Dimension-free error estimate for diffusion model and optimal scheduling
Authors: Valentin De Bortoli
Romuald Elie
Anna Kazeykina
Zhenjie Ren
Jiacheng Zhang
Venue: Accepted by TMLR ()

Abstract: Diffusion generative models have emerged as powerful tools for producing synthetic data from an empirically observed distribution. A common approach involves simulating the time-reversal of an Ornstein–Uhlenbeck (OU) process initialized at the true data distribution. Since the score function associated with the OU process is typically unknown, it is approximated using a trained neural network. This approximation, along with finite time simulation, time discretization and statistical approximation, introduce several sources of error whose impact on the generated samples must be carefully understood.
Previous analyses have quantified the error between the generated and the true data distributions in terms of Wasserstein distance or Kullback–Leibler (KL) divergence. However, both metrics present limitations: KL divergence requires absolute continuity between distributions, while Wasserstein distance, though more general, leads to error bounds that scale poorly with dimension, rendering them impractical in high-dimensional settings.
In this work, we derive an explicit, dimension-free bound on the discrepancy between the generated and the true data distributions. The bound is expressed in terms of a smooth test functional with bounded first and second derivatives. The key novelty lies in the use of this weaker, functional metric to obtain dimension-independent guarantees, at the cost of higher regularity on the test functions. As an application, we formulate and solve a variational problem to minimize the time-discretization error, leading to the derivation of an optimal time-scheduling strategy for the reverse-time diffusion. Interestingly, this scheduler has appeared previously in the literature in a different context; our analysis provides a new justification for its optimality, now grounded in minimizing the discretization bias in generative sampling.

Published in Transactions on Machine Learning Research (04/2026)
Dimension-free error estimate for diffusion model and opti- mal scheduling
Valentin de Bortoli Google Died
udebortoli@ggogle.com
Romuald Elic Google Deep
relie:kgoogle.com com
Anna Kazeykina Université Paris Saclay
Zhenjie Ren Université Evry- Paris Saclay
shenjic.ren/uunio-erry.
Jiacheng Zhang
jicchenghangleeuk.ccoo edu.
Chinese University of Hong Kong
Reviewed on OpenReview: https: opw.revveeereeeeee.e net/ forum? id= tdruto000
Abstract
Diffusion generative models have emerged powerful tools for producing synthetic data from an empirically observed distribution. A common approach involves simulating the time-reversal of an Ornstein Uhlenbeck (OU) process initialized at the true data distribu- tion. Since the score function associated with the OU process is typically unknown, it is approximated using a trained neural network. This approximation, along with finite time simulation, time discretization and statistical approximation, introduce several sources of error whose impact on the generated samples must be carefully understood. Previous anal- yses have quantified the error between the generated and the true data distributions in terms of Wasserstein distance Kullback-Leibler (KT. divergence. However, both metrics present limitations: KL divergence requires absolute continuity between distributions, while Wasserstein distance, (hough more general, leads to error bounds that scale poorly with di- mension, rendering them impractical high- high-dimensional settings. In this work, we derive an explicit, dimension-free bound on the discrepancy between the generated and the true data distributions. The bound is expressed in terms of smooth test functional with bounded first and second derivatives. The key novelty lies in the of this weaker, functional metric to obtain dimenssonn-nndeeendde functions. As an application, we formulate and solve a variational problem to minimize the time-discretization error, leading to the derivation ion of an optimal time-scheduling strategy for the reverse-time diffusion. Interostingly, this scheduler has appeared previously in the literature in a different context; our analysis provides a new justification for its optimality, now grounded in minimizing the discretization bias in generative sampling.
guarantees, at the cost of higher regularity the test
1 Introduction
Generative models (GMs) have attracted substantial interest in machine learning over the past decade. Their core objective learn the underlying probability distribution of observed data, enabling the generation of new samples that resemble the training set. Notable classes of GMs include Variational Autoencoders (VAFs), Generative Adversarial Networks (GANs), and diffusion-based models (DMs), each employing dis-

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tinct optimisation objectives to achieve faithful data synthesis and generalisation Goodfellow et al. (2014)
Kingma & Welling (2013): Ho et al. (2020): Rezende et al. 2014
Score-based generative diffusion models (SGMs) have recently gained particular attention, achieving state- of-the-art results in generative modelling Song & Ermon (201 Song et al. (2021). Roughly speaking, the functioning of SGMs can be described as follows. Tet mo denote the true data distribution that the model aims to learn. One first simulates an Ornstein Uhlenbeck (OU) process starting from ma; for sufficiently large times, its marginal distribution m, approaches the invariant mcasure of the OU process, denoted m", which is Gaussian. In the second step, time-reversed process is simulated, initialised at m², to obtain approximation of the original distribution mo- However, the score V In my required for the time-revorsal dynamics is unknown in practice and must. be approximated, for example, by a neural network trained via score matching. Furthermore, a dineediscretisation scheme is introduced to numerically simulate the reversed process.
This procedure inevitably introduces several sources of error: the statistical ervor, due to replacing the true data distribution with its empirical approximation from finite samples; the finite-time simulation error, stermiing from approximating the true marginal distribution of the OU process by its invariant measure at large times; the mathe-ng error, arising from training the neural network to approximate the score function; and the lime-discrelisation error, incurred when simulating the reversed process numerically.
From a practical perspective, it is crucial to understand how these errors propagate through the diffusion model and impact the generated outputs. Most existing analyses of error estimation in diffusion models rely on relatively strong regularity assumptions on the score function
its estimator Chen et al. (2023d) De Bortoli (023); ; De Bortoli et al. (2021): Chen et al. (2023a) : Lee et al. (2022) 2023). More recent work Conforti et al. (2025) has proposed introducing an early stopping rule in the backward process to derive explicit and sharp bounds on the Kullback-Leibler (KL) divergence between the data distribution and the
generative model output.
In the existing literature, the discrepancy between the generated distribution and the empirical data distribu- tion is typically evaluated using the KL divergence or the Wasserstein distance, each of which has limitations. In particular, since the empirical approximation may of the learned distribution mo is not absolutely contin- nous, the KL divergence is not applicable to quantify the statistical error. As for the Wasserstein distance, it yields an estimate of the form W(m,, m O(N1//)), where d is the data dimension Fournier & Guillin (2015). This curse of dimensionality on the estimate becomes very poor in high-dimensional settings common to worldworld applications.
In this paper, we propose to overcome these limitations by introducing a weaker distance, defined through a class of smooth test functions with bounded derivatives. This approach yields statistical error estimates that are well-defined for empirical mcasures and free from the curse of dimensionality.
The main contribution of this paper is the derivation of a dimension-free explicit bound on the error between the true data distribution and the distribution generated by the SGM. This error is measured via a distance defined through a class of smooth test functions with bounded first and second derivatives. Achieving a dimension- free estimate comes at the cost of requiring higher regularity of the test functionals.
We begin by deriving the dimension-fiee free error bound in the continuous-tima setting. The resulting upper bound decomposes naturally into ribuutono from the itaticl error, the finite-time simulation error, and the mctrenna error incurred during the diffusion process. We then extend our analysis to incorporate a time discrctisation scheme for simulating the time-reversal process. To improve the efficiency of the backward simulation, we introduce time scheduling into the reve-reversal diffusion and derive the resulting error estimate as a function of both the discretisation step size and the scheduling function.
Importantly, minimising the derived error bound yields an optimal scheduler that enhances the accuracy of the generated approximation of the data distribution. This optimisation problem is equivalent to variational problem for the scheduler, whose continuous-time solution can be obtained explicitly. Interestingly, the scheduler thus derived coincides with a scheduling scheme that has already been proposed in the SGM literature, where it has been empirically shown to improve generative performance Albergo et al. (2023b).
Other scheduling strategies have also been studied previously Strasman et al. (2025). Our theoretical results
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provide rigorous justification for the optimality of the obtained scheduler, demonstrating that it minimises the error associated with the time discretisation of the inverse diffusion process.
2 Background
We first introduce diffusion model following the Stochastic Differential Equation (SDE) point of view of Song et al. (2021). We highlight that this is not the only possible presentation of diffusion models and more recent ones include Flow Matching Lipman et al. (2022) and Stochastic Interpolant [llerggo et al. 2023a However, focus the SDE perspective as it is amenable to our thcoretical investigation.
We aim at generaling samples from a distribution vag- Let my denote the true underlying data distribution, and define the empirical distribution based on N available data points as my == N Ei=1 Ezg. where the samples z assumed to be i.id. draws from mg. Let X be forward diffusion:
dX, b(X,)dt + dW,.
Denote by m, == Law(X;) given Xo ~ mu, and by m. == Law(X,) given X, ~ m
In order to generate sample from the distribution mg we consider the lime-reversal of X. Under mild assumptions, see Cattiaux et al. (023) for instance, we have that the backward process (X (X))eetu,1 is given by the following SDE
dx, - (-bx)) + a² ++1 v log my (())) dt + adW, for a 2 0.
Denote by M (aw(X,) given Law (X,) Law(Xx)) m In particular, note that nt ma Next, introduce the parametrized function S to approximate the score function V log
[
[Vlogm] (x) St(x)) m. (dx)dl.
inf SE8
(1)
In practice, S is parameterized with an expressive neural network and the loss minimized using equation 1 using Stochastic Gradient descent derived optimizers.
3
Main result
3.1
Continuous-time model
In this work, we are interested in the accuracy of the generation of equation 2 More precisely, we are interested the error between the distribution of u which correspond to the distribution the generative process stopped at time T 6, and mp, the underlying larget data distributtin.. We begin by considering the case without time-discretization and extend our results to deal with the time-disccetization in Section 3.2
Assumption 3.1. Assume that the drift is Lipschitz, that the forward diffusion admits a unique invariant mcasure m", and that H (mmlm") S 2pt H (mo]m") for some p > O, where H denotes the relative entropy (Kullback-Liebler divergence) and, for measures I and such that μ < v, defined via.
[1og
du du
H(μ H(
du.
Assumption 3.2. Assume that there exists score matching function S such that
St (z: m is L-Liscchitz on e [6,T];
2. The score matching satisfies:
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R J." /IVlog (f (a) S7 (x)]² m (dx))dt S 2
Remark 3.3. Although m is singular at the Hessian of its log-density is bounded on any interval [6, with 8 >> 0. In the Ornstein Uhlenbeck case b(x) -x a direct computation gives
max,, 1 x/11?
| log llop - 1-e-20
E 20)2
Consequently, for any 0 < 6 S t T, the right-hand side is finite, and the bound depends only on 8, T and the initial diameter max;j || - = Therefore, when using St to approximate v log m it is justified to require St to be Lipschitz in a uniformly for E [],
Now consider the diffusion driven by the score matching function:
dx - ((x))
d + a²
2 S (x:))) dt + adW. for a 2 0.
(2)
Denote by pt (X;) given Law(X;) m² We are going estimate the error between na and 2 ma and further obtain the error between pr and mg.
Unlike the vast majority of related literature, which estimates the error between the target measure mo and the simulated measure μ² under some well- known metrics or divergences, such Wassertein distance, total variation and Kullback Leibler divergence, we shall consider the error through regular test functionals. As we shall see, the regularity of the test functionals helps to obtain a dimension-free upper bound on the error.
Let P2 be the set of probability measures on Rdd with finite second moment, and G : P2 - be a test functional. We say that G linearly differentiable, if there exists a linear derivative Avn. P2 Rd + such that for any m, m' P2
G(m')- - G(m) (''
8G (Am' + (1 x)m, m)dd))dd. dm
Similarly, we may define the second order linear derivalive intrinsic derivative:
Further, we recall the definition of the 6m
G,,,(m,z) = V 8G
(m, x))
where V denotes as usual the gradient in x.
Assumption 3.4. Assume that G admits bounded intrinsic derivatives Dr. and V G, as well as bounded lincar derivatives &G and 8GG 5m 5m2
Remark 3.5.
Here are some examples of linear derivatives. G(m) ʃ f (x)m(ddx) then, cylindrical functional of the form
&G
f(x), For a
8mm.
G(m) = ==((((l)mmaa)) fa())m(dx)... / fa(z)m(dx))
we have
8G 5m
Σ EaF f(())((d)), / fa(=)m(de);... f f((()m((z)) f((e).
2. Let < d and let A E
Rgd satisfy||Al| < For any & E CF(R)), define
G(m) = / m(d))
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This functional admits bounded first and second linear and intrinsic derivatives with bounds not depending on the ambient. dimension d. Since the metric only probes through test functions defined on low-dimensional feature space, it can be insensitive to high-frequency differences in the ambient space. Concretely, two distributions can agree on low-complexity observables while differing on fine-scale structure (e.g. narrow spiky components, high-frequency texture, or perceptual artifacts).
Thcorem 3.6 (Error for continuous time-reversed diffusion). Let and satisfy Assumptions 3.1 3.2
and 3.4 Then the error reads
[G( - G(mo)))]
1 -Gmm)) - C1 + +Cee
C2e 22TT + Cge ( +1) +G6², 2pT
where 1 = C1 de oo (1 - 1lc0 + ll 1 )oo)) C2 = (m))))m) C3 = CpT C₁ = (4 G |V Dm G1 C₁ C22 Co C4 are independent constants, and C' depends linearly the Lipschitz constants of b and Sr
Remark 3.7.
Note that S and - are randor due to the randomness of mg The law 15-8 is the law of the time-reversed diffusson process with the learned score function Si-; which is chosen to match the empirical score m as in Assumption 3.2
2. Note that functional of the type G(m) (é, m) for given does not separate measures. However, one can take a supremum over all the C2 bounded functions & with bounded V Aø to obtain an error estimate on functional that separates measures.
3. The proposed metrics is weaker than the Wasserstein metrics in the following sense. When measuring the distance between two measures in the Wasserstein-1 metrics, the difference between the measures testod against Lipschitz functions with Lipschitz constant smaller than 1:
Wi(He)> = sup (f.p-1). v).
11
Note that W2 is stronger, i.c (H, W2(H, For metrics defined via G the testing is performed against a narrower class of regular test functions. To measure the TV distance bet two functions one needs to consider even a larger set of test functions, namely, all bounded measurable functions: || - vllzv= = 2 suPlis] ( v).
The differences between the metrics considered above further leads to the following comparison between continuous measure μ with its empirical counterpart y² in terms of these metrics. Note, that due to the classical result of Fournier & Guillin (2015) we have Wi Wi(,,,
O O(1/N//d) We (does not docrease N augments). At the same time, with the
also have that || - N 1lzv metrics considered in the present paper we get G(u)) - G(rx²) O(1v/N) (see Lemna 6.2b
An example of a metric related to a functional G satisfying the assumptions of the paper is the MMD metrics:
(M) = ((ee,e)) d((lx) du(z)d du(x')
If [Ku.'' duly) du(s') - 22 /f k(z, du(a)) de(u).
is, for example, a Gaussian kernal (often used in practice), then G MMD (14, v) admits bounded 1st and 2nd order linear and intrinsic derivatives.
4. Note that if the functional G is chosen as in Remark 3.5 2 then the bounds on derivatives are dimension-indoopedeent otherwise they typically scale lincarly with d.
5. Theorem 3.6 gives decomposition of error, notably, (i) statistical or finite-sample error given by 2pTT" (iii) score matching error C3c2(a² - 1), and (iv) small-time
C2
N term, finite time error 2pT
cutoff C
Given desired error level err, can take 8 S
Ci and then choose T of order N
V
RIT C
log
am C5
and the score-matching error of order a
2p
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3.2 Discrete- time model and optimal scheduling
We further investigale the error when we discretize the lime-reversal diffusion. In order to sludy the optinal discretization scheme, we introduce the scheduling (g(t)), of the forward diffusion:
dX, (tt + Vo(t)dW.
(3)
Compared to the study in Section 3.1 restrict ourselves to the case b(x) = -.. Let (mn²) )a be its marginal laws given Xo m² Note that the invariant measure m' is Gaussian N00 [1a in this case.
Define p :=m; m /m", so that
Vinyf = v In (x)))(()))))))))))(() (u)+ 2x.
The continuous time-reversal diffusion reads:
dx, = -t t) - log log (x:) dt V)dW-t)
Denote as in Section 3.1 by n == Law(X, e) given Law( (Xx) my. We are going to study the discretized time-reversal diffusion, where V Voggp* is approximated by the score function: for E [tx, tx+1)
(x.)
dxx = -(T 1) -x: + -i -i (xx) dt - VG(T-ffdWt
=
with 4y = 0, 1+11 44 h 1x -T 8 and X3 m². Denote by ul = Law Law(X (x
Assumption 3.8. Assume that:
ar + S7 (z; m is L-Lipschitz on E [J;
2. The score matching error is well-controledd
K
1k1 9(T ttdt
x / Vlooogp -7-1,
(z) my (dx) E
(4)
Note that we assume that for given and g there exists a score-matching function S such that the property 4 holds, i.c. 5* depends on and g.
Remark 3.9. Compared to Section 3.1 we use the score function to approximate V logg rather than Voog m. This choice motivated by two main reasons. First, when b(x) -z, the invariant measure m² explicitly known and Gaussian. Second, the proof of our discreto-time result relies on key estimates involving V Vogge (X)). This dependence also explains why we restrict ourselves to the case a = 1, contrast Section 3.1 for a # the definition of X changes, and the required estimate no longer holds.
Take linear test functional
G(m) ().
We consider the error between G(u4_s) and G(mo). Assumption 3.10. Assuie that &o is bounded and of bounded derivatives Vip, Ap, and that g is increasing and convex with g(d)= = 8.
Theorem 3.11. Let G(m) (). Under Assumptions 3.8 and 3.10 we have
E G((AA - G(mo)) |0] z
1 (IVell& + ||A + 11ll113 (e alT)
Co(T)) VVpl +
N
K-1
Co(Ty Viells
Σ
l:(T -
O(T-t) P 266-0(T D(T -x-1)))
k=0

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Remark 3.12. Comparing the conclusions of Theorems 3.6 and 3.11
that the additional terms
K
txt1
((6-)21)))) + Ch
Σ k=
LØ(T - -)) 2g(T (x+1) dt
capture the discretization error. Since the first term is ((e-29(1)) and therefore negligible for sufficiently large T, controlling the discretization error reduces to choosing the scheduler g so as to minimize
K
inf "
5(T - 1)1 29(TT txx dt.
ا(أ)=6,
1mm
lx
Moreover, note that this discrete objective differs from its continuous-lime counterpart by O(h) term. Indeed,
K-1
de+
(TT -
-2((T -t+-1) di
lø(T - (1)3e 1
29(T-1) di
k=0
lØ(T - -o)]- ( -e dt O(h).
and a change of variables then yields
K-1
1Ā(T - - 20(T di L' (0(1)ee- 20(1))
-2(()) dt.
k=0
This motivates the following contimuuss-imme optimal control problem:
-(0)00 inf - [ e 22((+) dl.
(5)
Corollary 3.13. The minimization problem equation is solved by the following optimal scheduling:
1 -e-e TT"
9²(() = -- In
T
4 Discussion on the optimal scheduler
We give in Corollary 3.13 the schoduler g" to minimizer the discrotization error upper bound obtainod
Theorem 3.11 Here we compute a Gaussian example, to illustrate that the scheduler indeed often optimal.
Suppose the target distribution is N(H, o²), that is mo N(p,o²). Then, the dynamic
ax, = - Vg(t)dWWe
will have the distribution So we can compute V logmm(2) (x) by
m: N o((t) ( p4 -0(()) (0z (02- (0-
2(x -(()
log (1)
2020) - 1)
Therefore, assuming zero ostimation error of the score function, get the reverse discrcte dynamic:
ax; = G(T - 10( ( -
2 x h 1 9T +()
-
1 29 (x. (202 + 2 to di + Vō(T - -1jaW.
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for t E (x+2)) Since inside each time interval, it is OU process, explicitly write out the distribution of X given that -o) [d))
In order see the discretization error upper bound obtained in Theorem 3.11 an be sharp, we take o² 2 In this case, have
E X =
a(r)oll--1) (4) + 2p =[[xA]e
-(( e- - -210T -2((0--e--1 1e-1)- (
14
k
2pe
e (T +
Σ e -T- -t)
2pe (k)
20(T -1-1 le-1)h. (6)
Note that 0o e220(0)2((1dd - Therefore, the discretization error reads
K
d (e (e-2g(T-t)
11 - E
R
_ 20(T 4 2e- 20 te-1)dt
2=1 1
4uh² Σ
2g(T -0-2(TT-u-11001 4-111
where the right hand side is approximately the value aim to minimize in Corollary 3.13 In this sense, claim the scheduler in Corollary 3.13 can be optimal.
In the following we numerically evaluate the expectation E [ by equation according several common schedulers. Let T 1, we shall try:
The linear scheduler g(t) = T't;
The optimal scheduler prescribed in Corollary 3.13 g(() = - -Inn - 1(1 - e-T"));
The commonly used cosine scheduler g(t) = - (ft)) where a such that cos(#a) = C-
We compare the errors the computed expectation values for different values of a.
Figure Error on expectation with different variances and schedulers
As shown in Figure I the prescribed optimal scheduler stays optimal or almost optimal in different cases.
Finally, we present non-Gaussian experiment. We take to be mixture of Gaussians in R2; in that case the score function can be computed explicitly. We compare my and 14= in terms of mean log-likelihood, Wasserstein W2 Wasserstein W1 and MMD metrics with a Gaussian kernel.
The results of the experiment are presented Figures and 3 We take ma = 0.3N I) +0.NN(33 3), 0.57). We observe that the scheduler proposed in Corollary significantly outperforms the linear scheduler, and generally exhibits superior performance compared to the cosine scheduler.

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Samples m_
SMD, scheduler
SMD, scheduler
SMD, opt
-2
-2
-2
-2.5
0.0
2.5
5.0
-2.5
0.0
2.5
5.0
-2.5
0.0
5.0
-2.5
0.0
2.5
Figure 2: Target distribution my and the result of score-matching diffusion (SMD) with linear, cosine and optimal scheduler.
Mean log-likelihood
Wasserstein W2
Wasserstein W1
MMD
0.30
In
lin
1.2
0.25
100
0.20
opt
0.8
0.15
-344
0.10
0.05
-3.6
opt
h
Figure 3: Comparison of mo and 14-s for a Gaussian mixture model R.2 (known score).
Remark 4.1.
1. For the dynamics of equation 3 the relative Fisher information with respect to the invariant Gaussian measure m satisfies the sharp decay estimate
[[m, | m²) s -2((1) [(mo)) m²),
while the relative entropy H(m, m²) obeys the dissipation identity
dm m²) = (m, m²) ((t) dt s -mmmo mm)) 2g(t) g(t) dt.
In our framework, the optimal schoduler g" defined as minimizer of equation 5 This choice enforces uniform distribution in time of the effective entropy dissipation along the evolution on [0, T).
This criterion is closely related, at an intuitive level, to approaches that characterize optimal sched- ules via constant-speed geodesics in the Fisher-Rao metric Zhang & Syed (2025) While in Zhang & Syed 2025) the optimality is interpreted in terms of geometric interpolation in probability space, our perspective emphasizes temporal regularizatio of the dissipation rate. The two viewpoints coincide in special cases: for instance, when the initial law in equation 3 is Gaussian with the same variance as the invariant mcasure, mi N (H. 1). the scheduler g" also yields a constant-speed Fishor-Raa Rao geodesic. In general, however, the two notions of optimality do not coincide.
2. (Relation to varianco-preservingg and rectified-ffow parameterizzatinns) Consider the scheduled forward SDE
dx, - j(t) X, dt + g(t) dw,
where g [0, Taig] + [0,T" is increasing. Under the time change = g(t), this reduces to the standard OU process and yields the explicit coupling
N, e -0(0) Xo v1 e-0(() Z.

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Introducing the interpolation parameter TVP v1 V1-ea e e-(ttt,, we recover the spherical (variance- preserving) interpolation X ===1 V1 TVP Xo + Tvp Z, which coincides with the VP coupling used in Now-matching.
Rectified flow instead employs the linear interpolation X, = (1- +Z. Defining TRE :- 1 1-e 1-e-a(() a(t) linearizes the data coefficient a(t) e e but the corresponding noise coefficient becomes V Rr TRF); and thus does not match the linear RF path. Under the optimal scheduler g (1) proposed in this work, a (t) is affine in algorithmic time, so that TRF is linear in t. This is the sense in which the proposed scheduler aligns with petified-flow parameterizations; the induced coupling remains the VP (spherical) interpolant rather than the linear RF path.
5 Related works
Statc-of-the-art results There exists an extensive literature on the convergence of denoising diffusion
models. While early works such as De Bortoli et al. (2021); Pidstrigachi (2022) Block al. (2020); De Bortoli (2023) and more recent works such Choi & Fan (2025) proved convergence results with exponential dependencies in the problem parameters, those results have been refined in subsequent analyses.
Early works proved error bounds with polynomial dependencies with respect to the dimension or linear dependency with respect to the dimension but under strong assumptions on the regularity of the score function, see Li et al. (2023 2024a); Lee et al. (2023); Chen et al. (2023d); Gupta et al. (2024). Those results were also extended bound the probability flow ODE associated with diffusion models Chen et al. (202cc):
Benton et. al. (2023).
The first bounds the Kullback-Leibler divergence linear with respect to the dimension were proven in Benton et al. (2024ab; Conforti et al. (2025) and do not require any smoothness condition on the score. The dependency with respect to & was mitigated in Li & Yan 2024 Li et. al. (2024b) to ((df)) To the best of our knowledge the best results obtained so far are of complexity O(min (d, d 1/3 dd/33T))-2// in
Jiao & Li (2024) leveraging a relaxed Lipschitz constant. It is also possible to improve those results by assuming that the data distribution is low dimensional PPotaptchik al. (2025); Liang et al. (2025). To the best of our knowledge, the result most closely related to our contribution is Li al. [2025a] which identify dimension-free convergence rates for diffusion models in the case of Gaussian Mixture Models.
Those state-of-the- results can even be improved with dependency of O(dK upon using higher order samplers Li et al. (2025b) and (dd5/a1/22 using accelerated samplers, see also Pedrotti et al. (2023) We also highlight that it is possible to improve those convergence rates even further, cven deriving sub-lincar time complexity results, by considering parallel sampling strategies as in Gupla et al. (2024); Bortoli et al.) (2025). Those results have been partially extended the case of stochastic interpolants Albergo et al. 2023b in Liu et al. (2025) (with sub-optimal rates).
Convergence in other modes. The convergence of the diffusion models regarding other metrics has been invostigated in several papers. For instance in Mbacke & Rivasplata (2024) ; Cheng et al. (2024); Gao
ct al. 2023); Gao & Zhu 2025): ; Yu & Yu (2025 show the convergence of diffusion models with respect to the Wasserstein distance of order 2. Contrary to De Bortoli (2023) their results is not exponential in some parameters of the diffusion models. Other results on the convergence of diffusion models in Wasserstein distance including Lee al. (2022) which improves significantly improves upon [e Bortoli (2023). However it relies strong assumptions regarding the Lipschitz constant of the denoiser. While Reeves & Plister (025)) draws connection with information theory their analysis does not provide insights with respect to the discretization of the reverse process and its convergence.
Other thcoretical analyses. Finally, we highlight a few other arcas of theoretical exploration of diffusion models. Li & Cai (2025) explored the convergence of discrete categorical diff'usio models, while Biroli el al.
characterized different modes of the backward diffusion leveraging tools from statistical physics. Chen ct al. (2023b) provides sample complexity bounds for distribution estimation in low dimensional settings. Finally, Buchanan et al. (2025) identified memorization and generalization conditions for diffusion models.
2024h
Chen et al. (004)) Gatmiry al. (2024) focus the learning problem in diffusion models and provide result
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for general Gaussian mixtures. Combining learning guarantees with the aforementioned convergence results
several works have derived near minimax optimality results Cai & Li (2025 Oko et al. 2023
Azangulov
et al. 20250.
6 Proof of Theorem 3.6
Introduce
ml, ([Law(Xx slx m)).
By definition of U we have trivially that U(T 6G G(Law x f- - 5 ~ 14 = G((
Also
U0, vō) G(Law i Xxx ~ m²)) = G(17-8) by [2) so that
U(T 6, 1 8) (5).
On the other hand, the function U satisfies the backward Kolmogorov equation:
a,U (DmUU-
a² +1
a² V.
b+
ST
Dm dm
(7)
2
(see Theorem 7.2 Buckdahn et al. (017)))
Lemma 6.1 (Regularity bound for U). Under Assumption 3.2 (1) and 3.4 we have the following regularity estimate on U:
x) s Glllo.
-
8U (t, m, a)
- <c
8G
8
(t, m,
S
8aG dm
Em
8m
5m
Lemma 6.2. Assume that yo has bounded derivatives depending on the bounds on the first and second-order derivatives, such that,
dg 5m
and
82 8m
Then, there exists constant C,
E B oN [le(m))) - ((m)))
C N
The proof of Lemma 6.1 and 6.2
is postponed to the Appendix.
Proof of Theorem
3.0 The weak error between 14-8 and uf mg reads
G(p G(m
= UTT - 8, s5) - U(T 5, 6
= U(), m²) - (000m²)
- 1/1
T-
(ou (OoU+ +oU+DDml Dmn a2+1
v log m m + a² V2 (DD (1, mz 2) (i (ddddd
2
2
= U00
a² 2
I f (DmnU(Vloog@ ST) (t, m x)) r) (dr)ddt
where the second equality holds due to the Itt's formula for measure processes (see for instance Theorem 5.99 of Carmona & Delarue (2018)) and the third equality holds due to the backward Kolmogorov equation
Recall Pinsker's inequality: || v S H v), where || ul|rv denotes the total variation distance between μ and v. This inequality implies that (0, U(0 m )
At mm 80
(mmr(m²).
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Therefore, by Pinsker's inequality and Assumption 3.1 have
E ((( - (((})))
S CE (0. m3) u -00,)))))
8U
C'
H H(mTlm") + CIE (U, - U m3) 1? U + 1)²
dm
80
e H (moomm) + CIE (0, U mn) [2] + Cr 12c 2 (a² + 1)?
5m
where constant may vary from line line. Moreover, due to the boundedness of the linear derivative of G and -0,-) (Lemma 6.1
by Itô's formula and Lemma 6.2 we obtain:
/ / (Dn)( + 1 V D Dm Gmm)(m x)) me (dx) S CC8.
E [G(mo) (ms))mm)))))))))))))) =
E [[(ms) -aamm)--Giie G(mm))) & N
- N
C
Finally, the desired estimate follows.
7 Proof of Theorem 3.11
Let us define a family of functions
and fx( Ok=0,1,... by
For E [ta, tx+1), u (t,rr)) = E E Xh 1k
{ = x, X = m7)]
(8)
with
(=)== and (x) u (to, for k = 0, 1,1,2,.... -
Lemma 7.1. Given the definition of (o(*) J=-,,,...... and (u( 1 above, fix & E (0,1, K - 1), for all a E Rd, and e [tx, tx41) we have
(t, & llolloo, (tu - | (1m (o(T).
Moreover, for all k E 10, K}
| 1oo S lsplooo
( S ( (1-111""
The proof is postponed to the Appendix.
Further, on [te, tx+1) the function (*) (;) solves the PDE:
(t, + LT t) (1,2;)2) + +G( t) (+ St (Ix) (t,)()) ((()) (14+1,2;;44) = (x)).
(9)
Since
(x) (b) (14, 2; x:) = EE (((ttttt))))) X 1k |8 =x = F up( X
X =x) =x]
get v(0)) (x) Ex)= x 2) X3 =s], and (pxx) ((K) (x)) Therefore, we have
G(u4_s) G( = x .)) (1p, m = () (m()),,)) - k,(k),m²)))
Now we ready to prove the main result on the discrete-time model.
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Proof of Theorem 3.11 1 Note that
G(44 8) G(m = (ul0), mm)) - (vl0) ( + =(b(),m))))))))))
m++ (v(()) mm²)
By Pinsker's incquality and Lemma 6.2 the difference (ul0), m²) - (v(0), m2) is controlled as follows:
B Ba @N [100000 m²) [,o00,,m²) - (ell)) m 12
(10)
2 (b/0) mm)
(m')) - -(0(00 mmm 2 + Rolld so [[10000 mT) - ((o09) ; mm
(2
< || 10)0 1 TV , + Iolrd) sN ||0100 .m)) - (0010) m m 1"]
2 11 18 (mT (mnm ) + )) S 1191 (e (c-20(T) + NN)
(11)
To estimate the difference (g(k),,²) - (v mt), it follows from Ito's formula and the PDEs equation that
g
(X - = K-1
(olk) mm8) (111), mmf (x
_ 0/0) (Xx..)
K-1
(M) (1e+1, (rais:X(1)) - u - (1x, x X,m]]
I
km
K-
150's 8
(1, + +T( - +)) (2) (t, T,: xx.)
E
0==
+ + - ()) ( (-x, + (i.pp vu((1) (t, .,: xxx) )
dt
equation 9
5(T - 1) (1---1(Vnnpp ( vi (1, xtx
E
where Xo ~ mx. Therefore,
(1o0k)),, ,,²²' (0 .m²))* )²
K
12+1
In (x) S7- ( (xx)) Vulli
27 Nr: & dde
g(T - 1)
1k
G(T - +) v In (Xx1) - S7 -1 (xx,a) || ||//ttt/ dt
<
E
G(T- v (2) (1, 8,; X (. ||Par]] }}
k=0
By equation 22
we have
de+1
g(T - -t] (t) (u
Vu) (tx
(t, X,
xx: ] < |1 /'/ g(T t)dt
(12)
k=
$ <g(T))|V<ll (2+2115"/1mm0007
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In order to deal with the first term, using the assumption equation 4 annd the estimate (38) in Conforti et al. 2025
we get
K
txt1
g(T -
VI Si (X (xx) ['acu]
ad]
8x
K
1x+1
5(T - t)dt E V Vimpp (x... (.)) - S -7 -n (xxe)) ||?
< 2
0=0
K-1
-t) V In 1p (x.)) v Vnmpr to (X Pat
+ 2
K
S 26 + C
E
g(T - (I /m") - I(mm to 1m (m))))))) 1)dt
k=0
Moreover, as proved in Proposition 3 of Conforti el al. 2025 we have
I(m 1m²) S (1)0(-))-9(2) I [(m mm) S I(m lmm)) for t < s.
Therefore,
9(T - (( mm)) - Immm (mm)) ddt
K
<
(o(T - -ka) -g(T - ((++1))) (Iim |m*) Imm /mn))
<E
- Σ (o(T - -4+11) 2T
(x) + gT 1x-1)) ((m (mn) (9(T- - -((0)) mmm)
K-1 (g(T- - -xx+1) - 2T - t+) + g(T tx ())1 2(6- + g(T - tx-1) 6)
<FI(m;
(1mmmmmm))
(0(T- (t)) -9(T (x+1)) (666- - 2 2(0- + (g(T)) 9(T - e
2(6 ((16--6))))
<EI(mE Im²)
2(9(T -xx)-gl to+1)) 2 2
(T)
K
da+1
1))-d) o(T (0 + ((5 205 U(T))
<hBI(m (m*)
Together with equation
12 we finally have
(lulk), - (o!0) m
+1+1
<Cr 2 + + (m3)
Σ 2 K=0
lø(T - -1)-ddd (15 ((T-)) + ((TT) l
and combining with equation
10
we have
1g g gN
(c( - G(m!") )
)²]
S 11101:30 (e (e-s(T) + ) + Cr
K-1
1441
+ + Em (m y/1
Σ
lØ(T 1) 2 206 260 9 + ((T))2(6--0(1))
k=0
14

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Published in Transactions on Machine Learning Research (04/2026)
By convexity in the first argument we get that I (mg
m=)) s &े Ei= I (m3" Im)) where mi the
law of process (3) at time corresponding to mo = 5x,- Then a direct calculation gives that EZ (m² Imm -
C (Var[X.] +
Finally, the difference G(mg)) (( follows from Ito's formula, and we obtain the desired estimate.
Proof of Corollary 3.15
Denote G(g(t)) = [e( 20(0 Consider a variation ó, of g(t) verifying 6,(0)
0, (T) 0. Then we have
((g(t) + 6,) g(g(t)) Ag())) + o(o)), where
T
I
(,(),) / 2g(t)èEe 20(1) -2(()) dt 2la(t) 20(t) dt.
Integrating by parts and taking the boundary conditions into account, we obtain the following first order condition on the solution of the optimization problem gr:
9: 10t 12 =.
Integrating this ODE we obtain that gi = In(-t C1) + Cz. Finally, using the boundary conditions we obtain that C1 = T-e-rr, C2 In (---->) which yields the required formula.
Remark 7.2. Let of be the optimal scheduler. scheduler....rr. Consider small variation ó, of of verifying 6,(0) = 0, (7') 0. Then
G + 5,) -(9))) = f((G (Ci 1) (
This term gives mcasure of scnsitivity of the error with respect to deviations from g".
8 Conclusion
In this paper, we have proved the convergence of denoising diffusion models under a weaker distance than what is usually considered in the literature. More precisely, we have derived upper-bound the weak error of discretization of the denoising process under some regularity assumptions used in the computation of the weak errors and regularity assumptions on the noisy score of the empirical measure. Our main result is dimensionless upper bound on the weak convergence error of diffusion models. To the best of our knowledge this bound is new. We show how our proof can be extended to cover the case of discrete diffusion models. Finally, we show that we recover the well-known Flow Matching (or Rectified Flow) schodule by optimizing the upper bound of our weak error analysis.
the test functions
While our analysis is tight in the Euclidean case, in future work would like to extend our weak error bounds to the case of general Denoising Markov Models combining analysis with the infinitesimal generator framework of Benton et al. (002bb).
Acknowledgements
A. Kis research is supported by PEPR PDE-AI project. Z. R's research supported by the Finance For Energy Market Research Center, the France 2030 grant (ANR-21-EXES-000) and PEPR PDE-AI project.
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A Analog in finite-state generative model setting
Recently, the finite-state generative model driven by the time-reversal Markov jump process has been earning increasing interest. In this section, briefly explain how our method adapt to this context. Since it is not within the main scope of the paper, we will only focus on the continuous time model (without time discretization error) for an oversimplified one-token model.
Consider an example where the token can take values from X. := 10, As in the diffusion model, we initialize the forward process at the distribution mo IR² on K. Eventually we shall sample according to mo, using a time-reversal process. Let the forward process be jump process with jump rate X and transition kernel r(i,j) 2 for j.j E K. In this setting, the marginal distribution (mæ)e evolves the solution to the equation:
dt m,(1) 2 -m,(1), for all i e K. d
Therefore (1) = e (i) + (1 - e } and the marginal m, exponentially converges to the invariant measure, the uniform distribution on X,, denoted by m As studied in Pham et al. (0225,, by denoting the score function s1) : m1())m,(1)) the rime-reversal jump process is of the jump rate Xi (2 sc))
and of the transition kernel F, i) == 2 (0)( and Feli, = 25.0)* Let the time-reversal jump process be initialized by the distribution mr. One verify that its marginal distribution is 14 mr mr-t- The dynamics of (Me); reads
x.00)
dt (1)1 (1- (1- -10 (1-))(())(()))
In practice, due to limited access data, one may initialize the forward process by a distribution m instead of mo- We will denote by (m²)), the marginal distribution of the associated forward jump process, and by (n²",, ) that of the time-reversal process. On the other hand, one may use the data to estimate the score function by parametrized function s* and try to control the error so that
mm) ~100 (i)dt < 2
(13)
Denote by (ut); the marginal distribution of the time-reversal jump process driven by the function s* and starting from m² so that Hg = m and
d 1 (i) 2 (1--5-((1-1))
-1-5
(14)
di
Now, as in our result for the diffusion model, we aim at measuring the error of the generative model through smooth test. function G R IR by estimating [Gmm)) G.
19

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Similar to the diffusion case, again rely the function U(t, G(TT"")) where MT pr is the terminal value at time T of the solution to the equation equation 14 given the initial condition 14 == m. We have
G(mg") = UT, m²) = UT, , G(+t) = U0)m"", and more importantly that Utt, (ot,zo.z) x1) solves the backward Komologrov equation
X 0
Σ
87- -i)) (1 (0)=) O. Ozi U
(15)
2
1=0,
Now note that
G(mo) G(G)) G((m)) - (( UT, p²) U(0. (0, (O, m²).
While it is easy to obtain (m)) G(m)") the middle term relics on the PDE equation
|| AN and [U m U00m")) S - the estimate of
15 Provided that 0z, U is bounded, we can verify that
U(T, ) =00
Σ 2n U(b (( -1 ())=((-))) -i) (sr- (0) de ST. 1=0,1
2
where the last inequality is due to assumption on the score matching error equation 13 Therefore, the conclusion reads
1 + +
1
G[G())) (G(m)))G(pG))
VN
Remark A.1. The literature on error estimation for discrete-state diffusion models remains relatively sparse. We note in particular the following several works that contribute to the understanding of discrete space diffusion models. (Su et al. (2025) derive bounds on TV distance between target and generated distibution in terms of statistical error in the framework of discrete flow matching. Huang et al. (2025) develop the complexity theory for (masked) discrete diffusion providing bounds the number of discrete score evaluators necessary to achieve the desired TV accuracy. Wang et al. (2025) introduce training techniques for reasoning diffusion models that allow for estimating the KL. divergence accuracy in an analytically tractable manner. Wan et al. (2025) develop a theoretical framework for deriving non-asymptotic KL divergence and TV error bounds for discrete flows with generator matching. [umar et al. (2055) show that the visual autoregressive generation mathematically discrete diffusion model allowing to export riffusion-related tools to VAR.
The works that are closest to our approach are Pham et al. 22025 Li & Cai (2025), quantifying the error via the Kullback-Leibh (KL) divergence. Note, however, that when the state space K is large and the token vocabulary is rich, then the empirical or model distribution mo is typically highly sparse. In that regime, the KL divergence is often ill-posed (infinite or numerically unstable due to zero coordinates) unless ad hoc smoothing is introduced. Measuring generalization error through a test regular function, as above, avoids this issue.
B Proofs of Lemmas
Here we collect the proofs for the technical lemmas.
B.1 Proof of Lemma 6.1
Proof of Lemma 6.1 First, it follows directly from [Bukkdahn et al. 2017 Lemma 5.1) that
D Ut, = E X
[a Dm ( X T (.2)
where t,, the process defined equation 2 starting from x at time Then, the bound of the intrinsic derivative of U, U, follows from our Lipschitz assumption on and S* together with the boundedness of D.
20

---

Published in Transactions on Machine Learning Research (04/2026)
In order to estimate the linear derivative, perform the following lifting, enabling us to reuse the estimate for the intrinsic derivative. Define the operator T C(Rd) - C(R by
((x) + T y) = p(x)y,
and the adjoint operator T>: 1( + P (IR by
- T (p) = J up(drdy)
Define G: P(R(+1) R by
G(p)= G(T)
For P1.P2 E P(R(+1), it holds, by definition of lincar derivative,
G(pi) G(m) = CT (pi) ((T)
= 1.' 8G
(pT (pi) (1 (()) )d22)
5m
L.
(T dm
8G
(pi) + (1 - (x-)) - 02 )a.
Hence, we have for all p E P(
8G
8G
(0) TT
(T' p),
6p
8m
that is for all x E Rd, y E I,, we have
8G
8G
(p, (z,))) = 8m
(T',
p. z)
(16)
8p
Similarly, we have
P6 (p, (z, w), (2',1/)
5G (T x yal.
(17)
8p3
5m
Consider the dynamic:
dX, = (b((x)) + للع S (X)) dt adW,
dY, 0.
and (p) = Law(X, Yi) given Law(Xp, Yo)= = Ü(Ipp) = G(pr
Define Ú: [0, x P(Rd+1) + R by
Let p = m 2 81, where 8 is the Dirac measure at It is not hard see that (p( pf(m) 61, and
(r) m, T (pi( = 11. (m)
then
ū(t, = -(p = G(u = Ut,
Then, by equation 16 and equation 17 have
zū óp (z.
8
8U
(() 8m
(
(t, m. m,
8p
8m
and similarly
J (z.
8U dm3
m, x,
5p2
21

---

Published in Transactions on Machine Learning Research (04/2026)
Therefore, we establish the link between the linear derivatives AD 0 s 0g ap by 5p"
8U 8 6m
with the intrinsic derivatives
8U dm
U (1, m, x. x') og
6-Ù
(1, m, m, 2) = Oy 8p (1, m, (I),
56 (Z)
dm
Finally, again apply [Buckdahn ct al.
2017 Lemma 5.1, Lemma 5.2) obtain the desired bounds. m
B.2 Proof of Lemma 6.2
Note that
1 (é, m = N Σ o('), (6, m
Σ
[[ X x
where X1 x2,..,X' XN are ii.d copies following the law mo.
Proof of Lemma 6.2 This essentially repeats the proof of Lemma 5.10 in
Delarue et al. 2019
Since yp is
differentiable, we have the following expansion:
((m4))p(mm)) [
dip (dmi (1 A)mr.)) :). m mmr mn>da d
6m
= 1/' 84p 5m
(Amq + (1 -1)r.;)) 64 (mT, m m
( d④p 5m
mr )dr) +
(mr, m mr
5m
Step 1. We first deal with the term S2 This may be rewrillen as
1
N
dep
(mr, Xx) Xo x²
dup
(mr, Xx)) Xn))):
S²
E
E
N
6m
dm
where under E, dX, b( b(X,) + dW, and Law(Xo) mo. Therefore, we have
N
Xo = xx)) (mT (mT,
N
dup
(MT, X Xo -00=x0 x =x(]
E[S]] =
Var
E
Var
E
N
N2
ôm
1
dup
(mr, XT) Xo xxxxxxxx =xxxx]] -
86 5m
1))
2
E
E
(mr, Xx)
8m
N
8m
N
Step 2. We turn to the term S₁, which first rewrite in the form
S = pids
where we define, for i 1,... and A E [0, 1],
8ip δm
84 (mr, Xr) Xo x²
o
R
(Am +(1 A)mT, (omnT,x)=
8m
dsp (2m + (1 -X)mr, Xr)
óp (mrr, xm)
-
8m
8m
where E is taken independently with max (viewing may non-random) and X is given by the dynamic dx, = b(X,)dt + aW, with Law(X,) = mo. Now, we have
S N2 /
Σ 2
dd
1 N2
Σ
(x))²
+2 > pivl p1ss)an dn dr.
22

---

Published in Transactions on Machine Learning Research (04/2026)
Note that |px| $ 4
dm al
d2 have
1
Σ (
das -
16
8m
2
N
N
It remains to bound pid for + j. Define my
N.-i-j by
m (. b, = N- Σ
[[ = xx],
and ---11-- 2- by
1. jo
E
óip dm (Am?)
N. +(1
++(++++1111---- - A)MT, XT)
5④ (mT, XT) Xo =x"
ôm
-E
(Am
N.-i.- + +11 (1- X)mT,
(mT, Xr)
dm
dm
Then, it is not hard to see that 1x 1- is conditionally independent of xx -
i-j given (Xx Ixikiis and therefore
E
E
x O.
(18)
Observe that i - p can be calculated by
óip 8m
dep
(Am -.- + (1 A)mT. X7) x Xo x*
E
(Am +(1 A)mer Xr)
8m
(19)
dip (Ama + --)mr, Xr) du dm
(Am - + (1 A)mer, x5)] {)mer;x))
8m
and
dup (2mæ+ +(1 A)MT, XT) 8m
dip
(Ami -Si (1 A)MIT, Xr)
dm
S/'
824g
(Ymm N. + A(1 - 7m - N + (1 - A)mi, XT, Mm, - m (i )dr.
Jm
Recall that
(6, mi mT N.- = (B
= X =| =x'I)
2
Σ B(o(XT) x*],
N(N 2)
we have
86
82
21
82 dm
2X
8
(mm N dm
(1 A)mT, Xr)
N. (. + (1 - A)MT,
+
6m
3m
N
am
where the last inequality holds because without loss of generality, we could assume N 2 3 and 3(N -2) 2 N. Plugging back into equation 19 we get
16
82
10y - 4/
<
N
23

---

Published in Transactions on Machine Learning Research (04/2026)
and together with equation 18 we have
E
E
10
6 1
+ E
p (1p) Xx ) + E
(12
64
8º
62
p E
1
- x +
B
|1 -
1110 - X -
Nm
6m
Therefore, we have
16
6p
128
8²
6p
= B[S9]
8m
8m
N
and thus
[(((m))) (m(mm)))))))
= =E + +2117
C N
where C depends only on the bounds on the first and second-order derivatives.
B.3 Proof of Lemma
Proof of Lemma 7.1 The bound of ||u[[)|| and ||b(0]] || can casily be obtained by probability representation.
As for the gradients' bound, For & E tx+1,, given X X follows an OU process, then we can write
X ~ N X2 alT -10 + -7-1 (x))(( - e - - +)) (att))))--) 1- 2(g((- -((-(())x-1))
2
Then,
u (t, (B(o(t+((()))
)
S xkk
E (o(( + (t, xt);
and
000 (x) = E E
|
(o(tt))) + + 21).
where 2 ~ N(O, Ta) is standard normal and
(t()))((t) 1 - 2Lx(T -)) -)- 11))
(t,r,))) =e -(T--)++(( ((0-0))0((T(4+))4553 +87 (xx)(1-e I(T (TT)))((T-tt)))))
2
Then,
d (v(k+1) (g(()(tt)) (z()) (t2, 2;:))
00.0(() (x) E
dx;
E
e a(T to) az: (S) (Si-1(
Oz, 2e, (o + (k) (t,, 2;,)
te oT (( (kt)) (o())) + (1)) (th, (tb.
which leads to
( bo | (e (o (1 -o -xx))) 1| 1
(20)
24

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Published in Transactions on Machine Learning Research (04/2026)
By iteration, get
K-1
|ollt)|
1 Il (0(T-(er1)-0(11--a
II
1 + (cot)- - 1)1
fm
1
(1+1) - 1) ||5 (ltpp.1oo) }
= Villoo exp
-(x) -9( 1x) +
- -
In 1 + (c (0)(T-1)- 0(0
t=
R-1
1 Villoc exp
9(T - -tx) -9(T - (2) + || 1ip.
Σ
(1
C - -6411))
1=
K-1
Vpllx exp g(T -14) -9(T (x) ||5 1,
Σ
g(T (1)) g((T 1et1)
t=
< Villa exp (OCT
txt) -ta) g(T tx)) (1+ [|+1|||llipp...oo)
}
(21)
which leads to the desired bound of || Using the similar argument in equation 20 for all y E Rd.
can also get
[d (a (t) + 1 (t,
x: 8) E Oz, u (2)
dr.
E (TT ()- alT -x Oz 2( (a (k) (t) + (0 (te,
which means for all y E Rd.
Vul (k) (4, :1)llao I c0(T- a(T (n
Combining with equation 21 we have for all y E Rdd.
(1,3)l S (IS
(22)
which completes the proof of the lemma.
25


Reviews:
[{"review_id": "T2QGtxhQfw", "reviewer": "Reviewer_2Vhg", "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": "", "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": "This paper presents a dimension-free error estimate for sampling with diffusion models. The guarantees are formulated in both the continue time reverse process and the discrete time reverse process, in terms of a “smooth” functional G defined over probability measures, equal to zero at the target.\n\n\nStrengths:\n\nThe approach relies on backward Kolgmorov in a space of probability measures and is rather elegant\n\nWeaknesses:\n\nClarity. The approach is clean, but at the cost of losing the reader. The paper employs tools from Lions calculus etc which are not standard to the average TMLR reader. A bit of context is needed for me to check the correctness of the equations\n\nSmooth functional. The authors should spend time trying to instantiate the functional. For example, does it exist G such that G(mu) = G(m) implies mu = mu? With that separation property, the authors might derive (asymptotic) convergence from their bounds.\n\nDimension-free. It’s a bit over claiming, I would say that the bounds do not depend explicitly on the dimension.", "claims_and_evidence_yn": "No", "claims_explanation": "I could not understand parts of the paper, specifically:\n\n\nAssumption 3.1: H is not defined\nAssumption 3.2. 1: maybe recall that the true score is Lipschitz on [delta, T] \nTheorem 3.5.: Is N superscript missing in the l.h.s? Asking because there is N in the r.h.s.\nRemark 3.8 The linear part in the score (that comes from the Gaussian) could be integrated in closed form no? Why do you learn it? \nTheorem 3.10: Does it apply only for functionals G linear in m (G(m) = <phi,m>) ?\nRemark 3.12: typo in the first equation g_{T-t} vs g(T-t). Besides, I had hard time understanding this remark, please clarify. \nSection 4: I understand it is a restricted setting, but I am wondering, do you claim the last equation before remark 4.1 (with a formal proof), or can you rigorously prove it? The status of this inequality should be clarified “claim”/“formal proof”/“rigorous proof possible but not in included in the paper”\nSection 5: My understanding is that the scheduler is only “nearly” the optimal one. So “nearly optimal” is more accurate than “optimal”\nProof of Theorem 3.5 (also applies to Proof of Theorem 3.10). Some material recalling how Backward Kolmogorov works in space of probability measures would be appreciated\n“Note that by the flow property…” I think the statement is trivial but difficult to parse because of the notation. The authors could clarify to help the reader\nFirst equation where H appears: What is the constant C here (in front)?\nSecond and third = symbol: In general I was not familiar with Backward Kolmogorov in the space of probability measures. Did you use linearity for D_m U ?", "audience_yn": "Yes", "audience_explanation": "Theory for sampling with diffusion models is a relevant topic for TMLR", "requested_changes": "Just clarify the points above please. \n\nBesides I was wondering:\nDo you measure distance to m_0^N? Where does the need to use m_0^N come from?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Q5VBkOquu7", "reviewer": "Reviewer_g4b7", "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": "", "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": "This paper provides dimension-free weak error estimates for diffusion-model generation and uses them to derive an explicit “optimal” time schedule for discretized sampling in an OU/VP setting.\n\nMain contributions:\\\n1.\tWeak functional metric for generative error. Instead of KL (ill-posed for empirical measures) or Wasserstein (dimension-cursed for empirical convergence), the paper measures discrepancy via smooth distribution functionals $G:\\mathcal P_2(\\mathbb R^d)\\to\\mathbb R$ with bounded derivatives, and bounds $\\mathbb E|G(\\mu_{T-\\delta})-G(m_0)|^2$. This choice enables statistical 1/N-type rates without explicit d dependence in the stated metric.\\\n2.\tContinuous-time end-to-end error decomposition. Theorem 3.5 gives a bound decomposing error into (i) finite-sample $(\\sim 1/N)$, (ii) finite mixing $(\\sim e^{-2\\rho T})$, (iii) score approximation (an integrated $\\varepsilon^2$ term), and (iv) a small-time cutoff penalty $\\delta$ to avoid singularities at t=0.\\\n3.\tDiscrete-time weak error + schedule optimization. For discretized reverse OU and linear observables $G(m)=\\langle\\phi,m\\rangle$, the paper derives a schedule-dependent discretization bound (Thm. 3.10) and optimizes an approximation to obtain a closed-form schedule which matches commonly used schedules and is related to flow matching/rectified flow parameterizations.\\\n4.\tBrief discrete-state illustration of the same weak-metric philosophy when KL is problematic for sparse empirical distributions.\n\nKey strengths: \\\n1.\tclear motivation for the weak metric\\\n2.\tinterpretable decomposition of error sources\\\n3.\texplicit schedule prescription that plausibly justifies a popular heuristic.\n\nKey weaknesses: \\\n1.\tdimension-free hinges on strong smoothness assumptions on G (potentially missing perceptual/high-frequency errors and hiding d in constants)\\\n2.\tschedule optimality is for a specific proxy bound under restrictive settings (OU/linear observables)\\\n3.\tlimited empirical validation and unclear interaction with score training (since changing g changes the training time/noise distribution and thus $\\varepsilon$).", "claims_and_evidence_yn": "Yes", "claims_explanation": "The core theoretical claims are largely supported within the paper’s stated setting and weak metric, but several interpretations (dimension-free, optimal scheduling, and the flow-matching connection) are not yet backed by sufficiently clear scoping or empirical evidence.\n\nSupported:\\\ni) The continuous-time weak error result and its decomposition (finite-sample, mixing, score error, cutoff $\\delta$) are presented coherently under explicit assumptions, and the derivations appear consistent with standard weak-error tools.\\\nii) The discrete-time analysis yields a schedule-dependent discretization bound and the derivation of the closed-form $g^\\*$ is convincing as an optimizer of the chosen approximate bound in the OU setting.\n\nUnder-supported:\\\niii) “Dimension-free”: while d does not appear explicitly, dependence can re-enter through derivative norms/regularity constants (and Fisher-information-like terms). The paper does not clearly characterize when these remain dimension-stable for practically relevant observables.\\\niv) “Optimal schedule” as an end-to-end prescription: optimizing discretization alone ignores that changing g(t) changes the training time/noise distribution and thus the score error $\\varepsilon$ (likely $\\varepsilon=\\varepsilon(g)$). Without theory or ablations with matched training reweighting, “optimal” is only partially justified.\\\nv) Empirics are too limited to support broad practical claims about scheduling benefits with learned scores.\\\nvi) The flow-matching/rectified-flow connection is plausible but asserted more than demonstrated. The explicit time-variable mapping and the precise sense of equivalence should be spelled out.\n\n\nOverall: Solid theoretical evidence for the stated weak-metric OU results with weaker support for the broader practical and interpretive claims.", "audience_yn": "Yes", "audience_explanation": "Yes. The paper should interest a meaningful subset of TMLR readers, especially those working on (i) theoretical foundations of diffusion-based generative models, and (ii) principled design of time schedules.\n\nIn particular, the paper’s value to this audience is:\n1) A “dimension-free” perspective via weak metrics: It reframes convergence/error analysis away from KL/Wasserstein toward smooth test-functionals where empirical convergence can avoid explicit dimension dependence, which is conceptually relevant for high-dimensional generative modeling.\n2) An interpretable error decomposition: The continuous-time bound cleanly separates finite-sample, mixing, score approximation, and cutoff effects, which is useful for reasoning about where generation error comes from in theory.\n3) Scheduling as optimization of discretization bias: The derivation of an explicit schedule aligned with commonly used heuristics provides a potentially useful theoretical justification and could motivate more principled schedule choices.\n\nTo summarize, the audience interest will be mostly on the theory side and not on the applied side, unless the paper strengthens empirical validation and clarifies how the weak-metric guarantees relate to standard evaluation criteria and to training–schedule interactions.", "requested_changes": "Major: \n\n1) Scope “dimension-free” precisely:\nMake explicit what constants depend on (e.g derivative norms, information terms) and discuss how they may scale with d. Provide at least one concrete, nontrivial class of observables G for which the required bounds remain dimension-stable (e.g., low-dimensional feature/projection expectations), and note what discrepancies the weak metric may miss (high-frequency/perceptual artifacts).\n2) Address schedule–training interaction ($\\varepsilon$ likely depends on g):\nEither include theory/discussion acknowledging $\\varepsilon=\\varepsilon(g)$, or add an empirical ablation: baseline vs $g^\\*$ with unchanged training time sampling/weighting vs $g^\\*$ with matched reweighting sampling.\n3) Strengthen empirics beyond 1D toy: Add at least one higher-dimensional controlled experiment (e.g., known-score synthetic or learned score in moderate d) validating step-size scaling and schedule effects, ideally also reporting whether gains transfer to at least one standard metric.\n\nMinor: \n\n4) Make the flow-matching / rectified-flow connection explicit. Add the exact mapping between OU time s, algorithmic time t via s=g(t), and FM/RF interpolation $\\tau\\in[0,1]$; clarify in what sense the schedules coincide (e.g., linearizing $\\alpha=e^{-s}$ in $\\tau$, not full equivalence of coupling paths of VP spherical interpolant vs linear RF path).\n5) Discuss proxy-optimality vs practical optimality. How tight is the bound and how sensitive is performance to deviations from $g^\\*$? \n6) Guidance for $\\delta, T, T'$. Provide heuristics for choosing the cutoff $\\delta$ and time horizons balancing mixing vs score accumulation.\n7) Discrete state-space section: deepen or streamline. Currently, it reads as mainly illustrative. Clarify whether it is intended as conceptual motivation or a true extension. If it is an extension, add at least one nontrivial discrete example (e.g., larger state space, and whether any schedule implications carry over). Otherwise, consider moving it to an appendix.", "broader_impact_concerns": "I do not see major novel ethical concerns specific to this submission beyond those typical for work that can improve generative modeling (e.g., potential downstream misuse for generating deceptive or harmful content).", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "03cPNbq0JN", "reviewer": "Reviewer_S1gY", "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": "", "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": "## Contributions\n\nThis paper studies the convergence of score-based generative models under a novel criterion: the expected distributional error on a class of smooth functionals on the space of probability distributions. This is a weaker condition then the convergence in terms of KL or W2, and the authors obtained dimension-free convergence rate under finite training data samples.\n\nThe key technical tools to establish the convergence of continuous-time dynamics are from mean-field SDEs (which I haven't fully understood due to my limited background in this area), and to analyze the discretization error, the additional tool is from the renowned Conforti et al. 2025 paper. The authors also provided insights into the optimal choice of schedulers that minimizes an upper bound on the discretization error, which is the first rigorous study of this issue to the best of our knowledge. This optimal scheduler turns out to be tight in the Gaussian case, which is a nice property. (For discrete diffusion model, https://arxiv.org/abs/2508.04884 showed that the cosine schedule is optimal under Fisher-Rao geometry. I'm curious if your obtained scheduler is also Fisher-Rao optimal).\n\nOverall, I think this is a solid theoretical contribution to the understanding of score-based generative models, and I recommend acceptance at TMLR.\n\n## Weaknesses\n\nThe main weakness of this paper is that the authors haven't explained the criterion they used to measure the distributional error in detail. As discussed above, this is a weaker condition than the commonly used metrics, and the authors haven't discussed its connections to other metrics and its implications in practice. For instance, the metric $G(m)=\\langle\\varphi,m\\rangle$ for some smooth test function $\\varphi$ should be connected to the bounded Lipschitz distance if you assume $\\varphi$ has regularity up to the first order. Are there any other examples of $G$ beyond inner products with smooth functions? Also, if the authors can provide some examples showing such metric can be much weaker than W2 or TV distance (e.g., when TV is large while the error in $G$ converges to zero), that would be helpful for readers to understand the implications of this metric.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The claims made in the paper are supported by well-written proofs and experimental results.", "audience_yn": "Yes", "audience_explanation": "I myself feel the paper has provided an interesting perspective to analyze the convergence of diffusion model, and I believe it would be interesting to people in sampling theory and probabilistic methods.", "requested_changes": "- Could the authors provide some examples of the functional $G$ and how to compute their linear derivative?\n- In theorem 3.5, the authors haven't made it clear the constant $C$ and $C_T$. How do they depend on the bound of derivatives in assumption 3.4? The current statement seems to suggest that when these derivatives are all $O(1)$ then $C,C_T$ are also $O(1)$, but what about the case when these derivatives scale with dimension? Also, I suggest the authors to provide some brief discussion of the obtained bound in both theorem 3.5 and 3.10, and in particular what each term represents, although it seems obvious to readers familiar in the analysis of diffusion models.\n\nMinor comments:\n\n- This paper uses $m$ to denote the probability distributions, which is not a common practice in the literature and I suggest the authors to use more standard notation such as $p$ or $\\mu$. Also, the notation $\\nabla\\log p$ in the discrete-time setting is originally $\\nabla\\log\\widetilde{p}$ in the Conforti et al. 2025 paper.\n- Paragraph before Eq (1): $\\mathrm{Law}(\\overleftarrow{X}_0)=m_T^N$.\n- In assumption 3.1, the meaning of $H$ is not introduced.", "broader_impact_concerns": "NA", "additional_comments": "- Do the authors think the same approach can be used for analyzing the convergence of *equilibrium* MCMC samplers such as Langevin dynamics / Langevin Monte Carlo / proximal sampler?", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)