Pith. sign in

REVIEW 4 major objections 4 minor 42 references

Langevin Monte Carlo Beyond Lipschitz Gradient Continuity

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read Inexact Proximal Langevin Algorithm samples from super-quadratic, non-Lipschitz-gradient potentials by keeping the proximal-step error below $\kappa\tau^{1+\alpha}$, with iteration cost $d^{(q_V+1)/2} O(\varepsilon^{-2})$.

desk verdict New inexact-proximal LMC scheme with real convergence claims; the key estimates are outsourced to a companion preprint and one experiment overreaches, but the core is worth refereeing. read the letter →

arxiv 2412.09698 v1 pith:P2TY45IR submitted 2024-12-12 stat.ML cs.LG

Matej Benko , Iwona Chlebicka , J{o}rgen Endal , B{l}ażej Miasojedow This is my paper · ORCID
classification stat.MLcs.LG MSC 65C0560J22
keywords LangevinMonteCarloproximalsamplingnon-smoothpotentialssuper-quadraticgrowthinexactoperatorKLdivergenceWassersteindistancemomentbounds
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 introduces IPLA, an Inexact Proximal Langevin Algorithm, and claims it can sample from densities proportional to $\exp(-V)$ for potentials $V$ that are convex, strongly convex outside a ball, and have polynomial growth of order $q_V+1 \ge 2$, without requiring the gradient $\nabla V$ to be globally Lipschitz. This is the regime where ordinary Langevin Monte Carlo can become transient and blow up, and tamed versions remain stable but move sluggishly in the tails. The main theorems state that if the proximal step is computed to error $\delta \le \kappa \tau^{1+\alpha}$ with $\tau < 1/\lambda_V$, then the averaged chain converges in KL divergence at the rate of Theorem 5.5, and in the globally strongly convex case the chain converges in Wasserstein distance at the rate of Theorem 5.7. As a consequence, generating one sample with accuracy $\varepsilon$ costs $d^{(q_V+1)/2} O(\varepsilon^{-2})$ iterations, which for $q_V=1$ matches the best-known dimension dependence of Langevin Monte Carlo. The paper also proves uniform bounds on all moments of the chain and demonstrates the method on a quartic light-tailed target, a Ginzburg-Landau model, and a 360000-dimensional Bayesian image deconvolution problem.

What carries the argument

The object that carries the argument is IPLA itself, Algorithm 1: each iteration applies an approximate proximal map $x \mapsto \operatorname{prox}_{\tau V}(x)$, the minimizer of $V(y) + |y-x|^2/(2\tau)$, with output error bounded by $\delta$, then adds Gaussian noise $Z \sim N(0,2\tau I_d)$. The proof uses the variational splitting of the free energy $F = F_V + F_E$ into potential and entropy terms, whose Wasserstein gradient flows are respectively the proximal map and Brownian motion. Its core estimate is Proposition 5.4, a Wasserstein-space analogue of the classical convex-optimization descent inequality $2\tau(f(x_{k+1}) - f(x_*)) \le |x_k - x_*|^2 - |x_{k+1} - x_*|^2 + C\tau^2$, with the additional term $C(\nu)\delta + K(\tau)\tau$ measuring the cost of the inexact step. The constant $K(\tau)$ is explicit in equation (5), and the moment bound of Theorem 5.1, proved by induction with Lemmas A.2 and A.3, controls all moments of the chain so that the super-quadratic tails cannot push the chain to infinity.

What would settle it

Run IPLA on $V(x)=|x|^4/4$ in dimension $d=1000$ with a step $\tau < 1/\lambda_V$, an initial measure with finite fourth moment, and proximal error $\delta \le \kappa \tau^2$, then estimate the KL divergence between the averaged empirical chain and the target $\mu_* \propto e^{-|x|^4/4}$. Theorem 5.5 predicts $KL(\nu_n^0 | \mu_*) \le (2n\tau)^{-1} W_2^2(\rho_0,\mu_*) + C\kappa\tau + K(\tau)$; if the measured KL consistently exceeds that right-hand side, or if any moment $\mathbb{E}|X_k|^m$ grows without bound over $n$, the central bound is wrong.

Watch

Extended reading notes

Core claim

The central discovery is that the exact proximal map in the Proximal Langevin Algorithm can be replaced by an approximate one without breaking convergence, as long as the approximation error is tied to the step size by $\delta = \kappa \tau^{1+\alpha}$. Under assumption (V), Theorem 5.5 bounds the Kullback-Leibler divergence of the averaged chain by $$KL(\nu_N^n | \mu_*) \le \frac{1}{2n\tau}\bigl($W_2^{2}$(\rho_N,\mu_*) - $W_2^{2}$(\rho_{N+n},\mu_*)\bigr) + C(\mu_*) \kappa \tau^\$\alpha$ + K(\tau),$$ and Theorem 5.7, for globally $\lambda_V$-convex $V$, gives $$$W_2^{2}$(\rho_k,\mu_*) \le 2\left(1 - \frac{\tau\lambda_V}{2}\right)^k $W_2^{2}$(\rho_0,\mu_*) + \frac{4}{\lambda_V}K(\tau) + 2\$kappa^{2}$ \$tau^{{2+2\alpha}}$\left(\frac{1-$e^{{-\lambda_V\tau(k-1)}}$}{1-$e^{{-\lambda_V\tau}}$}\right)^2.$$ These bounds imply the claimed $d^{(q_V+1)/2} O(\varepsilon^{-2})$ iteration complexity for one sample, with the proximal accuracy requirement being $\delta \le \kappa \tau^2$ for the KL result and $\delta \le \kappa \tau^{3/2}$ in the strongly convex case. The authors present these bounds as the proof that IPLA extends Langevin Monte Carlo to potentials whose gradients are not Lipschitz, and they support the theory with experiments in which ULA blows up while IPLA remains stable.

Load-bearing premise

Two load-bearing premises hold the proof together: the contraction estimates for the exact proximal chain are imported as a black box from a companion preprint by the same authors, and the potential $V$ is assumed convex on all of $\mathbb{R}^d$, an assumption that the Ginzburg-Landau experiment in Section 6, as parameterized, appears to violate; if either fails, the stated rates no longer follow.

Editorial extensions

If this is right

  • IPLA remains stable on targets with faster-than-quadratic polynomial tails, where ULA diverges and TULA takes unnecessarily small steps, as demonstrated on the quartic and Ginzburg-Landau experiments.
  • With proximal error $\delta \le \kappa \tau^2$, KL accuracy $\varepsilon$ is reached in $O(\varepsilon^{-2})$ iterations with dimension factor $d^{(q_V+1)/2}$, and $q_V=1$ recovers the best-known dimension scaling for Langevin Monte Carlo.
  • When $V$ is globally strongly convex, the proximal step may be solved less accurately ($\delta \le \kappa \tau^{3/2}$) and the Wasserstein error still converges at $O(\varepsilon^{-2})$ order.
  • All moments of the IPLA chain are finite uniformly in time for every $\tau<1/\lambda_V$, so the chain does not explode even though the gradient is not Lipschitz.

Reading between the lines

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

  • A natural extension the paper does not pursue is a stochastic model of the proximal error; a Markov-noise version would clarify whether the deterministic bound $\delta \le \kappa \tau^{1+\alpha}$ is necessary or merely sufficient.
  • The dimension exponent is controlled by $q_V$, so the explicit constant in $K(\tau)$ could be tested directly by measuring KL against $\tau$ for $V(x)=|x|^4/4$ and comparing with Remark 5.3.
  • If the Ginzburg-Landau experiment lies outside the global-convexity assumption, its stability hints that the contraction mechanism may tolerate local non-convexity, and a dissipativity or local-convexity version of Theorems 5.5 and 5.7 would be the next testable step.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 introduces the Inexact Proximal Langevin Algorithm (IPLA), a splitting scheme in which the exact proximal step of the proximal Langevin algorithm is replaced by an approximation with error bounded by δ, followed by an additive Gaussian step. Under assumption (V) (global convexity, λ_V-convexity outside a ball, and a polynomial-growth majorant of order q_V+1) and a finite initial moment of order q_V+1, the paper proves uniform moment bounds (Theorem 5.1), an averaged KL error bound (Theorem 5.5), and a Wasserstein contraction bound under global λ_V-convexity (Theorem 5.7). Complexity corollaries claim d^{(q_V+1)/2} O(ε^{-2}) iteration counts in both KL and Wasserstein settings. The numerical section tests IPLA on a quartic-potential example, a Ginzburg–Landau model, and a Bayesian image deconvolution task, with a code repository provided.

Significance. If the main results hold, the paper makes a useful contribution by extending Langevin-type sampling to potentials with non-Lipschitz, super-quadratic growth while keeping dimension dependence comparable to the best-known LMC rates when q_V=1. The moment bounds and the inexact-proximal error propagation analysis are nontrivial and of independent interest. The paper also provides reproducible code and clearly structured experiments. However, the central convergence theorems currently rest on inequalities imported from a same-author companion preprint, so the contribution is conditional unless those dependencies are resolved.

major comments (4)
  1. [Sec. 5 and App. A, Lemma A.6] The main convergence results depend critically on Lemma A.6, which is stated as Benko et al. (2024, Lemma 4.5) and not proved in this manuscript. Proposition 5.4 and Theorem 5.5 invoke Lemma A.6(i)–(iii), and Theorem 5.7 additionally invokes the exact-PLMC contraction 'by the same reasoning as in Benko et al. (2024, Proof of Theorem 2)' and 'the same lines as ... Theorem 6'. These imported estimates provide the proximal-descent inequality, the Gaussian-smoothing bias K(τ), and the contraction that determine the allowable δ and the reported complexity. Since the companion preprint is by the same four authors and is not independently reviewed, the paper should either give complete proofs of these inequalities in the appendix or state and verify the precise hypotheses under which they hold; as written, the main claims are conditional on an external unpublished source.
  2. [Corollary 5.8] The stated target is W_2^2(ρ_{n_ε}, μ_*) ≤ ε, but the reported iteration counts O(ε^{-2}) and O(ε^{-α^{-1}}) correspond to a target of the form W_2 ≤ ε (i.e., W_2^2 ≤ ε^2). Substituting the corollary's own conditions, Remark 5.3 gives K(τ_ε) ≤ C τ_ε d^{(q_V+1)/2}, so K(τ_ε) ≤ λ_V ε / 12 forces τ_ε ≲ ε d^{-(q_V+1)/2}; with n_ε ≍ τ_ε^{-1}, this yields d^{(q_V+1)/2} O(ε^{-1}) iterations when α ≥ 1/2. For α < 1/2, the condition τ_ε^{2α} ≤ λ_V^2 ε/(96κ^2 log^2(...)) dominates and gives τ_ε ≤ ε^{1/(2α)}, hence d^{(q_V+1)/2} O(ε^{-1/(2α)}) iterations, not d^{(q_V+1)/2} O(ε^{-α^{-1}}). The target and the complexity exponents must be made consistent.
  3. [App. B, proof of Proposition 5.4] In bounding the term I_2 = F_V[ρ_{k+2/3}] - F_V[ρ_{k+1/3}], the proof applies Lemma A.8 with the arbitrary measure ν. However, Lemma A.8 applies to the measure being smoothed, which here is ρ_{k+1/3}, not ν. Thus the displayed bound on I_2 does not follow from Lemma A.8 as written. The constant C(ν) in Appendix C also does not reflect moments of ρ_{k+1/3} in that first part. This is likely fixable using Theorem 5.1, but the proof needs correction before the KL bound can be considered established.
  4. [Sec. 6, Example 2] The Ginzburg–Landau potential with the stated parameters υ=2, κ=0.1, ς=0.5 has a negative quadratic contribution (1-υ)/2 = -0.5, so along constant configurations the Hessian at zero is negative and V is not convex on R^d. This violates the global convexity requirement in assumption (V). The example is presented as a demonstration of IPLA, but the proven guarantees do not cover it as parameterized. The authors should either extend the theory to this setting, change the parameters, or explicitly state that Example 2 is outside the theorem's scope, as they already do for the image-deconvolution example.
minor comments (4)
  1. [Lemma A.1] The first displayed inequality in Lemma A.1 appears to be missing a square: the term should be |prox_τ^V(x) - z|^2, not |prox_τ^V(x) - z|.
  2. [Lemma A.6(ii)] The statement of Lemma A.6(ii) writes W_2^2(ρ_{k+1/3}, ρ_k) as the second Wasserstein term, but the proof of Proposition 5.4 uses W_2^2(ρ_{k+1/3}, ν); the lemma statement is likely a typo and should be corrected.
  3. [Sec. 6, Example 1] The text says 'dimension d = 10 3'; this should read d = 10^3.
  4. [Corollary 5.8 proof] The bound (1 - e^{-λ_V τ(n_ε-1)})/(1 - e^{-λ_V τ}) ≤ n_ε is very loose and is stated without justification; since τ λ_V < 1, the denominator is bounded below by a constant, so a constant bound also holds. The proof should be clarified.

Circularity Check

2 steps flagged · score 4.0 of 10

Core convergence bounds are outsourced to a same-authors companion preprint; not a pure tautology, but two main theorems rest on load-bearing self-citation.

  1. self citation load bearing [Appendix A, Lemma A.6; used in Appendix B, Proof of Proposition 5.4 and Theorem 5.5]
    "In the theoretical analysis, the key role is played by the following lemma from (Benko et al. 2024). Lemma A.6 (Benko et al. (2024), Lemma 4.5). Let the function V : Rd → R, d ≥ 1, satisfy assumption (V) with λV ≥ 0, LqV > 0, and qV ≥ 1."

    The KL-error bound of Theorem 5.5 is obtained by summing Proposition 5.4, and the proof of Proposition 5.4 explicitly bounds its third and fourth terms by Lemma A.6(ii)-(iii) while its first term uses Corollary B.1, which is built on Lemma A.6(i). Lemma A.6 is not proved in this manuscript; it is imported verbatim from Benko et al. (2024), a companion preprint by the same four authors. The decisive proximal-descent and entropy-contraction inequalities are therefore not discharged here, so the main convergence claim inherits its central content from a self-citation rather than from an argument contained in this paper. This is not a definitional equivalence, but it is load-bearing self-citation that is not independently verified.

  2. self citation load bearing [Appendix B, Proof of Theorem 5.7]
    "With the above notation by the same reasoning as in Benko et al. (2024, Proof of Theorem 2) we get W2^2(ϱ̂k, µ∗) ≤ (1 − λV τ/2)^k W2^2(ϱ0, µ∗) + 2/λV K(τ), where K(τ) is given by (5). Given the above inequality we finish the proof by the same lines as in Benko et al. (2024, Proof of Theorem 6)."

    Theorem 5.7 is the paper's Wasserstein convergence guarantee and the basis for the ε−2 iteration complexity in Corollary 5.8, yet neither the exact-PLMC contraction estimate nor the final error bound is derived in the manuscript: both are asserted by explicit reference to the same authors' companion preprint. The main strong-convexity result therefore reduces to a self-citation chain, and if the imported contraction theorem requires additional hypotheses beyond those stated in (V), Theorem 5.7 as stated is unsupported. This is the second main convergence theorem whose proof is effectively outsourced to Benko et al. (2024).

full rationale

The paper is not circular in the strict definitional sense: no parameter is fitted to a target and then relabeled as a prediction, and no theorem is obtained merely by renaming its own assumptions. The moment bound in Theorem 5.1 is proved in the manuscript, and the inexact-error decomposition in Lemmas A.7 and A.8 is original to this exposition. However, the two principal convergence results are load-bearing self-citations. Proposition 5.4, which drives the KL bound in Theorem 5.5, invokes Lemma A.6(ii)-(iii) verbatim from Benko et al. (2024), a companion preprint by exactly the same four authors, and Theorem 5.7's Wasserstein bound is not proved at all; its proof refers to 'the same reasoning as in Benko et al. (2024, Proof of Theorem 2)' and 'the same lines as in Benko et al. (2024, Proof of Theorem 6)'. These companion results are not machine-checked, code-reproduced, or otherwise independently verified under the stated criteria, so they do not count as independent support. The Ginzburg-Landau experiment's possible conflict with global convexity is a correctness-risk concern, not a circularity, and is not counted in the score. The numerical comparisons against ULA, TULA, and Metropolis-Hastings are external benchmarks, but they do not supply the missing contraction proofs. A score of 4 reflects the fact that the central claims still contain independent algorithmic and moment-analysis content, while the decisive convergence inequalities are outsourced to the authors' own unverified companion paper.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no fitted free parameters in the theoretical claims; q_V, C_V, L_q, lambda_V, R_V, kappa, and alpha are problem or user-chosen constants. The main nontrivial external input is the companion preprint's Lemma A.6, which supplies the smoothing and descent inequalities on which the central theorems rest.

assumptions (5)
  • standard math Standard Wasserstein gradient flow and optimal transport results are used without proof.
    Invoked in Sections 1 and 4 via Ambrosio-Gigli-Savare (2008), Villani (2009), and Jordan-Kinderlehrer-Otto (1998), including the JKO splitting and the gradient flow of the entropy as Brownian motion.
  • domain assumption Assumption (V): V is C^1, convex, lambda_V-convex outside a ball, satisfies the polynomial-growth bound (4), and attains its minimum at 0.
    This defines the entire class of potentials for which Theorems 5.1, 5.5, and 5.7 are proved. It fails for the Ginzburg-Landau experiment in Example 2 as parameterized with upsilon = 2.
  • domain assumption Assumption (rho0): the initial distribution has a finite (q_V+1)-th moment.
    Used in Theorem 5.1 and Proposition 5.4 to control all moments of the chain and to justify the moment-dependent constants.
  • domain assumption Lemma A.6 and the exact-PLMC contraction results from Benko et al. 2024 are correct and applicable.
    Proposition 5.4 uses Lemma A.6 parts (i)-(iii) and Corollary B.1; Theorem 5.7 follows the proofs of Benko et al. 2024, Theorems 2 and 6. This is an unreviewed preprint by the same four authors.
  • ad hoc to paper The inexact proximal error satisfies |Theta| <= delta almost surely with delta = kappa tau^{1+alpha}, and the proximal subproblem can be solved with per-iteration cost O(d log delta).
    Algorithm 1 defines the algorithmic error model, and Appendix E argues the logarithmic cost from strong convexity. All asymptotic complexity claims depend on this contract.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Langevin Monte Carlo Beyond Lipschitz Gradient Continuity." pith.science (2026). https://pith.science/paper/P2TY45IR

@misc{pith2026241209698,
  author       = {Pith},
  title        = {Pith review of: Langevin Monte Carlo Beyond Lipschitz Gradient Continuity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P2TY45IR}},
  note         = {Machine review of arXiv:2412.09698}
}
abstract

We present a significant advancement in the field of Langevin Monte Carlo (LMC) methods by introducing the Inexact Proximal Langevin Algorithm (IPLA). This novel algorithm broadens the scope of problems that LMC can effectively address while maintaining controlled computational costs. IPLA extends LMC's applicability to potentials that are convex, strongly convex in the tails, and exhibit polynomial growth, beyond the conventional $L$-smoothness assumption. Moreover, we extend LMC's applicability to super-quadratic potentials and offer improved convergence rates over existing algorithms. Additionally, we provide bounds on all moments of the Markov chain generated by IPLA, enhancing its analytical robustness.

Figures

Figures reproduced from arXiv: 2412.09698 by the authors.

Figure 1
Figure 1. Trajectory of the 1 st coordinate from Example 1 starting in a tail. Both plots are based on the same data. 0.000 0.025 0.050 0.075 0.100 τ 0 100 200 300 RE IPLA TULA (a) 0.000 0.002 0.004 τ 0.00 0.25 0.50 0.75 RE IPLA TULA (b) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Dependence RE of IPLA and TULA (for Exam [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Result of the Bayesian Image Denoising from Example 3. The original photo by Zbyszko Siemaszko 1955-56. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Example 1, starting in tail. Trajectory of the 1 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Example 1, starting in tail. Trajectory of the 1 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Example 1, starting in minimizer. Dependence of RE and CV of IPLA, TULA and ULA on stepsize [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Quantiles of the posterior distribution for the Bayesian Image Deconvolution from Example 3 starting in a tail, for [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 39 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    urich. Birkh\

    Ambrosio, L.; Gigli, N.; and Savar\'e, G. 2008. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Z\"urich. Birkh\"auser Verlag, Basel, second edition. ISBN 978-3-7643-8721-1

    work page 2008

  4. [4]

    Andrieu, C.; de Freitas, N.; Doucet, A.; and Jordan, M. I. 2003. An Introduction to MCMC for Machine Learning. Machine Learning, 50(1): 5--43

    work page 2003

  5. [5]

    Beck, A. 2017. First-order methods in optimization, volume 25 of MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA. ISBN 978-1-611974-98-0

    work page 2017

  6. [6]

    Benko, M.; Chlebicka, I.; Endal, J.; and Miasojedow, B. 2024. Convergence rates of particle approximation of forward-backward splitting algorithm for granular medium equations. arXiv:2405.18034

    work page arXiv 2024

  7. [7]

    Bernton, E. 2018. L angevin M onte C arlo and JKO splitting. In Bubeck, S.; Perchet, V.; and Rigollet, P., eds., Proceedings of the 31st Conference On Learning Theory, volume 75 of Proceedings of Machine Learning Research, 1777--1798. PMLR

    work page 2018

  8. [8]

    Brosse, N.; Durmus, A.; Moulines, E.; and Sabanis, S. 2019. The tamed unadjusted L angevin algorithm. Stochastic Process. Appl., 129(10): 3638--3663

    work page 2019

Show all 42 references
  1. [9]

    Cattiaux, P.; Guillin, A.; and Malrieu, F. 2008. Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields, 140(1-2): 19--40

    2008

  2. [10]

    Chambolle, A. 2004. An Algorithm for Total Variation Minimization and Applications. Journal of Mathematical Imaging and Vision, 20(1): 89--97

    2004

  3. [11]

    A.; Li, M.; Shen, R.; and Zhang, S

    Chewi, S.; Erdogdu, M. A.; Li, M.; Shen, R.; and Zhang, S. 2022. Analysis of Langevin Monte Carlo from Poincare to Log-Sobolev. In Loh, P.-L.; and Raginsky, M., eds., Proceedings of Thirty Fifth Conference on Learning Theory, volume 178 of Proceedings of Machine Learning Resea...

    2022

  4. [12]

    Cover, T.; and Thomas, J. 2012. Elements of Information Theory. Wiley. ISBN 9781118585771

    2012

  5. [13]

    Dalalyan, A. 2017 a . Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent. In Kale, S.; and Shamir, O., eds., Proceedings of the 2017 Conference on Learning Theory, volume 65 of Proceedings of Machine Learning Research, 678...

    2017

  6. [14]

    Dalalyan, A. S. 2017 b . Theoretical guarantees for approximate sampling from smooth and log-concave densities. J. R. Stat. Soc. Ser. B. Stat. Methodol., 79(3): 651--676

    2017

  7. [15]

    S.; Karagulyan, A.; and Riou-Durand, L

    Dalalyan, A. S.; Karagulyan, A.; and Riou-Durand, L. 2022. Bounding the error of discretized L angevin algorithms for non-strongly log-concave targets. J. Mach. Learn. Res., 23: Paper No. [235], 38

    2022

  8. [16]

    S.; and Tsybakov, A

    Dalalyan, A. S.; and Tsybakov, A. B. 2012. Sparse regression learning by aggregation and L angevin M onte- C arlo. J. Comput. System Sci., 78(5): 1423--1443

    2012

  9. [17]

    Durmus, A.; Majewski, S.; and Miasojedow, B. 2019. Analysis of L angevin M onte C arlo via convex optimization. J. Mach. Learn. Res., 20: Paper No. 73, 46

    2019

  10. [18]

    Durmus, A.; and Moulines, E. 2017. Nonasymptotic convergence analysis for the unadjusted L angevin algorithm. Ann. Appl. Probab., 27(3): 1551--1587

    2017

  11. [19]

    Durmus, A.; and Moulines, E. 2019. High-dimensional B ayesian inference via the unadjusted L angevin algorithm. Bernoulli, 25(4A): 2854--2882

    2019

  12. [20]

    Durmus, A.; Moulines, E.; and Pereyra, M. 2018. Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau. SIAM Journal on Imaging Sciences, 11(1): 473--506

    2018

  13. [21]

    A.; and Hosseinzadeh, R

    Erdogdu, M. A.; and Hosseinzadeh, R. 2021. On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness. In Belkin, M.; and Kpotufe, S., eds., Proceedings of Thirty Fourth Conference on Learning Theory, volume 134 of Proceedings of Machine Learn...

    2021

  14. [22]

    A.; Hosseinzadeh, R.; and Zhang, S

    Erdogdu, M. A.; Hosseinzadeh, R.; and Zhang, S. 2022. Convergence of Langevin Monte Carlo in Chi-Squared and Rényi Divergence. In Camps-Valls, G.; Ruiz, F. J. R.; and Valera, I., eds., Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, ...

    2022

  15. [23]

    B.; Stern, H

    Gelman, A.; Carlin, J. B.; Stern, H. S.; Dunson, D. B.; Vehtari, A.; and Rubin, D. B. 2014. Bayesian data analysis. Texts in Statistical Science Series. CRC Press, Boca Raton, FL, third edition. ISBN 978-1-4398-4095-5

    2014

  16. [24]

    Goldenfeld, N. 1992. Lectures On Phase Transitions And The Renormalization Group (1st ed.). CRC Press

    1992

  17. [25]

    Goldstein, T.; Li, M.; and Yuan, X. 2015. Adaptive Primal-Dual Splitting Methods for Statistical Learning and Image Processing. In Cortes, C.; Lawrence, N.; Lee, D.; Sugiyama, M.; and Garnett, R., eds., Advances in Neural Information Processing Systems, volume 28. Curran Assoc...

    2015

  18. [26]

    Jordan, R.; Kinderlehrer, D.; and Otto, F. 1998. The variational formulation of the F okker- P lanck equation. SIAM J. Math. Anal., 29(1): 1--17

    1998

  19. [27]

    Krauth, W. 2006. Statistical mechanics, volume 13 of Oxford Master Series in Physics. Oxford University Press, Oxford. ISBN 978-0-19-851536-4; 0-19-851536-7. Algorithms and computations, Oxford Master Series in Statistical Computational, and Theoretical Physics

    2006

  20. [28]

    Livingstone, S.; Nüsken, N.; Vasdekis, G.; and Zhang, R.-Y. 2024. Skew-symmetric schemes for stochastic differential equations with non-Lipschitz drift: an unadjusted Barker algorithm. arXiv:2405.14373

    2024 arXiv

  21. [29]

    C.; Stuart, A

    Mattingly, J. C.; Stuart, A. M.; and Higham, D. J. 2002. Ergodicity for SDE s and approximations: locally L ipschitz vector fields and degenerate noise. Stochastic Process. Appl., 101(2): 185--232

    2002

  22. [30]

    K.; He, Y.; Balasubramanian, K.; and Erdogdu, M

    Mousavi-Hosseini, A.; Farghly, T. K.; He, Y.; Balasubramanian, K.; and Erdogdu, M. A. 2023. Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincaré Inequality. In Neu, G.; and Rosasco, L., eds., Proceedings of Thirty Sixth Conference on Learning Theory, volume 195...

    2023

  23. [31]

    ksendal, B. 2010. Stochastic Differential Equations: An Introduction with Applications. Universitext. Springer Berlin Heidelberg. ISBN 9783642143946

    2010

  24. [32]

    G.; Scott, J

    Polson, N. G.; Scott, J. G.; and Willard, B. T. 2015. Proximal algorithms in statistics and machine learning. Statist. Sci., 30(4): 559--581

    2015

  25. [33]

    V.; Luiken, N.; Leblanc, O.; and Uruñuela, E

    Ravasi, M.; Örnhag, M. V.; Luiken, N.; Leblanc, O.; and Uruñuela, E. 2024. PyProximal - scalable convex optimization in Python. Journal of Open Source Software, 9(95): 6326

    2024

  26. [34]

    P.; and Casella, G

    Robert, C. P.; and Casella, G. 2004. Monte C arlo statistical methods . Springer Texts in Statistics. Springer-Verlag, New York, second edition. ISBN 0-387-21239-6

    2004

  27. [35]

    O.; and Tweedie, R

    Roberts, G. O.; and Tweedie, R. L. 1996. Exponential convergence of L angevin distributions and their discrete approximations. Bernoulli, 2(4): 341--363

    1996

  28. [36]

    I.; Osher, S.; and Fatemi, E

    Rudin, L. I.; Osher, S.; and Fatemi, E. 1992. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1): 259--268

    1992

  29. [37]

    Salim, A.; Korba, A.; and Luise, G. 2020. The Wasserstein Proximal Gradient Algorithm. In Larochelle, H.; Ranzato, M.; Hadsell, R.; Balcan, M.; and Lin, H., eds., Advances in Neural Information Processing Systems, volume 33, 12356--12366. Curran Associates, Inc

    2020

  30. [38]

    Santambrogio, F. 2017. E uclidean, metric, and W asserstein gradient flows: an overview. Bull. Math. Sci., 7(1): 87--154

    2017

  31. [39]

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

    2010

  32. [40]

    Vempala, S.; and Wibisono, A. 2019. Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices. In Wallach, H.; Larochelle, H.; Beygelzimer, A.; d Alch\' e -Buc, F.; Fox, E.; and Garnett, R., eds., Advances in Neural Information Processing Systems, volume 32...

    2019

  33. [41]

    Villani, C. 2009. Optimal transport. Old and new, volume 338 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. ISBN 978-3-540-71049-3

    2009

  34. [42]

    E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; van der Walt , S

    Virtanen, P.; Gommers, R.; Oliphant, T. E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; van der Walt , S. J.; Brett, M.; Wilson, J.; Millman, K. J.; Mayorov, N.; Nelson, A. R. J.; Jones, E.; Kern, R.; Larson, E.; Carey, C. J...

    2020

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.