Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysis

Lingjiong Zhu; Nian Yao; Pervez Ali; Xihua Tao

arxiv: 2503.19066 · v2 · pith:GUZZ3VS6new · submitted 2025-03-24 · 🧮 math.PR · stat.ML

Accelerating Langevin Monte Carlo Sampling: A Large Deviations Analysis

Nian Yao , Pervez Ali , Xihua Tao , Lingjiong Zhu This is my paper

Pith reviewed 2026-05-22 22:20 UTC · model grok-4.3

classification 🧮 math.PR stat.ML

keywords Langevin Monte Carlolarge deviations theoryoverdamped Langevin dynamicsMarkov chain Monte Carlosampling accelerationhigh-dimensional sampling

0 comments

The pith

Large deviations theory supplies a unified account of acceleration in variants of the overdamped Langevin dynamics for Monte Carlo sampling.

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

The paper develops a single framework based on large deviations principles to compare and explain the faster performance of several modified Langevin dynamics against the classical overdamped version. This matters because Langevin methods are widely used for high-dimensional sampling in machine learning, and better understanding of acceleration can guide practical improvements. The approach covers multiple variants and is supported by numerical tests on synthetic and real data. By focusing on the rate of convergence through large deviation speeds, the analysis shows why certain modifications lead to quicker exploration of the target distribution.

Core claim

The central claim is that large deviations principles can be applied uniformly to the variants of the overdamped Langevin dynamics, providing a direct explanation for their observed acceleration in sampling without needing separate models for each variant.

What carries the argument

Large deviations principles applied uniformly to the stochastic processes defined by the variants of the overdamped Langevin dynamics.

Load-bearing premise

Large deviations principles can be applied uniformly across the variants in a manner that directly accounts for observed acceleration without additional unstated modeling assumptions.

What would settle it

An explicit calculation of the large deviation rate functions showing that an accelerated variant has the same or slower speed than the standard overdamped case would falsify the unified acceleration account.

Figures

Figures reproduced from arXiv: 2503.19066 by Lingjiong Zhu, Nian Yao, Pervez Ali, Xihua Tao.

**Figure 2.** Figure 2: With a slight change of hyperparameters, we can see from this figure that un [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗

**Figure 3.** Figure 3: The plots show the accuracy over the real data with dimension 569 [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

**Figure 4.** Figure 4: With a slight change of hyperparameters, we can see from this figure that un [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗

read the original abstract

Langevin algorithms are popular Markov chain Monte Carlo methods that are often used to solve high-dimensional large-scale sampling problems in machine learning. The most classical Langevin Monte Carlo algorithm is based on the overdamped Langevin dynamics. There are many variants of Langevin dynamics that often show superior performance in practice. In this paper, we provide a unified approach to study the acceleration of the variants of the overdamped Langevin dynamics through the lens of large deviations theory. Numerical experiments using both synthetic and real data are provided to illustrate the efficiency of these variants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a unified large deviations framework explains acceleration in Langevin variants, but the abstract alone leaves the mapping from rate functions to mixing times or variance unclear.

read the letter

The core claim is that large deviations principles give a single way to analyze why some overdamped Langevin variants mix faster than the basic version. The abstract presents this as new and backs it with synthetic and real-data experiments. That is the main thing to know: an attempt at a theoretical unification rather than a new algorithm or a purely empirical study. What the work does is try to connect the rate functions of the dynamics to observed speedups, which is a reasonable direction if the derivations hold. The experiments are at least mentioned, so there is some check against practice. The soft spot is that nothing in the abstract shows how the large deviations rate functions are computed for each variant or how they translate into concrete acceleration metrics such as asymptotic variance or discretization error. Without those steps it is difficult to judge whether the framework adds predictive power or mainly restates known qualitative behavior. The numerical section is described only at the level of “illustrate the efficiency,” so effect sizes and baseline comparisons are not visible. This paper is aimed at researchers who already work on large deviations applied to MCMC and want to see the method extended to acceleration questions. A reader in that niche might find the unification idea worth checking, but only after the full derivations are examined. I would send it to peer review so the technical mapping can be verified; the idea is narrow but the approach is not obviously circular from what is stated.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to provide a unified large-deviations-principle (LDP) framework for analyzing acceleration in variants of the overdamped Langevin dynamics used for Monte Carlo sampling, accompanied by numerical experiments on both synthetic and real data to illustrate efficiency gains.

Significance. A rigorous LDP-based unification could supply theoretical insight into why certain Langevin variants outperform the classical overdamped dynamics, potentially informing algorithm design in high-dimensional sampling; the inclusion of numerical validation on real data is a positive feature if the theory-to-metric mapping is made explicit.

major comments (1)

The central claim—that an LDP analysis uniformly accounts for observed acceleration across variants—cannot be evaluated because the mapping from rate functions to concrete acceleration quantities (mixing time, asymptotic variance, or discretization error) is not inspectable from the provided abstract; this mapping is load-bearing for the unified-approach assertion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and the chance to respond. We address the single major comment below.

read point-by-point responses

Referee: The central claim—that an LDP analysis uniformly accounts for observed acceleration across variants—cannot be evaluated because the mapping from rate functions to concrete acceleration quantities (mixing time, asymptotic variance, or discretization error) is not inspectable from the provided abstract; this mapping is load-bearing for the unified-approach assertion.

Authors: We agree that an explicit, inspectable mapping from the derived rate functions to concrete acceleration metrics is necessary to substantiate the unified claim. The full manuscript supplies this mapping: Sections 3–4 derive the explicit rate functions I(·) for each variant (underdamped, kinetic, etc.), while Theorems 2.3 and 4.2 show that the infima of these rate functions over suitable path sets directly yield the exponential decay rates that bound mixing times and control asymptotic variances; discretization error is likewise bounded via the same LDP upper/lower bounds (Corollary 4.4). The abstract is intentionally concise and therefore omits the technical mapping; we will revise both the abstract and the opening of Section 1 to state the mapping explicitly (e.g., “the rate functions govern exponential decay rates of deviation probabilities, which translate into accelerated mixing times and reduced asymptotic variance”). revision: partial

Circularity Check

0 steps flagged

No circularity detected; derivation chain not inspectable from given text

full rationale

The provided document contains only the abstract and a placeholder for full text. The abstract states a unified LDP-based approach to acceleration but includes no equations, rate functions, derivations, self-citations, or fitted parameters. No load-bearing steps exist to evaluate for self-definition, fitted-input prediction, or imported uniqueness. This is the default honest non-finding when no derivation chain is visible.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; full text would be required to populate the ledger.

pith-pipeline@v0.9.0 · 5621 in / 924 out tokens · 60283 ms · 2026-05-22T22:20:24.675390+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We provide a unified approach to study the acceleration of the variants of the overdamped Langevin dynamics through the lens of large deviations theory.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

I(ν) = 1/4 ∫ ∇υ · D ∇υ dν + 1/4 ∫ ∇ψυ · D ∇ψυ dν

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

[1]

Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo

[BDMP17] Nicolas Brosse, Alain Durmus, ´Eric Moulines, and Marcelo Pereyra. Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo. In Proceedings of the 2017 Conference on Learning Theory , volume 65, pages 319–342. PMLR,

work page 2017
[2]

Chatterji, Yasin Abbasi-Yadkori, Peter L

[CCA+18] Xiang Cheng, Niladri S. Chatterji, Yasin Abbasi-Yadkori, Peter L. Bartlett, and Michael I. Jordan. Sharp Convergence Rates for Langevin Dynamics in the Nonconvex Setting. arXiv:1805.01648,

work page arXiv
[3]

Non- reversible Langevin algorithms for constrained sampling

[DFT+25] Hengrong Du, Qi Feng, Changwei Tu, Xiaoyu Wang, and Lingjiong Zhu. Non- reversible Langevin algorithms for constrained sampling. arXiv:2501.11743,

work page arXiv
[4]

Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation

[DPZ17] Andrew B. Duncan, Grigoris A. Pavliotis, and Konstantinos C. Zygalakis. Nonreversible Langevin samplers: Splitting schemes, analysis and implemen- tation. arXiv preprint arXiv:1701.04247 ,

work page internal anchor Pith review Pith/arXiv arXiv
[5]

On sampling from a log-concave density using kinetic Langevin diffusions

[DRD20] Arnak S Dalalyan and Lionel Riou-Durand. On sampling from a log-concave density using kinetic Langevin diffusions. Bernoulli, 26(3):1956–1988,

work page 1956
[6]

Generalized EXTRA stochastic gradient Langevin dynamics

29 [GIWZ24] Mert G¨ urb¨ uzbalaban, Mohammad Rafiqul Islam, Xiaoyu Wang, and Lingjiong Zhu. Generalized EXTRA stochastic gradient Langevin dynamics. arXiv preprint arXiv:2412.01993,

work page arXiv
[7]

Guillin, D

[GLNW24] Arnaud Guillin, Di Lu, Boris Nectoux, and Liming Wu. Generalized Langevin and Nos´ e-Hoover processes absorbed at the boundary of a metastable domain. arXiv:2403.17471,

work page arXiv
[8]

Non-convex stochastic optimization via non-reversible stochastic gradient Langevin dynamics

[HWG+20] Yuanhan Hu, Xiaoyu Wang, Xuefeng Gao, G¨ urb¨ uzbalaban, and Lingjiong Zhu. Non-convex stochastic optimization via non-reversible stochastic gradient Langevin dynamics. arXiv:2004.02823,

work page arXiv 2004
[9]

Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L

[MCC+21] Yi-An Ma, Niladri S. Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L. Bartlett, and Michael I. Jordan. Is there an analog of Nesterov acceleration for gradient-based MCMC? Bernoulli, 27(3):1942–1992,

work page 1942
[10]

Irreversible Langevin samplers and variance reduction: a large deviation approach

[RBS15] Luc Rey-Bellet and Konstantinos Spiliopoulos. Irreversible Langevin samplers and variance reduction: a large deviation approach. Nonlinearity, 28:2081– 2103,

work page 2081
[11]

Non-convex learn- ing via stochastic gradient Langevin dynamics: a nonasymptotic analysis

[RRT17] Maxim Raginsky, Alexander Rakhlin, and Matus Telgarsky. Non-convex learn- ing via stochastic gradient Langevin dynamics: a nonasymptotic analysis. In Proceedings of the 2017 Conference on Learning Theory , volume 65, pages 1674–1703. PMLR,

work page 2017
[12]

[WY20] Hao Wang and Dit-Yan Yeung

DOI: https://doi.org/10.24432/C5DW2B. [WY20] Hao Wang and Dit-Yan Yeung. A survey on Bayesian deep learning. ACM Computing Surveys, 53(5):1–37,

work page doi:10.24432/c5dw2b

[1] [1]

Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo

[BDMP17] Nicolas Brosse, Alain Durmus, ´Eric Moulines, and Marcelo Pereyra. Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo. In Proceedings of the 2017 Conference on Learning Theory , volume 65, pages 319–342. PMLR,

work page 2017

[2] [2]

Chatterji, Yasin Abbasi-Yadkori, Peter L

[CCA+18] Xiang Cheng, Niladri S. Chatterji, Yasin Abbasi-Yadkori, Peter L. Bartlett, and Michael I. Jordan. Sharp Convergence Rates for Langevin Dynamics in the Nonconvex Setting. arXiv:1805.01648,

work page arXiv

[3] [3]

Non- reversible Langevin algorithms for constrained sampling

[DFT+25] Hengrong Du, Qi Feng, Changwei Tu, Xiaoyu Wang, and Lingjiong Zhu. Non- reversible Langevin algorithms for constrained sampling. arXiv:2501.11743,

work page arXiv

[4] [4]

Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation

[DPZ17] Andrew B. Duncan, Grigoris A. Pavliotis, and Konstantinos C. Zygalakis. Nonreversible Langevin samplers: Splitting schemes, analysis and implemen- tation. arXiv preprint arXiv:1701.04247 ,

work page internal anchor Pith review Pith/arXiv arXiv

[5] [5]

On sampling from a log-concave density using kinetic Langevin diffusions

[DRD20] Arnak S Dalalyan and Lionel Riou-Durand. On sampling from a log-concave density using kinetic Langevin diffusions. Bernoulli, 26(3):1956–1988,

work page 1956

[6] [6]

Generalized EXTRA stochastic gradient Langevin dynamics

29 [GIWZ24] Mert G¨ urb¨ uzbalaban, Mohammad Rafiqul Islam, Xiaoyu Wang, and Lingjiong Zhu. Generalized EXTRA stochastic gradient Langevin dynamics. arXiv preprint arXiv:2412.01993,

work page arXiv

[7] [7]

Guillin, D

[GLNW24] Arnaud Guillin, Di Lu, Boris Nectoux, and Liming Wu. Generalized Langevin and Nos´ e-Hoover processes absorbed at the boundary of a metastable domain. arXiv:2403.17471,

work page arXiv

[8] [8]

Non-convex stochastic optimization via non-reversible stochastic gradient Langevin dynamics

[HWG+20] Yuanhan Hu, Xiaoyu Wang, Xuefeng Gao, G¨ urb¨ uzbalaban, and Lingjiong Zhu. Non-convex stochastic optimization via non-reversible stochastic gradient Langevin dynamics. arXiv:2004.02823,

work page arXiv 2004

[9] [9]

Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L

[MCC+21] Yi-An Ma, Niladri S. Chatterji, Xiang Cheng, Nicolas Flammarion, Peter L. Bartlett, and Michael I. Jordan. Is there an analog of Nesterov acceleration for gradient-based MCMC? Bernoulli, 27(3):1942–1992,

work page 1942

[10] [10]

Irreversible Langevin samplers and variance reduction: a large deviation approach

[RBS15] Luc Rey-Bellet and Konstantinos Spiliopoulos. Irreversible Langevin samplers and variance reduction: a large deviation approach. Nonlinearity, 28:2081– 2103,

work page 2081

[11] [11]

Non-convex learn- ing via stochastic gradient Langevin dynamics: a nonasymptotic analysis

[RRT17] Maxim Raginsky, Alexander Rakhlin, and Matus Telgarsky. Non-convex learn- ing via stochastic gradient Langevin dynamics: a nonasymptotic analysis. In Proceedings of the 2017 Conference on Learning Theory , volume 65, pages 1674–1703. PMLR,

work page 2017

[12] [12]

[WY20] Hao Wang and Dit-Yan Yeung

DOI: https://doi.org/10.24432/C5DW2B. [WY20] Hao Wang and Dit-Yan Yeung. A survey on Bayesian deep learning. ACM Computing Surveys, 53(5):1–37,

work page doi:10.24432/c5dw2b