Non-asymptotic Error Bounds in $\mathcal{W}_2$-Distance with Sqrt(d) Dimension Dependence and First Order Convergence for Langevin Monte Carlo beyond Log-Concavity
Authors: Bin Yang
Xiaojie Wang
Venue: ICML 2025 poster (2025)
Keywords: Langevin Monte Carlo
log-Sobolev inequality
$O(\sqrt{d}h)$ error bounds in W2-distance
non-globally Lipschitz gradient
Primary area: probabilistic_methods->monte_carlo_and_sampling_methods
TL;DR: We prove an error bound $O(\sqrt{d} h)$ in $\mathcal{W}_2$-distance with sqrt(d) dimension dependence for Langevin Monte Carlo beyond log-Concavity.

Abstract: Generating samples from a high dimensional probability distribution is a fundamental task with wide-ranging applications in the area of scientific computing, statistics and machine learning. This article revisits the popular Langevin Monte Carlo (LMC) sampling algorithms and provides a non-asymptotic error analysis in $\mathcal{W}_2$-distance in a non-convex setting. In particular, we prove an error bound $O(\sqrt{d} h)$, which guarantees a mixing time $ \tilde{O} (\sqrt{d} \epsilon^{-1})$ to achieve the accuracy tolerance $\epsilon$, under certain log-smooth conditions and the assumption that the target distribution satisfies a log-Sobolev inequality, as opposed to the strongly log-concave condition used in (Li et al., 2019; 2022). This bound matches the best one in the strongly log-concave case and improves upon the best-known convergence rates in non-convex settings. To prove it, we establish a new framework of uniform-in-time convergence for discretizations of SDEs. Distinct from (Li et al., 2019; 2022), we start from the finite-time mean-square fundamental convergence theorem, which combined with uniform-in-time moment bounds of LMC and the exponential ergodicity of SDEs in the non-convex setting gives the desired uniform-in-time convergence. Our framework also applies to the case when the gradient of the potential $U$ is non-globally Lipschitz with superlinear growth, for which modified LMC samplers are proposed and analyzed, with a non-asymptotic error bound in $\mathcal{W}_2$-distance obtained. Numerical experiments corroborate the theoretical analysis.

aon-asymptotic Error Bounds in Wy-Distance with Sqrt(d) Dimension Dependence and First Order Convergence for Langevin Monte Carlo beyond Log-Concavity
Bin Yang" 1 Xiaojie Wang" 1
Abstract
1. Introduction
Generating samples from a high dimensional prob- ability distribution is a fundamental task with wide-ranging applications in the area of scien- tific computing, statisties and machine learning- This article revisits the popular Langevin Monte Carlo (LMC) sampling algorithms and provides non-asymptotic error analysis in Wy-distance in a non-convex setting. In particular, we prove an error bound O(vdh), which guarantees a mix- ing time Ö( (ide
Sampling from complex, high dimensional target probabil- ity measure x =dxx) e-V(=) where U(.) R - IR is potential function, finds wide-ranging applications in the area of scientific computing, statistics and machine learn- ing (Durmus Moulines, 2019; Song & Ermon, 2019; Wibisono, 2018; Liang Su, 2019; Kakade, 2003; Neal, 1992: Welling Teh, 2011). The Langevin stochastic differ- ential equation (SDE) associated with the target probability measure is defined by:
(ide)) 1) to achieve the accuracy toler- ance e, under certain log-smooth conditions and the assumption that the target distribution satis- fies a log-Sobolev inequality, as opposed to the strongly log-concave condition used in (Li et al., 2019; 2022). This bound matches the best one in the strongly log-concave case and improves upon the best-known convergence rates in non- convex settings. To prove it, we establish new framework of uniform-in-time convergence for discretizations of SDEs. Distinct from (Li et al., 2019; 2022), we start from the finite-time mean- square fundamental convergence theorem, which combined with uniform-in in-time moment bounds of LMC and the exponential ergodicity of SDEs the non-convex setting gives the desired uniform- in-time convergence. Our framework also applies to the case when the gradient of the potential U is non- non-globally Lipschitz with superlinear growth, for which modified LMC samplers are proposed and analyzed, with non-asymptotic error bound in Wa-distncce obtained. Numerical experiments corroborate the theoretical analysis.
dX, = -UU())) di + V2dW, Xo = L 0, (1) =0,
where W. :- (W.3 W.,... w [0, Ω Rd is d-dimensional Brownian motion defined on the filtered probability space (S, (Fe)tzo. P), satisfying the usual conditions. The initial data Ip Ω + Id is assumed to be Fo-measurable. Under mild conditions, the Langevin SDE admits т as its unique invariant distribution (see, e.g., (Pavliotis, 2014)). However, sampling from the exact dy- namics (1) directly is usually not accessible and the numeri- cal discretization becomes a practical option. The popular Langevin sampler is based on the Euler-Maruyama (EM) discretization of (1), given by
Yn+1 = Yn VU(Y)) + 22h Yo = xo.
(2)
0 is the uniform timestep and Ce (Gt 22 ..) Gaussian vectors. By combining gradient information with noise, this algorithm, termed as Langevin Monte Carlo (LMC), allows faster convergence and more efficient ex- ploration of complex, high-dimensional distributions com- pared to the naive Markov Chain Monte Carlo (MCMC). The capability to generate high quality samples and improve model training makes popular in many applications such generative models.
where
-2)T " k E N, are i.i.d standard d-dimensional
'Equal contribution 'School of Mathematics and Statistics, Hu- nan Research Center of the Basic Discipline for Analytical Mathe- matics, HNNALLAMA Central South University, Changsha 410083, China. Correspondence to: Xiaojic Wang <xx.jwang @ews..ddueenn x.j.wang7 @ com>>
The study of LMC sampling algorithms has attracted considerable attention over recent years. A central problem to determine the number of iteration steps (termed as mixing time) guaranteeing that the distribution of the Markov chain is a good approximation of the target
Related literature.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Non-asymplotic Error Bounds in Wy-Distance for LMC beyond Log-Concavity
measure и. To obtain the mixing time, the non-asymptotic sampling error analysis is critical and has been extensively studied in the literature (Cheng & Bartlett, 2018; Dalalyan,
2025; Pagès & Panloup, 2023; Majka et al., 2020; Neufeld et al., 2025; Vempala & Wibisono, 2019; Lytras & Sabanis, 2025; Lytras & Mertikopoulos, 2024; Li & Wang, 2025),
2017a; Durmus et al., 2019; Dalalyan, 201 20; Dalalyan & Karagulyan, 2019; Durmus& & Moulines, 2019; Li et al., 2019; 2022; Sabanis & Zhang, 2019; Xu et al., 2018), par- ticularly under a strongly log-concave condition (m > 0):
On the condition that the potential is strongly-con side a ball but possibly nonconvex inside this ball, Cheng et al (Cheng et al., 2018) established upper bound Ö(de ²) on the number of steps (mixing time) required for LMC (2) to achieve the accuracy tolerance in W1-distance. Under convexity at infinity condition, Majka et al. (Majka et al., 2020) showed error bounds ((d//4/2/4) and ((22/2)//2)) in W2 and Wi-distacce, respectively, which guarantees mixing times O(de and Ó to achieve the accu- racy tolerance c. Under certain log-smooth conditions and the assumption that the target distribution satisfies a log- Sobolev inequality, Mou et al (Mou et al., 2022) obtained an improved Kullback-Leibler divergence bound, implying mixing times of O(de-') in total variation distance and Wyydistance. For some non-convex conditions, Pages and Panloup (Pagès & Panloup, 2023) also proved some error bounds for LMC, but does not explicitly specify the dimen- sion dependence. When the drift coefficient is non- non-globally Lipschitz continuous, (Li et al., 2025; Pang et al., 2025; Neufeld et al., 2025) examined non-asymptotic error analy- sis of (modified) LMC sampling algorithms under convexity at infinity condition.
out-
yl², (c v, VU (2) - VU (v)) 2 m|z ul,, Vx,
E R (3)
Under this condition as well as the gradient Lipschitzness, the seminal work of Dalalyan (Dalalyan, 2017b) explicitly showed non-asymptotic convergence of order ((v dh) to Langevin diffusion with warm start, implying that the LMC achieves c error, in total variation distance, in Ö( steps. The authors of (Durmus& & Moulines, 2017) improved and extended the results of (Dalalyan, 2017b). In Wy-distance, Dalalyan (Dalalyan, 20) Durmus et al (Durmus et al., 2019), Cheng and Bartlett (Cheng & Bartlett, 2018) ob- tained similar non-asymptotic error bounds. By additionally imposing the Hessian Lipschitz condition on U, Durmus and Moulines (Durmus & Moulines, 2019) provided im- proved non-asymptotic error bound (dhh) in Wa-distance for LMC, which guarantees mixing time Ō($). Moti- vated by a recent work (Chewi et al., 2021) that shows better dimension dependence for Metropolis- Adjjused im- provement of I.M., Li et al (Li et al., 2022) obtained an error bound O(vdh) in Wy-distance, under the strongly log-concave condition (3) and additional smoothness condi- tions on U. This leads to Ō vd) mixing time bound, which turns out to be optimal (in terms of order) for LMC in both dimension dl and accuracy tolerance e.
Our contributions. In this work, we aim to answer the aforementioned question (Q) to the positive and show an error bound vdh in Wy-distance for LMC (2), under certain log- smooth conditions and the assumption that the target distribution satisfies a log-Sobolev inequality (see Theorem 2.))), opposed to the strongly log-concave condition used in (Li et al., 2019; 2022). This bound guarantees mixing time Ō ")),,,, which matches the best one in the strongly log-concave case and improves upon the best-known conver- gence rates in non-convex settings. To achieve it, we estab- lish a new, non-convex theoretical framework of uniform-in time convergence for discretizations of general SDEs (see Theorem 3.4). Distinct from (Li et al., 2019; 2022), start from the finite-time mean-square fundamental convergence theorem, which combined with uniform-in inttim moment bounds of LMC and the exponential ergodicity of SDEs in the non-convex setting gives the desired uniform-in -nnttmme convergence. Our framework also applies to the case when the gradient of the potential U is non-globally Lipschitz with superlinear growth, for which modified LMC samplers proposed and analyzed, with a non-asymptotic error bound in Wy-distance obtained (see subsection 2.3). The main contribution of this work can be summarized as follows:
Nevertheless, the strongly log-concave condition is seldom satisfied in practice. An interesting and natural question thus arises:
(Q). Can the error bound O O(/dh) in Wi-distance still hold true for LMC without the strongly log-concave condition?
Before answering this question, we highlight that the dy- namies of the Langevin SDE (1) in non-convex settings is crucial but non-trivial (see, e.g., (Eberle, 2016; Luo & Wang, 2016; Wang, 2020) and references therein). Fol- lowing the idea of the reflection couplings, Eberle (Eberle, 2016) showed contraction in Wi-distance for Langevin SDE (1) with the potential U being strongly convex in the long distance. When the unique invariant probability measure sat- isfes the log-Sobolev inequality (LSS), the author of (Wang, 2020) provided Wygcoonvrgeenee the equilibrium of the Langevin SDE (1).
Recently, there has been some progress in the non- asymptotic error analysis for (modified) LMC, without the aforementioned strongly log-concave condition, see, e.g., (Cheng et al., 2018; Chewi et al., 2024; Mou et al., 2022: Mousavi- Hosseini al., 2023; Pang et al., 2025; Li al.,
Based on the finite-time convergence theorem and the exponential ergodicity of SDEs, we establish a new, non-convex theoretical framework of uniform-in- time convergence (in Wyydistance) for discretizations of general SDEs, without requiring the strongly log-
2

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Non-asymptotic Error Bounds in W3-iisance for LMC beyond Log-Concavity
concave condition. The uniform-in-time convergence rate is explicitly expressed in terms of the contraction rate of SDEs and local convergence rates in finite-time of the numerical scheme.
two probability measures V1, V2 P(R4) define coupling (or transference plan) e between in and v2 as probability measure on (Rd R4, B( x RR4) such that o(A x R*) = 1i1 (A) and o x A) = (2( for all A E B(R4). We then denote by [(1,, U2) the set of all such couplings and define the L Wasssesstttn distance p distance in short) between a pair of probability measures ii and M22 as
As one application of the new framework, we prove error bound dhv in W2-distance for LMC, guaran- teeing mixing time bound O(vde=") to achieve the accuracy tolerance e, under certain log-smooth condi- tions and the assumption that the target distribution sat- isfies a log-Sop Sobolee inequality. This bound matches the best one in the strongly log-concave case and improves upon the best-known convergence rates in non-convex settings.
Upe 10- vl (Et(UI:
1/P - d)(
(2)
inf
for p 1. Given filtered probability space F2. we use E to mean the expectation. In addition, denote by C,(IR (resp. (2(R)) the Ba- nach space of all uniformly continuous differentiable and bounded mappings (resp. Borel bounded mappings). For N, let C) be the subspace of C2 (IR") consisting of all I-times continuously differentiable functions with bounded partial derivatives. For any CR())) and E P PRR)), we denote v(Ø) == SR (())
As another application of the framework, we also con- sider the case when the gradient of the potential U is non-globally Lipschitz with superlinear growth. For this case, modified LMC samplers are proposed and analyzed, with a non-asymptotic error bound in Wz- distance explicitly revealed.
The structure of this paper is as follows. Section 2 presents main results for LMC sampling algorithms, with an overview of proof also given. In Section 3, new non- convex framework of uniform-in-ting convergence theorem is established, based on which the main results are derived. Several numerical examples are reported in Section 4 and some concluding remarks are given in the last section.
2.1. Exponential Ergodicity of the Langevin SDE
In this subsection, we first prove the ceffform-i-tiime mo- ment bounds under the dissipativity condition and then pro- vide the exponential convergence in Wy-distance under one- sided Lipschitz condition and the log-Sobolev inequality.
Assumption 2.1 (Dissipativity condition). There exist two positive constants and μ', independent of d, such that
2. Main Results
(2r, VU (x)) > natl - j'd, Vx ER..
(4)
This section presents notation, assumptions and main results for Langevin Monte Carlo (LMC) associated to gradient Lipschitz potential and modified Langevin Monte-Carlo (mLMC) associated to gradient super-linearly growing po- tential. Also, an overview of proof is given.
We highlight that all constants used here (i.c., Mr pc) and throughout this section do not depend on the dimensions d. Instead of a strong convex condition, we put a one-sided Lipschitz condition on -V.
Notation Throughout this paper, we use N to denote the set of all positive integers and let No NU {0}. For all E N. let [n]] := 21,2,.. ... n and [n]o == 1. m). For convention, we set 0 = 1. The symbols A and V mean "minimum" and "maximum". respectively. We write Ô(-) to mean that O(-) log (.)) We also use the notation (.-)) and |- to denote the inner product and Euclidean norm of vectors in Rd, respectively. Let || || and || || denote the operator and trace norm of matrices, respectively. For function f Rd -> R. we write f to denote the i-th partial derivative of ʃ. The gradient V / is the vector of partial derivatives (fr .... Oaf) and the Hessian 2 f is the matrix (8t, [) The Laplacian of f is denoted by Af :- 2 1 S.
Assumption 2.2 (One-sided Lipschitz condition). There exists a dimension-indeeeedeen constant L > 0 such that for all x, E Rd,
(5) Assumption 2.3 Loog-Sobolev inequality). Let (pelreo be the semigroups associated to Langevin SDE (1) admitting a unique invariant distribution æ. Assume that the invariant distribution satisfies the log-Sobolev inequality, namely, there exists constant p. independent of d, such that, for any di C)((R.).
(r y. (2) (V) (w)) --/ -
a( 0²) pT((( x(02= =1
(6)
Let ((R) be the Borel -file of 8d and P(Rd) be the space of all probability distributions on (Rd,, B(Rd).. For
We mention that, beyond log-concavity, the log-Soboler inequality (ISII) a widely used assumption for the target
3

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Non-asymplotic Error Bounds in WyDiistance for LMC beyond Log-Concavity
distribution of interest in the field of Langevin sampling (Vempala & Wibisono, 2019; Lytras & Sabanis, 2025; Mou et al., 2022; Lytras & Mertikopoulos, 2024; Chewi et al., 2024). As indicated by Appendix A in supplementary ma- terial to (Mou et al., 2022), strongly convex outside a ball implies LSI. As a consequence, two typical examples satis- fying LSI are the Gaussian mixture and double-well well poten- tial (see Section 4). For more details on LSI, one can also consult (Ledoux, 2006).
with L (O)) Moreover, using (9) and the Cauchy-Sehie -echar inequality ensures that Assumption 2.2 holds with L = L1.
Assumption 2.7. There exists positive constant 01, inde- pendent of d, such that
E [ []] oid.
(11)
In addition, we need linear growth condition of the 3rd- order derivative of U, which has been also used in (Li et al., 2022) in strongly convex setting, As verified in Appendix G, a Gaussian mixture meets such a condition and the double-well potential meets a similar condition in Assumption 2.14. More examples can be found in (Li et al., 2022).
Lemma 2.4 (Uniform-in-time moment estimate for Langevin SDE). Let Assumption 2.1 hold and let (Xe he2o be the solution of the Langevin SDE (1). Then there exists a constant e E (0, 2p), independent of d, t, such that holds for all p E [1, +00),
E B| [+] S e " [1roo 2r]
2(2p- 1+
2p-2 (2p-c)p)
d².
cp
Assumption 2.8 (Linear growth condition of the 3rd-order derivative). There exist two positive constants Lg and Lo such that for all x, y E Rd.
(7)
The proof of this lemma can be found in Appendix A. Let {pete2 be the Markov semigroup associated the solution {Xe of SDE (1). Then for any / E P P( vp, denotes the distribution law of X, starting from Xo zo i.c., vP: == C(),), t 2 O. Next we present a proposition on exponential ergodicity in W2-distance of the Langevin SDE (I) under LSI.
12 + Lalar V (z)) S 1/
(12)
As remarked by (Li et al., 2022), Assumption 2.8 is not necessarily stronger than Hessian Lipschitzness. By As- sumption 2.6, we have
Proposition 2.5 (Exponential ergodicity in Wy-distance). Let Assumptions 2.2 and 2.3 hold. Then for any? t 2 and [(xo). there exist two constants K. > 0, independent of d, t, such that the semigroup Pt associated to SDE (1) and its invariant distribution a satisfy
}U(M s L.u.l. Vzz y E Rd
(13)
initial distribution
Lemma 2.9 ceaiform-in-timee moment estimate for LMC). Let Assumptions 2.1, 2.6 hold. If the uniform timestep satis- fies h S 427 p
P A 1 A 1, then the LMC (2) satisfies
Wa(up; m) S ICe "2((,,, m)-
(8)
BiIy 2] S e -m Bllxol neNo
1+11, d. (14)
sup
"
Such an assertion follows from Theorem 2.1 (2) and The- orem 2.6 (2) in (Wang. 2020), where the parameter depen- dence was not explicitly provided. Noting that the Langevin SDE (1) is driven by additive noise (non-degenerate) and owing to the oie-sided Lipschitz condition (5) and LSI (6), one can follow basic lines in (Wang, 2020) to identify two constants I. := ( 2pt ( 1/2e2 " V e 21++ 3, 17 := o only
Its proof is put in Appendix A. The first main result of this paper is as follows.
Theorem 2.10 (Main result for LMC). Let (pr Inee be the semigroups associated to LMC (2) and let Assumptions 2. 2.3, 2.6-2.8 hold. If the uniform timestep satisfies h म प f distribution = L(xo), it holds
1 A 1, then for any n No and initial
depending on L, p. but independent of d, t.
2.2. Main Results for Langevin Monte Carlo
We(PPn, x) s C. + C2 C2Vde de Anh
(15)
Now we turn to the LMC and report its non-asymptotic error bound in Wy-distance without log-concavity.
where = log (22 and
Assumption 2.6 (Gradient Lipschitz condition). There ex- ists a dimenssion-nndependent constant L1 > 0 such that
C Cp= c,,,, Lo, T/y Ta, K,,
(16)
C2 :0 ,,, o, 14),
VU (a:) VU (v) < 1, vl. Vx,yE E Rd.
(9)
are explicitly given by (148).
Proposition 2.11. Let assumptions of Theorem 2.10 hold. To achieve given accuracy tolerance c >> 0 under Wi- distance, a required number of iterations of the LMC (2) is of order Ō vd
Owing to (9), one can apply the triangle inequality show that for all Rd.
IVU (x) 1/2 + L,/o/
(10)
4

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Non-asymptotic Error Bounds in Wy-Distacce for LMC beyond Log-Concavity
holds for all 2, E R&, where 71 2 0,72 71 + 72 7 7
obeying
Table 1. Comparison of mixing times in Wy-distacce.
M-T
S-C
A-A
For this case, we consider a kind of modified Langevin Monte Carlo, also termed as projected Langevin Monte Carlo (pLMC), introduced by (Pang et al., 2025) and given by. Yo 20.
PAPERS*
(
YES
G-L
PAPER"
(de-))
YES
G-L, H-L
PAPER
YES
G-R-L-L-----
PAPER"
0 (dc-))
NO
G-L
PAPER
( (
No
G-L
==1== = TT)(Y) VV)) + V2hG.+1 (22)
THIS WORK
No G-L, 3-RD-L-G
(CHENG BARTLETT, 2018; DALALYAN, 2017A;
where Cx = (2. -2) are i.i.d standard dimensional Gaussian vectors, Th Rd + R is a pro-
DURMUS ET AL., 2019).
(DURMUS & MOULINES, 2019).
jection operator, defined by, for a dimension-independenn parameter and any x E Rd.
(LI AL., 2022).
(MAJKA ET AL., 2020).
E (MoU ET AL., 2022).
M-T: MIXING TIME. STRONG CONVEXITY. A-A:
(x)x) = min {1, Ad भ |x| x, + 0,
ADDITIONAL ASSUMPTION. G-L: GRADIENT LIPS- CHITZNESS. H-L: HESSIAN LIPSCHITZNESS. 3-RD-L- G: LINEAR GROWTH OF THE 3-RD DERIVATIVE.
0. (23)
See Appendix E for proofs of Theorem 2.10 and Proposi- tion 2.11 1I. In Table we compare the number of iterations of LMC algorithm (2) required achieve eπror W2- distance in the literature. Clearly, error bounds matche the best ones in the strongly log- log-concave case and improve upon the best-known convergence rates in non-convex set- tings.
with y being given in Assumption 2.12.
Lemma 2.15 (Uniform-in-time moment estimate for
pLMC). Let Assumptions 2.1, 2.12, hold. For the uni- form timestep satisfying I 2p ++20, A 1, the pLMC (22) has mniform-in-tiie moment bounds, for any E [0, 00) n N,
sup BiJÝm| Se - E|00 2p [P] + Cid,
(24)
2.3. Main Results for Modified Langevin Monte Carlo
neNo
Next we consider the case when the gradient of the potential U is non-globally Lipschitz with superlinear growth.
where to == nh, Cj = C(p, p L,,,,,L,11
Assumption 2.12 (Gradient polynomial growth condition). There exists a positive constant L,, independent of d, such that for all z, Rd, O,
Its proof can be found in Lemma 3.4 of (Pang et al., 2025).
Theorem 2.16 (Main result for pLMC). Let (pnIneNo be the semigroups associated to pLMC (22) and Assumptions 2.1-2.3, 2. 2.12-2.14 14 hold. If the uniform timestep satisfies 2n
IUU (x) VU (v) < L1 |x| + 10 yl- (17)
h S 2 A 1 p +044+ 1, then for any n E No and initial distribution v/ = L(xo), it holds
Thanks to (17), the triangle inequality and the Young in- equality, we infer
W2(upn, T) S (,d(11+22 O2d (25)
VU (x) (v)) [ L/d +L,g| VzER Rd.
(18)
where 11/ = VU (0) -L1 Similar (13), we get
where = log ook++tn((2)) and
(>)())) < (1 + [x))
(19)
G C( m, 02, L,, Ta, Th Th,, T1 KC),
(26)
Assumption 2.13. There exists some positive constant @2(p), independent of d, such that
C :0= c, 7,00 8, LT)
E E[ [r] < <2(p)dd, P 11+11),
(20)
See Appendix F for the proof of this theorem. This theorem also help us to obtain the mixing time, whose proof is similar to Proposition 2. and thus omitted here.
where 7 comes from Assumption 2.12.
Assumption 2.14 (Polynomial growth condition of the 3rd-order derivative). There exist two positive constants L' and LL. independent of d, such that
Proposition 2.17. Let assumptions of Theorem 2.16 hold. To achieve given accuracy tolerance > 0 under W2- distance, a required number of iterations of the pLMC (22) is of order Ō 000
(AU (x)) S 1 (++1//1 + Lod /2 /2/0
(21)
5

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Non-asymptotic Error Bounds in Wy-Distncce for LMC beyond Log-Coneavity
2.4. Overview of Proof
3. A Non-Convex Theoretical Framework of Uniform-in-Time Convergence Theorem
In this subsection we present an overview of the above non- asymptotic error bounds. For approximation (Y, hec. to the SDE (Xe) the goal of long-ting time error analysis is to bound W2(1Pn: x), where
The aim of the present section is to construct a non-convex theoretical framework of uniform-in in-time convergence theo- rem for general SDEs, which will help us to easily analyze the non-asympiotic error bounds of schemes in Section 2. To this end, we set up a general framework by introducing general SDEs and their numerical approximation as follows.
P(RR)) is the invariant distribution of (pr J:20 and IneNo is the transition semi- groups associated to (YnJnEN, By the triangle inequality, we have
W2(vin: T) SWa(vän-n Pn.. Vpn-ni Pto.) + W2(upn-njPe niPPh, R), n 2 n1.
(27)
3.1. SDEs and Their Numerical Approximations
We consider the following Itō SDEs as follows:
Following the triangle inequality, we give an overview of five steps that comprise the proof of Theorems 2.10, 2.16.
m g (Xi) dw Xo = = (32)
dX2 = (X,) dt +
Step 1. As proved in Lemma 3.1, the solutions of SDEs have uniform-in-time moment estimates with the help of dissipativity conditions. In addition, we require the numeri- cal approximations to be uniform-in-ting moment bounded (cf. Condition (A2)). See Subsection 3.2 for details.
where t 2 0. f: Rd + R is a drift function, g = (g'. ........... :...
9 )) Rdd 4 Rdxm is a diffusion function and {W, (W). w?,..., ,..,..,""""" .W."))e2 is a m-dimensional Wiener process. Let x, =(ss, x;;
x,,(()),
Step 2. We establish the finite-time mean-square fundamen- tal convergence theorem for general SDEs (see Theorem 3.3), which is used to deal with the first term on the right- hand side of (27):
t, denote the solution to (32) at t, starting from the initial value at given by
Xa,((1) = =x f. [(x.)dr km
El gx,, (K,)) (33)
Wa Poy: Ptn-,, ) S Ct )bP
(28)
where we explicitly show the dependence of the error con- stant C(tn,) on fn (cf. (48)). See Subsection 3.3 for more details.
To approximate SDE (32), we introduce the one-step ap- proximation Y (t, h) for the solution X x; + h) of SDE (32) in the form of, for all Rd
Step 3. To estimate the second term on the right-hand side of (27), we rely on the exponential ergodicity of the SDE (see Condition (A5)) to obtain:
(t, + h) a + (t),
(34)
where h is uniform timestep and & is a random vector defined on FP with moments of sufficiently high order and + is function from [0, +oo) x Rd x (0, (0,+ x RT to Rdd Using the one-step approximation (34), we recurrently construct numerical approximations (Ya neNo on the uniform mesh grid (ln = nh) given by
Wy(MPo-nuPt., : T) S -e-ntin Wy(v T). (29)
See Subsection 3.4 for details.
Step 4. Collecting (28) and (29) together and choosing in = Xo such that Ix- = e. one can derive from the uniform-in-time bounded moments that
Y = 2ho Y = Yn + + (Ln, Yn, En),,,
(35)
....... where Sn is independent of Yo, Yı, ... Yn Eu. .n for all 1.
Sn-1
W2(Kpn, x) C(x))h=-
T).
(30)
By iteration and using Lemma D.I, one can deduce
3.2. cefform-in-Timee Bounded Moments
Wa(vin, T) S kn Aoohh
(31)
Under the following dissipative condition, the general SDE (32) have uniform-in-tine bounded moments.
See Subsection 3.5 for details.
Condition (A1) There exist some constants p" [1, A ,". E (0. oc) such that drift and diffusion coefficients of the SDE (32) satisfy, for all x E Rdd
Step 5. We verify all conditions required by the non-convex theoretical framework of uniform-in-time convergence the- orem in Section 3 and calculate all constants to obtain the expected non-asymptotic convergence rate of the underlying LMC (2) and pLMC (22) for Langevin SDE. See Appendix and F for details.
(a, f()) + 3r [9( S ps 1 10/20/
(36)
Moreover, the initial value obeys E. [|<x//277 < oc.
6

---

Non-asymptotic Error Bounds in Wy-Distacce for LMC beyond Log-Concavity
We highlight that constants used here (i.e., A", ,("))) and throughout this section might depend on the dimensions d, m. When all established results in this section are applied to the Langevin SDE and its numerical approximation, the dependence will be explicitly given. Thanks to this con- dition, one can easily derive the uniform-in- in-time bounded moments for SDEs (32) whose proof is put in Appendix B.
Lemma 3.2. Let Conditions (AI), (A3), (A4) hold and let 0) be two solutions of SDES (32) starting from a, y. Let the uniform timestep h > 0 satisfy reprexentations
(t, +0), Xe,
2h S 1. Then for any t 2 0, 0 > O. the following
(t, = + (t,(
(43)
Le (t (++0) := (t + 0) - (x - y).
Lemma 3.1. Let Condition (A1) hold and {Xe]ee J:20 denote the solution of SDE (32). Then for any p E [1, [.,"", it holds
satisfy
[|x, [+] S C3 ]]] + Cb (37)
sup 120
E 1S + m)]²] = (1 + 4T y (44)
where c* E 20) Cj 21 1 and
E gt (t + +) 2 <(6 + 6 + 2
+ ) y
(p( == ctp (2n -p)
2 2p-2 )
prp.
(38)
(45)
where Ō% is given by (79).
Moreover, we need the following assumption on bounded moments of numerical solutions.
Now, we derive the following finite-time strong convergence theorem.
Condition (A2) There exist some positive constants hg- G 2 1 and (p) such that the numerical solution (35)
Theorem 3.3. Let Conditions (AI)(A4) hold and let the uniform timestep h >> 0 satisfy h S IF A ho A 1. For any fixed nı E N. we let T := n h. Let (Xe and (YnIneln)) denote solutions of SDES (32) and numerical approximations (35), respectively. Suppose that the one- step approximation (34) has local weak and strong error of order L1 and p2. respectively, with p2 2 2/2 P1 2 p2 22 i.e. there exist some positive constants Ki Ki. Kš, > O
has uniform-in-time moments, i.e., for any p 2 1. h S hos
sup 20 s GE | + (p). neNo
(39)
3.3. The Finite-Time Mean-Square Fundamental Convergence Theorem for SDEs
and r 2 1 such that for all 0 S SIST T h, x E Rd
This section revisits the finite-time mean-square fundamen- tal convergence theorem. Following the idea of the strong convergence theorem originally proposed by Milstein (Mil- stein, 1988), we aim to reformulate a general mean-square convergence theorem for one-step approximations of SDEs (32). First, we assume the monotonicity and polynomial growth conditions as follows.
[[(X(,,21++ Y(,,, m]]
(Kt + kK|1a(r))) 1/2 h1
(46)
[B[ [B[x xt, hh) +)) 1/2
Condition (A3) There exists a constant L' > 0 such that drift and diffusion coefficients of SDEs (32) satisfy, Va, y E R.
(Kg + k3)00p0) 1/2 hps.
hps
Then
(r-v.
S - (40)
sup [E [[[x(002b6i40 (m) YO,, wbi (0, (n)) 2/2 nE(ma]
(47)
Condition (A4) There exists a positive constant L such that the drift coefficients satisfy, for all y Rd, ro E oo),
1/2
<C (T) (K** + + 11/2,
[f(x) f()) < L3 (1 | + 1z vl. (41)
where
C* = e20+1000
(48)
Condition (A4) immediately implies that
and K and K* are explicitly given by (103).
[((z) < i. + 1 Vx E Rd
(42)
This theorem is proved in Appendix C. Similar results were obtained by (Tretyakov& & Zhang, 2013), where the param- eter (e.g.. T) dependence in the error constants was not explicitly shown. We mention that both finite-time conver- gence theorems follow the original idea of (Milstein, 1988).
where = - |f(0)) + Lo. Based on these two conditions, one can easily obtain the next lemma, whose proof is put in Appendix B.
7

---

Non-asymplotic Error Bounds in W3-Distacee for LMC beyond Cog-Coonyyvity
3.4. Exponential Ergodicity in Wiz-iistacce
In the present subsection, we assume that the general SDEs (32) have exponential ergodicity. Let {Pr) be the Markov semigroup associated to the solutions {Xe) of SDE (32), which is defined by, for all x E RC and 0 S $
(ro(z)= == (xx, (1)), & E Ba(R)).
(49)
(a) Probability density of the first component for the Gaus- sian Mixture.
(b) Probability density of the first component for the Double-well potential.
We put some conditions exponential ergodicity in W2- distance of SDEs.
Condition (A5) Assume the semigroup {Pr)ezo associated to SDE (32) has a unique invariant distribution IT. There exists two positive constants K* and 1/" such that, for all t 2 0.
W2(PP,, S Kee Kee "22
(50)
(c) Strong convergence rates of LMC algorithm for the Gaussian Mixture.
(d) Strong convergence rates of pLMC algorithm for the Double-well potential.
where // = ((%)) is the initial distribution.
In (Wang, 2020), some sufficient conditions are provided on coefficients of SDEs such that the underlying SDEs have exponential ergodicity in Wy-distance. In particular, the Langevin SDE (1) has such an exponential ergodicity prop- erty in our non-convex setting.
We can directly show that the Gaussian mixture satisfies Assumption 2.1 with μ = 1.4 n'd 2/a Assumption 2.8
with 44 = 0. Lo = Slal" Assumption 2.6 with L1 = 1+
3
4|a|² and the double-wed potential with a = 3 = 1 satisfies
the Assumption 2.1 with 1 = 1, μ'd = 1, Assumption
3.5. The Uniform-in-Timee Convergence Theorem
2.2 with L 4v2 + 19/2, Assumption 2.12 with 7 2
and L, 1, Assumption 2.14 with L. 0, Lo 4.
Under the above conditions, we are able to formulate the uniform in-time convergence theorem as follows.
`/1 = 72 1. For these two cases, strongly convex outside ball is satisfied and thus the log-Sobolev inequality is satisfied. Assumptions 2.1, 2.2, 2.6, 2.12 have been already verified in Proposition 3.1 of (Neufeld et al., 2025) and Example 2.5 in (Pang et al., 2025). In Appendix G, we give brief proof of Assumption 2.8 for Gaussian mixture and Assumption 2.14 for the double-well potential.
Theorem 3.4. Let conditions of Theorem 3.3 and Condition (A5) hold. Let (Pa)neN, be the semigroup associated to the numerical scheme (35). Then for any n E No and any initial distribution v/ = L(xaa). it holds
Wa(v 11) < Rihh + Rize An
(51)
In what follows set zg 0, a E R with |a| 2 and a = 1, 3 = 16, where all components of the a are equal. We emphasize that the potential U, of the Gaussian mixture is non-convex in our setting (see Example I of (Dalalyan, 201 for a convex setting, i.e., |a] < 1).
where X" = log k n¹' ((br and the constunts K3, Rgg are given by (122).
The proof is postponed to Appendix D.
To study the probability distributions, we fix the dimen- sion d 100, T 6 and simulate 300 independent Markov chains using LMC algorithm for each step size h = {0.001, 0.005, 0.01, 0.05, 0.1}. In Figure ((a) and (bb), we present the density curves of the first components for Gaussian mixture and double-well potential, respec- tively.
4. Numerical Experiments
In this section, numerical results are performed to verify the above theoretical finding. To this end, we consider two target measures specified by the following two potentials:
a). Gaussian Mixture,
Moving on the convergence analysis, we run the LMC algorithm (2) and pLMC algorithm (22) for the Langevin SDE by different stepsizes h till T 6. Here numerical approximations are performed using five different stepsizes h E (2-5,2 2-7 2-7,2-5 2-,2_,8 2 --.. The eract solution is iden- tified as the numerical one using a fine stepsize hye/ = =2- 13 and the expectations are approximated by computing aver-
(x()) = f/z al2 log +e e x x ZERd (52)
for a given a E Rdd.
b). Double-well potential,
U2(x)= = - |||] E|of, x ERRd.
(53)
8

---

Non-asymptotic Error Bounds in W3-Dssance for LMC beyond Log-Concavity
ages over 3000 samples. From Figures I(c), ((d), it is observed that the convergence rate is of order 1.
dient descent. In Conference on Learning Theory, pp. 678-689. PMLR, 2017a.
Dalalyan, A. S. Theoretical guarantees for approximate sampling from smooth and log- log-concave densities. Jour- nal of the Royal Statistical Society Series B: Statistical Methodology, 79(3):651-676 2017b.
5. Conclusion and Future Work
In the present article, we provide an error bound O(v in W2-distance for the classical LMC without log-concavity. For the case when the gradient of the potential U/ non- globally Lipschitz with superlinear growth, modified LMC samplers are introduced and analyzed, with W2 enror bound obtained. These results are derived essentially based on a newly developed non-convex theoretical framework of uniform-in-tue in-tmee convergence for discretizations general SDEs. This framework can also applied to other metrics and higher order LMC sampling algorithms (Sabanis & Zhang, 2019; Li al., 2019), which is our ongoing work.
Dalalyan, A. S. and Karagulyan, User-friendly guar- antees for the Langevin Monte Carlo with inaccurate gradient. Stochastic Processes and their Applications, 129(12):528--5311 2019.
Durmus, and Moulines, Nonasymptotic convergence analysis for the unadjusted Langevin algorithm. Ann. Appl. Probab., 27(3):1551-158) 2017.
Durmus, A. and Moulines, E. High-dimensional
Bayesian inference via the unadjusted Langevin algo- rithm. Bermoulli, 25(4)) 2882, 2019.
Acknowledgements
The authors thank Lei Dai, Yingsong Jiang, Chenxu Pang, Xiaoyan Zhang, Yuying Zhao, anonymous reviewers and area chair for suggestions that significantly improved the quality of this paper. This work was supported by Natural Science Foundation of China (Nos. 12471394, 1207 1488, 12371417).
Durmus, A., Majewski, S., and Miasojedow, B. Analysis of Langevin Monte Carlo via convex optimization. Journal of Machine Learning Research, (0(73):1-466 2019.
Eberle, A. Reflection couplings and contraction rates for diffusions. Probab. Theory Related Fields, 166(3-4):851- 886, 2016.
Impact Statement
Kakade, S. M. On the sample complexity of reinforcement learming. University of London, University College Lon- don (United Kingdom), 2003.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Ledoux, M. Concentration of measure and logarithmic Sobolev inequalities. Seminaire de probabilites XXXIII, pp. 120-216. Springer, 2006.
References
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Cheng, X. and Bartlett, P. Convergence of Langevin MCMC in KL-divergence. In Algorithmic Learning Theory, pp. 186-211. PMLR, 2018.
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Chewi, S., Lu, C., Ahn, K., Cheng, X., Le Gouic, T., and Rigollet, P. Optimal dimension dependence of the Metropolis-adjusted Langevin algorithm. In Conference on Learning Theory, pp. 1260-1300. PMLR, 2021.
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Non-asymptotic Error Bounds in Wy-Distance for LMC beyond Log-Concavity
Luo, D. and Wang, J. Exponential convergence in LP- Wasserstein distance for diffusion processes without uni- formly dissipative drift. Math. Nachr, 28914--1))1909- 1926, 2016.
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10

---

Non-asymptotic Error Bounds in Wy-Distacce for LMC beyond Log-Concavity
A. Proofs of Lemmas in Section 2
Proof of Lemma 2.4. Under Assumption 2.1, one can easily check that, for any p² 2 1, Condition (A1) fulfilled with
i ===p'd= + (2p' 1)d, =²=
(54)
Then Lemma 2.4 is a direct consequence of Lemma 3.1.
Proof of Lemma 2.9. Taking square on both sides of the LMC (2), one can use Assumption (2.1), (10) and the Cauchy- Schwarz inequality to derive that, for h I 1 A 1,
Yn+1 =|X/|== A² x
+ 2h/s+112 (1, (U,))) + +2 Sn+1) - 2V 2VU.) Sn+1)
<|Yn|' + 2 + 1h 189--2200 2h(Yn, (U,). + 22222(22.. (n+1)
(55)
<| 41,1+41,1110 v. 2 4h G 1 2/hh Vn 2 + 2v 2h (Y,, + (4I7 + 21()dd
=1) 2 4// +1 2V 2V S + (4L 2p''ddh.
Taking expectations on both sides of (55) we deduce
B[J + [2] <(1 E Y 1 44h [| (4T + 2p')dh
(56)
<1 - p |2] + (4 4T,2 + 2p')dh.
By iteration and by the assumption h S n² one can use the inequality x S e 0,00 x S to arrive at
n
E || n+1
E 1 [2] + (4 + AL, + 24 dh |2] <(1 ph)
Σ (1 ph)'
(57)
Se wc- El |2 4+42,+22 +2p" d.
The proof is now completed.
B. Proofs of Lemmas in Section 3
Proof of Lemma 3.1. For every integer 1, we first define stopping time as follows:
Tx = inf { 2 0: |X, 2 n}.
(58)
By using the Itô formula and the e p(eATT)) (51 + |Xenta13) 13)
we can derive that, for any 1 2 0, 0 and c* (0, 2/²),
1AT
AAT
=(e1 + (zd1))p + ccp ec'ps ps (61 + (x,l3)
2p
e ps
(51+ + (xx,)
[(x))) ds
+p/ 0
thnn
en'ma + +1x.13) 8-1 (u(x)))
ds 2p(p - 1) /
AATo
(e1 + 2x 8-2
Xx. g (Xs)) dss
AAT
(59)
+ 2p
e
(C + +(oo1))
(S. g (X,)) dw*
<(E1 + 1161))) + tcpp ( MMa c c (1 + 2), ds s + 2p
m
(E1 (x1, Σ
Xex g (X.)) dw
+p / thTT (1 + 1x,, (2 (2 ((xx) -2p 1)1 ds.
Taking expectations both sides and letting e yiield
1 PATA
dhT ecpee 22-2
E p(tnttt))) XXana 20] <R <B(1=00 20] + c
ee ee'po| 2d + +pE
(60)
(2(X.. ((x,)) + (2p 1))lo(X))))) ds

---

Non-asymptotic Error Bounds in Wy-Distacce for LMC beyond Cog-Concyvity
where the property of the Itô integral was used that
thTo
E
Xa
2p
Σ
(X.,)((X)))
dW 0.
(61)
km
Taking (36) into account, one easily derive from (60) that
e
<B [ [
+
pE J. (c* 24- p pol
x ds
+ 2plE
EATT
1 p0 po
X2 |x 20 ds
(62)
It follows from the Young inequality that, for any E2 > 0
X
3p
E2 X
2p
+ ) A..
(63)
Setting 82 2p'-c" one can further insert (63) into (62) obtain
E e
p(tATT) Xearm 2p
S B
[ ]r] + 2 2p-2 p
P. E pltntr))
(64)
12p=-c" 1p
Thanks to the Fatou lemma, we then let n + oo to attain
E
Xx
2p
Se S [
+ 2
(2pt, - P.
2p-2
pp-
(65)
Observing C. 2 12 cppt one can easily obtain the desired assertion.
Proof of Lemma 3.2. We first prove (44), By using the Itō formula and the condition (40), we obtain for any > 0.
E 1 (t + h)1
E | yl ( + l:th B (X)= (x, (x)) - f (X (8))) ds
r4h
+ h+h E
(66)
[1 (x) (X (x)))))) ds
<
[ v|²] + 2L*
J.
r+h
E [()) ds.
The Gronwall inequality helps us to show
S (t [)]1 S 21 2 21
(67)
We notice that for any S 27,
2
(2L'A)*
1 + 2Lth
(2L h)-1
21+221(ee1111 1)h 11 hL"h.
(68)
Inserting this into (67) yields (44). Regarding E| Z (1 + 1)) by applying the Itô formula, we have
++h
(1
-2
E (X)
(x)) ds
(69)
Ith
+
(X 9 (x,( | ds.
Bearing this in mind and noting
(x) Xt,(1 + hi) Xty(s) (xr - v),
(70)
12

---

Non-asymplotic Error Bounds in Wy-Distncce for LMC beyond Concaviyy
one can further use the Cauchy-Schwarz inequality
inequality and (40) to derive
E 1 Z (t + h)]²]
th
((xt,=(8 (8) ( - ( (Xx,(()))) + (X),,
- 9 (x), ( ds
r+h
2 I.
E fa -f (X),(()) - f (X,, (Xx,, (s))) ds
(71)
Ihh
X (s) ds + 2/2 vl +hh (()
S 2L
B[ls (x)) (x), ds.
=1f2
In what follows, we handle the two terms in (71) separately. We first emply (67) as well as h S 2l to show
([)X (s,( s 22t11|10 u12 S elæ y1² S 3/x- v
(72)
which directly implies
1 S 6T | - y 2h.
(73)
Before coming to the estimate of 12, one can derive from (41) and the Hölder inequality that
(x) ( - [xx( B[I (x) - (X,, (+)) ST (1 + (+) To + (x) (o)
/ [B[(1 + X (8) + (+) 1/2
1/2
[ [[]) ()))
(74)
1/2 1x - 1/.
<313
where we used (72) in the last step. By noting
B[| {] =1 50 =0 0,
(75)
and
E (s)
< (L][
[5) S [E[[xe((s))]]) ro (0,
(76)
one derives from Lemma 3.1 and (a + 6)2 S a + a, 2 0. (0, 1) that
1.
ro = 0,
ro 0,
Cilo 20 + ())(1)
ro E (0, 1), E (s) S
c 2n + (1))
ro E (0, 1),
1 2 + (ro).
To E [1,
Oolyl 2no + Co(ro):
To E [o).)).
(77)
Inserting this into (74), one can easily see
Bll (s)) (Xer(s)) s 3LL (C, + 2 + (00) 1/2 2/2 y/.
(78)
where
L.
0,
O (V
to E (0, 1),
(79)
(v3((C((ro),
To E [1, oc).
This suffices to ensure
1 S 6I, (C; Cola| 2ro + 20o 1/2 |cc yr 2h.
(80)
Plugging (73) and (80) into (71) shows
El + h)]3] < (99' + (b2 + -/
+ C 200) 1/2) 10c 212
(81)
as required.
13

---

Non-asymptotic Error Bounds in W3-Dsstncee for LMC beyond Log-Concavity
C. Proof of Theorem 3.3
Proof of Theorem 3.3. By (33), we have
X0,56 (tx+1) - Yo.z (fx+1)= = Xta.x(ts) (fx+1) - Ye,Y' Yo (f++1) = X(w))) (t++1) - Xe (tx+1)) + (Nt,,y (fx-1) Ye,, (tx+1))-
(82)
The first difference the right-hand side of (82) is caused by different initial data. The second difference is the one-step approximation error. Taking square and expectation on both sides gives
Xo.zb (4x-1) - No.z (1x-1)1²]
E ((e) (tu+1) - Ne (tx+1) 1 Fix - LE Kex (tx+1) - Ne (t++1) |²] Fis ]
(83)
=id.
+ 2F E (Xa (1x+1) - Nenya (la+1)) (Xearve (lx+1) - Youna (M++1) Fin]
=:gg
In the following we cope with the above three items separately. By the conditional version of (44), we first get
s + (1+444")BB X (4x) Yul²
(84)
In light of the conditional version of the second inequality in (46) and Condition (A2), one can easily treat J2 follows
J2 -
K2 K2 ( P2
K2 Kz C 2
(85)
K2 (r)((r) K3CT
| (266)21) 1122.
In order to estimate J3 in (83), first note that
Xite) (t++1) Xea.r (1++1) (tx) Yu ( (t++1).
(86)
As consequence,
J/ 2 E (X (tx)
YY,)" "x))((X ((x+1) Ye (16+2)) Fix) ]]]
=idas
(87)
2 E (
(Xmar (U++1) (X Yu.ra (lx+1)) Fen
t.Jaa
Yv is Fr, -eeasurable the conditional version of the first inequality in (46), together with the Hölder inequality and Condition (A2), leads to
Since X(tx)
x (tx) - Yτ) T E
(Ximx (t++1) - Yeurs ((x++1) ( [i.
Ju =2 R
X (lx) Yil P)) 1/2
E [X (1++1) - Years (1x+1)) Fru]] ]() 1/2
22
R
E
(2
[x (ta) Yu ())
[KT 1i+KKH KG p-1) 1/2 hP:
(88)
|x (tx) Yu ()] (2T+ (-11)) 1/2 h
52 E
<R
x8(1x(1)) (1x) - Ya 2 ho + [ KTC1() 1:
14

---

Non-asymplotic Error Bounds in Wy-Distacce for LMC beyond Concavittyy
To handle J32, we employ the Hölder inequality get
Fiul) 1/2 ( (1e (Ux+1) Ye (1x+1) ( )
1/27
J32 I 2B
(89)
Here utilize the conditional version of (45) to show that
(E (I )
1/2 < ))²²
+6L + (())( 20 + (r0)) 200)
1/2
(x(tx)) - -Y (90)
With this at hand, using the inequality Va +b S vā + +b++ b,,,,,be. O and the conditional version of the second inequality in (46), one can derive from (89) that
+G 1/2 (((ta) (Rg KK|x2 (2x) 1/27
J32 S2E
6LL
(07.1) 1/2 x(tx) - KR 1/2
$2R
+ +OL;) 1/2 ((O;, 1/2 + cb 1122(1)))) - Yol (RS + K3 Kg11212 1/27
hps+1/2
(91)
S(6L* + 6L;)
+ K31V2 1220
CCC (k: -k]yk )1000 1222.
By Condition (A2), one can deduce
E Kg + KgTTWP*] < (k: + + 11)
(92)
With regard to the third term in (91), by using the Young inequality, we obtain that, for any Ti 2 1,
E +G)
C7 K2
+
Co Y2 ro + K3 2 K (+x) + KY 2r+
2r C . (d K3)) 2-11
s(O:)) 1/ + 2ーナto
2r
C
K2)
To 2rtro
Ca n E (1x)) 2rtro]
+ 2rtro
+ 2rtro
C. d E Yı!
2r+ra]
2r4ro K2
2 200
+ 2,
2r
[ ( + शे K3 | 2r+r
+ 2rtro
2r
K 0
1x 2rtro]
+ K C E Y& 2n+n7]]
=(O;)) K3 +
4r
Ci d t1 (d K2)
2rtrn 2r
+
ro
K2(O;,)
20+ म.न
+ 2rtro
ra
Cā (d- "1 + 2r
2rtro
2rtro
2r + grtto Co d irtro
K +
2r
K2C + K2C) E [
2r+
(93)
By virtue of Lemma 3.1, one can obtain that
E (rrxtt < C3E(Ia
2 + Og (2r + ro).
(94)
In the same way, Condition (A2) implies
E V 2n+ru] < (1 + C7 + ro)-
(95)
15

---

Non-asymplotic Error Bounds in Wy-Distacce for LMC beyond Concaviyy
Plugging these two estimates into (93) gives
(((())) 1/2 + (( -
[ + K2]1)( (2))
s <(,)) R3 + artro Cid Ki)
2ntra
2"
+ 2rtro K2 (0;))
2-tro Co kKi) |3 [
+ C + nol) +
Cp d
1
2r
K2 +
2r C8 + kラCH) ( + C1((2 + too)
2r-ro
2r-
2r+ro
(96)
=(0;)) 1/2 + ar+to Oid 1 (d K2)
2-1ma
2"
+ 2r+to
K2
2m+r0
2r0 + 2 0o(d (d + +k))+K 03 + mo)
+ 2rtro Cid d +
2r K2 + 2r K2Cū + k2C;) (2r + ro) + (C))
+ 2r+ro K2
2rtro
2r
+ 2r+ro
To C/ C d TM +
2r K2C Cf + 2r K2 + K3C 1166 2rtro)
in+to
Inserting this and (92) into (91) directly implies
J32 s (6L' + 611 I
x (tx) Yx|
hh- (K2 +
(r]) ) ) + (10%))
+
4r 2rtro
Cid Kī)) ボ
+
ra
K2 (C,))
2rim 2
+
ro
Co (d + Ki)) Ci (2r + +n)
2rtra
2r+ro
(97)
(2 C/
/ r + 2ーー
K2 + 2rtro K2C + K2C Ci (2r + ro) ( iu (C) 12r 2d
+
To 2rtro
+ K2 (2))² + prtro C. " + 2rtro K2 + 2r+ro C
B 2 ) n
Putting estimates of Jæ1 and Ja2 together, we derive from (87) that
J3 S(1 (1++ + 6L + +617) (tx) - Yel + (Ki + K2 + KG + K5C1()
(16
(KT + K2) [266] (01 ( A
+ (10)) R222 + 4r
2rtro Cād r: (d K2)
2r+ro 2r
+
2rtno
no
K2 (C,₁ )
ariro 200
+
2r
to
(d + K3} (2r
(2r + mo)
+ (artrr Cjd ti + 2r K2 +
2r
K2 + K2
Č (2 (2r To) + (eniro (d))
artro
2rtro
+ 2r+n
To H (²t)² +
to ir+ro
G +
2r
K3 + .ाナッ KKGG)) (3+)) )1303
ar+ro
(98)
Collecting estimates of J1, J2 and J3 together and noting pl 2 p2 + 1/2, we immediately derive from (83) that
( [[X (1++1) - Ya+ 1 [2] < 1 + (1 + 10L + 6L2;4)) (4x) - Yu Yo + [Ri + +2 (イ + 2kz
+ (Ki + ((E3))111111111 1:22 + (1;;,)1/2 + tr Cid (d' K3)
2ntro
+ 2rtro
ro K2 ) actme
+
(d + K2) (2r + + (
to
C +
2r K2
2rtro
2r
2r
2r
K2 0j + K3 C7 (2r + +ro) + 2r+to ( 20
m Kブ (C²}
2rtro
2r+ro
mo C Ci d " +
2r K2C + iptro 2rtto
K K + K2 KK
E1 (10 2 n
(99)
Again, using the Hölder inequality and the Young inequality yields
B BI 2r]
<
2r LE [1 (ro]
2--
+
2r E = 2r+ro]
ro 2 0.
(100)
20
2r4-0
Keeping this in mind and by setting
= (ta) - 11 == 1+ + +6L5,
(101)
we thus get
2 S (1 + n hn) + (R + ( 11200,
(102)
16

---

Non-asymplotic Error Bounds in W2-Dsstnce for LMC beyond log-Concavity
where for any r 2 1,
K = + 2K 2K (r) 2230
(K1 + 2K2) + (
2ntro
2n+ro Cid (d' K2)
2
nu
K2 (O+))
2etrr Trg
+ 2n4ro
Ci Ci(d + (5)
2rtro
2rtro Ci d +
2r
K₂ +
2r
K2 K2 ()) C2 + ro).
(103)
2r+ro
2n+ro
K
2r (Ki + 2Kg Ci +
In (C)) +
vy K2 (0)² +
To irtro
2rtro
2rtro
K2C 20t.o
C + K.C.C..
2r+ro
Noting Eg 0 and by iteration one obtain the desired assertion.
D. Proof of Theorem 3.4
Before proving Theorem 3.4, we quote a lemma from (Ye& & Zhou, 2024).
Lemma D.1. Given n, E N. Eg > 0. and E (0, 1). If a non-negative sequence (anIneno satisfies
an S Es + qan-n11
(104)
for n S n E N, then
On S 53 + Coq 1-9
V No,
(105)
where co = max 1/o {a:}.
The proof of this lemma can be found in Lemma 3.18 of (Ye & Zhou, 2024).
Proof of Theorem 3.4. By using the triangle inequality, we obtain that, for any given n 2 n1.
(o H) I (W Pr,, V Pn- Pt.,) + Wa Ptn,: II)
(106)
With regard to W2 Pn- Pt,., 11), by using Condition (A5), we derive
W W2 PV Pt., MT) < Kre nnch Pn-nyy I), vn 2 m-
(107)
For given stepsize h > 0, we choose
71
log logk""1> +1 +1 g
log +1
h
(108)
such that
K'e 1m 1
(109)
As result,
P II) < W/ (n-pp I),,
vn n1
(110)
Next treat the first term in (106). Noting that recalling the definition of Widdistance and employing Theorem 3.3 well Condition (A2) lead us to, for any 2 n1.
P'P P., C(Y (o- v P'Pn-n Pan, L(X(0, (b-n))tnn)))
(111)
W (v'P. Pa-n, Poy: v/ Pn-nnj Pron)) <B||x(0, Yn-njimn)) - (0, Y, in-njibm)) (m,) ]²] <(C)(In.)) R + + n- prtoo]] 1,200-1
(112)
s(C K + K (c (r + 7) + Ca | 1:22-1
=(0" K + K* ćr(r + +)+ K Ci 2 1,200-1
17

---

Non-asymptotic Error Bounds in Wi-Distance for LMC beyond Cog-Concyvity
which directly implies, for any n1.
1/2
Pa Pa (tm) K + Kr +)+ x' 122-1/2
(113)
As h S 2 one can derive that
fm nih S [oo +1hh h log +*+
21 =
(114)
17
Recalling (48), one can easily see that C* (tra) S (x*x), and therefore, for any n n2.
(2) 1/2
W2(1' Pn, Pm,, P/ P...)) [C( (K' K 2) (T hP2-1/2
(115)
Putting (110) and (115) together, we derive from (106) that, for any n N
W2(1/P,, s0*(x) (K + (r +)+ + 1/2 h
1 (W (1))
(116)
Utilizing Lemma D.I acquires, for any n E No.
Wa(1/P,, M) < (x) K (r 2) K 1 1/2 1/---112 1/2 c'
W2(1 Pk. II). (2(1/Pp.1)
sup
kefmi 1elo
(117)
Recalling the definition of W2-distance and using Lemma 3.1 and Condition (A2) lead to
sup W2( Pe, II) ke(o1-1jo
< (2 sup
[| 2
+2
/ RR'
|x|
1/2 (T(((d)))
(TT0))) +T)
1/2
keNo
(118)
According to (108), one can easily see that
11" nh
Xtnhi
(119)
log K +1
log +++ 21
1
implying
1I S -Ain
(120)
Plugging this and (118) into (117) yields
Wa(v'P,, II) s k3122-1/2 Rze-h
Vn Vn 0,
(121)
where
(< 2r+ro] 1/2
)
1/2
(122)
We thus get the desired assertion.
E. Proofs of Main Results for LMC
In this section, we list auxiliary lemmas which will be used to prove our main results. We now carry out the error analysis of LMC (2). The one-step LMC approximation Y is defined by
(I, + h) x VU(x) V2(We+h W.).
(123)
Equipped with Assumptions 2.6, 2.8, one can prove one-step strong and weak errors as follows.
18

---

Non-asymptotic Error Bounds in Wy-Distacce for LMC beyond Log-Concavity
Lemma E.1 (One-step errors analysis of LMC). Let Assumptions 2.1, 2.6, 2.8 hold and let (1,, (0,,,,41) h) denote the solution the Langevin SDE (1) at t h, starting from the initial value x at t. Then the one-step LMC (123) has local weak and strong errors of order 2 and 1.5, respectively, i... for any 0, < h < 1 and E R.
(t, h) (t, h3]] <( + Ka|x|
(124)
(B[x((,,=;+1))) h)) -1,,,) 1)) 1/2 < (k2 + K2|z,/) 1/
where
Ki ::4 (222 (2 D + 102 + 2+2 L1 + dd
Ki == 4(13 + Li).
(125)
K₂ : L 21 + +[)dd
K₂ = 4L1.
(126)
Proof. We first notice that
f.th 1t x(t,x;t+h) ((t,z;+++) --- (VU((((t)))))) - VU(x)) ds,
f-
(127)
where the lóō formula gives
J: (UUUXKUCEE/VVVXXXX::)))AAXXXXX dr
(128)
+ + / (U
Then taking expectation leads to E [t, h)- (t,, h)]
1th [ + drds
IE
.th [.
|UUUUUXUEEE/DCXXXXXXXXXAXXXXXXXD drdss.
(129)
Thanks to (10), (13), Assumption 2.8 and Lemma 2.4, we arrive at
E [[X(,,,;L+ h) ((,,; h)]
J. 1th
2E [1 [
1/2
drds
4th j [2R[21]TRa X1,, +1 222, x(,,,))))) 1/2
drds
(130)
14h 1: (4(13 + 1 X(t, r) + 4(12 + L1(R)dd 1/2
drds
1x (2) + 4 + 1B)dd 1/2 12
A(T2 +
d (
2+2
(11(2t (( L + L8 2+ 1 + L + 4 (1))= 12
n2
The first assertion of (124) thus validated. Recalling (127), one can use the Hölder inequality, the Lipschitz condition (9) derive
r+h
E X(t, h) (t,, h) <h
E
(U(((t, xt; (x))) ds (131)
<Tihh tth
B([x(1,,;;)) x 3
where
x(t,z;s) - l" - (X.,... dr S V2aw,
(132)
19

---

Non-asymptotic Error Bounds in Wy-Distacce for LMC beyond Jog-Concavity
As result, using (10) yields, for any [t, 1 + ht..
6)) 52 E /. - VUUXX (r)) dr + 1. v2aw. (T)
22h
E
(X (r)))²] dr + 4dh
(133)
<4TRh [." E [[ (r) dr + +L, + 4dh
<(4(2 (4 D + L 1)d /
Keeping this in mind, we derive from (131) that
E x(1,; ((, 1D] S (1(2t2al Li 21341 + L3)d 41/1013)16
(134)
as required.
Armed with Lemma E.1 and by verifying all Conditions (A1)-(A4) for the Langevin dynamics (1), can derive the following error bound of LMC in finite time from Theorem 3.3.
Lemma E.2 (Error analysis of LMC in finite time). Let Assumptions 2.1, 2.6, 2.8 hold and let {Xi X2h denote the solution I A 427 A A then for fixed E N and denoting T == ht the LMC (2) has global mean-spp mean-square error of order one in finite time, i.e.,
of the Langevin SDE (1). If the timestep satisfies h S 2E
[B[[X,.. - Ya 1))
1/2
S ((T))(Ke (x 1/2 h,
sup ne(mi]
(135)
where (TT) and K2, KL are given by (143) and (144), respectively.
Proof. To apply Theorem 3.3, we have to verify all conditions imposed there. By Assumption 2.1, Condition (A1) holds true with
i = p'd + (2p" 1)d, 1r' = 1
(136)
for any p" 2 1. Recalling Lemma 2.9, one can easily see that Condition (A2) is fulfilled with
ho # AI, =2= (1)
4+41 d.
(137)
#
By Assumption 2.2, Condition (A3) holds true with
L -==h L.
(138)
Similarly, Assumption 2.6 implies that Condition (A4) satisfied with
= O,
L - 444
(139)
Moreover, by Lemma 2.4, we obtain that Lemma 3.1 holds true with
C = e cpt (p(2) = 2(2p 2(2p- 1+N/1"
2p ) d.
p-1
(140)
[2 c)p
In light of Lemma E. one can verify that (46) in Theorem 3.3 holds true with
1,
R Kg Ki K1, pr 22 Kラ R₂
K2 K2,
P2 3
(141)
All in all, can derive from Theorem 3.3 that
[B||X,, - -x,19) 1/2 < 2104+100+21111 ((R) + 5ks) + [4+41;722n'd (K1 + 5K²) + (Kr + 5K2)
1/2 h.
(142)
20

---

Non-asymplotic Error Bounds in Wy-Distncce for LMC beyond log-Concavity
As consequence, the desired assertion (135) is validated by setting
((T)) == (1+1002+))
(143)
and
Kr := + 5ka) + (4+41 Id (Ki + 5K2)
(4+42,+2/''d - -(1( (4(22t (222222 Li + Lib 2+2n' 1++1,,7 + 20 2+22 L1 +22244 + 13)
(AL3 2AL:))L d.
K2 = : + 5K2) =44 24TH).
(144)
Next we aim to prove the main result of the LMC, by using Theorem 3.4.
Proof of Theorem 20. Except for Condition (A5), all conditions in Theorem 3.4 have been verified previously. In view of Proposition 2.5, Condition (A5) holds true with
=== K. 1)¹ = n7.
(145)
By means Theorem 3.4, we have
W2(vPn; n) S k2gh + Rze Amb, Vn eo,
(146)
where
A log log 14 (2L)
7
RS = 20(x)
[Rr (Rr
44487422 K,d + K E1 P1) 1/2
(147)
|اا'" 2 + 2+22 ) B| 1/2
X "
logK++1
+ 2L
K = 20
In view of Assumption 2.7 and (143), (144), one can obtain the desired assertion (15), by setting
E 22 20+100022)) lox K+1 + 2bn (4
2++22 L. + I + 2+2x' I + 13TT) + 20 2+2 1f + 21 + L1))
4412/+2p' (812 + 48T,)) + (4Th + 24T.))0)) 1/2
(148)
C2 :2 I+al 2421,+t1 + +oi
1/2
We thus complete this proof.
Proof of Proposition 2.11. Given an error tolerance e > O, by means of Theorem 2.10, one can choose k to be large enough and h to be small enough such that
Civde-Me Abh Sg C2vith S S
(149)
ensuing
(2(pDH, Se.
(150)
Solving the first term of inequality (149) shows
22 v c
Nh log
(151)
and second part of inequality (149) shows Inserting this into (151) yields as claimed.
1 h
20,
(152)
k:
20,vd log
28, 2C, = - =(()a)
(153)
21

---

Non-asymptotic Error Bounds in Wy-Distacce for LMC beyond Log-Concavity
F. Proofs of Main Results for Modified LMC
The organization this section is similar to the previous section. Clearly, The one-step pLMC scheme given by
x1; h) (x) (U(T
v2(W+++ W.).
(154)
In the following we prove some useful properties of the pLMC algorithm (22). Lemma F.1. Let Assumption 2. 2. hold and let Th be defined as (23). Then, for all TR.,, the following estimates hold true
(T)))) S min |zl|, (d h x641
[VU(T())) s (L) + 2L1 1 /2/2
(155)
The proof is straightforward, which can also be found in Lemma 3.3 (Pang et al., 2025).
Lemma F22 Let y > 0 be given Assumption 2.12 and let Th be defined as (23). Then, for all x E Rd, we have
x (x)) 2
(156)
The proof of this lemma can be found in Lemma 5.2 of (Pang et al., 2025).
Thanks to Assumptions 2.12 and 2. 14, we can prove one-step strong and weak errors as follows.
Lemma F.3 (One-step errors analysis of pLMC). Let Assumptions 2.1, 2. 2, 14 hold and let ht; h) denote the solution to the Langevin SDE (1) at ++ h, starting from the initial value x at t. Then the one-step pLMC (154) has local weak and local strong errors of order and 1.5, respectively, i.e., for any t 2 0, < h < 1 and x E R
E x h) - at, -)]l <(; + Kj|æ| 10 1/2 n2,
(157)
E X(t, ( 1)) 1/2
<(₁, /gll|100+100 1/2 1/2/2
where Ki = Cdbrt1, Ki == Cd-1, R₂ = Cd K₂ Cd-4 with the uniform constant C Clup'' 0, Lo, L6, Li, Li) not depending on d.
Proof. Noticing that
- ds, -o)) ff.
x(t,z,t+h ( +t, x (T))))
(158)
one can use the Itō formula derive
x(,,;1+1) - Y 21,2; L+ 1+h S: (UUUUXU drds
+ /2 f.th/, ['VUU(((,):)) (x- (
1hh
(159)
Taking expectation on both sides and using the Hölder inequality, Lemmas F.I. F22, Assumption 2.12 yield
18 [t I; + xt,
+ x - (x(x) VU)
*th
drds + .th J. ([V(AU(X((( drds
+ 20
ر1 [ |-| (T)) x0 T
447 1. E
(tx drds + [ beth S: (E (E(VVAXXXXXXXX)))))) 1/2
drds
+ 20 4--4 22 + 2L 1 |x|" + od da )हें)) ) - 1
(160)
22

---

Non-asymptotic Error Bounds in Wy-Distncce for LMC beyond Jog-Concavity
By using (18), (19), Lemma 2.4 and the Young inequality, we have
E [
sT. (1 + [X
$2 (1
(161)
SC(d (z|
where C' == ,, Similarly, can utilize Assumption 2.14 to attain
E (| s (2 L33d" E (t, s C(d + (z)2rt))
(162)
where C C'u, c, Lo Lo). Thanks the fact ब d and the Young inequality
d x 47+5
& تيية 2 + 4 57+5 |x| 50
(163)
one can deduce
20 20-40-1 d 1 h² + 20L1 |x|1 1f ब of
शाा.. 1f -- 1----4 d 22) 4
sC d 2| 4145 + d - ///0// 5745 + d 2d क |x| 1255 1i2
(164)
<C(d + | | 1/2
where C == C(,,
Putting these estimates together, we derive from (160) that
E[x(t, h) (t, h)]] S Od + + 1z=10000+110 1/2//2
(165)
where C
== C(u Hé, 7,7, 0, Lo, L1, L1). The first assertion of (157) is thus comfirmed. Next, taking square and then expectation both sides of (158), one use Assumption 2.12, the Hölder inequality and Lemma F.2 to obtain
E
x ((tt h) (t, +)1
<2 x - (x) + 2E [. +h
(xU(s(,,x;))) - vU(T)))))))))) ds
E (u(x) vU(T (x)) "] E (U(((,)), vU(x) ds (166)
<40 88 d 1/2 81410 n + 4h
-40-87 87-8,
1/20 878 10 + 16 + | +
8 87+ 10
+ +441
E
(1
+ " [X(t 1 /dds.
Using the Young inequality yields
40 40-8 88-888 4d *|x 11+ + 161,
+ | + 2 1ー 7:. 9-8 -8 1|x 11+10
<C(d 12/ + d " 12/ 10 d d 10) 3
(167)
<C(d + |1/00/1000
where C. := C(,,0, 0, L1). Now, we estimate the third term in (166). It follows from Lemma 2.4, (18) that, for q 2 1,
E X (1;
x 20]
+ E S. V2aw, (20 (
2-
E
(VU(X,, dr
(168)
52 1/24-1
dr 23 (2q
<C(d' + +|x 22(+++1)))
23

---

Non-asymptotic Error Bounds in Wy-Distacce for LMC beyond Log-Concavity
where C = C(u, e, L,, 1 q). Bearing this in mind and using the Hölder inequality and Lemma 2.4 give
| c+h
8 (1 +
E
|2| |
+ (x 2;; (t)
- x=
ds
rth
+ E (1+ | 2t
$242
E
X x(t
4o
85
ds
(169)
<C(d 1/
where == C(u,p' e, L1, Collecting (167), (169) together, one can derive from (166) that
(1, 1, +1,,,;t++)) 1 c(d d 100===010+000 h²²,
(170)
where == Cl e7, 0 Lo, 1.: L,, L4). Now the second assertion in (157) is proved.
Equipped with Lemma F.3, one can follow a similar line to Lemma E.2 to prove the enror analysis of pLMC in finite time.
Lemma F.4 (Error analysis of pL.M in finite time). Let Assumptions 2.1, 2.2, 2.12, 2.14 hold and let (X1)120 denote the solution of the Langevin SDE (1). If the timestep satisfies h S म.. A 2ي A p42(1,22.) A 1, then for fixed N and denoting T h, the pLMC (22) has global mean-square error of order one, i.e.,
2v
1/2 s ((T)( (1Ellool))tt0) 1/2 1/2h, h, 1n117) (B[IX, -
sup ne(m) (m1]
(171)
where C(T), K, and K are given by (178).
Proof. Owing to Assumption 2.1, Condition (A1) holds true with
i = j'd + (2p" - 1)d, 1 = μ
(172)
for any p" 2 1. In view of Lemma 2.15, Condition (A2) is fulfilled with
ho
2p A 1, +2L
Ci = =1
(i(p) CidP
(173)
By Assumptions 2.2 and 2.12, Conditions (A3) and (A4) hold true with
L = L,
=
(174)
Additionally, taking Lemma 2.4 into account, Lemma 3.1 holds true with
C = 1 2 cpp (p(p) = 2(2p 1+/1
( (20-0)p) (2-
) dP.
(175)
cp
In light of Lemma F.4, one can verify that (46) Theorem 3.3 holds with
5- + 5, ri = 4, KJ
Ki.
KJ = Kj,
Pi = 2,
Kj = Kg.
K = K2,
(176)
With these arguments at hand, one can apply Theorem 3.3 to acquire
[ - Yn 3)) 1/2 < e 11 (Cdd11/2t1 + 11100 1/2 h
(177)
where Č == C(u, e,7, ,,,, 0, I, Lo, The T.,, TT). Evidently, the desired assertion follows, with
C(T) = ed0+10040L111 k,= C4l12/2+1, Kp Cd-
(178)
24

---

Non-asymptotic Error Bounds in Wy-Distacce for LMC beyond Log-Concavity
Proof of Theorem 2.16. Using the technique the proof of Theorem 2.10, we can obtain the non-asymptotic error bound of pl.MC (22). First, by Proposition 2.5, Condition (A5) holds true with
K² 17 77-
(179)
Using Theorem 3.4 shows
Wa(vpa, RT S Kihh + KGe Andh Vn E No,
(180)
where
Kj == 2((x) ( Kp Cj K (11--101 (pd 10)/2 + KpB(oo/ 1/2
X = log (2L))
(181)
k= = e c(((2t2n' + 4B| 1/2
X Ing K++
"
21
In view of Assumption 2.13, one can obtain the desired assertion (25) with
Õ Ce Cea (114) K* + totn), O2: : (2= 20 + 403(1)) 1/2
(182)
where C. == Clu, pl, e, 7, 0, 02 L, Lo. Lo. L1, L1). Thus, the proof is completed.
G. Proof of Auxiliary Results for Gaussian Mixture and Double-well Potential
We now verify Assumption 2.8 for the Gaussian mixture. For all a E Rd, the gradient of U1 is given by
(U)(2) = a a
2a amm
(183)
and the Hessian matrix is given by
(x) 2(r.n) (1+oole.a)p200) an
(184)
By the definition of the Laplacian operator, we have
(r(())=t((Vuuu) Σ
dagee(orna) (1+e2 ) =d= E(1- 1=1
(185)
Then
((AU(()) 8/a/2 (112(()())))) - 2(x))))
8l (1-a 2(aaa)))
8/o/2 1--e 2(x,o))
(1+e21000333
143e2(a
43e4(a.a) +3e
360tn.n) e e-diana) +3e 2(0,0) 23 a.
(186)
By the inequality 1 < and e O. for R. we have, by taking |a| = 2,
(AAU(()) < ( < Slal |xc|
128 3
|zc|
(187)
required.
In what follows, we check Assumption 2.14 for the double-well potential. One immediately obtains
(U2(x) = | x, (2U2(x) (|z|2 - 1)Ia+ 2x1, (U)(x) = tr (V(VU(())) = (d 1) + d,
(188)
which further implies
v(AU((2)) = -(dd + 1)x.
(189)
Thus, we have
((AU(())) & 4d/=//
(190)
validating Assumption 2.14.
25


Reviews:
[{"review_id": "n57GkRg4rh", "reviewer": "Reviewer_DL9W", "summary": "The authors derive a sampling error bound in Wasserstein-2 for a discrete time discretization of Langevin Monte Carlo. The bound contains two parts, an error term due to finite time truncation of the Langevin dynamics, and an error term due to discretization over a finite time horizon. The innovation of this work appears to be in the analysis of the finite time discretization term, which is done entirely through moment-based calculations. Notably, they bound the error accumulation over short time intervals (Lemma E.1) and long time intervals (Lemma E.2) using rather many assumptions on the regularity of the drift, plus dissipativity of the target measure.\n\nThe dissapitivity condition implies some uniform-in-time moment estimates (Lemma 2.4) on the true and discretized processes, and the regularity assumptions on the drift are used to transfer these moment estimates into bounds on the discretization error over short time intervals. \n\nThis analysis is similar in style to bounds on ODE discretization error, where $O(h)$ step size dependence is typical. However, to carry out this approach for stochastic dynamics the authors must apply Itô's formula to the drift $\\nabla U(x)$, forcing them to use Assumption 2.8 linear growth of $|\\nabla(\\Delta U(x))|$ which appears to be uncommon in the literature.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to Reviewer DL9W  \n\nWe sincerely thank the reviewer for constructive suggestions and comments. Next we address all comments point-by-point and  will revise the manuscript to incorporate these suggestions.\n\n### About *Summary*\n> However, to carry out … Assumption 2.8 … which appears to be uncommon in the literature.\n\n**Response**: Thanks. We would like to mention that, Assumption 2.8 comes from Li et al. 2022, where the authors remarked that, it is not necessarily stronger than the widely used Hessian Lipschitz condition (see Section 4 in Li et al. 2022). In fact, as pointed out by Li et al. 2022, there exist potentials, e.g. $U(x)=x^{4}$, that satisfy Assumption 2.8 but violate the Hessian Lipschitz condition. Moreover, it is shown that a class of Gaussian mixtures satisfy Assumption 2.8.\n\n### About *Essential References Not Discussed*\n>1. Please discuss how the proof in this work differs from that of Li et al. 2022.\n\n**Response**: Thanks. To show the difference between our work and Li et al. (2022) clearly, it is worthwhile to illustrate the main idea of Li et al. (2022). Indeed, their arguments are based on a direct long-time mean-square convergence analysis of the numerical method, which essentially relies on the use of the log-concavity condition. To see this fact, we provide a key idea behind their error analysis. Suppose we get the following error estimates:\n$$E[|X_{t_{k+1}}-Y_{k+1}|^2]\\leq E[(1+\\epsilon h)|X_{t_k}-Y_k|^2-2h\\langle X_{t_k}-Y_k , \\nabla U(X_{t_k})-\\nabla U(Y_k)\\rangle]+c_2 h^3,\n$$ where $\\epsilon >0$ can be sufficiently small. Then the log-concavity condition, i.e. $$\\langle x-y,\\nabla U(x)-\\nabla U(y)\\rangle \\geq c_1 |x-y|^2, c_1>0\n$$ is essentially used here to arrive at the contraction: $$E[|X_{t_{k+1}}-Y_{k+1}|^2]\\leq[(1-(2c_1-\\epsilon)h]E[|X_{t_{k}}-Y_{k}|^2]+c_2h^3,\n$$ for $0<\\epsilon<2c_1$. Armed with the contraction, one can easily get the uniform-in-time error bound by iteration. However, without the log-concavity, this framework does not work in obtaining uniform-in-time error bounds.\n\n In contrast to Li et al. (2022), we develop a new framework of uniform-in-time error analysis without the log-concavity. The proof consists of two key components. First, we derive the finite-time mean-square convergence error bounds for the LMC, which grow exponentially with respect to the time length $T$. Arguments for this step require only Lipschitz condition and avoid any convexity assumption. Second, we obtain uniform-in-time error bounds by relying on the exponential ergodicity of the Langevin dynamics, which is available under one-side Lipschitz condition and log-Sobolev inequality (see Subsection 2.4 for details). Moreover, our framework also works for the case of non-globally Lipschitz continuous $\\nabla U$, which is not even investigated in Li et al. (2022).\n\nTo summarize, the approach of error analysis is essentially different from Li et al. (2022) and more relaxed conditions (non log-concavity and non-glabally Lipschitz conditions) are used to cover more problems.\n\n>2. How (if at all) is the log-sobolev condition … due to truncation at a finite time?\n\n**Response**: Thanks. The LSI is not used in the finite-time error analysis of our paper. It is only used/required to obtain the exponential ergodicity of the Langevin dynamics (see Proposition 2.5 and Step 3 in Subsection 2.4 for details). As explained above, the authors of Li et al. (2022) did not do finite-time error analysis but used the log-concavity essentially to directly carry out a long-time error analysis and derive an infinite-time convergence theorem.\n\n### About *Weakness*\n>1. The major weakness … are largely the same.\n\n**Response**: We deeply apologize for not making the originality clear, to make you confused. As explained in our response to your previous comments, the approach of error analysis in our paper is essentially different from Li et al. (2022). As commented by the reviewer cxoz: \"The authors establish this result through a new discretization analysis for SDEs which combines uniform-in-time LMC moment bounds with a finite-time fundamental mean-square convergence theorem\".\n\n>2. Another weakness … of this analysis?\n\n**Response**: : Thanks. We apologize for the current presentation. \nIndeed, Section 3 is a general framework of uniform-in-time convergence for discretizations of general SDEs, independent of Section 2. So Theorem 2.10 does not use assumptions in Section 3.\nSome assumptions in Section 2 for particular Langevin SDEs can be regarded as particular ones in Section 3. But they are not redundant and we agree that the current presentation can be improved in the revision for better readability. Due to the work limitation, please refer to our responses to *About weakness* of the reviewer H65o for details.\n\n### About *Questions*\n**Response**: Thanks. These concerns have been carefully addressed previously. Hope you are satisfied with our answers. Please tell us once you have further questions.", "claims_and_evidence": "The theoretical claims made in this work are supported by clear and convincing evidence in the form of proofs.", "methods_and_evaluation_criteria": "N/a", "theoretical_claims": "I read the proofs of Theorem 2.1, Lemma 2.4, Theorem 3.7, Lemma E.1, and Lemma E.2.", "experimental_designs_or_analyses": "Figures 1(c) and 1(d) are a convincing proof of concept for this work.", "relation_to_broader_scientific_literature": "According to Table 1, this work proves an optimal error bound of $\\tilde{O}(d^{1/2} \\epsilon^{-1})$ without log-concavity. A bound of the same order was already shown in (Li et al. 2022) using both log-concavity and the third order growth condition on $U(x)$ required by this work.", "essential_references_not_discussed": "Please discuss how the proof in this work differs from that of Li et al. 2022. How (if at all) is the log-sobolev condition used in the discretization error bound over a finite time horizon? I ask because, it is important to clarify which parts of the present work are original relative to Li et al. 2022, which uses similar assumptions and a similar technique (Gronwall-based discretization error bounds via direct control of moments). Does the finite time discretization error analysis of Li et al. 2022 make use of log-concavity? If not, would their approach work equally well using only a log-sobolev inequality to bound error due to truncation at a finite time?", "other_strengths_and_weaknesses": "The major weakness of this work is that it is rather unclear what is the originality, given the significant similarities between it and Li et al. 2022. Relaxing the assumptions of Li et al. 2022 from log-concavity to merely log-sobolev is rather incremental if the proof techniques are largely the same. Another weakness of this work is that it is hard to read because of many different assumptions introduced throughout. I counted 9 assumptions spread across Section 2, only 5 of which are required by Theorem 2.10, but then four more assumptions stated in Section 3, which contains Theorem 3.7 that is essential in the proof of Theorem 2.10. Does Theorem 2.10 also require the assumptions in Section 3? Are any of these assumptions redundant? Could the presentation be simplified so it is easier to keep track of the many requirements of this analysis?", "other_comments_or_suggestions": "N/a", "questions_for_authors": "1. Does Theorem 2.10 require Assumptions 3.1, 3.3, 3.4, 3.5 indirectly through its use of Theorem 3.7? If so, please add them to the statement of Theorem 2.10 so that the statement is correct. Are any of the assumptions stated in Section 3 redundant? \n2. How does the proof technique in this work differ from that of Li et al. 2022? What are the original contributions contained in the techniques used by the present paper?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pGa4H7wgle", "reviewer": "Reviewer_cxoz", "summary": "This paper establishes an almost optimal convergence rate of $\\tilde{O}(\\sqrt{d}/\\epsilon)$ in $W_2$-distance for Langevin Monte Carlo (LMC) when the target measure satisfies the log-Sobolev inequality, along with dissipativity and smoothness conditions. The authors establish this result through a new discretization analysis for SDEs which combines uniform-in-time LMC moment bounds with a finite-time fundamental mean-square convergence theorem. For non-smooth settings where the gradient norm may grow super-linearly, the authors study a projected version of LMC and establish $W_2$ convergence bounds using their discretization analysis.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to Reviewer cxoz\n\nWe really appreciate your carefully reading and insightful comments. We will respond to each comment below and revise the manuscript according to these suggestions.\n\n### About *Weakness* \n\n>1. There is not sufficient discussion on the implications of Assumption 2.2, …, discussed below.\n\n**Response**: Thanks. You are absolutely right. We apologize for your confusion caused by a typo here. The correct condition in Assumption 2.2 should be $$\\langle x-y,\\nabla U(x)-\\nabla U(y)\\rangle\\geq -L|x-y|^2.$$ We confirm that the analysis throughout the paper is based on the correct form above, and we will fix this typo in the revision.\n\n>2. The dependence on most constants is implicit, …. The dimension-dependence of some constants is also not clear.\n\n**Response**: Thank you. We apologize for any confusion caused by the unclear of dependence to the constants. We will discuss the dependencies of the parameters in detail and explicitly explain how they depend on the number of dimensions $d$ in the revision.\n\n### About *Comments or Suggestions*\n>1. I believe there is an alternative approach to prove Proposition 2.5 which does not rely on Assumption 2.2. …, which only required Assumption 2.3.\n>2. Using the above argument, …, these approaches can lead to new error bounds for LMC in sub-exponential or heavy-tailed settings.\n\n**Response**: Thanks for your insightful and interesting discussion. The direction you suggested is really promising for extending the analysis to more general settings. However, essential difficulties still exist. On the one hand, we believe Proposition 2.5 can hold without Assumption 2.2, but the analysis of finite-time mean-square convergence of numerical methods with convergence rates essentially rely on the use of Assumption 2.2, which is nothing but the one-sided Lipschitz (also called monotonicity) condition on the drift coefficients $-\\nabla U(x)$. As far as we know, without the monotonicity condition, obtaining mean-square convergence rates is highly non-trivial for numerical SDEs (see Hutzenthaler and Jentzen, Ann. Probab., 2020). On the other hand, our approach relies on two key properties: (i) the chosen metric must satisfy the triangle inequality; (ii) the metrics on both sides of the inequality must remain consistent. Since the KL divergence does not satisfy the triangle inequality, inequalities such as KL$(\\nu p_t,\\pi)\\leq e^{-4t/\\rho}$KL$(\\nu,\\pi)$ do not work seemingly. \n\nWe sincerely thank the reviewer for bringing these two works of Chewi et al. (2024) and Mousavi-Hosseini et al. (2023) into our notice. Ideas in these works are very inspiring, and we will definitely aim to extend our framework to incorporate weaker functional inequalities and alternative metrics  for our future research. This is not trivial and we need more time to do it. \nIn the revision, we will cite and make some comments on these two papers.\n\n>3. For better readability, the authors can change the list of assumptions in line 231 to Assumptions 2.1-2.3 and Assumptions 2.12-2.14.\n\n**Response**: Thanks for your suggestions. We will do this in the revision.\n\n### About *Questions*\n>1. I don’t see how Assumption 2.2 follows from Assumption 2.6. …. There should be a discussion on when this assumption is satisfied.\n\n**Response**: Thanks a lot for your helpful comments. As mentioned to the first question in *About the weakness*, this issue comes from a typo in the previous manuscript, which will be corrected in the revision.\n\n>2. The LSI constant $\\rho$ should be missing somewhere in Proposition 2.5 and Theorems 2.10 and 2.16, as it controls the convergence rate.\n\n**Response**: Thanks a lot for your constructive suggestions. In the revision, we will explicitly show it in corresponding statements.\n\n>3. How does $C_{\\nabla}$ of Lemma 2.15 depend on dimension?\n\n**Response**: Thanks a lot for pointing out this issue. The constant $C_{\\nabla}$ is independent of the dimension, as it can be written $C_{\\nabla}=L_1'+L_1$. Please refer to Lemma 3.3 in Pang et al.(2025) for details.", "claims_and_evidence": "The claims are supported by rigorous statements and proofs.", "methods_and_evaluation_criteria": "Not applicable.", "theoretical_claims": "I didn't verify the correctness of the proofs, but the overall strategy seems sound.", "experimental_designs_or_analyses": "Not applicable since the paper is mostly theoretical.", "relation_to_broader_scientific_literature": "The analysis of Langevin Monte Carlo is a fundamental problem in sampling and of interest to many researchers in the ICML community. Beyond establishing the optimal $W_2$ convergence rate for LMC under the log-Sobolev inequality and smoothness, the discretization analysis introduced here may be used in future analyses of LMC and its variants as well.", "essential_references_not_discussed": "Most relevant references are cited. There are additional references in the *other comments or suggestions* section below whose discussion can help give a broader picture of LMC analysis. They are presented below.", "other_strengths_and_weaknesses": "**Strenghts**:\n\nAs mentioned above, the discretization analysis can open the room for novel analyses of LMC under different assumptions or for analyzing different variants of LMC. The paper also handles locally Lipschitz potentials which is a valuable contribution that has received less attention in the literature.\n\n**Weaknesses**:\n\n* There is not sufficient discussion on the implications of Assumption 2.2, and I think certain statements about when it holds might be incorrect. In fact, this assumption may even not he necessary, I think there are alternative approaches to prove continuous-time exponential ergodicity in $W_2$ under a log-Sobolev inequality, discussed below.\n\n* The dependence on most constants is implicit, which makes interpreting the results for certain examples complicated. The dimension-dependence of some constants is also not clear.", "other_comments_or_suggestions": "* I believe there is an alternative approach to prove Proposition 2.5 which does not rely on Assumption 2.2. Specifically, the log-Sobolev inequality guarantees that $\\mathrm{KL}(\\nu p_t, \\pi) \\leq e^{-4t/\\rho} \\mathrm{KL}(\\nu, \\pi)$. Moreover, a log-Sobolev inequality implies Talagrand's transport inequality with the same constant, i.e. $W_2(\\nu p_t, \\pi) \\leq \\sqrt{\\rho \\mathrm{KL}(\\nu p_t, \\pi)}$. Therefore $W_2(\\nu p_t, \\pi) \\leq e^{-2t/\\rho} \\sqrt{\\rho \\mathrm{KL}(\\nu p_t, \\pi)}$, which only required Assumption 2.3.\n\n* Using the above argument, in fact we can weaken Assumption 2.3 to only a Poincaré inequality (as done in Chewi et al., 2024 for LMC) or even to weak Poincaré inequalities that covers heavy-tailed distributions (as done in Mousavi-Hosseini et al., 2023 for LMC). While the transport inequality no longer holds here, $W_2$ can be bound with Rényi distances (see Chewi et al., 2024 and Mousavi-Hosseini et al., 2023 for respective settings). Combined with the discretization analysis here, these approaches can lead to new error bounds for LMC in sub-exponential or heavy-tailed settings.\n\n* For better readability, the authors can change the list of assumptions in line 231 to Assumptions 2.1-2.3 and Assumptions 2.12-2.14.\n\nReferences:\n\nS. Chewi et al. \"Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev.\" Foundations of Computational Mathematics 2024.\n\nA. Mousavi-Hosseini et al. \"Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincare Inequality.\" COLT 2023.", "questions_for_authors": "1. I don’t see how Assumption 2.2 follows from Assumption 2.6. A simple application of Cauchy-Schwartz and triangle inequalities results in\n$$\\langle x - y, \\nabla U(x) - \\nabla U(y)\\rangle \\geq -(2L’_1 d^{1/2} + L_1 \\vert x \\vert + L_1 \\vert y \\vert)\\vert x - y\\vert$$\nBut $\\vert x \\vert, \\vert y \\vert$ are not bounded. Also, dependence on the initial Wasserstein distance is not clear.\n\n    It seems more like Assumption 2.2 is a form of relaxed convexity. For $L = 0$, it exactly implies convexity of negative log-density. There should be a discussion on when this assumption is satisfied.\n\n2. The LSI constant $\\rho$ should be missing somewhere in Proposition 2.5 and Theorems 2.10 and 2.16, as it controls the convergence rate.\n\n3. How does $C_\\nabla$ of Lemma 2.15 depend on dimension?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "o6yUwukidN", "reviewer": "Reviewer_H65o", "summary": "This paper addresses the challenge of sampling from non-log-concave distributions, including those that satisfy a dissipativity condition or a log-Sobolev inequality. The authors approach this problem by discretizing the Langevin dynamics and establish a state-of-the-art convergence rate of d^{1/2}\\varepsilon^{-1} in the W_2 distance, under the assumptions of gradient Lipschitz continuity and linear growth of the third derivative. The theoretical findings are further verified by numerical experiments.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to Reviewer H65o\n\nWe sincerely appreciate your time and effort in reviewing our manuscript. Your insightful comments and constructive suggestions have greatly helped us improve the quality of our work. Below, we provide a point-by-point response to each of your concerns, along with the corresponding revisions in the manuscript.\n\n### About *Claims and Evidence*\n\n> However, there are areas where the evidence is less convincing: the optimal error bound sounds confusing since this paper gives improved bound with stronger assumptions and there is no lower bound compared to their upper bound.\n\n**Response**: Thank you for pointing this out. We apologize for any confusion caused by this statement. You are right that there is no lower bound here and we will remove the word \"optimal\" and revise this statement throughout the paper, following your comments. \n\n### About *Weakness*\n\n> There are too many assumptions and the presentation could be clearer.\n\n**Response**: Thanks. We agree that the current presentation can be improved for better readability. Indeed, Section 3 is a general framework of uniform-in-time convergence for general SDEs and Section 2 is focused on the particular Langevin SDEs. In the revision, we plain to reformulate \"assumptions\" in Section 3 as several “conditions” (e.g. $H_1$, $H_2$,...). Then Section 2 put some assumptions on Langevin SDEs so that conditions in Section 3 can be satisfied and theoretical results there can be applied to Langevin SDEs. If you have any other good idea, please let us know. Thanks a lot.\n\n### About *Comments Or Suggestions*\n\n> - Line 330, Assumptions 3.1, 3.4, 3.5, 3.3 -> Assumptions 3.1, 3.3, 3.4, 3.5.\n> - Line 331 assssumed -> assumed.\n> - In Assumption 2.6, There exists a dimension independent constant -> There exists a dimension‐independent constant.\n> - Line 424 This framework can also applies to -> This framework can also be applied to.\n> \n**Response**: Corrected. Thanks!\n\n### About *Questions*\n>1. Can your framework be extended to other metrics?\n\n**Response**: Thanks for your question. Yes! Our framework can be extended to other metrics, which relied on 3 conditions below:\n\n- the metric satisfies the triangle inequality, such as total variation (TV), $W_p, p\\in[1,\\infty)$ distances;\n- the underlying Langevin dynamics is exponential ergodic with respect to the chosen metric;\n-  the sampling algorithm  admits finite-time convergence and uniform-in-time moment bounds.\n\nOnce these properties are verified, the general framework in Section 3 remains applicable. This would be an interesting direction for our future research.\n\n> 2. Beyond the difference in time horizon, how does your framework in Section 3 differ from that of Tretyakov and Zhang (2013)?\n\n**Response**: Thanks a lot. To be honest, the finite-time convergence theorem (Theorem 3.7) in our paper, as well as Tretyakov and Zhang (2013), follows the original idea of Milstein (1988). The difference is that we need to provide explicit dependence on time and other parameters in the error bound, which are not done in Milstein (1988) and Tretyakov and Zhang (2013). Such explicit dependence, particularly on time $T$, is essential, as it allows us to combine the finite-time error estimates (Theorem 3.7) and the exponential ergodicity of the SDEs to establish the uniform-in-time convergence result (Theorem 3.9). As commented by the reviewer cxoz: \"The authors establish this result through a new discretization analysis for SDEs which combines uniform-in-time LMC moment bounds with a finite-time fundamental mean-square convergence theorem\".", "claims_and_evidence": "The paper’s primary claims are supported by a rigorous theoretical framework and are backed by numerical experiments on controlled examples. However, there are areas where the evidence is less convincing:\n\n- the optimal error bound sounds confusing since this paper gives improved bound with stronger assumptions and there is no lower bound compared to their upper bound.", "methods_and_evaluation_criteria": "The proposed methods are well-suited to the problem. The paper develops a robust uniform-in-time convergence framework and provides optimal error bounds in W_2 distance, which are standard and relevant metrics for evaluating sampling algorithms. The use of synthetic benchmarks, such as Gaussian mixtures and double-well potentials, is appropriate for initial validation, though further testing on diverse or real-world datasets could enhance the evaluation.", "theoretical_claims": "I only read the proof in the main body.", "experimental_designs_or_analyses": "I only check it in high level.", "relation_to_broader_scientific_literature": "Their framework closely resembles traditional SDE discretization methods. However, while most analyses assume a bounded time horizon, in the context of sampling the relevant time horizon scales as \\log d. In this paper, the authors explicitly characterize how the error depends on T.", "essential_references_not_discussed": "I think the related works appear to be appropriately cited and discussed in the paper.", "other_strengths_and_weaknesses": "There are too many assumptions and the presentation could be clearer.", "other_comments_or_suggestions": "- Line 330, Assumptions 3.1, 3.4, 3.5, 3.3 -> Assumptions 3.1, 3.3, 3.4, 3.5\n- Line 331  assssumed -> assumed\n- In Assumption 2.6, There exists a dimension independent constant -> There exists a dimension‐independent constant\n- Line 424 This framework can also applies to -> This framework can also be applied to", "questions_for_authors": "- Can your framework be extended to other metrics?\n- Beyond the difference in time horizon, how does your framework in Section 3 differ from that of Tretyakov & Zhang (2013)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "T4jM9qHyLs", "reviewer": "Reviewer_rDrb", "summary": "When generating samples from a target distribution $\\pi$ from a large\ndimension $d$ -- including when the normalization constant is unknown -- one\noften employs Langevin Monte Carlo (LMC). This method starts by constructing\na Langevin diffusion where its invariant distribution matches the desired\ntarget distribution and then runs a discretized version of the diffusion to\ngenerate samples. Since the discretization stepsize $h$ introduces some\nerror, these samples are not exactly from the target distribution. This\npaper, along with a vast literature before it, aims to quantify the rate of\nconvergence of LMC samples toward the target distribution in terms of the\n$L_2$ Wasserstein metric and terms $h$ and $d$. The paper argues under a\nwide set of assumptions, e.g. the target distribution satisfies a\nlog-Sobolev inequality and dissipaivity condition, that the error rate is\n$\\tilde{O}(\\sqrt{d}h)$ which is state of the art. Crucially, this improves\nupon past work that assumes the target distribution must be strongly log\nconcave. The authors also provide a framework for proving non-asymptotic\nresults for other samplers, namely the projected LMC sampler pLMC.\n\n## update after rebuttal\nBased on the authors feedback and other reviews, I'm inclined to keep my score of a weak accept.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to Reviewer rDrb\n\nThank you for your valuable feedback on our paper. We are grateful for your thoughtful comments, which have guided us in refining the manuscript. Here, we address each of your questions in detail and highlight the changes made accordingly.\n\n### About *Weakness*\n\n> This does require that the target distribution satisfy a log-Sobolev inequality and dissipaivity condition, which may be a larger assumption than previous work. This may be the work's biggest weakness.\n\n**Response**: Thanks a lot for your valuable comment. The log-Sobolev inequality and dissipaivity condition is indeed more strict than the Talagrand transport inequality and the Poincare inequality. However, the error bound $O(\\sqrt{d}h)$ was obtained for LMC under a strongly log-concave condition, see, Li et al. (2022). The main aims of this work are to answer the key question: *Can the error bound $O(\\sqrt{d}h)$ still hold true for LMC without the strongly log-concave condition?* As discussed above, compared with the strongly log-concave condition, the log-Sobolev inequality and dissipaivity condition are weak. We aim to do the error analysis under more relaxed conditions for further work.\n\n### About *Comments Or Suggestions*\n\n> L431: It should read \"probability distributions\".\n\n**Response**: Corrected! Thanks!\n\n> Figure 1 (c) and (d): Should the x-axis be \"stepsize\" instead of timesteps? It seems that the discretization error should decrease with the number of time steps, but perhaps I'm confused by the terminology.\"\n\n**Response**: Thanks for pointing out this issue. You are absolutely right that the x-axis is \"stepsize\". We are sorry for this and will fix this typo in the revision.\n\n### About *Questions*\n\n> While most assumptions laid out in this paper are relatively simple to understand, Assumption 3.3 seems less obvious. How severe of an assumption is this?\n\n**Response**: Thanks for your comment. Assumption 3.3 means that moments of a numerical algorithm should be uniform-in-time bounded, which is essentially used in the infinite-time convergence analysis. In Section 2, concrete numerical methods (such as the LMC algorithm and the pLMC algorithm) are provided satisfying Assumption 3.3 under some assumptions (see Lemma 2.9 and Lemma 2.15). In the revision, we will add some comments and discussions following  Assumption 3.3 for better readability.", "claims_and_evidence": "Yes, the simplified theoretical argument in the paper seems reasonable, and\nall assumptions required to obtain the desired bounds on the LMC convergence\nare clearly laid out. The theoretical arguments are corroborated by some\nempirical experiments to demonstrate that their rates of convergence are\nfollowed in the case of a Gaussian mixture model target distribution.", "methods_and_evaluation_criteria": "The core contribution of the paper is theoretical, although the empirical\nexperiment offered in Section 4 does align with their theoretical\nprescriptions.", "theoretical_claims": "I briefly investigated some of the proofs for the arguments supplied\n(Appendix B and C) and saw no obvious errors.", "experimental_designs_or_analyses": "The empirical work in this paper is light, but what the authors have\nprovided appears valid.", "relation_to_broader_scientific_literature": "The theoretical work to quantify the convergence rates of LMC is mostly of\ntheoretical interest, but there is a good chance that ideas from this work\ncould lead to a practical variation of LMC that achieves better convergence\nproperties.", "essential_references_not_discussed": "To the best of my knowledge (which in this field is not up to date) the\nauthors have included necessary references.", "other_strengths_and_weaknesses": "For the most part, the paper is very thorough and well written. The\ntheoretical arguments are carefully laid out and a simplified version in the\nmain paper offers guidance on how to read the proofs. I found the comparison\nto other work also eludicating, as this field has many papers which all have\nrelatively small but significant differences in the error bounds they can\nprovide.\n\nThe main novelty of the paper appears to be the way that the authors\ndecompose the problem into two pieces: a finite time mean square fundamental\nconvergence theorem for SDEs that handles quantifies discretization error\naccumulated from LMC, and an appeal to erdogicity that bounds the error from\nonly running a finite number of LMC steps. This does seem to be a\nsignificant approach for tackling LMC convergence when the target\ndistribution is appropriately behaved.\n\nThis does require that the target distribution satisfy a log-Sobolev\ninequality and dissipaivity condition, which may be a larger assumption than\nprevious work. This may be the work's biggest weakness.", "other_comments_or_suggestions": "There are a couple possible typos:\n\nL431: It should read \"probability distributions\"\nFigure 1(c) and (d): Should the x-axis be \"stepsize\" instead of timesteps?\nIt seems that the discretization error should decrease with the number of\ntime steps, but perhaps I'm confused by the terminology.", "questions_for_authors": "[Q1] While most assumptions laid out in this paper are relatively simple to\nunderstand, Assumption 3.3 seems less obvious. How severe of an assumption\nis this?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper analyzes the convergence of Langevin Monte Carlo. There are several contributions: 1) convergence in W2 under LSI instead of commonly assumed, stronger assumption of log-concavity; 2) sqrt(d) dimension dependence, which was what people tried for a long time and a substantial progress in the literature, however established under stronger conditions; 3) a proof that quantitatively shows that a recent modification of LMC (Pang+ 25) works under superlinear growth condition instead of global Lipschitzness, which is weaker. All reviewers and I agree this is a good paper. Although no machine learning application has been discussed, this is a work on the theoretical foundation of machine learning. I hence recommend acceptance as poster (not higher due to insufficient machine learning components), however under the condition that the authors remove the overclaimed word "optimal" as some reviewer pointed out (and I agree).
Decision: Accept (poster)