Pith. sign in

REVIEW 4 major objections 4 minor 1 cited by

kTULA: A Langevin sampling algorithm with improved KL bounds under super-linear log-gradients

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read kTULA reaches near-optimal KL sampling rates on super-linear-gradient targets, with an error exponent arbitrarily close to 2.

desk verdict kTULA's claimed KL rate is not established: the proof leans on a false t-uniform score bound in Lemma B.1, so the main theorem currently rests on an internal error. read the letter →

arxiv 2506.04878 v1 pith:H6ZXHZVV submitted 2025-06-05 math.ST cs.LGmath.PRstat.MLstat.TH

Iosif Lytras , Sotirios Sabanis , Ying Zhang This is my paper
classification math.STcs.LGmath.PRstat.MLstat.TH MSC 60J2565C0565C30
keywords kTULAtamedLangevinalgorithmKullback-Leiblerdivergencenon-asymptoticboundssuper-linearlog-gradientslog-Sobolevinequalitynon-convexoptimization
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

kTULA is a tamed Langevin sampling scheme built for targets whose log-density gradient grows faster than linearly, and the paper proves that its non-asymptotic Kullback-Leibler error to the target decays like the stepsize $\lambda$ raised to a power arbitrarily close to 2. That near-quadratic rate is the best known among tamed Langevin methods for such targets: earlier schemes such as sTULA and mTULA achieved a KL rate of 1 or a Wasserstein rate of 1/2. The proof works under four assumptions covering initial conditions, polynomial local Lipschitz growth of the Hessian, dissipativity, and a log-Sobolev inequality for the target; the last of these is the genuinely load-bearing one. As corollaries, the KL bound yields a Wasserstein-2 error with rate arbitrarily close to 1, and an expected excess-risk bound for the optimization problem whose proxy is sampling from $\pi_\beta$.

What carries the argument

The engine is the splitting-tamed gradient $$h_\$\lambda$(\$\theta$)=a\$\theta$+\frac{h(\$\theta$)-a\$\theta$}{(1+\$\lambda$|\$\theta$|^{(l+1)/\epsilon_h})^{\epsilon_h}},$$ which keeps the dissipativity of $h$ while making the bias $|h-h_\lambda|^2$ of order $\lambda^2$ and the global Lipschitz constant of $h_\lambda$ only $O(\lambda^{-\epsilon_h})$. The analysis hinges on a functional identity for the time derivative of $\mathrm{KL}(\pi^\lambda_t\|\pi_\beta)$ along the continuous-time interpolation, imported from the sTULA analysis, and a decomposition of the conditional expectation $\mathbb{E}[h_\lambda(\theta^\lambda_n)-h(\theta^\lambda_t)\,|\,\theta^\lambda_t=\theta]$ into four terms. Three of those terms are directly $O(\lambda^2)$; the fourth, involving the score of the discrete density, is controlled by Lemma 4.5's bound $J_n\le C_{J,\epsilon_h}\lambda^{-\epsilon_h}$, which is derived through an entropy-power and Fisher-information recursion. This combination is what pushes the final exponent from $1$ to $2-\epsilon$.

What would settle it

Using the explicit constants from Appendix C, simulate kTULA on the double-well potential $u(\theta)=|\theta|^4/4-|\theta|^2/2$ at $\beta=1$, with $\lambda=10^{-4}$, and estimate $\mathrm{KL}(\pi^\lambda_n\|\pi_\beta)$ by fine-grid numerical integration. If the measured KL ever exceeds $C_1\lambda^{2-\epsilon_h-\epsilon(1-\epsilon_h/2)}$ for the stated $\lambda_{\max}$ constraint and any allowed $\epsilon_h,\epsilon$, the theorem's constants are wrong; if, across a range of $\lambda$, the measured bias exponent is strictly below $2-\epsilon$, the claim of a near-$\lambda^2$ rate is not supported.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 2.4: for any $\beta>0$, $\epsilon_h\in(0,1/2]$, and $\epsilon>0$, there are explicit constants $C_0=3C_{LS}/2$ and $C_1$, given in Appendix C, such that for all $0<\lambda\le\lambda_{\max}$ and $n\in\mathbb{N}_0$, $$\mathrm{KL}(\pi^\lambda_n\|\pi_\$\beta$) \le $e^{{-C_0\lambda n}}$\,\mathrm{KL}(\pi^\lambda_0\|\pi_\$\beta$) + C_1\$lambda^{{2-\epsilon_h-\epsilon(1-\epsilon_h/2)}}$.$$ Because $\epsilon_h$ and $\epsilon$ can be chosen arbitrarily small, the stepsize exponent is $2-o(1)$, and the Wasserstein-2 corollary inherits an exponent $1-o(1)$. The authors view this as the first near-$\lambda^2$ KL guarantee for Langevin-type algorithms under super-linear log-gradients; the comparison section shows prior tamed schemes only reached Wasserstein or KL rates of $1/2$ or $1$. The proof interpolates kTULA in continuous time, derives an exact differential identity for $d\mathrm{KL}/dt$, and bounds each term; the key term is controlled through a uniform Fisher-information estimate $J_n \le C_{J,\epsilon_h}\lambda^{-\epsilon_h}$.

Load-bearing premise

The target must satisfy a log-Sobolev inequality with a constant $C_{LS}$ that appears directly in the contraction coefficient and in every error constant.

Editorial extensions

If this is right

  • For any prescribed KL accuracy $\delta$, choosing $\lambda\approx(\delta/(2C_1))^{1/(2-\epsilon)}$ and $n\approx(1/(C_0\lambda))\log(2\mathrm{KL}_0/\delta)$ delivers $\mathrm{KL}(\pi^\lambda_n\|\pi_\beta)\le\delta$, with the iteration count growing linearly in $\lambda^{-1}$ up to log factors.
  • The Wasserstein-2 guarantee $W_2(\pi^\lambda_n,\pi_\beta)\le C_2(e^{-C_0\lambda n}\mathrm{KL}_0+C_1\lambda^{2-\epsilon})^{1/2}$ carries a near-first-order stepsize rate, improving the $1/2$ rate of mTULA and sTULA.
  • For $\beta$ sufficiently large, kTULA provably finds near-minimizers: $\mathbb{E}[u(\theta^\lambda_n)]-\inf_{\theta}u(\theta)\le C_3e^{-C_0\lambda n/2}+C_4\lambda^{1-\epsilon}+C_5/\beta$, so sampling-based optimization inherits the improved discretization rate.
  • Under the stronger small-$\beta$ and convexity-at-infinity regime, the same algorithm attains an exact Wasserstein-2 rate of 1, as the paper notes in its comparison with prior work.
  • The tamed coefficients guarantee uniform-in-time moment bounds, so super-linear-gradient targets no longer cause finite-time explosion of the iterates.

Reading between the lines

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

  • The $\epsilon$ loss is an artifact of the Young and H\"older steps used to control the score term; a sharper Fisher-information estimate might achieve exactly $\lambda^2$, at the price of a different dimension dependence.
  • The log-Sobolev assumption is the real bottleneck: for multi-modal or high-dimensional targets $C_{LS}$ is typically exponential in the barrier height or dimension, so the displayed constants would be vacuous even though the algorithm itself may continue to mix; replacing LSI with a Poincar\'e inequality would forfeit the KL contraction but might retain a Wasserstein statement.
  • Because the taming and bias-order arguments are independent of the Gaussian noise structure, the same $2-\epsilon$ KL rate plausibly transfers to stochastic-gradient and mini-batch versions of kTULA, which would directly matter for deep-learning applications; this is not shown in the paper.
  • The near-$\lambda^2$ KL rate matches the best possible first-order Euler discretization rate, suggesting that further gains would require higher-order or underdamped schemes rather than better taming.
Share X Bluesky LinkedIn Reddit HN

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 a tamed Euler–Maruyama-type Langevin algorithm, kTULA, for sampling from targets with super-linearly growing log-gradients. Under dissipativity, polynomial local Lipschitz conditions on the gradient/Hessian, initial-density regularity, and a Log–Sobolev inequality for the target, Theorem 2.4 claims a KL-divergence bound with discretization rate λ^{2−ε_h−ε(1−ε_h/2)}, arbitrarily close to λ^2, and Corollaries 2.5 and 2.7 transfer this to Wasserstein-2 and expected excess-risk bounds. The proof route is a continuous-time interpolation, a functional inequality (12), a decomposition into three error terms (14), a novel Fisher-information bound (Lemma 4.5), and explicit constants in Appendix C.

Significance. If the main rate claim is correct, the paper would give the best-known KL discretization rate for tamed Langevin algorithms under super-linear log-gradients, improving the O(λ) rate of [23] and the W2 rates of [28] and [3]. The constants are explicit, the assumptions are clearly stated, and the two applications (double-well potential and a neural-network objective) give concrete targets satisfying the assumptions. The manuscript contains no fitted parameters and the rate claim is falsifiable. However, the proof as submitted relies on an invalid uniform score bound in Appendix B, so the significance is conditional on a substantial repair of the proof.

major comments (4)
  1. [App. B] The claimed uniform linear bound |∇log π_t^λ(θ)| ≤ C0(1+|θ|) is false under Assumptions 1–4. Take d=1, β=1, u(θ)=θ^4/4, so h(θ)=θ^3 and H(θ)=3θ^2. This target satisfies Assumption 2 (l=2), Assumption 3 (a=1, b=1), and Assumption 4 (the density e^{-θ^4/4} satisfies LSI). Let θ0 be drawn from πβ, so π_0^λ=πβ satisfies Assumption 1. For t>0, π_t^λ = πβ ∗ φ_s with s=2λβ^{-1}t. For fixed λ and t sufficiently small, set θ=s^{-1/8}. Since θ ≫ √s, the convolution is concentrated near θ, giving ∇log π_t^λ(θ) = -θ^3(1+o(1)). Hence |∇log π_t^λ(θ)|/(1+|θ|) → ∞ as t→0 along this sequence. No finite C0 independent of t and θ can exist. Thus the proof step replacing (λ(t−n))^{-1}(e^{c1|θ|}+e^{c2}) by C0(1+|θ|) is invalid, and the lemma statement itself is contradicted by a model satisfying the paper's assumptions.
  2. [Sec. 4.4] The functional inequality (12) is the starting point of the proof of Theorem 2.4, and its derivation is attributed to [23, Theorem 7.8] with Lemma B.1 supplying the regularity conditions. Since Lemma B.1's bound (53) is false, the present manuscript does not establish (12). Moreover, the target log-gradient in the admissible example is cubic, so a linear score bound cannot hold; any correct proof of (12) must use polynomial-growth estimates compatible with super-linear log-gradients. Until this is supplied, Theorem 2.4 and Corollaries 2.5 and 2.7 rest on an unproved identity.
  3. [Sec. 4.4] The displayed definition reads J(π_t^λ∥πβ) := (1/β)∫ |∇log π_t^λ − ∇log πβ|^2 dπβ. The subsequent application of Assumption 4 in the proof of Theorem 2.4, however, requires the integral to be with respect to π_t^λ, as in the standard LSI form KL(μ∥ν) ≤ (C_LS/2)∫ |∇log(dμ/dν)|^2 dμ. If the printed dπβ is literal, the contraction step does not follow from LSI. This is likely a typo, but it must be corrected because the stated measure is load-bearing for the argument.
  4. [Sec. 4.3] The statement of Lemma 4.3 bounds π_n^λ at grid points, but the displayed proof works with t∈(n,n+1] and σ²=2λ(t−n)β^{-1}, then concludes a bound for π_{n+1}^λ. The proof does not clearly justify expunging the (t−n) dependence; the natural bound from [29, Proposition 2] at time t would scale like (λ(t−n))^{-1/2}, not λ^{-1}. Since Lemma 4.3 is used in Lemmas 4.4 and 4.5, the proof needs to be rewritten so that the induction step and the t-dependence are consistent.
minor comments (4)
  1. [Abstract] The abstract states a rate 2−\bar{ε}, while Theorem 2.4 gives 2−ε_h−ε(1−ε_h/2); the relation between ε_h, ε, and \bar{ε} should be stated explicitly to avoid confusion.
  2. [Sec. 2.3] The text says Assumption 1 is easily satisfied by 'constants and Gaussian random variables'. A constant initialization is a Dirac measure, which does not have a density with exponential decay and polynomial log-gradient; the statement should say Gaussian (or absolutely continuous) initializations with sufficiently light tails.
  3. [Eq. (12)] Equation (12) is written as an equality, but the right-hand side with coefficient −(3/4)λJ and 4λβ is obtained via Young's inequality and is an upper bound, not an exact identity. The label should be corrected to avoid a false equality claim.
  4. [Throughout] There are typographical issues: 'minimzers' (Section 2.2), 'the second therm' (Appendix B), 'straight forward' (Appendix A.2), and inconsistent use of π_t^λ versus π_t^λ in the interpolation definitions. These should be cleaned up.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the KL/W2 rates are derived from explicitly stated assumptions (LSI, dissipativity, polynomial Lipschitz growth) using external tools; the only self-citations are background theorems, not fitted predictions.

full rationale

The claimed derivation is not circular. Theorem 2.4 is obtained by combining the functional equality (12) with Lemmas 4.4-4.7 and Assumption 4; each lemma is proved in the appendices from Assumptions 1-3 and standard external results ([24, Lemma 7], [29, Proposition 2], [31, Proposition 3]). The tunable parameters epsilon_h and epsilon are proof knobs chosen in the analysis, not fitted to data, and no input quantity is renamed as a prediction. The constant C1 is explicit in Appendix C and depends on known problem parameters (d, beta, C_LS, a, b, KH, Kh, L, initial moments); there is no statistical fit. Self-citations to [22, 23, 28] are used as published theorems and tools rather than as unverified premises that force the conclusion; for instance, the paper sketches the derivation of (12) in Appendix B instead of merely asserting it, and the LSI sufficient conditions from [23, Theorem 5.3] are used only in the application sections. The skeptic's objection about Lemma B.1's uniform bound |log pi_t^lambda| <= C0(1+|theta|) is a potential correctness gap: the manuscript replaces (lambda(t-n))^{-1} by a constant, which is not valid uniformly as t approaches n from above. However, a false or unproved lemma is not circularity: it does not show that the theorem is equivalent to its inputs by construction. Hence the circularity score is low, while the correctness concern should be assessed separately.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim rests on four structural assumptions and several prior theorems used as tools. The analysis parameters epsilon_h and epsilon are arbitrary proof knobs; they improve the lambda-exponent at the cost of constants. There are no fitted parameters and no new unobservable entities.

free parameters (2)
  • epsilon_h
    Taming exponent in h_lambda, equation (5). It can be chosen arbitrarily in (0,1/2]. The final rate and every constant in Appendix C depend on it; making it small improves the lambda-exponent but blows up constants and dimension dependence.
  • epsilon
    Young's inequality parameter introduced in the proof of Lemma 4.4, equation (37). It is arbitrary and positive, appears in the final rate, and controls the tradeoff between the rate and the constants C_D,epsilon,epsilon_h.
assumptions (6)
  • domain assumption Assumption 1: the initial density pi_0^lambda has exponential decay and its log-gradient and log-Hessian have polynomial growth.
    Used in Lemma 4.3, Lemma B.1, and to ensure J_0 is finite. It is stronger than a finite moment assumption but is satisfied by constants and Gaussian initializations.
  • domain assumption Assumption 2: the Hessian H is polynomially Lipschitz with degree l-1 and H and h grow at most polynomially.
    Used throughout Lemmas 4.1, 4.2 and the proof of the functional inequality; it replaces global Lipschitz smoothness.
  • domain assumption Assumption 3: dissipativity <h(theta), theta> >= a|theta|^2 - b.
    Gives contraction of moments, the dissipativity of h_lambda in Lemma 4.1(i), and controls lambda_max in (6).
  • domain assumption Assumption 4: pi_beta satisfies a Log-Sobolev inequality with constant C_LS.
    Directly produces the contraction term in Theorem 2.4 and the Talagrand inequality used in Corollary 2.5. This is the most restrictive structural assumption.
  • standard math Functional equality (12) from [23, Theorem 7.8] and the decomposition from [24] are used as black boxes.
    The paper says (12) can be obtained by the same proof as [23, Theorem 7.8] and then adopts the decomposition (13)-(14) from [24]. These are prior theorems with published proofs.
  • standard math Entropy power and Gaussian smoothing bounds [31, Proposition 3] and [29, Proposition 2].
    Used in Lemma 4.5 to bound the Fisher information of the discrete iterates and in Lemma 4.3 to bound the score of the smoothed density.

how reviews work

0 comments
Cite this review

Pith. "Pith review of kTULA: A Langevin sampling algorithm with improved KL bounds under super-linear log-gradients." pith.science (2026). https://pith.science/paper/H6ZXHZVV

@misc{pith2026250604878,
  author       = {Pith},
  title        = {Pith review of: kTULA: A Langevin sampling algorithm with improved KL bounds under super-linear log-gradients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6ZXHZVV}},
  note         = {Machine review of arXiv:2506.04878}
}
abstract

Motivated by applications in deep learning, where the global Lipschitz continuity condition is often not satisfied, we examine the problem of sampling from distributions with super-linearly growing log-gradients. We propose a novel tamed Langevin dynamics-based algorithm, called kTULA, to solve the aforementioned sampling problem, and provide a theoretical guarantee for its performance. More precisely, we establish a non-asymptotic convergence bound in Kullback-Leibler (KL) divergence with the best-known rate of convergence equal to $2-\overline{\epsilon}$, $\overline{\epsilon}>0$, which significantly improves relevant results in existing literature. This enables us to obtain an improved non-asymptotic error bound in Wasserstein-2 distance, which can be used to further derive a non-asymptotic guarantee for kTULA to solve the associated optimization problems. To illustrate the applicability of kTULA, we apply the proposed algorithm to the problem of sampling from a high-dimensional double-well potential distribution and to an optimization problem involving a neural network. We show that our main results can be used to provide theoretical guarantees for the performance of kTULA.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. RELTA-SGLD: Relative-Growth Localized Taming for Nonconvex Stochastic-Gradient Langevin Learning

    stat.ML 2026-07 conditional novelty 6.0 of 10

    A relative-growth, threshold-localized taming denominator for SGLD achieves O(λ) stationary W1 and (in the potential case) W2 accuracy for nonconvex superlinear stochastic-gradient oracles.

Reference graph

Works this paper leans on

38 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [23]

    Taming under isoperimetry.Stochastic Processes and their Applications, page 104684, 2025

    Iosif Lytras and Sotirios Sabanis. Taming under isoperimetry.Stochastic Processes and their Applications, page 104684, 2025

    2025

  2. [28]

    Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting.Journal of Mathematical Analysis and Applications, 543(1):128892, 2025

    Ariel Neufeld, Matthew Cheng En Ng, and Ying Zhang. Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting.Journal of Mathematical Analysis and Applications, 543(1):128892, 2025

    work page 2025

  3. [3]

    $L^2$-Wasserstein contraction of modified Euler schemes for SDEs with high diffusivity and applications

    Jianhai Bao and Jiaqing Hao. L2-Wasserstein contraction of modified Euler schemes for SDEs with high diffusivity and applications.arXiv preprint arXiv:2411.01731, 2024

    work page Pith review arXiv 2024

  4. [1]

    Smooth sigmoid wavelet shrinkage for non-parametric estimation

    Abdourrahmane M Atto, Dominique Pastor, and Gregoire Mercier. Smooth sigmoid wavelet shrinkage for non-parametric estimation. In2008 IEEE International Conference on Acoustics, Speech and Signal Processing, pages 3265–3268. IEEE, 2008

    work page 2008

  5. [2]

    Towards a theory of non-log-concave sampling: first-order stationarity guarantees for Langevin monte carlo

    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 Langevin monte carlo. InConference on Learning Theory, pages 2896–2923. PMLR, 2022

    work page 2022

  6. [4]

    On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case.Bernoulli, 27(1):1–33, 2021

    Mathias Barkhagen, Ngoc Huy Chau, ´Eric Moulines, Mikl´os R´asonyi, Sotirios Sabanis, and Ying Zhang. On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case.Bernoulli, 27(1):1–33, 2021

    2021

  7. [5]

    The tamed unadjusted Langevin algorithm.Stochastic Processes and their Applications, 129(10):3638–3663, 2019

    Nicolas Brosse, Alain Durmus, ´Eric Moulines, and Sotirios Sabanis. The tamed unadjusted Langevin algorithm.Stochastic Processes and their Applications, 129(10):3638–3663, 2019

    work page 2019

  8. [6]

    On Stochastic Gradient Langevin Dynamics with Dependent Data Streams: The Fully Nonconvex Case.SIAM Journal on Mathematics of Data Science, 3(3):959–986, 2021

    Ngoc Huy Chau, ´Eric Moulines, Miklos R´asonyi, Sotirios Sabanis, and Ying Zhang. On Stochastic Gradient Langevin Dynamics with Dependent Data Streams: The Fully Nonconvex Case.SIAM Journal on Mathematics of Data Science, 3(3):959–986, 2021

    work page 2021

Show all 38 references
  1. [7]

    Sharp convergence rates for Langevin dynamics in the nonconvex setting.arXiv preprint arXiv:1805.01648, 2018

    Xiang Cheng, Niladri S Chatterji, Yasin Abbasi-Yadkori, Peter L Bartlett, and Michael I Jor- dan. Sharp convergence rates for Langevin dynamics in the nonconvex setting.arXiv preprint arXiv:1805.01648, 2018

    2018 arXiv

  2. [8]

    Analysis of Langevin Monte Carlo from Poincar´e to Log-Sobolev.arXiv preprint arXiv:2112.12662, 2021

    Sinho Chewi, Murat A Erdogdu, Mufan Bill Li, Ruoqi Shen, and Matthew Zhang. Analysis of Langevin Monte Carlo from Poincar´e to Log-Sobolev.arXiv preprint arXiv:2112.12662, 2021. 25

    2021 arXiv

  3. [9]

    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

  4. [10]

    Nonasymptotic convergence analysis for the unadjusted Langevin algorithm.The Annals of Applied Probability, 27(3):1551–1587, 2017

    Alain Durmus and Eric Moulines. Nonasymptotic convergence analysis for the unadjusted Langevin algorithm.The Annals of Applied Probability, 27(3):1551–1587, 2017

    2017

  5. [11]

    High-dimensional Bayesian inference via the unadjusted Langevin algorithm.Bernoulli, 25(4A):2854–2882, 2019

    Alain Durmus and Eric Moulines. High-dimensional Bayesian inference via the unadjusted Langevin algorithm.Bernoulli, 25(4A):2854–2882, 2019

    2019

  6. [12]

    Convergence of Langevin Monte Carlo in chi-squared and R ´enyi divergence

    Murat A Erdogdu, Rasa Hosseinzadeh, and Shunshi Zhang. Convergence of Langevin Monte Carlo in chi-squared and R ´enyi divergence. InInternational Conference on Artificial Intelligence and Statistics, pages 8151–8175. PMLR, 2022

    2022

  7. [13]

    On the diffeomorphisms of Euclidean space.The American Mathematical Monthly, 79(7):755–759, 1972

    Warren B Gordon. On the diffeomorphisms of Euclidean space.The American Mathematical Monthly, 79(7):755–759, 1972

    1972

  8. [14]

    Martin Hutzenthaler, Arnulf Jentzen, and Peter E. Kloeden. Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients.Proceedings of the Royal Society of London A: Mathematical, Physical...

    2011

  9. [15]

    Martin Hutzenthaler, Arnulf Jentzen, and Peter E. Kloeden. Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients.Ann. Appl. Probab., 22(4):1611–1641, 08 2012

    2012

  10. [16]

    Laplace’s method revisited: weak convergence of probability measures.The Annals of Probability, 8(6):1177–1182, 1980

    Chii-Ruey Hwang. Laplace’s method revisited: weak convergence of probability measures.The Annals of Probability, 8(6):1177–1182, 1980

    1980

  11. [17]

    Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity–the Strongly Convex Case.arXiv preprint arXiv:2301.08039, 2023

    Tim Johnston, Iosif Lytras, and Sotirios Sabanis. Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity–the Strongly Convex Case.arXiv preprint arXiv:2301.08039, 2023

    2023 arXiv

  12. [18]

    N. V . Krylov. A Simple Proof of the Existence of a Solution of It ˆo’s Equation with Monotone Coefficients.Theory of Probability & Its Applications, 35(3):583–587, 1991. doi: 10.1137/1135082

    1991 doi

  13. [19]

    Non-asymptotic estimates for TUSLA algorithm for non-convex learning with applications to neural networks with ReLU activation function.IMA Journal of Numerical Analysis, 2023

    Dong-Young Lim, Ariel Neufeld, Sotirios Sabanis, and Ying Zhang. Non-asymptotic estimates for TUSLA algorithm for non-convex learning with applications to neural networks with ReLU activation function.IMA Journal of Numerical Analysis, 2023

    2023

  14. [20]

    Langevin dynamics based algorithm e-TH ε O POULA for stochastic optimization problems with discontinuous stochastic gradient.Mathematics of Operations Research, 2024

    Dong-Young Lim, Ariel Neufeld, Sotirios Sabanis, and Ying Zhang. Langevin dynamics based algorithm e-TH ε O POULA for stochastic optimization problems with discontinuous stochastic gradient.Mathematics of Operations Research, 2024

    2024

  15. [21]

    Taming neural networks with tusla: Nonconvex learning via adaptive stochastic gradient langevin algorithms.SIAM Journal on Mathematics of Data Science, 5(2):323–345, 2023

    Attila Lovas, Iosif Lytras, Mikl ´os R´asonyi, and Sotirios Sabanis. Taming neural networks with tusla: Nonconvex learning via adaptive stochastic gradient langevin algorithms.SIAM Journal on Mathematics of Data Science, 5(2):323–345, 2023

    2023

  16. [22]

    Tamed Langevin sampling under weaker conditions

    Iosif Lytras and Panayotis Mertikopoulos. Tamed Langevin sampling under weaker conditions. arXiv preprint arXiv:2405.17693, 2024

    2024 arXiv

  17. [24]

    Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity.Bernoulli, 28(3): 1577–1601, 2022

    Wenlong Mou, Nicolas Flammarion, Martin J Wainwright, and Peter L Bartlett. Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity.Bernoulli, 28(3): 1577–1601, 2022

    2022

  18. [25]

    Supplement to ”Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity”

    Wenlong Mou, Nicolas Flammarion, Martin J Wainwright, and Peter L Bartlett. Supplement to ”Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity”. Bernoulli, 28(3):1577–1601, 2022

    2022

  19. [26]

    Towards a complete analysis of Langevin Monte Carlo: Beyond poincar´e inequality

    Alireza Mousavi-Hosseini, Tyler K Farghly, Ye He, Krishna Balasubramanian, and Murat A Erdogdu. Towards a complete analysis of Langevin Monte Carlo: Beyond poincar´e inequality. InThe Thirty Sixth Annual Conference on Learning Theory, pages 1–35. PMLR, 2023

    2023

  20. [27]

    Non-asymptotic estimates for accelerated high order Langevin Monte Carlo algorithms.arXiv preprint arXiv:2405.05679, 2024

    Ariel Neufeld and Ying Zhang. Non-asymptotic estimates for accelerated high order Langevin Monte Carlo algorithms.arXiv preprint arXiv:2405.05679, 2024

    2024 arXiv

  21. [29]

    Wasserstein continuity of entropy and outer bounds for interference channels.IEEE Transactions on Information Theory, 62(7):3992–4002, 2016

    Yury Polyanskiy and Yihong Wu. Wasserstein continuity of entropy and outer bounds for interference channels.IEEE Transactions on Information Theory, 62(7):3992–4002, 2016. 26 I. LYTRAS, S. SABANIS, AND Y . ZHANG

    2016

  22. [30]

    Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis

    Maxim Raginsky, Alexander Rakhlin, and Matus Telgarsky. Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis. InConference on Learning Theory, pages 1674–1703, 2017

    2017

  23. [31]

    Information theoretic proofs of entropy power inequalities.IEEE transactions on information theory, 57(1):33–55, 2010

    Olivier Rioul. Information theoretic proofs of entropy power inequalities.IEEE transactions on information theory, 57(1):33–55, 2010

    2010

  24. [32]

    A note on tamed Euler approximations.Electron

    Sotirios Sabanis. A note on tamed Euler approximations.Electron. Commun. Probab., 18(47):1–10, 2013

    2013

  25. [33]

    Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients.Ann

    Sotirios Sabanis. Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients.Ann. Appl. Probab., 26(4):2083–2105, 2016

    2016

  26. [34]

    Higher order Langevin Monte Carlo algorithm.Electronic Journal of Statistics, 13(2):3805–3850, 2019

    Sotirios Sabanis and Ying Zhang. Higher order Langevin Monte Carlo algorithm.Electronic Journal of Statistics, 13(2):3805–3850, 2019

    2019

  27. [35]

    A fully data-driven approach to minimizing CVaR for portfolio of assets via SGLD with discontinuous updating.arXiv preprint arXiv:2007.01672, 2020

    Sotirios Sabanis and Ying Zhang. A fully data-driven approach to minimizing CVaR for portfolio of assets via SGLD with discontinuous updating.arXiv preprint arXiv:2007.01672, 2020

    2007 arXiv

  28. [36]

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

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

    2019

  29. [37]

    Global convergence of Langevin dynamics based algorithms for nonconvex optimization

    Pan Xu, Jinghui Chen, Difan Zou, and Quanquan Gu. Global convergence of Langevin dynamics based algorithms for nonconvex optimization. InAdvances in Neural Information Processing Systems, pages 3122–3133, 2018

    2018

  30. [38]

    Nonasymptotic esti- mates for stochastic gradient Langevin dynamics under local conditions in nonconvex optimization

    Ying Zhang, ¨Omer Deniz Akyildiz, Theodoros Damoulas, and Sotirios Sabanis. Nonasymptotic esti- mates for stochastic gradient Langevin dynamics under local conditions in nonconvex optimization. Applied Mathematics & Optimization, 87(2):25, 2023. ARCHIMEDES/ATHENARESEARCHCENTRE...

    2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.