient L angevin dynamics and H amiltonian M onte C arlo. Journal of Machine Learning Research , 22(239):1--69, 2021

    2021

  18. [26]

    Breaking reversibility accelerates L angevin dynamics for global non-convex optimization

    Xuefeng Gao, Mert G\" u rb\" u zbalaban, and Lingjiong Zhu. Breaking reversibility accelerates L angevin dynamics for global non-convex optimization. In Advances in Neural Information Processing Systems (NeurIPS) , volume 33, 2020

    2020

  19. [27]

    Global convergence of S tochastic G radient H amiltonian M onte C arlo for non-convex stochastic optimization: Non-asymptotic performance bounds and momentum-based acceleration

    Xuefeng Gao, Mert G\" u rb\" u zbalaban, and Lingjiong Zhu. Global convergence of S tochastic G radient H amiltonian M onte C arlo for non-convex stochastic optimization: Non-asymptotic performance bounds and momentum-based acceleration. Operations Research , 70(5):2931--2947, 2022

    2022

  20. [28]

    Generalized EXTRA stochastic gradient L angevin dynamics

    Mert G\" u rb\" u zbalaban, Mohammad Rafiqul Islam, Xiaoyu Wang, and Lingjiong Zhu. Generalized EXTRA stochastic gradient L angevin dynamics. arXiv preprint arXiv:2412.01993 , 2024

    2024

  21. [29]

    A class of W asserstein metrics for probability distributions

    Clark R Givens and Rae Michael Shortt. A class of W asserstein metrics for probability distributions. The Michigan Mathematical Journal , 31(2):231--240, 1984

    1984

  22. [30]

    calculus : High-dimensional numerical and symbolic calculus in R

    Emanuele Guidotti. calculus : High-dimensional numerical and symbolic calculus in R . Journal of Statistical Software , 104(5):1--37, 2022

    2022

  23. [31]

    Topics in Matrix Analysis

    Roger A Horn and Charles R Johnson. Topics in Matrix Analysis . Cambridge University Press, 1994

    1994

  24. [32]

    Asymptotics of the spectral gap with applications to the theory of simulated annealing

    Richard A Holley, Shigeo Kusuoka, and Daniel W Stroock. Asymptotics of the spectral gap with applications to the theory of simulated annealing. Journal of Functional Analysis , 83(2):333--347, 1989

    1989

  25. [33]

    Machine learning: the basics

    Alexander Jung. Machine learning: the basics . Springer Nature, 2022

    2022

  26. [34]

    Norms on direct sums and tensor products

    Peter Lancaster and Hanafi K Farahat. Norms on direct sums and tensor products. Mathematics of Computation , 26(118):401--414, 1972

    1972

  27. [35]

    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 N esterov acceleration for gradient-based MCMC ? Bernoulli , 27(3):1942--1992, 2021

    1942

  28. [36]

    High-order L angevin diffusion yields an accelerated MCMC algorithm

    Wenlong Mou, Yi-An Ma, Martin J Wainwright, Peter L Bartlett, and Michael I Jordan. High-order L angevin diffusion yields an accelerated MCMC algorithm. Journal of Machine Learning Research , 22(42):1--41, 2021

    2021

  29. [37]

    Almost sure contraction for diffusions on R ^ d

    Pierre Monmarch \'e . Almost sure contraction for diffusions on R ^ d . A pplication to generalized L angevin diffusions. Stochastic Processes and their Applications , 161:316--349, 2023

    2023

  30. [38]

    Ergodicity for SDE s and approximations: locally L ipschitz vector fields and degenerate noise

    Jonathan C Mattingly, Andrew M Stuart, and Desmond J Higham. Ergodicity for SDE s and approximations: locally L ipschitz vector fields and degenerate noise. Stochastic Processes and their Applications , 101(2):185--232, 2002

    2002

  31. [39]

    The Maple Handbook: Maple V Release 4

    Darren Redfern. The Maple Handbook: Maple V Release 4 . Springer Science & Business Media, 2012

    2012

  32. [40]

    Non-convex learning via stochastic gradient L angevin dynamics: a nonasymptotic analysis

    Maxim Raginsky, Alexander Rakhlin, and Matus Telgarsky. Non-convex learning via stochastic gradient L angevin dynamics: a nonasymptotic analysis. In Proceedings of the 2017 Conference on Learning Theory , volume 65, pages 1674--1703. PMLR, 2017

    2017

  33. [41]

    Introduction to Numerical Analysis , volume 1993

    Josef Stoer, Roland Bulirsch, R Bartels, Walter Gautschi, and Christoph Witzgall. Introduction to Numerical Analysis , volume 1993. Springer, 1980

    1993

  34. [42]

    Support Vector Machines

    Ingo Steinwart and Andreas Christmann. Support Vector Machines . Springer Science & Business Media, 2008

    2008

  35. [43]

    The randomized midpoint method for log-concave sampling

    Ruoqi Shen and Yin Tat Lee. The randomized midpoint method for log-concave sampling. In Advances in Neural Information Processing Systems , volume 32, 2019

    2019

  36. [44]

    Inverse problems: A B ayesian perspective

    Andrew M Stuart. Inverse problems: A B ayesian perspective. Acta Numerica , 19:451--559, 2010

    2010

  37. [45]

    Consistency and fluctuations for stochastic gradient L angevin dynamics

    Yee Whye Teh, Alexandre H Thiery, and Sebastian J Vollmer. Consistency and fluctuations for stochastic gradient L angevin dynamics. Journal of Machine Learning Research , 17(1):193--225, 2016

    2016

  38. [46]

    Hypocoercivity

    C\' e dric Villani. Hypocoercivity. Memoirs of the American Mathematical Society , 202(950):iv+141, 2009

    2009

  39. [47]

    Air Quality

    Saverio Vito. Air Quality . UCI Machine Learning Repository, 2008. DOI : https://doi.org/10.24432/C59K5F

    2008 doi

  40. [48]

    Generating multivariate mixture of normal distributions using a modified C holesky decomposition

    Jin Wang and Chunlei Liu. Generating multivariate mixture of normal distributions using a modified C holesky decomposition. In Proceedings of the 2006 Winter Simulation Conference , pages 342--347. IEEE, 2006

    2006

  41. [49]

    A variational perspective on accelerated methods in optimization

    Andre Wibisono, Ashia C Wilson, and Michael I Jordan. A variational perspective on accelerated methods in optimization. Proceedings of the National Academy of Sciences , 113(47):E7351--E7358, 2016

    2016

  42. [50]

    Nonasymptotic estimates for S tochastic G radient L angevin D ynamics under local conditions in nonconvex optimization

    Ying Zhang, \" O mer Deniz Akyildiz, Theodoros Damoulas, and Sotirios Sabanis. Nonasymptotic estimates for S tochastic G radient L angevin D ynamics under local conditions in nonconvex optimization. Applied Mathematics & Optimization , 87:25, 2023

    2023

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.