On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates
Authors: Stefano Bruno
Ying Zhang
Dongyoung Lim
Omer Deniz Akyildiz
Sotirios Sabanis
Venue: Accepted by TMLR ()

Abstract: We provide full theoretical guarantees for the convergence behaviour of diffusion-based generative models under the assumption of strongly log-concave data distributions while our approximating class of functions used for score estimation is made of Lipschitz continuous functions avoiding any Lipschitzness assumption on the score function. We demonstrate via a motivating example, sampling from a Gaussian distribution with unknown mean, the powerfulness of our approach. In this case, explicit estimates are provided for the associated optimization problem, i.e. score approximation, while these are combined with the corresponding sampling estimates. As a result,  we obtain the best known upper bound estimates in terms of key quantities of interest, such as the dimension and rates of convergence, for the Wasserstein-2 distance between the data distribution (Gaussian with unknown mean)  and our sampling algorithm.
Beyond the motivating example and in order to allow for the use of a diverse range of stochastic optimizers, we present our results using an $L^2$-accurate score estimation assumption, which crucially is formed under an expectation with respect to the stochastic optimizer and our novel auxiliary process that uses only known information. This approach yields the best known convergence rate for our sampling algorithm.

Published in Transactions on Machine Learning Research (02/2025)
On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates
Stefano Bruno
sbruno@ed.ac..m
School of Mathematics University of Edinburgh
Ying Zhang
yingzhangg@hhust--g..du.... edu.cr
Pintech Thrust
The Hong Kong University of Science and Technology (Guongshou)
Dong- Young Lim
@liistuiis.oo oc.kr
Department of Industrial Engineering Artificial Intelligence Graduate School Ulsan National Institute of Science and Technology
Ömer Deniz Akyildiz
deniz.okkildiz@@@ppriid..
Departmend of Mathematics Imperial College London
Sotirios Sabanis
s. s.sabamis Wed
School of Mathematics University of Edinburgh
The Alan Turing Institule
National Technicul Universily of Athens
Reviewed on OpenReview: https: // openreview. net/ forum? id= zjækrb4ehr ehr
Abstract
We provide full thcoretical guarantees for the convergence behaviour of diffusion-based gen- erative models under the assumption of strongly log-concave data distributions while our approximating class of functions used for score estimation is made of Lipschitz contimuous functions avoiding any Lipschitzness assumption on the score function. We demonstrate motivating example, sampling from a Gaussian distribution with unknown mean, the powerfulness of our approach. In this case, explicit estimates are provided for the associated approximation, while these are combined with the corre-
optimization problem, i.e. sponding sampling estimates. As result, obtain the best known upper bound estimates in terms of key quantities of interest, such as the dimension and rates of convergence, for the Wasserstein-2 distance between the data distribution (Gaussian with unknown mean) and our sampling algorithm. Beyond the motivating example and in order to allow for the of diverse range of stochastie optimizers, we present our results using an aa-accurate score estimation assumption, which crucially formed under an expectation with respect to the stochastic optimizer and novel auxiliary process that uses only known information. This approach yields the best known convergence rate for our sampling algorithm.
1 Introduction
Diffusion-based generative models, also known as score-basod generative models (SGMs) (Song & Ermon 2019 [Song et al. 2021; Sihl-sickstein al., 2015: Ho al.) 2020)) have become over the past few years one of the most popular approaches in generative modelling due to their empirical successes in data generation

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tasks. These models have achieved state-of-the-art results in image generation (Dhariwal & Nichol 2021
Rombach et al. 2022), audio generation (Kong et al. 2020) and inverse problems (Chung et al. 2022 Song
et al. 2022] Cardoso et al. 2024 Boys et al.] 2024) outperforming other generative models like generative
adversarial networks (GANs) [Goodfellow et al. 20114). variational autoencoders (VAEs) (Kingma & Welling
2014), normalizing flows (Rezende & Mohamed 2015) and energy-based methods (Zhao et al. 2017).
SGMs generate approximate data samples from high-dimensional data distributions by combining two dif- fusion processes, a forward and backward in time process. The former process is used to iteratively and smoothly transform samples from the unknown data distribution into (Gaussian) noise, while the associated backward in time process reverses the noising procedure and generates new samples from the starting un- known data distribution. A key role in these models is played by the score function, i.e. the gradient of the log-density of the solution of the forward process, which appears in the drift of the stochastic differential equation (SDE) associated to the backward process. Since this quantity depends on the unknown data distribution, an estimator of the score has to be constructod during the noising step using scorc-matching techniques (Hyvärinen] 2005 Vincent| 2011). These techniques have the advantage of not suffering from known problems of traditional pushforward generative models, such mode collapse (Salmona et al. 2022).
The widespread applicability and success of SGMs have been accompanied by a growing interest in the tcalretical understandings of these models, particularly in their convergence theory e.g. in Block et al.
20200g ; De Bortoli et al. (2021); Bortoli (2022]; Lee et al. (2022 ; Yang & Wibisono (2022) Kwon et al. (2022
g Liu et al. (2022); Oko et al. (2023 Lee et al. (2023) Chen et al. (2023a d) Li et al. (2024); Pedrotti
et al. (024); Conforti et al. (2025); Benton et al (2024). with further works appearing after the first version of
our preprint e.g. in Tang & Zhao (024);; Strasman al. 2025]; Mimikos-Stanme Staankoo-SSammtoouullos et al. (2024). At its
core, this new generative modeling approach combines optimisation and sampling procedures specifically, the approximation of the score and the creation of new samples, which make its theoretical analysis both an
interesting and a rich challenge.
Some of the recent advances in the study of the theoretical properties of SGMs concentrate around the sampling procedure by assuming suitable control for the score estimation procedure. For instance, the analysis in Lee al.] (2022); Chen al. (2023d) assumes that the score estimate L'-accurate, meaning that the L error between the score and its estimate is small, and provides estimates in total variation (TV) distance. Under the same assumption, the more recent contribution in Conforti et al. 2025) establishes non- asymptotic bounds in Kullback Leibler (KL) divergence by assuming finite relative Fisher of data distribution with respect to a Gaussian distribution.
The main drawback of the aforementioned T.2-accurate (and in some cases I accurraa)) score estimation assumption is that the corresponding expectation is given with respect to density of the solution of the forward process, which depends on the unknown data distribution.
Our approach introduces novel auxiliary process that relies solely on known information and uses the density of the solution of this process in the 1--accurate score estimation. To further highlight the powerfulness of our approach, we present a motivating example on the case of sampling from a Gaussian distribution with unknown mean. Full theoretical estimates for the convergence properties of the SGM are provided in 2 distance, while our choice of stochastic optimizer for the score approximation is a simple Langevin-based algorithm. Our estimates are explicit and contain the best known optimal dependencies in terms of dimension and rate of convergence (Theorem []. To the best of the authors' knowledge, these are the first such explicit results with transparent dependence on the parameters involved in the sampling and optimization combined procedurcs of the diffusion models. By connecting the diffusion models with the theoretical guarantees of machine learning optimizers via standard stochastic calculus tools, the results in Theorem together with the bounds achieved in the more general setting in Theorem 10 provide the theoretical justification for the empirical success of the diffusion models.
We close this introductory section by highlighting some other, alternative approaches which were recently developed. One may consult Yang & Wibisono/ (2023) for an approach based the assumption that the score estimation error has a sub-Gaussian tail. This is stronger assumption than L'-accurate. In Chen et al. 2023bb,, non-asymptotic bounds in TV and Wassertein distance of order 2 are derived when the data distribution is supported on a low-dimensions linear subspace. Finally, convergence guarantees in TV were
2

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developed in Chen et al. (023c) for the probability flow ordinary differential equation (ODE) implementation of SGMs under the I acccuuaaee score estimate assumption.
Notation. Let (2, F, IP) be a fixed probability space. We denote by [[X] the expectation of a random variable X. For 1 p < 00; LP is used to denote the usual space of p-integrable real-valued random variables. The I--integrability of a random variable X is defined as [[[[X]]] < xp. Fix an integer M 2 1. For an TR random variable X, its law on B(RR"), i.e. the Borel sigma-algebra of I denoted by [(X). Let T' > 0 denotes some time horizon. For positive real number a, denote its integer part by [a]. The Euclidean scalar product is denoted by (,-),, with || standing for the corresponding norm (where the dimension of the space may vary depending on the context). Let R>o == far E R x > 0}. Let f R. + R be continuously differentiable function. The gradient of f is denote by V.. For any integer q 2 1, let P(R9) be the set of probability measures on (()). For μ4, / E ()R"), let C(p, v) denote the set of probability measures on M(TR2M such that its respective marginals are j and v. For any μ and v E (TRM), the Wasserstein distance of order is defined as
W)(u, inf (EEC(H,)) J J/nn J.m la - y1 dc(x; v)
2 Technical Background
In this section, we provide brief summary behind the construction of score-based generative models (SGMs) based on diffusion introduced in [Song et al. (2021). The fundamental concept behind SGMs centers around the use of an ergodic (forward) diffusion process to diffuse the unknown data distribution TD E P(RMM) to a known prior distribution and then learn a backward process to transform the prior to the target distribution To by estimating the score function of the forward process. In our analysis, we focus on the forward process (Xx))e(o,T) given by an Ornstein- Uhlenbeck (OU) process
dX2- = -x, dt + V2 dB, X, ~ xD,
(1)
where (B.) is an M-dimensional Brownian motion and we assume that EE [o is chosen to match the forward process in the original paper (ong al. 2021), which is also referred to as Variance Preserving Stochastic Differential Equation. The noising process can also be represented follows
2] < co. The process
X X, + ozli, my = of -1e
Z, ~ ~(0,
(2)
where R.5. denotes almost sure equality and INs denotes the identity matrix of dimension M. Under mild assumptions on the target data distribution TD (Haussmann & Pardoux 1986 Cattiaux et al., 2023 the backward process (Yi)eeu.ry = (Xr-)) is given by
dY; (Y; + 2V log dt + / dB. Yo ~ L(XT).
where helorr is the family of densities of (C(X)))e(0o.) with respect to the Lebesgue measure, B, is an another Brownian motion independent of B, in [ defined on (D,F,P). However, the sampling is done in practice from the invariant distribution of the forward process, which, in this case, is a standard Gaussian distribution. Therefore, the backward process 3 becomes
dÝ. = (% 2 (Vo) dt + /2 dB. Vo ~ Tx = (O,
(4)
Here, we have Yo Zr. Since TD is unknown, the score function V loggt in 3 cannot be computed exactly. To address this issue, an estimator s(., s(.,0".-) is learned based on family of functions s: [o, X - R parametrized in 0, aiming at approximating the score of the ergodic (forward) diffusion process over a fixed time window [0, Tj. In practice, the functions s are neural networks and in particular cases like the motivating example in Section 3.1 the functions s can be wisely designed. The optimal value 0* of the parameter 0 is determined by optimizing the following matching objective
/
Rdd 50+B
V Vlogp((X)) - s(1,0, dt
(5)
3

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To account for numerical instability issues for training and sampling at 0 as observed in practice in [Song
et al. (2021 Appendix C) and for the possibility that the integral of the score function in may diverge
when f = 0 (see Appendix A a discretised version of the score-matching optimization problem is usually
considered
[h T k(t) minimize 30 U(0)
Vlogp((z)
(z( dr dt,
(6)
T T- R
where e > 0 and x [0, + R>0 is a weighting function. The score-matching objective U in can be rewritten via denoising score matching Vincent 2011) as follows
( -)][() s(T, S(T, m, Xu x,,)))
(7)
where the expectation is over Uniform((e, Xo TD and Z ~ (N0, Im); and where C' E R is constant independent of 0 (see Appendix A for the derivation with the OU representation 30 The stochastic gradient
H H.Rd x R.m + Idd of 6 deduced using 7
is given by
M
H(0, x) = 2x((t)
Σ i=1
or 20 + (t, ma + t12)) (ts 0, m, + to2),
(8)
where x = (t, vo. =) E T with m 2M + 1. As contribution to this analysis, we introduce an auxiliary process (Y,aux) denoted by 0, for t E [0,
containing the approximating function s depending on the (random) cstimator of d
dy (Y,"xx + +2 (T ),uN)) dt /2 dB,,, YOux Too = N(0 LM).
(9)
The process will play an important role in the derivation of the nonasymptotic estimates in Wasserstein distance of order two between the larget data distribution and the generative distribution of the diffusion model. Indeed, it connects the backward process the numerical scheme II which facilitates the analysis of the convergence of the diffusion model (sce Appendix C.2 and Appendix D.2 for more dctails). For this reason, we introduce a sequence of stepsizes (7H ) (o such that Σ " = T. For any k E (0...., K), let tx-1 = Yo with to and tk+1 T. Let 7 = 7 E (0,1) for each k O...., K. The discrete process (Y (oE of the Euler Maruyama approximation of 9 is given, for any k E (O....,K) as follows
YRM = FM + 7YEM + - -,, + / Z YFM T = Tm),
(10)
where | is a sequence of independent M-dimensional Gaussian random variables with zero mean and identity covariance matrix. We emphasize that the approximation I0 the one chosen in the
original paper (ong al. (2021). Finally, the continuous time interpolation of 10 for t E [0, 77, is given by
dYP.m_ =(P (1//11 +2 s(T Lt/nln, 0, EM (1/1)))) + 22 dB..
EM Too = =V(0, Iar)
(11)
where ((PEM)- = ((YEM) at grid points for each & E (0...., K+ + 11).
3 Main Results
Before roducinng the main assumptions of the paper, we present a motivating example.
3.1 A Motivating Example: Full Estimates for Multivariate Gaussian Initial Data with Unknown Mean
In this section, we consider the case where the data distribution follows a multivariate normal distribution with unknown mean and identity covariance, i.c., X, TD AV4,, Ia) for some unknown j e Rd with M d. We show that, by using diffusion models, we are able to generate new data from an approximate distribution that is close to ap- More precisely, provide non-asymptotic convergence bound with explicit constants
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in Wasserstein-2 distance between the law of the diffusion model and TT, which can be made arbitrarily small by appropriately choosing key parameters the upper bound. In this example, the estimation of the score function reduces to the estimation of the unknown mean by the methods of convex optimization with optimal dependence of the dimension combined with the most efficient sampling method for high-dimensional Gaussian data.
In this setting by using we can derive the score function given by
v (z)))(()) -x + m4He
(12)
which can be approximated using
stt, (],00x)==00 m (t, 10, x R Rd
(13)
To obtain approximated score, i.c., to obtain optimal value of 0 in 13 opt for popular class of algorithms called stochastic gradient Langevin dynamics (SGLD) to solve the optimisation problem G In
addition, we choose the weighting function x(t) = of as in Song & Ermon (2019) and we set e = in
Using the approximating function
13 in
S.
we can obtain the following expression for the stochastic gradient
H(0, x) 20 Σ6
0z (0,1(00) (-m+a)) (2) 0,2 (0) + m m.e;
(14)
20 (or - = - 022 + m,,))
where x (f, R" with m 2d + and e; denotes the unit vector with i-th entry being 1. Fixing the so-called inverse temperature parameter 3 > 0, the associated SGLD algorithm is given by
0 : 80:
8 = + AH 1) + 22X/3 Ent1, E No.
(15)
where À > 0 is often called the stepsize or gain of the algorithm, (En)nEN, is sequence of standard Gaussian vectors and
(Xn)eNo = (Tn X Zn)neNo
(16)
is sequence of ii.d. random variables generated as follows. For each n E No, we sample Tn from Uniform((o, TT) such that L(T.)) = L(T), sample Xo.n from कn N(n, Ia) such that L(Xo,)) = L(o)), and sample Zn from (0, Ia) such that ((Z,)) = ((Z). In addition, we consider the case where fo (En)nENo: and (Xn)xEN, in 15 are independent and we have E (0, Xn+1)l (U).
Throughout this section, fix
0 O<As
(17)
Theorem states the non-asymptotic (upper) bounds between the generative distribution of the diffusion model L(YEM, FM ) and the data distribution TD. An overview of the proof can be found in Appendix C.2
K+1)
Thcorem 1. Under the setting described in this section, then, for any T > 0 and E 1/2],
W2(C(EEM EM K+1), TD)
E I| + Vd)
(18)
+ (v + 2V 33) 2 E 8. + dCscuD.. + / /XCSGLD.)
+ (VIId + 13)
where CSGLD, and CSGLD, are given explicitly in Table [ In addition, the result in 18 implies that for any > 0, if we choose T > To. B2 3o, 0 < X S \o, 2 ns, and < 7 Yo, then
W2(C(P (LLCCRR41,,
RM mD) < 6,
where Ts. Bb, Ao. ng and vs are given eaplicilly in Table 7

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Table 1: Explicit expressions for the constants in Theorem
CONSTANT
PUL. EXPRESSION
Csa
Csa
4B(o,, (o, 12] m-- Xoo 10110
T3
2-
lu
1v2
E[I.Xo/3]+ Vd
144d( / 4/3 2/33) /(6 [(6
min
A6
x(776( 4/3 2v33) 2lo (o, 1x + a. 12] m- )²
(AE(o? In
/ 2v/3) /
WITH o- 19]/88
(S Aa)
78
min
6/14 13210"13)1/2)
Remark 2. The result in Theorem achieves the optimal rate of convergence of order one for Euler Milstein schemes of SDEs with constant diffusion coefficients. Furthermore, one notes that the dependence of the dimension on the upper bound in 18 is in the form of vd. To the best of the authors' knowledge, the result in Theorem | is the first converyence bound where the parameters involved in the sampling and optimization steps of the diffusion models appear explicitly. In the optimization procedure, we use SGLD algorithm 15 to solve the problem D Since the stochastic gradient H in in strongly conver by Proposition 13. it has been proved, for instance in Barkhagen et al. (2021), that, for large enough 0, the oulput of SGLD is an almost minimizer of when is large. Thus, the diffusion model, we set Aº indicating V log (x)) z s(l,0,xz) = s(1, 02,, x)) for large values of and for all and Crucially, this allows us to use the established convergence results for SGLD to deduce a sampling upper bound for (2(C(V K+1), TD) with explicit constants in 18 Consequently, this bound can be controlled by any given precision level 8 by appropriately choosing T, X, and 7.
This motivating example has focused on exploring the convergence of diffusion-based generative models using Langevin-based algorithm, specifically SGLD, which is well-known for its theoretical guarantces in achieving global convergence. However, in the general case discussed in Section 3.2 we do not prescribe a specific optimizer to choose to minimise the distance between and l"
3.2 General Case
In this section, derive the full non-asymptotic estimates in Wasserstein distance of order two between the target data distribution TD and the generative distribution of the diffusion model under the assumptions stated below. As explained in Section
2 (see also Appendix A),. it could be necessary in the general setting
to restrict & E [c. 7] for e E 1) in 6 Therefore, we truncate the integration in the backward diffusion at T- - e and run the process (Y.)
3.2.1 Assumptions for the General Case
In the motivating example in Section 3.1 we have chosen the SGLD algorithm to solve the optimisation problem Other algorithms, such as ADAM Kingma & Ba 2015), TheoPouLa (Lim & Sabanis, 2024) and stochastic gradient descent (Jentzen et al. 2021 can be chosen as long as they satisfy the following assumption. Fix
Assumption 1. Let 6* be minimised of and let be the (random) estimator of 0- obtained through some approcimation procedure such that 0 < co. There exists AL such that
E 0 p- EAL-
The score-matching optimization problem
necessarily (strongly) convex.

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Remark 3. As consequence of Assumption 1 we have B||0|] < 2E + 2
We consider the following assumption on the data distribution.
Assumption 2. The data distribulion ap has fönile second moment, is strongly log-concave, and J. V log (O) < 00.
Remark 4. As a consequence of Assumption and the preservation of strong log- cog-conavvtyy under convolu- lion, see, e.9.; Saumard & Wellner (2014 Proposilion 3.7); there erists Laro : [0, - (0, oo] such that for all t [0, and 2, 12M, we have
(Vogg(()) Vlog B) (M)o(t) -
(19)
The function (t) in 19 has a lower bound for all E [0, which we denote by L Moreover, Assumplion with the estimate 19 implies that the processes in and have a unique strong solution, see, e.g., Krylov 1991 Theorem 1).
Next, we consider the following assumption on the approximating function s which is used in Remark 12 Assumption 3.a. The function s [0, x Rd RM R is continuously differentiable in x E RR Let D1 Bd xIR + RR, D2: j0, T) x [0, 7] - RR and D3 [0, Tj (0, T + RR+ be such that Se Se (2(t, dt di <
oo and Je Je D) ae di < co. For E [] and for all t, f E [0, x. E and 0, 0 E RR,, we have that
-(T,0,)) D1(0.0)L-11 D2(1,2)0 Da(1,D)x F)
where D1, D2 and D3 have the following growth in each variable: i.e. there crist K1, K2, and K3 > such that for each i, T [0, and 0,0 Rd,
(1(0000 0) S K1 + 0 10), [D((t f) K3(1 + |t| + 189))
(2((t 1810 111))
By adding a further condition on the gradient of s in Assumption 3.a we are able to achieve the optimal rate of convergence in Theorem 10 below.
Assumption 3.b. Let be as in Assumption 3.a and there erists K₁ >> such that, for all a, xt E R and for any k I,...M,
(L, 0, - Vgs (1, (1, S (1 + 2 x). V V (K)
Remark 5. Let K Kroaa
K1 + K2 + K3 |s(0,0,0) > Using Assumption 3.b we have
st, 0, x) < Kroaa(1 + (1 + |0| + |x])
We postpone the proof of Remark to Appendix D.1
Remark 6. Assumption 3.a and 3.b impose Lipschitz continuity
famāly of approzimating functions
(() with respect to the ingul variables, I and x, well as the parumelers 0. It is well-known that the continuity properties of neural networks with respect to t. and x are largely determined by the activation function at the last layer. For instance, neural networks with activation functions such as hyberbolic tangent and sigmoid functions at the last layer satisfy the Lipschitz continuúly with respect to and x [Virmaux &
Seaman 2018 Fazlyab al. 2019).
For the following assumption on the score approximation, we let i be as in Assumption 1 and we let ) be the auxiliary process defined in Assumption 4. There exists sSN > 0 such that
E
V log Pr (Y.uu) s(T -r, V.u.u dr < <SSN.
(20)
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Remark 7.
We highlight that the expectation in Assumption 4 is taken over the auriliary process This densily is known since the approzimaling funclion 8 and the estimator 0 are known. To the best of authors knowledge, this is a novelty with respect to previous works (Bortoli 2022 Chen et al. 2023d; Lee et al.) 2022, 2023: Chen et al. 2023a; Conforti et al.) 2025; Benton et al. 2024 which consider the unknown density of the forward process (or ils numerical discrelization..
Remark 8. Assumption 4 is satisfied, along with Assumption 3.6 for data distributions satisfying Assump- tion beyond the multivariate Gaussian case discussed in the motivating erample. Indeed,
T-c
E
V loggrr (Y,rur s(T -r, (r, V,aum)) dr
S 2 E
V VlogpT- (Y,aur) - s(T - -r, -r,un, dr y,uu))
(21)
T-e
+ 2 E
|s(T -r, Y, -r,0", s(T - -, dr.
J
If 0* is such that s(1, 00,00 x) = V Vlggp(<)) in the molivaling example in Section 3.1 then the first term on the right-hand side 21 vanishes. Otherwise, we expect that the first term on the right-hand side above to be small. Indeed, by the definition of strong log-concaviity, see e.g. SSamardd & Wellner (2014. Definition
2. 9), we have
V ll log(g(z)) + V (l)((,(()),
(22)
where g is some log- nog-concave function and oi is multivariate Gaussian density. If g is multivoriate logistic distribution P [Malik & Abraham 1973), then its score function is given by
a log(g(z)) -1 - (M 1) exp(-xxn) 0zk
IK 00,
k = 1, M,
(23)
1 k- cxp(- xx2)
while the Hessian of lz)( is bounded (see Appendir for more details). Therefore, the score function of the mallivariate logistäc distribution 23 is Lipschilz, and, as consequence, V log pt(x) in 22 is still Lipschilz. Thus, we expect to have a good control on the first term on the right-hand side of 21 since the function s salisfying Assumplion 3.0 approzimales Lipschita score function. In general, there erists a function
dl,x)) := [00:=Vloggpan Vlggp(()) - x))
1>0,
(24)
and therefore one has define log- log-concave function g such that the first term on the right-hand side of 21 is small. Clearly, this is problem specific challenge and it may not always have good solution. The
second term the right-hand side of 21 is controlled by Assumption 3.b and Assumption (see Appendix
B for more details).
Remark 9. We conduct numerical experiment to show the convergence of diffusion--used generative
models under Assumplion 7
3.0 and 4 We consider the case where Xo TD = Nuu, Ia) with 2
μ (-1.2347, - 0.89244) and d = We use SGLD algorithm 15 to solve the optimization problem 7 with k(t) = o 0, T A = 5 10-5, 3 1022 and n = 4 10a At each iteration, 128 mini- batch samples are used to estimate the stochastic gradient. Then, we generate samples using the Euter-Maruyama approzimation 10 with 7 10-3 and s given 13 At each ileration n, we evaluate the quality of 100, 000 generated samples using the Wasserstein distance of order two. In addition, we compute the L error between the score function and the approzimated function s with 02, using the auciliary process as in Assumption 4 Figure 1 (a) demonstrates the error in Assumption vanishes as the generative model converges, where the degree of convergence is measured by W2(C(YEM) TD). This empirical observation. justifies Assumption 4 Moreover, we explore the relationship between the quality of generated samples and the error using 7 via
denoising score matching in Figure 0 (b).
TThe multivariate logistic distribution example elliptical distribution widely used portfolio risk management Xiao
& Valdes 2015 Owen Rabinovitch 1983).
8

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20
Wg/e(Y2 3.7g
15
WXIc(P2 (.)
10
10
0.5
05
0.0
0.0
0.5
0.4
0.3
0.2
0.1
0.0
0.75
0.70
0.65
0.60
0.55
0.50
Error (Assumption
denosing matching loss
(a) Error with Assumption
(b) Error using (()) in
Figure 1:
The quality of generated samples with respect to (a) the error with Assumption and (b) the error obtained through denoising score matching using (0) in 7
3.2.2 Full Estimates for the General Case
The main result under the general setting is stated as follows. An overview of the proof can be found in Appendix D.2
Theorem 10. Let Assumptions 1 such that for any T and 7, E 1),
2 3.6 and hold. Then, there exist constants C1, C22, Css and C₁ > 0
W2(C(VRM)) TD) & Cive+ + Oze -uo(--o Ca(T, + CC(TT
(25)
where C1. C2 C3 and C4 are given explicitly in Table 4 (Appendix B addition, the result in implies that for any > O, if we choose 0 S < es, T > To, ESN < ESN,5 and 0 < 7 < 7o with es, To, 8SNS8, and 7s given in Table 4 then
W,(L(YEM), TD) 6.
Remark 11. The error bounds in 25 are not as good as the ones provided in Theorem due to the general form of the approzimating function s. We emphasize that the optimal rate of convergence of order E [ for the Fuler or Milstein scheme of SDEs with constant diffusion coefficients is achieved using the Lipschitz continuity on the derivative of s in Assumption 3.8 the explicit expression of C4 in Table the dependence of the dimension is O(M) due numerical techniques from Kumar & Sabanis Theorem I0 to achieve the optimal rate of convergence.
(201 used in the proof of
Remark 12. If we replace Assumption 3.6 with Assumption 3.a in Theorem 10 then the bound 25 becomes
W2(C(YEM), TD) S CIVE Cee
2L -)
where C1, C2. C3 are the same in Theorem 10 and Ca(T, contains better dependence on the dimension, namely M), than the one achieved in Theorem 10 und il is explicilly provided in Table A (Appendir h Although this relazation achieves the same dependence on the data dimension as in the motivating example in Theorem 1 it leads to a worse rate of convergence of order 1/2.
4 Related Work and Comparison
We describe assumptions and results of various existing works in our notation and framework to facilitate the comparisons with our results which are provided in 2- Wasserstein distance, Beyond being theoretical relevant, the choice of the use of this metric is motivated by applications in generative modeling. A popular performance metric currently used examine the quality of the images produced by generative models is the Fréchet Inception Distance (FID) which was originally introduced in Heusel et al. 2017 This
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metric measures the Fréchet distance between the distribution of generated samples and the distribution of real samples, assuming Gaussian distributions, which is equivalent to the Wasserstein distance of order two. Our results provided under this metric are therefore also relevant for practical applications.
We can classify the previous approaches based on their assumptions on two key quantities: (i) score ap- proximation error and (ii) assumptions on the data distribution. Based on these approaches, we give a brief overview of some of the most relevant recent contributions in the field.
4.1 Score Approximation Assumptions
The carly work of De Bortoli et al.
(2021) requires b.ond on the score approximation, which is in contrast with the L nature of the score-matching optimization problem
6 Most of the recent analysis of
score-based generalive models, see, e.g.. Chen el al. (2023d); Lee et al. (2022 2023
Chen et al. (2023a)
Conforti et al. 20225 Benton et al. 22224 have considered assumptions (in L
on the absolute error of the
following type, i.c. for any lo E {0, .........
- 17, there exists E > 0
E
V log Pt,, (Xt,)) s(tx, 0 Xt, )1] S.
(26)
where the expectation is taken with respect the unknown (ejory) and Ô is a deterministic quantity. In Chen et al. (2023a): Conforti et al. 2025;; Benton et al. (2024) the assumption 26 is written in integral
(averaged) form. In Lee et al. (2023) the bound in 26 is not uniform over t, L.e. E. == S/f and this allows
o7 as t - 0. The authors in Conforti et al. (2025) use the relative 1-2-score the score function to grow as approximation error such as
E 2 P. (x.,) s(tx, 0, Xx,) S E [ log (x.
(27)
where the expectation is with respect to the unknown density of the law of X, against the Gaussian dis- tribution oo (2), i.e. pe(x)= == (o and s(t, 0.x) ) in 27 is the approximating function for the score
function of Pt- pointwise assumption in Bortoli 2022). given by
V loggr(<) - stt, 0, x)1 S ((1++1=)//
(28)
for C 2 0, is considered under the manifold (compact) setting. The assumption 28 takes into account the explosive behaviour of the score function as - O. In an attempt to obtain a weaker control than 28
the
assumption Bortoli
2022
A5) is used, namely
E V log pr (Y s(T - t, 0,,
C [(1 [1 /ot
(29)
We note that, unlike 26 and 27 the expectation in 29 is taken with respect to the algorithm Y given by the exponential integrator (EI) discretization scheme which has known density. However, the bounds of Bortoli 2022 Theorem H.1) in Wasserstein distance of order one derived under this assumption scale exponentially in problem parameters such as the diameter of the manifold M and the inverse of the early stopping time parameter 6, i.c. exp(O(diam(I)) /c)). Furthermore, an assumption similar to 29 is considered in a very recent and concurrent result") Gao et al. (2023):
V Pog (Y s(T (I 1)n, 0 1. ())))) 1/2 SC.
sup I=
E
where 7) > 0 is the stepsize and T Kn with K N.
We emphasize that the existing results in the literature do not take the expectation with respect to the stochastic optimizer Ò. This, logether with the use of our novel auxiliary process 9 that uses only known information, allows us to deduce state-of-the-art bounds in the Wasserstein distance of order two in the following sense. The bounds scale polynomially in the data dimension M, i.c. O(VM) as shown in Theorem and Remark [2 and achieve the optimal rate of convergence: of order one in Theorem [ and of order a E [1. in Theorem 10
"This analysis be extended to the Buler-Maruyama numerical scheme in YEM 10
4The concurrent paper Gao al. 2023) appeared few days earlier when the first draft of this work was made available
online.
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4.2 Assumptions the Data Distribution
The vast majority of the results available in the literature are in KT. divergence and TV distance. For two general data distributions μ and v, there is no known relationship between their KL divergence and their W2. However, for strongly log-concave data distributions, a bound on the Wasserstein distance of order two in terms of KL divergence follows from an extension of Talagrand's inequality Gozlan & Léonard 20100 Corollary 7.2).
Some convergence results in different metrics can be deduced under the following lypes of assumptions. Convergence bounds in Wasserstein distance of order one with exponential complexity has been obtained in [Bortoli (2022) under the so-called manifold hypothesis, namely assuming that the target distribution is supported lower-dimensional manifold or is given by some empirical distribution. Moreover, the results in TV distance in Lee et al. (2022) and in KT. divergence Yang & Wibisono| (2023) assumed that the data distribution satisfies a logarithmic Sobolev inequality and the score function is Lipschitz resulting in convergence bounds characterized by polynomial complexity. By replacing the requirement that the data distribution satisfies a functional inequality with the assumption that TD has finite KT. divergence with respoct to the standard Gaussian and by assuming that the score function for the forward process is Lipschitz, the authors in Chen et al. (2023d) managed to derive bounds in TV distance which scale polynomially in all the problem parameters. By requiring only the Lipschitzness of the score at the initial time instead of the whole trajectory, the authors in Chen et al. (2023a) Theorem 2.5) managed to show, using an exponentially decreasing then linear step size, convergence bounds in KL divergence with quadratic dimensional dependence and logarithmic complexity in the Lipschitz constant. In the work by Benton et al. 2004)) the authors provide convergence bounds in KL divergence that are linear in the data dimension, up to logarithmic factors, using carly stopping and assuming finite sccond moments of the data distribution. A careful examination of Benton et al. (2024 Proof of Theorem 1 and Corollary 1) reveals that the authors still require the uniqueness of solutions for the backward SDE 3 For instance, either measurability and boundedness of the drift (Stroock & Varadhan 1997) or integrability of the drift (Röckner & Zhao 2023) should be imposed if the uniqueness of weak solutions is considered. Therefore, additional conditions on the score function, depending on the type of uniqueness of solutions considered, are st.ill needed in their bounds in Theorem 1 and Corollary 1.
Assuming the finiteness of the second moment of the data distribution and using an EI discretization scheme with constant and exponentially decreasing step sizes, the authors in Conforti et al. 2025 Corollary 2.4) derive KL divergence bound with carly stopping, which scales linearly in the data dimension up to loga- rithincc factors. In terms of dependence of the data dimension, this bound is slightly worse than the bound in W2 provided in Remark 12, which does not include the logarithmic dependence of M. Bounds in KL without carly stopping have been derived in Conforti ct al. (2025) for data distributions with finite Fisher information with respect to the standard Gaussian distribution. The bounds in Conforti et al. (2025) Theorem 2.1 and 2.2) scale linearly in the Fisher information when an ET discretization scheme with constant step size is used and logarithmically in the Fisher information when an exponential-then-consss
step size [Cofforti et al.] (2025 Theorem 2.3) is chosen. We note that Conforti et al. (2025 Theorem 2.1 and 2.2) cannot achieve the optimal rate of convergence of Theorem in the motivating example.
We summarise the results of Chen et al. (2023a) Benton et al. (2024); Conforti et al. (2025 and compare them to ours in Table 2 and Table 3 making the distinction based on whether the early stopping rule is applied. In addition, a careful examination of Chen et al.] (2023a Proof of Theorem 2.2) reveals that the
authors require the uniqueness of solutions for the backward SDE 3 For instance, when strong solutions are considered, this implies that the score function should be (at least) monotone in the space variable, and a
suitable integrability condition in t (similar to our Assumption 2 is still needed to guarantee uniqueness of the solution (sce, o.g., Krylov (1991 Theorem 1)). For weak solutions, we refer to the discussion of Benton et al. 2024 Proof of Theorem and Corollary 1) above.
We emphasize that in Theorem 10 we do not assume the score function to be Lipschitz continuous with a uniformly bounded Lipschitz constant. This is particularly useful for future work in nonconvex set- tings, where the upper bound estimates will be independent of the (potentially large) Lipschitz constant of the score function, which could otherwise hide additional dimensional dependencies. The requirement
11

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1." V log (0) in Assumption 2 is weaker than the Lipschitz assumption the score function, but it is still difficult to verify in practical applications.
As pointed out in Chen et al. 2003d Section 4), some type of log-concavity assumption on the data distri- bution is needed lo derive polynomial convergence rales in 2-Wasserstein distance. This justifies the need for our Assumption 2 The concurrent result [ao et al. (2023) has similar assumption. For completeness, we also mention that the results the same metric, which have appeared after the first version of our preprint arXiv (e-g.; [Tang& Zhao [2024] Strasman et al. 2025 continue to assume strong log-concavity of the
data distribution.
Table Summary of previous bounds without early stopping and our result in Theorem Bounds expressed in terms of the number of steps required to guarantee error of at most ō in the stated metric. The relative Fisher information of the largel an against standard Gaussian measure is denoted by FT( All the bounds assume that has finite second moments.
Optimization
Regularity condi-
Metric
Complexity
Reference
problem solved tion
No
Vt, Voggtt L- Lipschitz
Ô ( M
Chen al
2023a The-
orem 2.1)
δ ( M+E (ooEl log (2)
No
Conforti
et
al.
Conforti
(2025
H2), i.e.
et al. 2025
with 2 = M -FI(TT
FI(TDD [oo) < 00
Theorem
2.3)
Yes
TD ~N(,,,))
W2(C(Y k+1) TD)
M Ō
Theorem
12

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Published in Transactions on Machine Learning Research (02/2025)
Table 3: Summary of previous bounds with early stopping and our results in Remark 12 The results in Chen
ct al. (2023a Theorem 2.2), Benton et al. (2024 Theorem 1), and Conforti et al. 2025 Corollary 2.4) are
stated for the smoothed version of the data, denoted by *b and using the score approximation assumpttio
26 with g. Therefore, an additional error should be added to their bounds, as the distance between T and
TD scales with VM in W2 (see Appendix D.2).
Assumption on the
Metric
Error bound
Reference
data
Finite second mo- ments of WD and
(KLCC(F))ITTT
[El M)e-T VTg
M(T
Chen al 22023a The- orem 2.2)
loge 1 ")/VN, (+ additional bounds between Tb and Tp)
26
See Section
422
for conditions
on V Po for the
uniqueness of Y,
V VO(M/N) VMe-22, (+ additional bounds between Th and TD)
Finite second mo- ments of TD and
Benton el al
22024 Theo-
26 See Section for conditions on V log for the uniqueness of Y,.
rem
Finite second mo- ments of TD and 26
(K(CC(P)MISS)
+ + VTg + [c[[M + E[|Xo| with e, E (0, 1/2] (+ additional bounds be-
Conforti
+og((")))))
et al. 2025
Corollary
tween Th and TD)
2.4)
Assumption 2 and Assumption
W2(C()EM),,)) TD)
O(VM)E + O(VM)e -T=
Remark 12
0(((4CC2Bxx0)TT-0 VESN
O(VMe p {
Acknowledgments
This work has been supported by The Alan Turing Institute through the Theory and Methods Challengo Fortnights event "Accelerating generative models and nonconvex optimisation", which took place at The Alan Turing Institute headquarters. This work was made possible by Besearch-in-Groun programme funded by the International Centre for Mathematical Sciences, Edinburgh. This work is supported by the Moangzhou-HHKSTTUU
Joint Funding Programs (No. 2024A03J0630 and No. 2025A03J3322), This work has received funding from the Ministry of Trade, Industry and Energy (MOTIE) and Korea Institute for Advancement of Technology (KIAT) through the International Cooperative R&D program (N0.500552.)). S.S. was supported by the Alan Turing Institute under the EPSRC grant EP EP/N510 /91110112//
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Appendix
A Objective Function via Denoising Score Matching
In this section, we show that the objective function U in can be written into using denoising score matching (Vincent 2011 and the OU representation
X; m,Xo o,Z, my = of = e Z (M),
(30)
We start by noticing that where d denotes equality in distribution.
VV log x)x pz( dr RM
(V))( slt,0, (0,00x)). dr
RRM
po Po dž, s(t.0.z)) dr (Spe Pu Pu ( VVV log(poo dž, s(t, s(t, 0) dr
(31)
JRM
JRM
{ (L, 3) V log(p: stt, (1)) dz dē.
JRM JRM
where Pro is the density of the transition kernel associated with and Pr.0 is the joint density of X and Xo- Using 31 in the objective function given in and the OU representation 30 have
U(0) = {:
T k(t)
( ([Vog)((z) 2 (tt, |s(t, 0,(t,0,x)1{) Po(x) dr dt T T
T k(t)
[ ( Vlogp)(t)) 2 (= 6(t,0,
T
+ +|s(t, peo( 2) dx dë dt
K((t)
( /
V loggoo(2)) s(t,0,,)) pr,(o, 2) dr dā df
T JRN JRN
(32)
xd())
(x) da de
K(())
[T
[
v log (ejo pr,,o,,, T) dæ dā dt
T
[
T <(()
( | (logp(())(Pp(())) dr dt
E[K(T)l0 ' Z s(T,0, m, 0,,
T T-
k(I) T T Jud
V log (z)
3) dr dā dt.
We emphasize that we may choose to restrict t E [e with E (0, 1) to prevent the divergence of the integrals on the right-hand side of 32
B Additional Discussions about Assumption 4
In this section, provide additional details about Remark in which used the multivariate logistic distribution (Malik & Abraham
1973
-(M+1)
M
M exp(
g(x) = M! exp
Σ
rk
1+
Σ
k= ==1,... 1...., M, -00 < 34 < 00,
k=1
k=
17

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Published in Transactions on Machine Learning Research (02/2025)
and derived its score function in 23 The Hessian of log(g(z)) is
8²
exp(-xx) (1 Σ k== exp( (-Ix)))))
exp(-22 -xkk)
log(g(>))
- (M 1))
(1 Σ
M k= exp(-3rx))
(33)
8
exp(-IT) exp( Fz})
0zzōzz log(g(z)) 11)
M, t
(1+E exp(-Ix)))
Since the Hessian 38 is bounded, we can conclude that the score function of the multivariate logistic distri-
bution defined in 23
is Lipschitz continuous. By using the same arguments as Remark g we can conclude
that V Poggp(()) defined in 22 is Lipschitz.
Using Assumption 3.b and Assumption 1 have
E
V log - (Y,ux) - s(T - -ruNN dr
T-c
T-c
< 2E
21+ r1 Blld 0-]3] dr V log logpr-r(2 (Faux) s(T - 0 y.ux)) dr + 2K
Ju
< 2E
V PT PY (YFux) s(T - y.ux)|² dr + - + 47 < ESN,
where the error of the first term on the right-hamd side above is expected to be small since the Lipschitz V (logp((>) is approximated by the function s satisfying Assumption This satisfies Assumption
In addition, we remark that the motivating example in section 3.1 is special case of the general setting, Indeed, Assumption i is satisfied due to Lemma [14 with Ô 8 being the nthhiiterate of the SGLD algorithm 14 and 0" == μa. Assumption
is satisfied by the score function 12. Assumption 3.b is satisfied by the 10), (2, == e-', Ds(t, == 1, for any 0, E R4, 1, E [), and K1 = K2 Ka = K4 = 1. Furthermore, both functions satisfy Assumption with
approximating function 13 with d = M, = (0,
= M. Indeed, by Assumption I with H" = μu, we have
V RogpT- I (Y,
s(T' - -r.-r, r,0, 0,00 y.ax dr
<
V po (YPaux) - s(T - r. 0r, y dr
T-c
2 E
|s(T 0- y s(T - r, 0, Y.uux) ) dr
J0
(e
2T) Ell0 0-13) (e 2x EAL ESN-
C Proofs of the Results for the Multivariate Gaussian Initial Data with Unknown Mean
In this section, we provide the proof of Theorem We start by introducing the results which will be used in the proof of Thcorem
C.1 Preliminary Estimates
We provide the results for the SGLD algorithm 15 with À given in 17 3 O, OSGUD, and CSaLD, given in Table well as for the auxiliary process
g the discrete process 10] and the contimuuu- time interpolation
with 8 given in 13 and E (0,1/2)
Proposition 13. For any 0, 0 E Rd and x E R"
H x) H(0xx) =20/m 0 -0. (H(0,x) - ((0,)), 0) =20, 0 2
18

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Proposition 13 is derived from the definition of the stochastic gradient 14
The proof of the following lemmas are postponed to Section C.3 Lemma 14. For any E No,
E 0 0 (1- (1=)) [100 8-13] dCSGLD,1 SC
As consequence of Lemma we derive the following corollary.
Corollary 15. For any No,
E [10 & 2e 2m [100 8-²] + 2dC 2ACsauD 2
Lemma 16. It holds that
sup E [
Cauxs
120
where Caux == (8/3)(e 2nAB[o mi [100 - 1-] + dC dCISGLD, + ACSGLD.2 10013) + 2d.
Lemma 17. It holds that
sup E [ E [2]
CEM:
KEN
where CEM 3d 20(e 2nAE:o2A [10. [100-0% 1-] + dCSGLD,1 + ACSGLD,2 + 10" (2).
Lemma 18. For any 1/2, one obtains that
sup E
[
EM
BB [c/1l>
BM
1
S 7CCEMssss
120
where CEMose Sd 56(e AnABlop m JE
[10 - [3
dCSGLD, ACSGLD.2 10-13).
Lemma 19. Tt. holds that
sup E s [2]
CEM:
120
where ÕEM 18d + 128(e 2m [100
0- dCsauD, + ACSCLD, 10-13).
C.2 Proof of the Main Result in the Motivating Example
Proof of Theorem
We derive the non-asymplotic estimate for W2(C() EM TD) using the splitting K+1).
We(C(YEM), TD) W2(TD, [(Y W2(C(Y(x+1) C(Y ((Yix))))
(34)
+ W2( ( ((Vaux)))) + W2 W2((aauu (x+1) ((YRM))
Since tK+1 T,, we have W2(TDD [(Yxx+))) 0. We provide upper bounds on the error made by approximat- ing the initial condition of the backward process Yo L(XT) with Yo Too, i.e. W2(C(Y c(Yi..)) )), the error made by approximating the score function with s, ic. We(C(F(k.n)) L(Youx }), and the discretisation error, i.e. (C ), C(Y EM )), separately.
Upper bound on W2(C(Y(x..) L(Yix..)) 1)). Applying Itō's formula and using and with the score function given in 12 we have, for any E [0.
d|Y, - 7112= 2(Y: - Y, Y, - Y. + ((Voogrr-(()) V Vlog Plogpr-(())))
(35)
= - - V/1.1.
Integrating and taking expectation both sides in 35 yields
Blly; = = E Yol? 1-12-22
-2 2/ E Y.1ids
(36)
S 2E 2.
19

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the representation with Zr d Yo and 1. S m,, we have
Using 36
E|[YeK11 Yoxus [2] S 2xx+ B Y
e - my Xo + (or 1)
(37)
< 2e #k [
Using 37
we have
(2( ((Yext)))
S VEIl
8extil? S / (vB[[ vd).
(38)
Upper bound on W2(4(Yr ). ((Y()))) Applying Itô's formula and using the process with the score have, for any 1 [0, TT.
function 12 and the process with the approximating function 13
dB 2 2(Y y Y, Y, + 2(V log - sT - t, Y, ))
(39)
- yaux dr 4(y yaux mm--(uu 0))dl.
Integrating and taking expectation on both sides in 39 and using that the minimiser H = -4, we have
BII Y [1 = - / EJiÝ, y Ids + pl
/ B((Y, Y -m-=(00 0)
(40)
Differentiating both sides of 40 and using Young's inequality, we obtain
d
B yaux [2] 2 y.ux []] 4B((Y, yaux -00 8))1
dt
s - Elji - y + 4e 2T e 2010000 - -0.
which implies that
d (c'Bljy: y (2) s 4e 2T " - 1.21.
Integrating both sides and using Lemma yields
Elji, - Y:aux
I (4/3)e 27 (6
1E 1
(41)
< (4/3)e-2(1-))) -2 2nAB ( El10 00 + dC ACSGLD,2).
Using 41
we have
W2(C(Vien)) C(Yaa) S E Exiiexe Yaux [2]
(42)
< /4/3(e
0-( dCsGLD, + \CsGLD.2)
Upper bound on W2(
(xx)) ), BM EM
)) Applying Ité's formula and using the processes and 11 with we have, for any t [0,T],
K+
the approximating function 8 given 13
/ PPMP 2 2(Yauux P.EM Y F.
(t/
+ 4 P.M s( - t, 0, V.SUX) s(T 1t/r/,, PEM
(1/
(43)
-22(aaux PPFM yaax (RMM))de
+ 4(yaaux PEM (mr-+ - MT -1-15/11,))) 10)dt.
20

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Integrating both sides and taking expectation in 43 yields
E | P.M
=- oo
/
|yaux PE ds /
ff"n IR (vvauxe PEM P.em 8 11//17/ ( ds
(44)
nt
/
(yaux P.M (mr-s mr -T-L0//17)0) ds.
+ +
E
Differentiating both sides in 44 using Young's inequality and mr-t - mT -T-14/177 S -/ we have
d
E
y
21E
(Y P.MM (2]
21E
BBM v2 dB, Y
dt
( (YYIIx P.EM,
2E
(-Y "//dd> + 2mmr 11/111//710)dk
Le/717
+ 4B[ D.EM (MTT
mh -mr-111112))1
y,aux D.EM. V2 S" dB.
(45)
2E 1 -B||ya P.aMPs
10/717
+ + 1PP(--PMMM + 2m - -1/7111 8- 2(T 1011101110
y.aux -aaux_PPM V2 S dB.
--Ely PEMPP 2E
1e/700
+ 44 PM
EM [5//]/ 12] 24-211
We derive an upper bound for the second term on the right-hand side 45 Itô formula applied to tB, and Young's inequality, have
Using Cauchy- inequality,
Yaux B.M V2 L" dB,
2E
16/7/7
V2 I" dB,
PEM (Yaux yaux (1/7))) (D.EM
2E
[1/17)1
10/7l7
(-Yaux + 2m 0)ds, v2 /' dB,
-2E
1t/111
11/-17
1' dB,)
21E
DEM 18/717 + 2m 10/7170/s
V2
11/107
2F
14/7/7 (Yaux - PEM
(0/2l7)ds dss V2
dB,
11/717
2E
14/1l7 Y dds, 22
(Yaux
dB,
11/117
[V S (-yau)) V2
2E
dB,
19/761
18/214
[ v2 v2
1' dB.
+ 21E
V
dBrds, v2
1//
11/1/7
21

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W/nn I Trrnndderi (-Ya dddds
2
1)
1/2
2
1/2
2v
E
E
dB,
L0/717
(." dB, - 11/717 sdB,, f dB,
-
}}
(46)
41
Je//11
L0/7Jy
v2-5/2
sup EE' [|Yaux []] d 2d7?.
120
Substituting 46 intoo 45
yields
l nux 1 = - -PEM V27 5/2
sup [E| |X,
dl
12:
+ 22 + 2dy + 4
EM
| + 241
[
Thus,
d (e'BllYy - (1 S e'
E| ( 120
(22
3d 4 -
(47)
dt
Integrating both sides in 47 and using Lemma
16
Lemma
and Corollary
15 yield
E| y GEMB2 2]
(2d (8/3) 2nABBo (2) Bl10 0-1² dCCaL, AC + +001)))
+ 3dy + 4-2 + 20 - 2n EE - 1- dCSGLD. + ACSGLD,2 + 18" (2))
(48)
+ 4872 2nABBo, 0, dC x
+ (0*)3)
< 18dy2 + 132-2 (e 2n |F( 0- + dCscL... ACSGLD,2 + 13)
By using
48
we have
W2 W2(C(V
(x11) ). L( ((P41)) K+1))
F.M !2] $ / /BILYNU
(49)
1x++
< (118 + 132(c- 222
E[ 10:00-000 -0010 + dCCGLD.1 + ACSGLD, + 207
Final upper bound W2(C()EMM) k+1): TD). Substituting 38
42
49
into
34
yields
W2(C(PPM ( TD)
S /2e- (VE X Vd)
+ V4/3(e 2nA m Bl16o 0 [2 + dCsGID, + XCSGLD.2)
+ 718d + 132(c 2n - 0- + dCssLD../33 ACCcuu
ACSGLD.2 + 10413))1/2
(50)
< 22 (T | Vd)
+ (V4/3 2 2/ (e 2 0- + dCSGLD.I + ACSCLD,2)
+ x18d+ 132
The bound for W2(C(E K+1 ). mD) in 50 can be made arbitrarily small by appropriately choosing parameters including T, X, and 7. More precisely, for any ō > O, we first choose T > Ta with T3 given explicitly in Table such that
12e- 2T (v
Xol Val < 5/4.
(11)
22

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Next, choose B 2 Bs and 0 < À S As with Bs and A5 given explicitly in Table and, in the case where À = X8, we choose n 2 ng with given explicitly in Table such that
1 0- dC + ACSGLD.2) + 2nAB m2
$ (V4/3+2//3) V (3E
+ +334/3+2v33 10 (o, 2 + m- + (011
(52)
+ + o
E 0- 21
- 6/12 + 6/12+ - 8/4.
Finally, we choose 0 < < 7 with given explicitly in Table 1 such that
r(1d + 132 13) < 8/4.
(53)
Using
51
52 and
53
50
we obtain W2(C( K+1) RD) < 8.
C.3 Proof of the Preliminary Results
We provide the proofs of the results of Appendix
C.1
Proof of Lemma
14
By 15 have for any n E No,
8 0" - -HH(O, AH(O,,,,.tt) + 22/B 5n+1 2
10a
1²1
O2 0-2² +2 +2(0h -0,---A' Xn+1) 22/3 Ent1)
-AH(O) X,+1) + V2X/3En+1 1
(54)
00 20 -2 (0,, - 0", H(,,, Xn+1) - H(0", H(0",X,+t)) ,,+1)))
- 2A -A) - 0", H xx,1)) + +2 22/3 (0) - 0", (n+11) + X2 [H Xn+1)l²
22/22/3 (0,, X,+1), En+1) + (22/8) 1
Taking conditional expectation on both sides in 54 yields
E |10 x+ - 0-12 1 = 10; 1²1
2AE (00, - ,1, (O,,, X,+1)
+ 2
[(()) X,+1) - H('0 xx+1))²² |
+ 2
|11(0", x00,x,n)10 02| + +2Ad/B.
Recall that 0 < A <
Using Proposition
13 and the stochastic
gradient 14 we have
E [10a x++1
0-/21 AN < (1 - 22F [o, -0 -
22d/3
2XE [o~m?²] 00 0 [o- 10 0-1²
SE [o?m² (0- + m,/ + m0 )²]
< (1 2XE [ /0) 0 22d/B
+ SXPE ? (,- | + x + +a, 1 + 10,
This implies that
E [100 8-13] < <(1 22E [o?m?]) n+1
[100 - 001 + dC + AC
23

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Proof of Lemma 16 I E [0, 7
Applying Ito's formula and using the process with given in 13 we have, for any
d|y,uupp= p + 2(Y, 2mr- + 2 v2dB.) + 2ddt.
Integrating both sides and taking expectation, we have
E [| 2 / || P] ds + f./ B[ 2mr [x + 2dt.
Then, differentiating both sides, we have
[
= 2E [
["] + 2B[ 2mm- 00 + 2d
dp
E ||a + 4 2d,
which, by rearranging the terms, yields
(e'B [ < 4e-21 2 + 2de".
Integrating both side and using Corollary 15 we obtain
E | | se "X + (4/3)c 2T (c 1) 2d(c' 1))
d2 + (4/3)e 27 (e - -
$ 2d + 2( (8 2n 1²1 dCSGLD, + ACSGLD, + 10 ())
Proof of Lemma
17
Using the process
10 with the approximating function given in 13 we have
Y BYEM 2
IEE + --(-YEM + 22m 22mT= 2> 2
(55)
+ 2 FM + +-(-Yt FM + 2mmT= v 27 Zu+1).
Taking conditional expectation on both sides in 55 using the independence of YFM and Z Young's inequality and 0 < 7 S 1/2, we obtain
y 2 E
Y [o]
MY 2(Y --YE + 2 0))
+ EM + 4- m 22,
(Y.EM 2mm 0) + 2nd
(1-2>) 2 10
2nd
+ x, - EM 2
< (1 -
2nd
> FM 2 RM x FM
(r
< (1 7) y 2M 47 + 22)002 2yd.
Thus,
2 S (1 (1- -+ + 10102 + 2d.
(56)
Using Corollary 15 in 56
yields
E |
EM |||
< -
dd 2d + +0(ee 2nxE(o3m²)
E |100 - P1 p] --
dCScLD, + AC + 10(1))
24

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Proof of Lemma 18 Using the process 11 with the approximating function given in 13 Lemma
17
Corol-
lary 15 and
E (0, 1/ 1/2,, we have, for any E [0,
P FM
EM [t/791
f dB,
E
PPM 18/7|7
22mr-1o/117)ds v2
11/777
Le/311
||
-SPEM 14/717 -2mr-1e/>17
21 2dy
2 [X//T17 [3] + 8 |003] +dd
6y (c-2e(e- (
2nA
[100 0-
dCSOLD, ACSGLD,2 10)
(e
2n 10 m E [10 0- + dCCSLDD ACSGLD + 10* 2d
CEMose
where CEMose Sd + 56(e 22n2Bl0 [00- -0-0% 1-13] + dCSGLD,1 + 2
+ 10*13).
Proof of Lemma 19
Applying Itô's formula to the process with the approximating function given in
we have, for any E [0,
y EM 2(Y.EM, - + - 0) + 2(Y.EM, v2dB.) + 2ddt [e/51/ 12IPeMMe 2(P.EM P.EM PEM
11/5) 2(P.EM 2
2(PPEM v2dB,) 2ddt.
Integrating both sides and taking expectation, we have
E
= -2
/
[
EM 2 P]]
ds + 2 E (DEM PEM -PEM
(9/739) ds
jn
S/"
(PAM 2m 19//1/) ds + E | [2]
+ 2 TE
+ 2dl.
Then, differentiating both sides and using Young's inequality, yield
2
2E
Y.EM +
2E
(ReEM, FEM PEM
dt
+ 2E
(Y.EM
2m
0)/
+ 2d
[2]
[1,EE
PEM [t/7)>/ [2] 8 + 2d.
E
2
Using Lemma
18
we have
d df
E (2)) c <2 + 8E[ 2d).
Integrating both sides, using 7 E (0, 1/2] and Corollary 15 yields
E [2 < se- (1 e (2) EMase + SE[[017 2d)
< 18d 128(e 2n 23 m2 1100 2] + ddscan,, + AC + 10-12))
25

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D Proofs of the Results in the General Case
In this section, we provide the proof of Theorem 10 We start by introducing the results which will be used in the proof of Theorem 10
D.1 Preliminary Estimates for the General Case
Throughout this section, we lix e E (0, 1). The following auxiliary results will be used in the proof of Theorem and their proofs are postponed to Appendix D.3
We provide an upper bound for the moments of (Y,EM) )re(o.T] defined in
Lemma 20. Let Assumptions and 3.6 hold. For any p E [2, 4] and E [0, -e,,
sup E PEM S (lM,
where
CEM,p(t) = 13p--11- 22200 KK (14701))
x [pa + 230 KK (1 + (((0P))( + ToP) + 2 (pM + +p(p -1(p-2))81)
The following result provides an estimate for the one step error associated with (YEE)) )eefo,21 defined in 11
Lemma 21. Let Assumptions and 3.8 hold. For any p E [2, 4] and E (0,T-cc)
E |
RM F.M
[1/7771 S x C.Mosssspp
where
CEMose.p 2- (CEM,, KP Torrall T2 (EE()))>(CEM,pTT) 3 B((ŌP()))
+ (Mp(p-11) 1))
The following result is a modification of Kumar & Sabanis
2019
Lemma 4.1).
Lemma 22. Let Assumption 3.b hold and let b [0,2] Rd R - R such that
b(1, 0, x) = x (= 0, x)).
(57)
Then, for any I, E R LE [0, 0 E Rd, E [ 1|, and k = I,.
M (b(k) ab()) (x' - -x') s k((1++) - =
(t.0,)) -b00
Oy'
Lemma 23. Let Assumption 3.8 hold and let be as in 57 Then, one obtains that, for any (t, (t,0,)) e [0, x Rd RM and k = M.
(0) (1,0,z) S + 2K 2Ka + 2
(58)
D.2 Proof of the Main Result for the General Case
Proof of Theorem
10 We proceed as in the proof of Theorem using the splitting
W2(C(Y EM), TD) MD)) I Wz(mD, ((++))) + W2(C(Y,g)) (()x)) + Wy(C(Y)x) C( Ws(C(YY""), C(Y
(59)
where the first term on the right-hand side in corresponds the error made by the carly stopping and the remaining terms have the same interpretation of the corresponding ones in 34
26

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Upper bound WT(TD, ((xx)) For any E [0,T], note that
m S 11
of = e 21.
(60)
Recall that tx= = T - c. Using the representation of the OU process 2 and the inequalities 60 we have
((Y(x)
W2(TDD ((XT-x)
< v E [|Xo XT [2]
- v2 (1 mu 1x Xol + or-ok VM
(61)
I VOr (v E Xo 2| + VM)
S 2 Xol + VM).
Upper bound on W2(C(Yix), (((xx)))) )). Using Itò's formula, we have, for any t E [0, c).
d - 2/ = 2(Y - Y2, Yt + 2V log - Yz - 2V log log dt
(62)
Y 7.12 de + 4(v Y,, Vog Vlogpr- dt.
By integrating, taking expectations on both sides in 62 and using Remark 4 with the lower bound EMO in the estimate 19
we have
/ dk
B[[Vix
VEX
12] $ B 9017+ +
2(1- 2mm0) E Y - 8:12 dt
(63)
$ B 20 222so0
Using 63
the representation with Zr d y and 60 we have
El[Yxx 11xl < Bljy ee 21 2Txx0)
Bl|mmr Xo + (or - 1)88 e 2111-
(64)
s 2 [E| 2 + M) 2 2
Using 64
we have
W2(C((,,,) 2(x,)) 5 / Bl|Y(x Yex [2]
(65)
s 22( X]]]+]++ VM)e 2Txo(T--)--
Upper bound on (2(C(()x), Using Itô's formula, we have, for t EE 10, c),
di, yaux 2 =2(Y; Yaux Y, log (Y:) yauxx s( - 1, yaaux) di
=2| y.uux2 2 dt + 4 ( Y. V log logpr-:(Yt) V log Pr ( (Y,.ux)) dt
(66)
+ (Yr yauxx V log Pr (Ysux) s(T 1, ylax)) dt.
27

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By integrating and taking the expectation on both sides in 66 and using the Remark with the lower bound Tmo in the estimate 19 Young's inequality with Ç E (0, 1) and Assumption 4
with we have
T- Y [2] ds
B| T--cl
2
B
T-r
+ 4
BRYY - Y logpT-s - Pr [X] ds
+4
EIYY - Y V Po, - s(T - -, y ds
T--
2(1 + 5 5-25 -2ī - Yaux ds
(67)
T-
+2
E| log Pr- (Y.UX) s(T - s,0,Y.uxx )(² ds
2(1 + -tīmo) Eljř, - yaux [) ds + FESN T-
s 2e ( -20
Using 67 and tx- = - 6, we have
- / Biiix Yaax
(68)
I -2toool VESN.
Upper bound on W2(C(YY ((YMM). The following bound is derived by modifying Kumar & Sabanis 2019 Lemma 4.7). For the sake of presentation, let b: [] x Rd RM + RM such that
-() = x 2s
(69)
Consequently, (YEM))Ej))))
keju, can be expressed using 69 as
dحep - Lt/7l,,
PEM ( dt v2 dB,
γEM Tx =N IM).
(70)
Using Itó's formula, have, for EE (0, -c], c),
d|yaux 2EMR2
2(Yaux -PEM yaux + 2 s(T - 1,0, Yaux)
PPM [1/7]7 2 s(T 11/717,0, PPM (1/117)) di
-2/yraux P.EM2 dt + 4(yaaux - P.M s( 1, V.uux s(T - t, D.D.M dt
(71)
+ 4(Y.aux 8PEM sT - -,, PEM >(T 11/717, 0, PEM {) di
+ 2(Y,aux P.E b(T t,0, Y.EM) b(T 0, YEM [1/))))))) dt.
28

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Integrating and taking the expectation on both sides in using Cauchy-Schwarz Schwarz inequality, Young's inequality with Ç E (0, 1), Assumption 3.b and Remark 3
yield
E y
12
yaaux PEMP]] ss + 4 1 E (Yauxx -.EM, -8,0, Yaux) -s(T 8, 0, (.0MMMM)) }) ds
E 2
0
E
(Yaux PEm s(T s, 0, 8.m (1/711) s(T 15////7, 0, 9TEM (1/51/111 ds
E (Y P.a b(T 0, P.M b(T s, 0, P ((07)11))) ds
+
2(1 + 5 21 + 2/T - )| |xaaux PEM [2] ds
+ 2 / E (1sTTT - s, (pM))))) - s(T - Ls////1,1 0, PEM
(t/>]> )|] ds
(72)
E (Yaux P.M b(T 0, PEM) - b(T - s, 9 ds
+ 2
2(1 ++ (1 + 2T E y pEMPp] ds
+ -22- E (1 (1+2|88) 2 S( Is- 1s-1s) |s/>1>/ ds
KG
+ 2 E (Yaux PEM b(T 0, PEMD b(T - s, 0, PBM (111//1111] ds
<
2(1 + + +1 + 2) E. | prMMp]
ds + 45 -K((1 + 8(EAL + 1001))))200
+ 2 /' E (Yaaxx PEM b(T D.EM) - b(T - s, ((MM))))) ds.
We proceed estimating the third term on the right-hhad side of 72 Ç (0, 1), Lemma 22 and Lemma 21 yields, for any & [0, T -c]
Using 70
Young's incquality with
E (Yaux PEM b(T -,,0, D.E. 6(T -, 0,0,Y MEM 1))> ds
s,
M
/'' (aa)x.(8) ).MM()) (1 (T s, D.EM 6(4) (T 5, 8EM (377)T)
E
M 0ilh) (T 0, PEM ( (PEM.)) - (OMM)) ds
Oxl
1" (yaux)())) (PEM.()))
(b( (T O. PEM P (PEM.G) - (EMM
E
Jg
Ox
(4/717
ds
]2MMa] ds
2
3
M
+
E
(6()) (T - s. 0, PEM) - (()) s, 0, ()MM)))
2ç
k=
29

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M 064 (T' 5, S.M. (e/ss1) (PEM..)) - s( 1 ds
Y.M
dx)
M
(a) (..M.(())
M (T s, 0, & EM
x
10/717
"
(T - 1r/-111 0, PEM ((//19)) dr
ds
Omd
10/717
M
M Obt 8, 0, V.eeM
[ Le/rin
(a)
V2 dB
ds
dx
K2
Jn
E
Y Faux pEMP?
ds +
2Ç
(1+ 2 -19)
1IEE PEM
| ds
(x)xx.(k) (.M.
(73)
M (T( 0, EE
10/7
- (0/(T-Lr7)), ppMM) (r/>1>) dr
ds
Oxt
14/7/7
M
M Oiik) (T 8, 0, PEM
yaaux (EM,(.)
(e/7)>)
V2 dB
ds
dx
R=
K2
E
Y yaux PEMP [2]
ds 72
t(1
27 CEMose,
5
(T 8, O. PEM
(0/7)7)
-OO(TLL/71pp LT/717,0, (1/5]9) dr ds
Ox
abik (T' 0, PEM
(a)) - (EM.(.)
(e/711)
V2 aB
ds
Oxl
1e/7ln
We proceed estimating the third and fourth term on the right-hand side of 73 separately. The third term estimated using Young's inequality with Ç E (0, 1), Lemma Lemma 20 and for any E [0, - c), we obtain that
Remark Remark and
M
10 (k)x,(k) (.E,
1=1
M (T(()((T -, 0. PEM 18/717)
1"
(00 - 1r/1dn 0, PRM
P.P (PM))))dr do
x
Oxd
18/7/7
< E yaux
[2EM2]] ds
2 3
27
1 fr
M
+
M(1 + 81 (1 + 27))0) E
Σ
(T
1r/11% 0, ( ds
14/217
30

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Published in Transactions on Machine Learning Research (02/2025)
< f B | Srraux_Ptt0 P.MP] ds
2 0
47
M 8K3(1 2
(1+ 16K7 nota (1+ + (1/7)/ [r/7]>] 121 32K fota + (1 (1 E| [1])
dr ds
5
E |Yaux - p2 ds
2 2
7²
4M(1+ 8K3(1 + 472a))
x1
(1 + 16K + +2aa)
sup usast
B[[Y,MM]] 32K Pota + (1 + A²AL + 2100
C
E ||aaux [.aaa ds
2 0
+ 1 4M(1 + (1 + 4)
5
(1 + 16K2 Poeal (1 + 20 22 + 32Keotal (1+ (12220)111 + AL +2(07
x
(74)
We proceed with the estimate the fourth term on the right-hand side of 73 For k I..., M, we note
(k)
(K)
(aux.(K)
-P/
(RM,(R))
(b(K) - r, y.ux) ((() r, 0, P.EM dr
19/7
L (b(R) rr PEM) ((() - r, 0, V.M {)) dr
(75)
+
19/7l7
f (()(
(1) (T iM [1/>]>] - 1r/w// 0, ()))))) (F dr.
19/7/7
By using 75
Cauchy-Schwarz Schwarz inequality, Young's incquality, Assumption 3.b and Lemma 23 have
M
(a)) - (EM.(K)
M (b())(T s,
(1/717)
fr
V2 1200
ds
Ox.
18/177
1=1
M
ab(k) (T - 0,0, PEM
IE
V2 d
ds
Ozt
19/717
M
1'- (
(6RA) (T - C
Y.uux) 6(0) -, r, 0, p.PM)) dr
1=1
M
(T 8, 0. VEM (
[u V2 dB!!)
ds
Ozi
31

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Published in Transactions on Machine Learning Research (02/2025)
M
!.
(Sem (T - -r, -r,0, P.MM - (T) r,
E
9PM (1/517)) dr
1//717
M
Ob(()(T 8, 0, VEM
19/7//
V2 dB()) ds
x
Oz
10/717
M
ΣΣ
18
2(s( (T' 0. (/pp1 (1/717)
(T
1r//17 0, / (1//1=)) dr
M
(b(k)(T s, 0,
8EM
10/117
v2 diB()) ds
Oxd
10/717
E
11//11(14(T - r, y.ux)) b(0) (T r, 0, {rEM))dr
M Oh(k) (T 8, 0, VEM
x x/1/2
V2 dB()) ds
Oxi
M
1B (T - r, 0 p.BM) (T r. 0, F.M
1r/71,)/dr
L8/751
k=1
M Ob(h (T - s, 0, (3/21)
V2 cB(()) ds
Oal
M
E
(1) - r, (
- -T
PEM
O, PEM (
18/717
k=1
M ab(h) (T s, ô,
PEM
[
V2 dB(()) ds
dzi
E
[B<K) (T 0. y.uux) - 12b) r, ).P.MM) Pdr ds
2
10/717
1=1
2
M
M Ob(h) (T s, 0, γPE
[3/ v2 cB
+
I
ds
Or.
j=1
27 1/2
M
E
(6(0(T- - D.P.MM) - ((0) (T) -r, r, (PMM))))
18/7/7
k=1-
27 1/2
M 0b(k) (T s, 0, FM
(5/717)
X
/2 aBa
ds
Oxt
27 1/2
V 2)s(1) (T r. 0, PEM (1/11))) ((( - 11//l1, 8-EM [175]>]] dr
E
19/7/7
1/2
M ( (T 0, V.M (
X
E
V22 dBb)
ds
1a/71n
Ozi
3=1
32

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Published in Transactions on Machine Learning Research (02/2025)
< E
(1 8K3(1 + 1)
P.EMpad Pdr ds
15/7|7
M
0 (T s, 0, 11/7)17
+ 27 2
ds
drd
+ 1/211 + 8K + 2/
1/2
M
1/2
M 06 (T 0,
TEM
Σ k=
BIDEEM
PEM
Lr/717 PI dr
18/7/7
dr
ds
12/717
Or.
2
1/2
M E
M Obckk (T s, 0, EM
+ + (K + SEAL + 11007331122
E
[
dr
ds
1e/7ln
Oxs
(76)
$ (11+ S1 + =4T20)) /
sup
y
P.M [2]
ds + 422 + 8K3(1 + 47200)
0515$
1/2
M
M ab(k) (T - s, G.GY F.
E/'
+ + + 8K3 + 2) 1/2
1/2
E
dr
ds
EMose,
Oxt
1/2
M
M ab (T s, 0, PEM {
+ + 1lle0eiie (1 + AL + 110
Σ
E
dr
ds
18/717
Oxs
< (1 (1+882(1 + 4T22m) // sus E y.aux
p.EMp? [2]
ds + 442ttM + 8K3(1 + 472m))
OSTS$
+ + 8K
11 + 47 20 1/2
3/22 4K, (1 - SEAL + 181071)11121
EMose,
x |tMv2 s1 + 27 20 1/2
Using
and
in
we have
E
y
aux
D. SPEMp]]
4(1 5 Ks(1 + 27 (K+ 4T(a))) sup E y - (PEM)]] ds
U5r5s
+ + +8-216-1 M(1 + (1 )
x 161 Total (1 + 22a) C) + 32K? 32
(1 + +220)
+ AE + 2)0
+ +222 + 47 20 ( JEEMsse + 2tM + (1K(11+477) +
-214455 8(EAL + +10-!3)
+ +4 SK3 + 2/2 EMose,2 + 7 3/3//2222 (1 SEAL + 88/00
x [ 8K3 + (TT22))
Thus,
yaux - [
sup E.
o5+5(
4(1 Ks + 2 4Ks 2T2))))
sup 05r58
y {EMM2) ds
33

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Published in Transactions on Machine Learning Research (02/2025)
+ 22042
+ (15 44 4 CEMose, + +M + ( +4T2a)) + 2 8(EAL + +10-
+ (1 + 47
x [(1 + 16K [1++66] (1 + 2 CEM,2 (T) + 32K Total (1 + T + AL + +1107²1]
+ +[[ + SK + 22 2 + 2K (1 + SE + 1/021)
x [M + 8K3 + 1//
(77)
$ 2e + Mey
(1+ 47 2c C 1Mose,1 + 41 + 8K 4T + 25-- S(EAL + +10
+ +4<- + SK + 47
Total x [[1+ 16K (1 + T 202 () + 32K 3oeal + (2 + AL 210
+ 2[1 + 813 + 2)
1/2 /2 EMocs,, + 2K (1 + 8t + 1/07)))1/21
[ 1K 47 272
Using 77
and tK = - E, we have
W2 ),
E | -PEM
DEMB
S v 22 + (14 (1+472 (1 VT
(1+ 1 + 4M - 8K3(1 + ))) + 25 2 + 8(EAL + +100)1)
(78)
1:40-1((1 + 8K3(1 + 272a))
x [(1 + 16K (1 + 22 CEM, (T) + 32K Total (1 + T2aa + A 2(0
+ 2(( + 8K3(1 47 1) 1/2 1/2
EMase,2 + 2K, (1 + A5AL + 111213)
1/2
1722 44420)))))
Final upper bound on W2(C(YEM) ), mD).
Substituting 61 65
and
78
into
59
we have
Ws(C(VEM ) TD)
$ [VEi[Xol VM)2VC
+ V2(VE VM) 2T
+ 1 VESN
34

---

Published in Transactions on Machine Learning Research (02/2025)
x Kis- (1 (1+ (0T22))EMmas + (K) +
+ -10013)
+ +45- (1 + 8K3(1 + 47
x (1 + 16K (1 + T22) C) (T) + 32K + TTa0) - Aē + 210
(79)
+ +[ + SK + 2T 1/2 011/2
2 EMase, + 2K2 (1 + SE + 810713811/21
x | 8K3(1 (1 4 2TT
The bound for W2(C(5MM)) (ML TD) in 79 can be made arbitrarily small by appropriately choosing parameters
including e, T, ESN and 7. More precisely, for any 8 0, we first choose 0 S é < 6s with es given in Table
such that the first term on the right-hand side
79
x + V
(80)
Next, choose T> T3 with T3 given in Table 4 such that the second term the
side of 79 is
2 E X
v MMe -[mo((-t))- 8/4.
(81)
Next, we turn to the third term on the right-hand side of 79 We choose 0 < ESN < ESN,5 with EsN given in Table 4 such that
(0-Tc(tt-22xo)01 VESN 5/4.
(82)
Finally, we choose <<> < 75 with 76 given in Table such that the fourth term the right-hand side of is
222
V
Ki5-1(1 (1+ 20x CeeRoOCCMeeee/// + 4M SK3(1 + 4T + (1 + 10^13))
x
+ 1 8K3(1 + 4T2a))
x [[1+ 16K2 + 2 (22 (T) + 32K T2a)(1 (1 (1 + AL + 2107
+ 2( 81 27 )) 1/2 1122
EMose.2 + 2K, (1 + SEAL + 12/1/2)
(83)
1/2
x | 1 + 1//2)
< 6/4
Using
81
82 and
we obtain W2(C( TD) 8.
D.3 Proof of the Preliminary Results for the General Case
We provide the proofs of Section 3.2 and Appendix D.1
Proof of Remark 5 Using Assumption
3.b
have
|s(t, x))
|s(t, 0, s(0 0, (,0,00,
(1 + 10) K2 141))01 + Ks + +19)] |
< Krotal (1 + (1 + [0] +|]),
where KTotal : K1 Kg + K3 (s(0, 0, 0)
35

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Published in Transactions on Machine Learning Research (02/2025)
Proof of Lemma 20
Using Itô's formula, we have, for any t E [0, c) and p E [2,4]
d d|PEMp
p MpEa, ( 1 ) dt + p p 2p... 2 ( - |t/>|7 1r, 0, & [t/>1>> )) dt
(84)
+ p p
poeeappas w. v2 dB, 2
P
1 dt + p(p - 2)
2 8 p.MPa 2 pdr.
Integrating and taking expectation on both sides in
84 using Young's inequality and Remark
we have
E y p.EMp]
1 p
+ +P [." E (19-EMp-epe
BPEM ((MM) ds
= E
+ +pp / E (1 s(( 13/717, 0, S (1/>11)1 ds
+ +pMM +p- - 2) L'" E -aaaaaaa ds
pPEMp)) pp] + 3(p 1) 1/- E [ ds + L' E
S
DEM (1771111
PP]
ds
+ //" E |T - 18/>17 0, (( ds
2
(pM + p(p - 2)) p-2 2 ! E PEMp
+ p
ds
< E [ [pEMp))
+ 3p 2 - D / E
|
ds + //' E
BM [ ds
/ (1 IT' |s/>|>| /
+ Potal
[ p|
ds
+
Kitnaal (1+ k
(1110PP))
+ T'- |>/>>>/ ds + (pM + p(p - 2)) 2 &t
< E PPPMpp pp]
2
+ 2 Khotaal (1 + TTP)) /"' sup E [(pppp]] ds
+ (3p
p
ogrss
+ 2 2pM + p(p - 2t
+ 23 KP (1+ (1] + Toptt
Using Grönwall's inequality, we have
sup E 05e5!
p.EEM p
E [p pp] + (3p 1
2
+ 222- 'KP KPotall (1 (1+T TTOP)) l" sup E
[ ds
p
o5r5.
+ 2 2 2K Total (1 + E1 + + 2PM +p(p - 2))
2
$ "333- 1-30-1-3+20 ---222--KK (1+
(E
|EMP] + 230 2 2K
(10++)
+ (pM +p(p- ((pp-2))$(). p
2
2
Proof of Lemma
21 Using 11
Lemma 20 and Remark 5 we have, for any t E [0, T - c] and p E [2, 4)
y EM
EE [t/Tj>
PBM ||] +E [ v2 dB,
< E
[
[e/>1> +2 s(T -1/T-10
PPM
PEM
36

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Published in Transactions on Machine Learning Research (02/2025)
S 2 70 E
EYM
EM
+2 E -IsTT[L Lt/nl,, 0, 0, ((/)))))))) + + 1))
[t/T]>]
2" (CCM
(t) + 33pp-2KPond (1 (t) 2 Kootaal (1 + 1 (T))
7 (Mplp- 1))
<9 CEMose.p
where
24 3(1 B((OP))))))))))))0)))) + (Mp(-11)3 CEMose.y = (CEM, (T) + Toca (1 (EM, 2
Proof of Lemma 22 By the mcan value theorem, for any 1: = ......M M, we have,
(1,0 06(h) (t, 0, qx + (1 q)
(x' - =z()),
Ozy
for some (0, Hence, for a fixed E (0, 1), we have
(1,00, (1,0 Σ
(xt x')
Oz
M (b1)) (t, qx + (1 o)
M
Ol((K (1, 0, .))
(x' - z(())
(x x²)
Oy
Oyt
M
(b) (1, 0, qu + (1 9)
(b(h) (1, 0,
1,e - x²
Oy
Oy
The proof is completed using Assumption 3.b
Proof of Lemma
23
At E R M and for any E RM, we have, for any k = 1.... M,
V (k. (1,0,z),, lim (t, r + vh) s(k) (t, (t,
Using Assumption 3.b
wC have
5 ( 0, + +h) ((,) 01,
(V (t,
lim
h-40
Da t) 10 + vh x1
(85)
lim
|h| Ka(1 + ///
h-+0
Taking (t,00) (L.
in 85 we have
V (A (t,0,2)) (3(1 + 2
(86)
Using 57 and 86
obtain
(1b( (t,8,z) 11+ 2/2 ( (1)
1 + 2Kg 2111%).
37

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Published in Transactions on Machine Learning Research (02/2025)
E Table of Constants
Table 4 displays full expressions for constants which appear in Thcorem 10 and Remark 12
Table 4: Explicit expressions for the constants in Theorem 10 and Remark 12
CONSTANT
DEPENDENCY
FULL EXPRESSION
C
O(VM)
20
B||xo/] vM)
C2
O(vM)
V2
VElIXx/11 VM
Ca(T,
0(6446-0/M0X0-01
V25 1of+(--/xx))T-0
7(4 Toin (1+720))
O(Me 11 (patt
(T),
(E[[Y) B] + 10K2 7(1 + 2FAL + 2/0 210 72 2MT)
29 1129 (1 (1177))
7 p (1at1)
(T))
(E([Yo PEM 1 1024 T 7T
8 4M 4)T)
CaMoxe2
O(Mc 1 (ALt
2(CEM,a(T) Kotaa (16CeM +2
32 +70 AEAL
[]) 2M
CEMam,4
O(M rdots (o0tl)
8(CEM,((T) + Ki + 7 ( (1024CcM. 8192(1 I1|0a ID) 144M
V2 1
k1+ CEMon 4M 11
+25 'Kf(1 + 8(EAI. 10"
O(Me pdutl 1oMattty
+ M( + 8K3 + 27
c)(T,)
[(1 + 16K T C)E..2(()
32 i + )1 )(1 2 + 2/0'
+ + SK 47 1/2 2/1/2
+ 2K, SEAL + 1/0"13)111
1/2
[MV 27
Olv Me p2at1 120 120-121/23
x (6 C +88 1/1 12 ")
+ +25-1/2K + )T 1) 1//00//22 Opitonn
///( ||| VM)3
Ts
(2T
In 1v2
VElIXo 7 VM 16
(ç82) /32)e -2146-22000000
88..8
(6/(AV)))/<(T--)) 1/(2a), (/ 14K5t 47
Kis (11 22 1M(( 883 17
+ 'kā( + a(EAL + 10719))
+ 44 -M M(1 + skg(s 27
73
[(1 + 16k (1 + ( ()eM
32 (1 + 22a) (1 + 25 + 2/0" [3)]
+ 2[1 + sKŽ(1 472 c 2K: (1 + REAL + 8 1/3
11(2a)
x [MV2(1 skŽ(l 1)
38


Reviews:
[{"review_id": "Ywe6soKsbk", "reviewer": "Reviewer_3tvY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength\n\n-optimal dependence in the dimension of theoretical guarantees\n-W_2 guarantees\n\nWeaknesses\n-Strong hypothesis hard to verify in practice", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The article under consideration provides with theoretical guarantees of convergence for score-based generative models in W_2 distance assuming, among other things, that the data distribution is log-concave. This problem has been recently  deeply investigated by several other works. The main interest of this work is that guarantess of convergence are in W_2. The article suffers of several limitations as well. However, I believe it is a contribution that would interest the readers of TMLR and I cautiously recommend publication provided the authors address the comments below in full detail.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Major comments\n1)I must say it seems hard to me to fulfil both assumptions 3 and 4 at the same time. How can one be sure that the L^2 approximation error can be small if the class of approximating functions s(.,.,.) satisfies the regularity properties imposed by (3)?  To convince the reader of this, one should show that the score function of a log-concave distribution meets the requirements imposed by Assumption 3, with the exception of the regularity on \\theta, which I am ok with assuming. One sided Lipschitzianity of the score is granted by Log-concavity, but what about the Lipschitz upper bound? What about time regularity? And the Lipschitzianity assumption on the derivative of the score?\n Thus, I don't feel comfortable saying that Theorem 10 does not applies as such to log-concave distributions and that the title of the article is fully justified. It may be possible to verify these requirements, at least for some distributions other than the Gaussian, but it is not done here. \n2)The table in section 4.2 comparing the findings of this article with other relevant contributions in the field is incomplete and not clear. First of all, it does not distinguish between result obtained that compare directly the law of the generative model with the data distribution and articles that make use of an early stopping rule, that is to say articles that compare the law of the algorithm with a smoothed version of the data. This distinction is  imporant. In fact, having a KL against smoothed data does NOT provide with a KL or W_2 bound against the data unless further (strong) assumptions are made. Moreover, even ignoring all contibutions that came after this paper first appeared, there seem to be some gaps. For example, the referenced paper by Conforti et al obtains results that are better than those of Chen et al without having to resort to early stopping. Moreover, it also contain a result that is equivalent to Benton et al. in the early stopping regime. \n3) I don't understand Remark 7. I agree the assumption of the auxiliary process is new but its density is also unknown. It can be simulated easily, this I agree. However, given samples from data, even the time reversal process can be simulated by simulating the forward and changing time. So I see a clear advantage in using the auxiliary process if one can bypass the use of samples from the data. Samples from data are indeed required to train the neural network that approximates the score, so what is the real gain?\n4) Assumption 1 seems to assume implicitly that there is a unique  optimal parameter \\theta^*. It does not seem very realistic. Can the hypothesis be formulated differently to avoid this problem? Also, what is the significance of this hypothesis? Why is it needed? Since other papers in the field don't need it, the authors should clarify its role.\nMinor comments\n1) It seems to me that is not the process that has the representation (2) but rahter its marginal distributions.\n2) Why not integrating directly the linear part in (11) in the Euler scheme?\n3) I don’t understand the sentence at the beginning of sectionI 3.2 . I agree that the score may not be computed at t=T. However, by using the score at time T-\\varepsilon one can easily define Y^{EM} at time T.\n4) Pg 4: the process (9) instead of the process 9\n5) Eq:10 it should be T-t_k instead of T-k\n6) The notation for the auxiliary process is not the same in (9) to (20)\n7) I would put the Gaussian result as example of application of theorem 10, emphasising that better constants can be obtained in this case, rather than a main result.", "broader_impact_concerns": "No broader impact concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "oFRWNiWdMw", "reviewer": "Reviewer_Jfhr", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n* Improved upper-bounds on the Wasserstein distance between generative and target distributions.\n\nWeaknesses:\n\n* I feel that the justification on the impact of the novel auxiliary process is unclear. The authors mainly claim that the difference is that \"the approximating function s and the estimator $\\hat{\\theta}$ are known\". I do believe this falls short in explaining the reason why this process is useful. What is the implication of considering the auxiliary process on the theoretical derivation? How does it lead to improved convergence bounds?\n\n* Also, the link between Assumption 1 and Assumption 4 should be explained with more depth. Since Assumption 4 says that the estimator of the score is L2-accurate, why does one need Assumption 1? Is it because Assumption 4 is only defined on the auxiliary process defined in Equation 9?  \n\n* The writing could be made more direct and more concise.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Disclaimer: I am not an expert in the convergence theory of diffusion models. Thus, I do not have a good knowledge of the literature on the topic and could not verify the proofs carefully.\n\n* State-of-the-art convergence bounds for diffusion based models in terms of Wasserstein distance between target distribution and generative distribution (Theorem 10). This is achieved thanks to 2 contributions.  \n\n* The main technical contribution concerns the use of an auxiliary process (Equation 9) in the study of convergence bounds. This auxiliary process is simply defined as the backward process, where the true score is replaced by its approximation learned by stochastic optimization.  \n\n* The authors provide a toy example where they apply their analytical framework to bound the Wasserstein distance between a target multivariate Gaussian with unknown mean, and a distribution generated by a diffusion model. In this case, they consider the score to be learned by SGLD, and use known results of SGLD convergence.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See weaknesses: more discussion about the impact of the novel auxiliary process, and about the links between Assumption 1 and Assumption 4.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "30yEYYJzQB", "reviewer": "Reviewer_Kv1D", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper provides comprehensive study about the properties of diffusion model for log-concave distributions.\n\nHowever, I am very suspecious of the gain from diffusion model for log-concave case. All the constant numbers are hidden in the estimation of the score function. For previous diffusion proofs (such as Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions), they claim the complexity of sampling is mainly about the learning process of the score and consider the general case.  To the best of my knowledge, diffusion models works well in highly multi-modal distributions with ill log-Sobolev constant. There are not even sufficient empirical evidence showing the better performance for log-concave case. More evidence and better interpretation for the assumptions are needed.\n\nThe discussion with other related work is very limited. For example, the proximal sampler based methods [1-3] and recent diffusion-based sampling works [4-6]. Some of them explicitly consider the complexity within the estimation. Some claims to be the \"first\" might be reviewed.\n\n\n[1] Structured logconcave sampling with a restricted gaussian oracle.\n\n[2]  Faster high-accuracy log-concave sampling via algorithmic warm starts.\n\n[3] Improved analysis for a proximal algorithm for sampling. \n\n[4] Reverse Diffusion Monte Carlo\n\n[5] Linear convergence bounds for diffusion models via stochastic localization\n\n[6] Faster Sampling without Isoperimetry via Diffusion-based Monte Carlo", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors provide detailed convergence estimates for diffusion models under strongly log-concave data distributions, improving understanding of their theoretical stability and reliability.\n\nUsing Lipschitz continuous functions for score estimation, the paper claims to achieve fast convergence rates by controlling the Wasserstein-2 distance between generated samples and the target distribution.\n\nBy introducing an $L^2$-accurate score estimation assumption and an auxiliary process, the paper supports a range of stochastic optimizers, broadening the method's applicability in practice​.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Futher justify the log-concave case. For example, how can you provide a implementable algorithm to be faster than Langevin based ones.\n\n2. Estimation of score function. How to estimate the complexity there?\n\n3. More discussion with the related works.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The main reason for the rejection of the paper was that its references were not up-to-date. I agree with this assessment and request that the authors expand and update their literature review accordingly.

The authors state that "A careful examination of [...] is still needed." I request that they provide additional details and clarification regarding this claim. Moreover, Conforti et al. also present results on early stopping, which should be acknowledged and discussed in the paper.
Decision: Accept (poster)