Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients
Authors: Stefano Bruno
Sotirios Sabanis
Venue: Accepted by TMLR ()

Abstract: Score-based Generative Models (SGMs) approximate a data distribution by perturbing it with Gaussian noise and subsequently denoising it via a learned reverse diffusion process. These models excel at modeling complex data distributions and generating diverse samples, achieving state-of-the-art performance across domains such as computer vision, audio generation, reinforcement learning, and computational biology. Despite their empirical success, existing Wasserstein-2 convergence analysis typically assume strong regularity conditions--such as smoothness or strict log-concavity of the data distribution--that are rarely satisfied in practice. In this work, we establish the first non-asymptotic Wasserstein-2 convergence guarantees for SGMs targeting semiconvex distributions with potentially discontinuous gradients. Our upper bounds are explicit and sharp in key parameters, achieving optimal dependence of $O(\sqrt{d})$ on the data dimension $d$ and convergence rate of order one. The framework accommodates a wide class of practically relevant distributions, including symmetric modified half-normal distributions, Gaussian mixtures, double-well potentials, and elastic net potentials. By leveraging semiconvexity without requiring smoothness assumptions on the potential such as differentiability, our results substantially broaden the theoretical foundations of SGMs, bridging the gap between empirical success and rigorous guarantees in non-smooth, complex data regimes.

Published in Transactions on Machine Learning Research (09/2025)
Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients
Stefano Bruno
sbruno@ed.ac.
School of Mathematics
University of Edinburgh, United Kingdom
Department of Industrial Engineering
Ulaa National Institute of Science and Technology, Republic of Korea
Sotirios Sabanis
s. s.sabanis@ed.cce
School of Mathematics
University of Edinburyh, United Kingdom
National Technical Universily of Athens, Greece Archimedes, Athena, Research Center, Greece
Reviewed on OpenReview: https: // openreview. net/ forum? id= uS9a VSR77F
VRB7XF
Abstract
Score-based Generative Models (SGMs) approximate a data distribution by perturbing it with Gaussian noise and subsequently denoising it via a learned reverse diffusion process. These models excel modcling complex data distributions and gencrating diverse samples, achieving slate-of-the-art, performance eration, reinforcement learning, and computational biology. Despite their empirical success, existing Wasserstein-2 convergence analysis typically assume strong regularity conditions- such as smoothness or strict log-concavity of the data distribution- that are rarely satisfied in practice. this work, we establish the first. non-asymptotic Wasserstein-2 convergence guarantees for SGMs targeting semiconvex distributions with potentially discontinuous gra- dients. Our upper bounds are explicit and sharp in key parameters, achieving optimal dependence of O(va) on the data dimension d and convergence rate of order one. The frame- work accommodates wide class of practically relevant distributions, including symmetric modified half-normal distributions, Gaussian mixtures, wellle-well potentials, and elastic net potentials. By leveraging semiconvexity without requiring smoothness assumptions on the potential such as differentiability, our results substantially broaden the theoretical foun- dations of SGMs, bridging the gap between empirical success and rigorous guarantees in non-smooth, complex data regimes.
domains such computer vision, audio gen-
1 Introduction
Score-based Generative Models (SGMs), also known as diffusion-baase based generative models (Song & Ermon
2019 Song et al. 2021 Sohl-Dickstein et al. 2015 Ho al.] (020), have rapidly emerged over the past few
years as popular approach in modern generative modelling due their remarkable capabilities in generating complex data, surpassing previous state-of-the-art models, such Generative Adversarial Networks (GANs)
(Goodfellow et al.
2014 and Variational AutoEncoders (VAEs) (Kingma & Welling 2014). These models
are now widely adopted in computer vision and audio generation tasks Kong et al. 2020 Chen et al. 2021
Mittal et al. 2021 Avrahami el. al.] 2021 Kim et al. 2021 Bansal el al. 2023 Saharia et al. 2022 Po el al.
2023 Zhang et al. 2023), text generation (Li et al.
2022
Yu et al.
2022
Lovelace et al.
2023). sequential
data modeling (Alcaraz Strodthoff 2023 Tashiro et al.
2021 Tevet et al.
2023). reinforcement learning
and control (Pearce al. 2023 Chi et al. 2023 Hansen Estrucc et al.
2023
Reuss al. 2023 Zhu al.
2023
Ding & Jin 2024 as well life-science Chung
Ye 2021
Jing et. al.
2022
Watson al. 2023

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Published in Transactions on Machine Learning Research (09/2025)
Song et al.
Weiss et al. 2023 We refer the reader to the survey papers Yang et al. 2022
22033
Chen
et al. (2024) for a more comprehensive exposition of their applications.
The primary goal of SGMs is to generate synthetic data that closely match a target data distribution TD. given a sample set. In particular, these models generate approximate data samples from high-dimensional data distributions by combining two diffusion processes, a forward and a backward process time. The forward process is used to iteratively and smoothly transform samples from the unknown data distribution into (Gaussian) noise, while the associated backward process reverses the noising procedure to generate new samples from the starting unknown data distribution. A key role in these models is played by the score function, i.c. 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 with the backward process. Since this quantity depends on the unknown data distribution, an estimator of the score must be constructed during the noising step using score-matching techniques (Hyvärinen 2005
Vincent 2011
The widespread applicability and success of SGMs have been accompanied by a growing interest in the theoretical understandings of these models, particularly in the convergence analysis under different metrics
such Total Variation (TV) distance, Kullback Leibler (KL) divergence, Wasserstein distance, e.g., Block
ct al. 2020): : De Bortoli et al.
20021: 9: Bortoll (202)); Lee et al. (2022): Yang & Wibisono (2022): ; Kwon et al.
(2022): : Liu et al. 2002)); Oko et al.
(2023):][aa Lee Lee et al. (2023); ; Chen et al. (2023a
b):
Li et al. (2024); Pedrotti
iet al. 2024); Conforti et al. (2025); Benton et al. (2024); Strasman et al. (2025
; Bruno et al. 2022 : Tang
& Zhao (2024);
Swwnikkos-Staaattouuuoo et al. (2024); Wang & Wang (2024);
; Silveri & Ocello (2025); Yu
& Yu (2025). In this work, we provide a non-asymptotic convergence analysis in Wasserstein distance of
order two, as this metric is often considered more practical and informative for estimation tasks (sce e.g., equation 4 and is closely connected to the popular Fréchet Inception Distance (FID) used to assess the
quality of images in generative modeling (see, e.g., Section 4 A significant limitation of prior analysis in
Wasserstein-2. 2, e.g., Strasman et al. (025 ) Gao et al. (2025) Bruno et al. (2025): Tang & Zhao (2024); Wang
& WWnn (0044: Yu & Yu (2025), is their reliance on strong regularity conditions-sin such as smoothness or striet concciicyvvty of the data distribution and its potential. These assumptions facilitate mathematical
tractability but limit the applicability of theoretical results to more general settings, especially when the
data distribution is only semiconvex and the potential's gradient may be discontinuous. The only exception outside the strict log-concavity regime is the recent contribution in (Silveri & Ocello (2025). where the authors assumess that the data distribution is weakly convex. However, their analysis still requires the potential to be & Ocello (2025 Proofs of Propositions B.1 and B.2)), and the stepsize of their generative algorithm must be bounded by a quantity inversely proportional to the one- sided Lipschitz constant of the potent ial Silveri & Ocello (2025 equation (30))) Still, such conditions on TD in existing Wasscrstein-2 convergence analysis do not fully reflect the complexity of real-world data, which often exhibit non-smooth or non-log-concawe distributions. Therefore, the aim of this work is to address the
twice continuously differentiable (see, e.g., (Silveri {.g.eSiivveri
following fundamental question:
Con Score-bused Generalive Models be guaranteed to converge in Wasserstein-22 distance when the data distribution is only semiconvez and the potential admits discontinuous gradients?
We provide a positive answer to this question by combining recent findings in non-smooth, non-log-concavs sampling, with standard stochastic analysis tools, thereby presenting the first contributions in the based gencrative modcling litcrature for non-smooth potentials. We establish explicit, non-asymptotic Wasserstein- convergence bounds for SGMs under semiconvexity assumptions on the data distribution, accommodating potentials with discontimuous gradients. This framework covers a variety of practically relevant distributions arising in Bayesian statistical methods, including symmetric modified half-normal distributions, Gaussian mixtures, double-well potentials, and elastic net potentials, all of which satisfy our relaxed assumptions.
In addition, our estimates are explicit and exhibit the best known optimal dependencies in terms of data dimension, i.c., O(vd) in Theorem 19| and rate of convergence, i.c., 0(7) in Theorem [1 In contrast. to prior works under the same metric Silveri & Ocello (2025: Gao et al. (2025): Strasman et al. (2025): Tang
& Zhao 2024), our estimates in Theorem 19 and Theorem 21 derived without imposing any restrictions
2

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Published in Transactions on Machine Learning Research (09/2025)
the stepsize of the generative algorithm[ making them more suitable for practical implementation. By circumventing the need for strict regularity conditions on the score function and allowing discontinuities in the gradients of the potentials, our work significantly expands the theoretical foundation of SGMs. This advancement not only bridges the gap between empirical success and thcoretical guarantecs but also opens new avenues for the application of diffusion models to data distributions with non-smooth polentials.
One source of error in the construction of the generative algorithm arises from replacing the initial condition of the backward process with the invariant mcasure of the forward process. To ensure this error remains small, the drift terms of both SDEs must satisfy, for instance, a monotonicity property with time-dependent bound that meets an appropriate integrability condition (sce, e.g., equation 19 and equation 23 below). To address this, we identify time horizon for the generative algorithm that ensures the paths of the two backward processes become contractive. Notably, the integrability condition on the monotonicity bound depends only on the known constants in Assumption [ making it signíficantly casier to verify in practice compared to the analogous condition in Silveri & Ocello (2025 Appendix C), which relies on weak convexity constants that are often difficult to estimate.
In conclusion, we present the first explicit, dimension- and parameter-dependern Wg-convergence guarantees for Score-based Generative models operating on data distributions having potentials with discontinuous gradients. Our results mark an important step forward in the rigorous analysis of SGMs, providing both theoretical insights and practical tools for advancing generative modeling in challenging, non-smooth regimes,
Notation. Let (S,, F, P) be a fixed probability space. We denote by E[x] the expectation of a random variable X. For 1 p < 00, TP is used to denote the usual space of p-integrable real-valued random variables. The I'-integrability of random variable X is defined as E[|X]]] < co. Fix an integer d 1. For an R--valued random variable X, its law B(R²), i.e. the Borel sigma-algebra of Rd is denoted by L(X). Let T > 0 denote a time horizon. For a positive real number b, we denote its integer part by [b]. 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 f Rd + I be a continuously differentiable function. The gradient of f denote by V.. For any integer q 2 1, let P(TR)) be the set of probability measures on B(TR)). For μr, P(IR"), let C(u,v) denote the set of probability measures ( on ((IR2d) such that its respective marginals are a and v. For any μ and P(RR), the Wasserstein distance of order 2 is defined
( inf
(CCC())) //// u/² dc(x.)))
W2(u,))
Table 4 (Appendix E lists the main symbols used throughout this work along with references to where they are delined.
2 Technical Background for OU-based SGMs
In this section, we briefly summarize the construction of score-based generative models (SGMs) via diffusion processes, as introduced by [Song el al. (2021). The core idea behind SGMs is to employ an ergodic (for- ward) diffusion process that gradually transforms the unknown data distribution TD E P(Rd) into a known prior distribution. A backward (in time) process then learned to transform the prior back to the target distribution an by estimating the score function of the forward process. In our analysis, we consider the forward process (Xt)teJo.T) to be an Ornstein- Uheenbbek (OU) process, consistent with the choice in the original paper Song et al. (2021)
dX, = -x, di + 2 dB, X D TD,
(1)
where (Bo)eeto,1y is d-dimensional Brownian motion and we assume that E| 2] oc.
The results in Silveri & Occllo 2025 Cao al. 2025 Strasman et 2025): ) Tang Zhao 2024 require the stepsize
to be controlled in terms of the Lipschitz constant the strong convexity constant the target data distribution; see, c.g.,
Table helow. The only exceptions the results Bruno al. 2025 Remark 12 and Theorem 10), which, however, requires
stronger assumptions the data distribution than Assumption below.
3

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Published in Transactions on Machine Learning Research (09/2025)
For target data distributions TD that are absolutely continuous with respect to the Lebesgue measure, and whose densities are continuous and integrable, the backward process (Yt)eE(o.T] (Xeeo) is well defined Millet et al. 1989; Haussmann & Pardoux 1986), and is given by
dy; = (Y, - 2V Po dt + 2 dB,, Yo ~ ((XT),
(2)
where {pe (e is the family of densitics of {((X,) (eEo,T) with respect to the Lebesgue mcasure, B, is an another Brownian motion independent of B₁ in I defined on FPP. In practice, however, the initial distribution is taken to be the invariant measure of the forward process, which corresponds to the standard Gaussian distribution. As a result, the backward process in 2 becomes
d² = (Y, + 2 V Pog dt + V2 dB, Po The = =
(3)
Since the target distribution TD is unknown, the score function V Vlogp, in 2 cannot be computed exactly. To overcome this limitation, an estimator s(., 0", -) is learned based on a family of functions s [0, x xR" + H paramctrized in 0, aiming at approximating the score of the ergodic forward process over a fixed time window [0, In practice, 8 are neural networks and in particular cases, e.g., the motivating example in Bruno et al. 2025 Section 3.1), the functions s be carefully designed. The optimal value 0* of the parameter 0 is determined by optimizing the following score-matching objective
Rdd 3 + E L VVlgpi(X)) - s(1, 0,), di
(4)
An explicit expression of the stochastic gradient of derived via denoising score matching (Vincent 2011 is provided in Bruno et al. (2025) equation (8), Section 2). Following Bruno et al. (2025 Section 2), we define an auxiliary process ({""))reoom) that incorporates the approximating function 8, which depends on the (random) estimator of 0r denoted by 0. For & E [0,T], this process is given by
dy (Y + s(T - 0,,0, ))ux)) dt + v2 dB,, youx To =x :
(5)
The auxiliary process] serves as a bridge between the backward process 3 and the numerical scheme 7 and it facilitates the analysis of the convergence of the diffusion model (see the upper bounds involving Y in the proof of Theorem 19 in Appendix C for further details). We now introduce the numerical scheme. Let
the step size 7j- = 7 E (0, 1) for each 0, 0,....J, where J E N. The discrete process ) of the Euler- Maruyama approximation of is given, for any j JJ,,....J,, , as follows
y ++1 = yj + (Y, FM + 2 sT ty, Y,,.)) 24 Zy+1. YO Tox N(O,
(6)
where {2 (2)..E(o......11))) is a sequence of independent d-dimensional Gaussian random variables with zero mean and identity covariance matrix. The continuous-time interpolation of 6 for t E (0, is given by
drra (141 +2 + L1/71,, P,M () di v2 d
PrE xx = =N(00
(7)
where 2(PEM)- (())EM) at grid points for each j {0.................. + +1}.
3 Wasserstein Convergence Analysis for SGMs
In this section, we provide 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 discussed in Bruno et al. (2025 Section 2 and Appendix A), it may be necessary to restrict E [c, T] for e E (0, 1) in 4 to account. for numerical instabilities that can arise during training and sampling near f = 0 as also observed in practice in [Song et al. (2021) Appendix C), and for the possibility that the integral of the score function in 4 may diverge when t= Therefore, we truncate the integration in the backward diffusion at. T and consider the process (Y:)
2The regularity of the Orustein Uhlenbeck semigroup for all (0.T. (sce Appendix A. and Conforti ct al. 2025 Proof of Proposition 3.1)) ensures that the necessary and sufficient conditions for the reversibility of the diffusion process are satisfied; Thearem 2.1). These conditions mp included
see, e.g., Millet al. 1989 Theorem 2.2) or (Hausemann & Pardoux 1986
in Assumption
4

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Published in Transactions on Machine Learning Research (09/2025)
3.1 Assumptions
We begin by stating the main assumptions of our setting. The optimization problem in can be solved using algorithms such as stochastic gradient descent. (Jentzen et al. (021), ADAM (Kingma & Ba 2015 Stochastic Gradient Langevin Dynamics (Bruno et al., 2025, Section 3.1), and ThcoPouLa (Lim & Sabanis, 2024), provided they satisfy the following assumption.
Assumption 1. Let 0* be minimiser" of and let 0 be the (random) estimator of 0r obtained through some upprosimation procedure such that |100
co. There exists At such that
E 0 <
Remark 1. As consequence of Assumplion 1 one oblains E[ < AL + 2/0
In this work, we consider the potentials to be semiconvex functions a broad generalization of convex func- tions that includes non-convex functions whose eurvature is bounded from below. This class allows for discontinuities in the gradient while retaining key analytical properties of convex functions, such as the existence of well-defined subgradients.
Definition 2. A function U is semiconvex if there exists K such that U + 1
-2 is convex.
Semiconvexity has received significant attention in the machine learning community (Davis et al.
2018 Sun
& Yu
2019 Richards & Rabbat 2021 Liu et. al. 2021: Rafique et al. 2022), optimization (Li et al.
2020 Ma
2020 Li & Xul 2021] Hu ct al. 2025), optimal control (Cannarsa & Sincstrari, 2004). and the study ct al.
of fully nonlinear partial differential equations (Braga et al. 2019 Payne & Redaelli 2023 We refer the
reader to Duda & Zajucek (2009): Cattiaux & Guillin (2014) for comprehensive overviews of the mathematical challenges associated with this class. Importantly, semiconvex functions may admit discontinuous gradients
which are characterized using the Fréchet subdifferential (Bazaraa et al. 1974 Alberti et al. 1992).
Definition 3. For a function U Rd + E, we define the subdifferential (U(x) of U at E Rd as
(U(x) p RdR lim inf U(2) - U(x) (5, x) 0). 2-5
1
The set B is closed and convex, and may be emply in general. We say that U is Fréchet subdifferentiable at if (U(x) 0. Any element hxx) (U(x) is called Fréchet subgradient of U at Rd. When U is differentiable at x, the subdifferential reduces to (U(x) (vU))). Crucially, semiconvex functions are
Fréchet subdifferentiable at every points in R4, i.e. @U(x) * 0 for all x dAlberti et al.
1992 Cannarsa
& Sinestrari 2004). In this case, h(x) + Kx corresponds to the classical convex subgradient (Vial 1983
Proposition 4.6). Moreover, every clement of the subdifferential of a semiconvex function satisfics a ono-sided
Lipschitz condition, ensuring the existence of h(x) E (U)).. The following lemma- adapted from (Alberti et al. 1992 Proposition 2.1) and presented in (Johnston al. 2025 Lemma 1) and its subsequent corollary formalize this property.
Lemma 4. (Alberti al. 1992 Modification of Proposition 2.1) Let a semiconvez function. Then, is locally Lipschitz continuous, the subdifferential set OU is non-empty, compact, and 5 E OU (x), if and only if
U(z)-UP - -)) K
x!?
for all x, E d
Corollary 5. [Johnston et al. 2025
Corollary 1) Let xr, 2 E Rd,, p E @U()), and q e (U)... Then,
(p q. x 2) K x 2/2
The scoro-matching optimization problem is not necessarily (strongly) convex.
4Sme of these references refer semiconvex functions weakly convex. We avoid this terminology to prevent, confusion with the nation weak convexity introduced in Definition helow.
5

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We state the assumption the target data distribution TD below. Recall that h(x) E aU(x) is the Fréchet subgradient of U at E IRd.
Assumption 2. The data distribution To has a finite second moment and it is absolutely continuous with respect to the Lebesgue measure with TD(dr) = exp(-U(x)) dr for some U : Rd - RR. Morcover,
(i) The potential U is continuous and its gradient erists almost everywhere.
(ti) The potential U is K-semiconver (on ball). That is, there crists K, R 2 0, such that for all 2. E ERA,
(h(x) - -()), - B) K| EP. when x x- F < R.
(ti) The potential U is p-strongly conver at infinita) That is, there exists μ > 0 such that for all E, Rd.
(h(z) h( - 2 u/x xf?
when r R
(9)
where R is the same as in point (ti).
Remark As a consequence of Proposition 23 due to Conforti el al. 2025 Proposilion 3.1), and As- sumption 2 (i), we have that, for any t. E (0, the map x + (p) is continuously differentiable, and for any x E Ra, the map ++ pe(x)) is continuously differentiable on (0, T]. Moreover, Assumption 2 implies that the processes in and 3 have unique strong solution.
Next, we consider the following assumption on the approximating function 8, which is also adopted in Bruno et al. 2025 Assumption 3.a).
A Assumption 3.a. The function s: [0, [] x RM M x Rd + Rd is continuously differentiable in E Rd, Let D RM AR RR,, D2 [0, x [0, 7] I,, and D3 [0, x] x (0, 7] RR, be such that S:" Je D2(1, di dř do and Se_ Set B) dt dē < co. For 1 1] and for all t, f [0, 2, Rd, and 0, 0 E RM, we have that
|s(1, <T,0,,) < (0,00) 70 (2(1, 0 Da(1,2)xx FI,
where D1, D2 and D3 have the following growth in each variable: i.e., there exist K1, K2, and K3 0 such that for each t, E [], and 0,0 E RM,
[D, 0) < K1 + 10] + (0), (Da(t, 8) & Ka(1 + 1 + (7%).
[D2(t,0 S K2(1+ +1 + 111)),
Remark 7. Assumption 3.0 requires that the approzimating function is Lipschitz continuous in both the imput variable x and the parameter 6. In time t, it allows to be either Hölder continuous (for E [2. 1)) or Lipschilz continuous (for a a= =1= 1). This relazed continuily in I for the drift lerm of 7 is slandard for the Mader-Maruyame schemes for SDEs. Crucially, we show in Theorems 19 and 21 that this weaker condition still guarantees convergence of the generative algorithm to .D. As noted by Bruno al.) 2202 Remark 6) in the context of neural network-bused approzimations, Assumplion 3.a. when a 1, covers the case where neural network with hyperbolic tangent. or sigmoid activation function at the final layer. Moreover, Assumption 3.0 implies that the process in
is implemented
5 6 and have unique strong solution.
Remark 8. Let Kreeal = K1 + K2 + K3 1s(0,0,00) 0)1 0. Using Assumption 3.a one obtains
st 0)x) S K (1 + (1 + [x]).
The proof of Remark 8 can be found, e.g., in Bruno el al. 22025 Appendix D.3). By imposing an additional
condition on the gradient of s in Assumption 3.a as done in Bruno et al. (2025 Assumption 3.b) we obtain the optimal convergence rate established in Theorem 21 below.
Assumption 3.b. Lot s be as in Assumption 3.a and there exists K. > 0 such that, for all 2, E Rd and for any k = l,
(H) V (t,0, x) Ka 2/ z/.
Intuitiivlyy outside sufficiently large region, bends upwards least as much as quadratic function.

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Published in Transactions on Machine Learning Research (09/2025)
For the following assumption the score approximation, let 1 be in Assumption
1 and we let
(EE00))(()77) be the auxiliary process defined in 5
Assumption 4. There exists eaN > 0 such that
V log Pr - 8(T -1,,-,,, -,,,,,,,, Y,unn)p dr < ESN.
(10)
Remark 9. Assumption 4 is now a standard assumption considered in the literature, see, e.g., Gao et al. (2025); Bruno et af. 2025 Strusman al. 2025); Silveri Ocello (2025), and its theoretical and pructicul soundedness discussed, e.g., in Bruno ol. 22025 Remark 7, 8, and 9).
3.2
Assumption 2 and Weak Convexity of the Data Distribution
We start by introducing the definition of weak convexity, concept that has been widely used in conjunction with coupling techniques to analyze the long-time behavior of gradient flows SDEs (Conforti 2023 Conforti et al. 2023b and we extend its application here to subgradients of U.
2024
Definition 10. The polential U IRd R is weakly convex if its weak convexity profile ky 10, (0) - R
defined as
(M(x)) h(T), - 5)
Ky (r)
inf a, [--a|=r
(11)
where h(x) E (x) is the subgradient of U at x E Rdd satisfies
Key
- B-)
for all r >>> 0,
(12)
for some constants B,L 0, where the function ft [0, oo] + 0o] defined as
fL(r) = 22 tanh (
(13)
Remark 11. The weak convesity profile Ky in 11 serves as an ameraged/integrrtlen converily lower bound for the potential U, evaluated over pairs of points separated by a distance r > 0. Unlike the standard converity condition KU 2 0, which characlerizes convex (or log-concave)) polentials, Desinition 10 allows Kt to vary with r, thereby admitting non-uniform lower bounds. This generolization yields a broader and more flexible
notion of converity that extends beyond classical log-coneavity,, making Definition 10 substantially weaker
than log- conccviiyv (Conforti 2023 2024- Conforti et al. 2023b Silveri & Ocello 2025).
We modify Conforti el al.
(2023) Lemma 5.9) to our setting, namely when 3 expression of the weak convexity constant at each E (0.
have an explicit
Lemma 12. CCofortii et al. 2023b Modification of Lemma 5.9) Assume that U is weakly conver as in Definition 10 and fir (0,7). Then, the function x + - log (r(x) is weakly conver with weak converity profile K_ log (x)) satisfying
B
e
log (r)
-2 B (1 3) Sr.
B
+
B)
In porticular, the score function satisfies
(elggp((a) V Vlogp( z) -,,(z xp. for x, E
(14)
with
e 2t
Ō 3 B) 2t
Be 21 L.
(15)
(8
Recently, this notion has been used in the context of score-based generative modcls Silveri & Occllo
2025
Definition
3.1).
See Silveri & Ocello 2025 Lemma R.4) for similar statement.

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We show that Assumption (ii) and Assumption 2 (iii) are related to the notion of weak convexity (Definition
10p in the sense made precise in Proposition 13 below. An overview of the proof of Proposition 13 below can
be found in Appendix [
Proposition 13. Let the data distribution ao be in Assumption 2 and let fr and L be as in Definition
10
Then the potential U is weakly conver in Definition 10 with
(r) 2 p fz(r,, for all r O,
(16)
where μ > 0 in 16 is the strong converity at infinity constant from Assumption 2 (tii). Conversely, if U is weakly conver as in Desinilion 10 with lower bound 10 for some known constants μ and L >> 0, then
I. The potential U is ji-strongly conver at infinity with ÿı μ R- (R) B. E Rdd, we have
such that for all
(h(x) h()), 8) 2 u x/?
when
rr
R.
(17)
which holds for all R > 0 when μ > L and for R 2 Ro =
250 1i/2
with being the solution of 45 when
S L.
The potential is K-semiconwez, such that there exists x (h(x) h(ā), - x) 2 k| 7/.
2 0 for all x, x E Id,
when
Jx F < R.
(18)
where R the same as in point (2).
As consequence of Proposition 13 and Lemma 12 one obtains the explicit form of Ō2 in 14 in our setting, which is given in the following corollary.
Corollary 14. Let U be K K-semiconvez as in Assumption tii and be p-strongly conver at infinity as in Assumplion (ii)) and fir & (0, (0, Then
(o)gg(())) - V Vo
(T), x- - x) S -3,00 F for E.
(19)
where
e 2t
305
J + (1 21 (1+ (1 4) L,
(20)
for some L > satisfying 16 Remark 15. By Corollary and the proof of Proposition 13
we have
lim 3 = L <- [
(21)
which shows that -3.0 is not the lowest bound for the left-hand side of 19 We emphasize that the gap between the limit on the left-hand side 22 and the semiconveaity constant K due to the particulor choice of fr. in 1I in Proposition 13 This gap may vanish if we replace fr. with an appropriate function f F Conforrti et al. 2023a Section 2.1.2), (Conforti et al., 20236 Section 5.3.1), where
8:= (E0)((,))). r r f(22122 non-decreasing, concave, bounded such that
sag.
limrf(r= =0 f 20, 2f" + +II + +40
Note that F is non-emply and contains r + 2 Stanh(r/2), For this reason, we use the constant K + l as prozy of the constant and replace 20 with the following monotonicity bound
e 2t
8
K...K.P
(n (1- 2 (K + 11).
(22)
11 (1 u)--21
Moreover, it holds that
lim 3 KS
We refer to the proof of Proposition
13
in Appendix below for the derivation of this constant.
Indeed, this shows that 50 K.p
20S for the hand side of 11 Corollary 14
8

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Published in Transactions on Machine Learning Research (09/202))
0.8
Pl.g))
Vlogp((0.5)
0.7
0.6
0.4
0.3
0.2
0.0
-0.1
-044
0.5
Time
Figure Score function 25 for fixed x and time f.
and
lim 3,00 lim 8 0.S.K.H =
1-30
t+000
which is consistent with Too ~ NN(0, La), the invariant distribution of the OU process.
Using the explicit expression of 22 are able to find time for which the integral of the monotonicity
bound 10 Bt K is positive. The proof of the following result is postponed to Appendix B
Proposition 16. Let 2> and 2 The time integral of K. from Remark
15 is
/'"
1
20
20)2 (K + m)) ds 20
B(1, 0, K)
11 25
(n + (1
(23)
1
log (u(c?2 - 11) +
K
+1 P
1u(22t - - +1 +1 - 1)] >0, 0,
1
2
when > t" > t:= In (V1+ (2) with t* = inf (t > 0: Bt, H, K. > 0}.
Remark 17. 1f we consider the case when K = 0 in Assumption (ii), then is satisfied for all > O.
Remark 18. We provide a numerical ilustration of the critical time of Proposition 23 when non-log-
concave distribution becomes log-comcon cogcooce in the case when TTD is a one-dimensional Gaussian minture with lwo equi-weighled modes 1) = 2, each mode having same variance 8² 9, namely
(ddr) (2(18T)1/2)11 ( exp (-1x-
1x-22
|
2/2
+ exp (- [1x 2/2 21p) )) dr.
(24)
18
18
For this choice of TD. the semiconverity constant conslant 22220 (57) .ं. Here, R 2 0. The score function of 24 is given by
2n²
81 and the strong converity at infinity
(52)2
(2 2mm )
- 2m,)
2m
exp
2(9m +
exp
29m tof)
logg:(() 9m + of 9m + of exp
E [0, x E RR
(25)
2mz-2m² 2(9m +
2
+ exp
70mp+03
Note that 8 K08,Kyt is function of time.
9

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Published in Transactions on Machine Learning Research (09/2025)
where my = and of 1 - e comes from the representation of the OU process in 52
Figure displays
the time behaviour of score function 25
over the time interval [0, 10] for fixed values
0.8 and = 0.5.
We also indicate the time = in
K = 3.2377 < t" from Proposition 25 The score function in Figure
converges to V oo = -, where Toc = 1).
3.3 Main Results Optimal Data Dimensional Dependence and Rate of Convergence
The main results are stated as follows. An overview of their proofs can be found in Appendix C
Theorem 19. Let Assumptions
2 3.0 and hold. Then, there exist constants C1, C2N C3 and C4 0 such that for any T > and 7, 00 1),
Wa(C(YFMM) TD) S CIVE+ Oze -2
SS. Oze- g. Cc((Tc)VESN C,(T,))7
(26)
where C1, C2, O3 and Ca are given explicitly in Table (Appendir E B K.S.K. is defined in 22 and its
integral compuled in Proposition [ In addition, the resull in 200 implies that for any > 0, o we choose << < < 68, T> To, 0 < ESN < ESN,A and 0 << < <8 with 68, Ts, SsN,6, and 78 given Table
then
W2(C(YFMM) TD) 8.
Remark 20. The constant CT(T,c) in the error bound in 26 contains the optimal dependence of the data
dimension, í.e. O(vd), which has been found under the more strict assumption of strong-log concavity of aD
Bruno al. (2025) Theorem and Remark 12). However, the optimal dependence of the dimension is achieved at the expenses of worst rate of covergence of order 1/2.
The optimal rate of convergence of order [2 1] for the Euler or Milstein scheme of SDEs with constant diffusion coefficients can be attained in Theorem 19 provided that B[]01*] < and that Assumption 3.a. is replaced by Assumption 3.b as stated in Theorem below.
3.b and 4 hold, and assume that B[(0(1] < do Them, there exist conslants C1 C2, Cs and C4 such that for any T > O and E
Theorem 21. Lel Assumplions
E (0,
WM( RD) S CIVE+ Oze -2 S.. 200,8 de-e C2(T, (T(T)))7*
(27)
where O1 C2, Cs and Co are given explicitly in Table (Appendir E B. oS. is defined in 22 and its integral is compuled in Proposition 16 In addition, the resull in (27 implies that for any > 0, if we choose <<E< < c,, T > To, 0 < <SE < ESN,5 and 0 < 78 with Es, Ts. ESN,6, and 7s given in Table then
(2 FM) TD) 8.
(28)
Remark 22. The conslant Č₁ explicilly given in Table (Aypendia exlvibils a linear dependence the data dimension, i.e., O(d). This scaling arises from the erplicit Milstein scheme developed in Kumar & Sabanis 2019), which relies Assumption 3.b and is leveraged in the proof of Theorem 21 to achieve the optimal convergence rate of order a E 12. This explicit Milstein scheme requires control on the fourth moment of the one-slep discretization, see for instance, Lemma 25 (Appendir O which enables a convergence rate in W₂ consistent with the known optimal rate of convergence for the Euler or Milstein scheme of SDEs with conslant diffuusio coefficients. However, this comes at the cost of a worse dependence on the data dimension.
3.4 Examples of potentials satisfying by Assumption
We present several examples to demonstrate the wide applicability our Assumption 2 to a broad class of data distributions, some of which are not covered by previous results Wasserstein distance of order two
Siiverr & Ocello 2025 Strasman al. 2025 Gao et al. 2025 Bruno al. 2025 Tang & Zhao 2024 Yu
& Yu 2025).
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3.4.1 Symmetric modified distribution
We consider the case of a one-dimensional symmetric modified half-normal distribution
TD(dr) = exexp (-Ex²
(x])
dæ, x E R,
(29)
V v
for some unknown { > 0 and normalizing constant
L
+ + (-1)-g-m/2 ((n)
V
m
where [(n) is the Gamma function. We refer the reader Appendix D for additional details about the
derivation of 29 As highlighted in Sun et al. (2023 Section 2), the modified half-normal distribution
posterior distribution to sample from in Bayesian Binary regression, analysis of directional data, and Bayesian graphical models.
appears in several Bayesian statistical methods
Assumption
(i) is satisfied for ((x) = Ex² + |x|. In addition, we have, for all x, E
(M(a) M(F), - 3) = 2 (g) - . + (or 1>1,,, a<o - (x - 1 E20)
(30)
2 2 2
which shows that Assumption (ii) is verified for any K 2 0, and Assumption (ii) is verified for 25. Therefore, we can conclude that 29 satisfies Assumption 2
3.4.2 Multidimennional Gaussian mixture distribution
We consider multidimensional Gaussian mixture data distribution with unknown mean and variance, i.c.,
*D(dr) E (2 exp (- Jar nil 2s
dr, x RR.
(31)
with > 0, 1 E Rd and & [0, for i E {l,.... 1 such that Σ = 1. The authors in Silveri
& Ocello 2025 Appendix A) show that the score function of 31 is Lipschitz continuous and log TD is weakly convex. Thercfore, Assumption 2 is satisfied. In addition, the distribution 31 covers also case of the
weuble-well potential:
U(x) | x E Rd
(32)
which is 2-semiconvex and strongly convex at infinity.
3.4.3 Multi-dimensional Potentials
Similarly as Section 3.4.1 one can proves that the elastic net potential:
U(x) = |xc + Lodl
Rd.d
(33)
satisfies Assumption Moreover, the following potential
U(x) max { [x|3),
E Rd.
(34)
verifies Assumption 2 with K- 0, R 1, and p well as the following non-convex potential presented
in Johnston et al. 2025 Example 4.2):
U(x) = max |x], [x|
x E Rd
(35)
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Published in Transactions on Machine Learning Research (09/2025)
4 Related Work and Comparison
In recent years, there has been a rapidly expanding body of research the convergence theory of Score- based Generative Models. Existing works for convergence bounds can be divided into two main approaches, depending on the divergence distance used.
The first approach focuses on a-divergences, particularly the Kullback Leibler (KI.) divergence and Total
Variation (TV) distance (c.g., Benton et al. (2024]; Conforti et al. 2025)] Yang & Wibisono (2023): Li &
Cail (2024); Block el. al. [2020]; De Bortoli el al. (2021); Lee el al. ( Li et al. (2024); Lee el al. (2023); Chen al. (2023a b): Oko et al. (2023); Liang al. (2025): Yang & Wibisono| (2022)). which are the vast majority of the results available in the literature. Crucially, bounds KL divergence imply bounds
TV distance via Pinsker's inequality, strengthening their wide applicability. We provide a brief and selective overview of some of the findings following this first approach. The results in TV distance in [ee et al. (2022) and in KL divergence Yang & Wibisono/ (2023) established convergence bounds characterized by polynomial complexity under the assumption that the data distribution satisfies a logarithmic Sobolev incquality and that the score function is Lipschitz continuous. By replacing the requirement that the data distribution satisfics a functional incquality with the assumption that TD has finite KL divergence with respoct to the standard Gaussian and by assuming that the score function for the forward process is Lipschilz, the authors in Chen et al. (2023b) managed to derive bounds in TV distance which scale polynomially in all the problem parameters. By requiring only the Lipschitaness of the score at the initial time rather than along the full trajectory, the authors in Chen et al. (2223a Theorem 2.5) managed to establish, using an exponentially decreasing then linear step size, convergence bounds in KT. divergence with quadratic dimensional dependence and logarithmic complexity in the Lipschitz constant. Later, [enton et al. (024) provided KL convergence bounds that are linear in the data dimension, up to logarithmic factors, by assuming finite sccond moments of the data distribution and employing early stopping. However, both the results of Chen et al.] (2023a Theorem 2.5) and Benton et al. [2024 Theorem 1 and Corollary 1) still require the uniqueness of solutions for the backward SDE 2 and therefore additional assumptions on the score function are needed. For further discussion on this point, we refer the reader to Bruno et al. (2025) Section 4.2). Assuming finite second moments and using an exponential integrator (FT) scheme with both constant and exponentially decaying step sizes, the authors in Conforti et al. (2025 Corollary 2.4) derive a KL divergence bound with early stopping, which scales lincarly in the data dimension up to logarithmic factors. Bounds in KL without early stopping have been derived in Conforti et al. (2025) for data distributions with finite Fisher information with respect to the standard Gaussian distribution. We note that this condition on TD stated in Conforti et al. (2025) Assumption H2) still requires that the potential U E (Rd).. The KL bounds provided in Conforti et al. (2025) Theorem 2.1 and 2.2) scale linearly in the Fisher information when an EI discretization scheme with constant step size is used, and logarithmically in the Fisher information when an exponential-ther constant step size Conforti et al. 2025 Theorem 2.3) is employed.
The second approach focuses on convergence bounds Wasserstein distance, a metric which is often consid- ered more practical and informative for estimation tasks. We can relate results following this approach with the results of the first approach only when TD is a strongly log-concave distribution. In this case, boubounds in terms of KT. divergence follow from an exlension of Talagrand's inequality (Gozlan & Léonard 2010 Corollary 7.2). However, for two general data distributions, there is no known relationship between their KL divergence and their Wz. Therefore, we cannot compare our findings in Theorem with the results derived following the first approach. One line of work within the second approach assumes
19 and Theorem 21
(at least) strong log-concavity of the data distribution (Strasman et al., 2025 Gao et al.
2025
Bruno et al.,
2025 Tang & Zhao 2024]] Yu & Yu, 2025). Under this (strict) assumpttonn, Bruno et al.
2025 Remark 12)
achieved optimal data dimensional dependence, i.e., reaching O(vd). The recent bound in Silveri & Ocello
22025 Theorem D.1) exhibits similar scaling in d while relaxing the strong log-concavity assumption
(.g..Siivver & Occllo (2025, Proof of Proposition B.1 and B.2)). Our Assumption 2 is much weaker than this requirement. and it allows lo consider the case of potentials with discontinuous gradients covering a wider range of distributions as outlined in Section 3.4 Another line of work following this approach focuses on specific structural assumptions of the data distribution. For instance, convergence bounds in Wasserstein distance of order one with exponential dependence on the problem parameters have been obtained in Bortoli (2022) under the so-called manifold
to wcakly log- conccviav,,y, but still requiring that the potential A² exists (sce, c.g., /iilveri
12

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Published in Transactions on Machine Learning Research (09/2025)
hypothesis, namely assuming that the target distribution is supported lower-dimensional manifold is given by some empirical distribution. Under the same metrie, the authors in Saimiko-SStaattooulle et al. (2024) provide a convergence analysis when the data distribution is defined on a torus. Under the W2 metric, Wang & Wang (2024) derive convergence bounds assuming that the tail of Tp is Gaussian and that U E C², which is a stronger condition than merely requiring VU to be Lipschitz. We summarize in Table | and the best results obtained in W2, i.c., Bruno et al. (2025); Silveri & Occllo (2025) and compare with our best result, which scale polynomially in the data dimension, i.e. O(Vd) in Theorem 19 As mentioned in the Introduction, previous W2 bounds (Silveri & Ocello 2025 Gao et al. 2025 Strasman et al. 2025 Tang & Zhao 2024) require the stepsize " of the generative a algorithm to be tuned based on quantities that are often difficult to compute in practice, such as the Lipschitz or strong convexity constant of the data distribution, which can lead lo very small stepsizes. In contrast, Theorem and Theorem 21 impose no such restrictions, making them more suitable for practical use. Table summarizes the assumptions on y in prior works and compares them with ours.
We close this section by briefly commenting the choice of deriving our results in Wasserstein distance of order two. Beyond its theoretical relevance, this choice is motivated by practical considerations in generative modeling. First, the Wasserstein distance is often regarded as a more informative and robust metric for estimation tasks. Second, a widely used performance metric for evaluating the quality of images produced by generative models is the Fréchet Inception Distance (FID) Heusel et al. (2017). which measures the Fréchet distance between the distributions of generated and real samples, assuming Gaussian distributions. In particular, this Fréchet distance equivalent the Wasserstein-2 distance. Thus, providing convergence results under the Wasserstein-2 metric enhances the practical relevance of our theoretical findings.
Table 1: Summary of previous bounds for W2(C(FFM)) TD) and our result in Theorem 19 assume that (dxx)) x e=U(x)d has finite second moments.
All the bounds
Assumption on Error bound
Reference
TD
O(viyve + O(Vd)e 2Lxo(T-e)-c -VV))-=uxo((-e)- 0(eltt522mmo)(--00 VESN O ( p2a113/////111111 1/22
U convex, V (0)
strongly
+
Bruno et al (0025
Re-
mark 12)
L ([e, T),, and with TMo 0 lower bound of the strongly convex constant of U, see Assumption
c.g., Bruno al. (2025 Remark 4).
U E CC((Rdd) weakly convex, and Assumption
LL +5)
+5) 2/2) W2 (TD, TT)) + 4T (3, T,, I,,)) +
Silveri
(2 +vd+ vd+ (D L,))],
Ocello 2025
Theorem
with Ly 0 one-sided Lipschitz constant for VU, see e.g., Silveri &
D.1)
Ocello 2025 Assumption H1), n13, L, defined in Silveri & Ocello 2025
equation (29)), and 2/(L 5)2
Assumption and Assumption
Theorem 19
O( -2 S." 808, de-e
+0 (000)0))-)))221 K.8. dt "ESN + O(Vde 220 7 700t122/2)
ALL )t,
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Published in Transactions on Machine Learning Research (09/2025)
Table 2: Summary of restrictions on the stepsizes of the generative algorithm YJM used in the previous
bounds for (2(C(YFM)) mD) and our results in Theorem 19 and Theorem 21 All the bounds assume that <D(d)) & e (u(()) dæ has finite second moments.
Assumption on Restriction on the stepsize
Reference
TD strongly convex, VU Lipschitz con- tinuous, and Assumption and
U
Gao et al.
e
21 -
2025 As-
0
min OST
(1+e
21
1 mo
1))(1 + ((L(t)) 2M1
sumption 4)
used in Gao
et al. 2025
Theorem 2)
1)e
0 7 OSIST
min
mig
mg 1)e
with mo > strong convexity constant for U, see e.g.. Gao et al. Assumption 1), M1 > defined in Gao et al. 2025 Assumption 2),
(0025
L(t) = minogegr(( - --- 2a La) Lipschitz constant of V log pa (z)
and Lo > 0 Lipschitz constant for VU, see e.g., Gao et al. 2025 sumption 1).
As-
U strongly convex, V
Bruno al.
7 E (0, 1).
2025
Re-
(l)
mark 12)
([E, and
Assumption
U strongly convex, and
Strasman
C(T-1) 0 0<7< ( L(T - s)) L(T 11,+1
et al. 2025
VU Lipschitz continuous
Proposition
C.3) used
with it,, OSiSJJ J} regular discretization of [0, T), Cr-s and LT- are strong log-concavity and Tipschitz constant, respectively for V Voog Pt.
in Strasman
8,2"
et al. 2022
where Poa is the density function of mean zero Gaussian distribution with variance o² Ta--
Theorem
4.2)
U strongly convex, VU Lipschitz con- tinuous, and with >> 0 strong convexity constant. Assumption U E C²((R,))
Tang & Zhao
0 <<> < min (2. )
K 2T + m))
(2024 Theo-
rem 3)
Silveri
&
weakly convex, and Assumption
0< (2L + 5)2'
Ocello 2025
Theorem
with Lu 0 one-sided Lipschitz constant for VU, see e.g., Silveri & Ocello (0025 Assumption H1).
D.1)
Assumption 2 and Assumption
Theorem
E E (0, 1).
19
and
Theorem
Acknowledgments.
This work was supported by Innovate UK [grant number 10081810). 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 (No.P002582)). This work has been partially
14

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Published in Transactions on Machine Learning Research (09/2025)
supported by project MIS 5154714 of the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the Next GenerationEU Program.
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Appendix
A Regularity of the Score Function
We recall the following result to justify the smoothness of the map
(0, 1] Rd (t, + p((x) E R+,
(36)
where Pt density of the forward process defined in Section 2
Proposition 23. [Conforti et al. 2025 Proposition 3.1) Let To be absolutely continuous with respect to the Lebesgue measure, and denote its density by Pa- The map defined in 36 is positive and solution of the sollowing Fokker Tokker-Planckk equation on (0, T) x Rd:
0.p((x) - dir( p.) Apr(x) =0, 0, for (1, E (0, x Rdd.
Moreover, il belongs to ((, (00,0,,)x8) xRR; i.c. for any E (0, T), + pr(x) is tuice continuously differentiable, and for any E Rdd, t pe()) is continuously differentiable (0,T).
B Further Details on Assumption and Weak Convexity of the Data Distribution
We provide the proofs of Section
3.2
Proof of Proposition
13 We begin by considering that To satisfies Assumption 2 Recall that fL is defined
as in 13 Note that + r- fr(r) is non-increasing on (0, oo) and (0) L r fr((r) for E (0, R). We
look for L > 0 satisfying
= f = 2 tanh (2L = K + p. inf r - f rE(0.R)
(37)
Equivalently, we look for I ///2 > 0 such that
+ 1 R², subject to x> VK+H R.
x ttanh(x)
(38)
4
2
20

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Published in Transactions on Machine Learning Research (09/2025)
VK+a
L K + n.. Note that ta)h(x) S for all 2 0.
Therefore, if we choose
R then
2
VK +
VK+
VK+
K + n².
R tanh
2 < 4
(39)
2
Using 39 and go x tanh(x) oo, we deduce that there exists xt such that
tanh(x*)) K R²
(40)
4
with x= > v K+ R, since æ + x tanh(x) is non non-decreasing on (0, oo). By Assumption 2 and 37 we have
(r(r) 2 μ - (K + 11)
(41)
2 μl (
for r r R.
Moreover,
ku (r) 2 4
2 μ 2 - - for > R.
where it is used that fr)fr(r) > 0 for all >>>> 0. This proves the first part of the statement in Proposition
13 i.e. the lower bound 16
Conversely, assume that U weakly convex as in Definition with lower bound 16 for some known constants # and O. We look for such that
(t)(r) 2 1 ---
2 4 Rt
(42)
> 0,
Vr > R
where it is used that r-1 r-1fz(r) (r) is decrcasing on (0, oc). Let pt μ R (R(),, so 42 becomes (r pr for all > R. One notes that
μ tanh (RT.1(())))))222 0.
(43)
(RL)/2)/2
If p > L, 43 is satisfied for all R> p S L,
43
holds for R 2 Ru, where Ro the unique solution
2/2
1/2
tanh
(44)
R
2
250 Let R/2 then Ro 7,72,
where 20 solves
tanh
(45)
L
Since tanh(=) monotonically decreases from 0 as increases, a unique zo solving 45 exists for μ
250 Therefore, 42 is satisfied for Rg = L.72- This proves that U is Jisttrongly convex at infinity, and therefore
Using the assumption that U is weakly convex in Definition 10, one obtains that
(r)
2 - fr(rr)))
(46)
2 4 L.
for r R.
If p L, then (r) 2 KK for r S R for all R > 0 and K 2 O. If μ < L, then, by setting K I, in 45 we have (r)rr) 2 - for r S R for all > 0. This proves that is K-semiconvex, and therefore statement in Proposition
We distinguish two cases for the lower bound in 46
18 This concludes the proof for the second part of the
21

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Published in Transactions on Machine Learning Research (09/2025)
Proof of Proposition 16
We look for t" satisfying
Bit K) = 2 log (u(e" - 11)+ +
m +1)
-1)
1
-
1) 0.
(47)
+)+1
Equivalently, we look for e2t" -1 such that
K
μ.E pa + > 0.
gape + 1) log(ur + 1)
+1
(48)
Note that 48 is satisfied for all x > when = In addition, we have
lim g(ux 1) 0, x x-0+
(49)
lim g + 1) = oe, ++c
and
K + 11
K
9 (pt + 1) =
2 0 when =
dx
112 +
(uex + 1)2
112
(50)
d²
1
+ 2(K + 1)
2K 112
dx dzz 9 + +1
2 0 when
S
+
(JuT 1)2
(μx 1)3
By 50 the function g in 48 has minimum at
and
1²²
K =1o0=
+1)
K - log
+11
+1
< 0.
1
1
for all K, 0. By 49 and 50
there exists K
such that ns
48 is strictly positive. Therefore, there exists
t In + such that 47 holds.
C Proof of the Main Results
In this scction, present the proofs of Thcorem bound the moments of the process ) defined in 7 along with estimate for its one-step discretization error. These results will be instrumental in the subsequent proofs.
19 and Theorem 21 We begin by recalling upper
Lemma 24. Brano al. 2025, Lemma 20) Let Assumplions) and 3.a hold, and suppose that B[j0pt] < oc
for any p E [2, 4). Then, for any t [0,T- -c.
sup E
[ p < (t),
where
CEM.p(t) = 113p-1 1 2--2222--11 11222-11p
kP Toai (1+r0P))
E 1
PM
+ 23p-2ppp kP
(1tar +E( + TaP)
2nd p(p - 2))51)
x
and Tottal is defined in Remark
Lemma 25. Bruno et al., 2025 Lemma 21) Let Assumptions and 3.a hold, and suppose that E[0ppl] < oc for any p E [2, 4). Then, for any 1 E [0,T - =t),
| BM
PMM BM 1//Jm
$ 2 CEMose.p-
where
CEMose.p := : 20 (CE,,, + 1200 2 (1+ B((PP)))
+ (dp(p- 1))8
CEM,p and K Total defined in Lemma 24 and in Remark respeclively.
22

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Proof of Theorem 19 We derive the non-asymptotic estimate for W2(C(Y))M) TD) using the splitting
W2(C(Y)MM) ) TD) & W2(TD. (())) + W,(C(Y..) +
(51)
+ -W2(C(),) + (), )()
We provide upper bounds on the error made by the early stopping, i.c. W2(TD, ((),), the error made by ap- proximating the initial condition of the backward process Yo ((Xx) with Yo Too i.e. W2(C(Y,)) ((Y,, the error made by approximating the score function with s, i.e. W2(C((Y,) ((YT)))))) and the discretisation error, i.e. (2(C() ((YEM)) separately.
Upper bound on WDDD ((Y,)) This bound can be established by following the same argument as in Bruno et al. 2025 Proof of Theorem 10), which relies on the representation of the OU process
X, a.s. m, Xo o,Ze. m. m-=e-, - Ze ~ ~N0,
(52)
where 0.5. denotes almost sure equality. Therefore, we have
Vd),
(53)
where t = T c.
Upper bound on W,(CCYY,) Using Itô's formula, we have, for any t E [0, T c),
d| - 2/ = -(Y Y, Y, 2V log Ý, 2ologrr-PPoo dt
(54)
Y 7,12 di + 4(2 Yr, V Vlogpr Poggr-+(())) dt.
By integrating and taking on both sides in 54
we have
E Mr, y2,13
=E
No Yol2
+ /
2E |r Y:p] dt
(55)
//² 4E (Y, Y., V log pr (Y:)
log pr (Yi) dt.
By integrating, taking expectations on both sides in 55 using Corollary 14 the representation 52 with Zr d Yo (where denotes equality in distribution), the incquality ae S my for any t e [), we have
E [Ye, 12,1
< E [1o - F/12 + j E Yr [iV Yd1l]
di //' 2os E Yr 7.P
< Eljy 2 -2 f 50s de)
(56)
= El + (or - 1) ee 2 S soo, atl
2 [E[ 2] + d) 2(t 3 808 2T
Using 56
Remark 15 and and t, = T c, have
W2(C(Y.)), - ElIY,, Y2,1² 12
(57)
Vdle -2 g08.K. dt-c
x V2( | X1
23

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Published in Transactions on Machine Learning Research (09/2025)
Upper bound on ((Y Using Itô's formula, we have, for E [0, c),
d y =(YY y Y V VlogPr-((Y)) y - -,,0-,,,,,"" Y,ux)) dt
=2|.; - y p dt + 4 (Y, - y logpr - Vog i (Y( dt
(58)
+ 4 (Yr Y Y,ax, v log pr pr (X, _ s sT t, 0, y,uu)) dt.
By integrating and taking the expectation on both sides in 58 using Corollary
Young's inequality with
Ç (0, 1) and Assumption wc have
T
Bl
T-e
2
B[ yaux ds
T-c
+ 4
E(YY, - y Pogpr-- V log Pr-pr-s(2) (Y ds
+4
B(Y, - Yaux V log (V,uux) s(T -8,0
y ) ds
(59)
T-
2(1 + () Bii. y []] ds
100 E I Y-uup
dt 25 lesN
< 2c (1+c)((--e-41 1o 2.th 5- ts-es
Using 59
Remark
15 and t, T c,, we have
W.(C(())) < B[jy,, Y [2]
(60)
(0-) 800,Ki0 VegN-
Upper bound on W2 WC(C))) L(Y.MM)).
Using Itô's formula, we have, for E [0. T c),
d 1
2(Yauxx -PEM yaux + sTT - 1,0,Y. Y.Ux) PRM 2 s(T 11>11,,0
PE ((/117)) di
[1/7
22/yraux P.EM2 dt + 2(yaaux - PEM PEM (1/7J)) dt
(61)
+ +4(Yaux FPM (T - 1, 1, y,ux)) s(T 1, 1,0, P.RM)) di
+ 4(Y,aux P.EM, s(T 0, P.EM) -sTT Lt/nln, 0, PEM () dt.
24

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Published in Transactions on Machine Learning Research (09/2025)
Integrating and taking the expectation both sides in 61 using Young's inequality for (0, 1), Cauchy Schwarz inequality, Assumption 3.a
Lemma
25 and Remark
we have
E 1
s (2+ + 3()
Elly PPMP di + 5-
[t/217
[2]
dt
+ +4Ka(1 + 2
E| -Y, S. jjdt
T-
+
B[[s(T 1,0, P.MM - s(T - 11/J11,0 0, 9 10/711 17d
(2+ 36 + 4K3(1 2l
Ellyauxx 9.MMis dt
+5- c)CEMose,2 + 85 T--E
2a (T - -e) + 4E 01
+ 2T
ElDEEM B. [t/>1> 19
(62)
$ +2+ 36 + 4Ka(1 + 27(h))
Eljyaux - preme [2] di
+ 7 <) C) 2 + 4K3(1 + 27
+ 80 8 200 (T - () + AL + 1)
(2
5 7TT CEMos, 2 + 4K3 + 22
86 (T e)K A5 + 880013)
Using 62 and 1, =T =T-
-e, e, we have
(( S -6)
(63)
x EMose, 2Kg
(C1/2
2 AL + 1107)))113)
Final upper bound on W2(C(YEM), MD). Substituting
53
57
60
and
63
into
51
we have
W2(C(YFM), mp) < ( E Xol 2 + Vdj2ve
V2(v El XD vdje S K. di-c
/ (1()) ())- 2 1 K., dt
(64)
Vesn
+ > 1/25-1/2 (122,200000122242222+12111111111
1C1/2 (C + 2K + 27 + 2/2K + 1/04/231/2)
The bound for (2(C(EMM) TD) in 64 can be made arbitrarily small by appropriately choosing parameters
including 6, T, eSN and 7. More precisely, for any 5 > O, we first. choose 0 6 < 6s with es given in Table 3
such that the first term on the right-hand side of 64 is
(v X + VaJ2VE< 8/4
(65)
Next, we choose T> > T3 with T3 given in Table g such that the second term on the right-
right-hand hand side of 64 is
V2(vE([XoPP + vdye S." K0 dt-r 6/4.
(66)
25

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Published in Transactions on Machine Learning Research (09/2025)
Next, turn to the third term on the right-hand side of 64 We choose 0 ESN ESN,6 with ESN,6 given in Table B such that
1 S. g. dt VESN 6/4.
(67)
Finally, choose 0 < 7o with 7o given in Table such that the fourth term the right-hand hand side of 04 is
(68)
(C EMose,2 (1 (1 27 + 222K 8E +810413)1/3 5/4.
Using
67 and 68 we obtain W2(C(Y)MM) =D) 8.
Proof of Theorem 21 Using the splitting 51 the proof follows along the same lines of the Proof of The-
orem 19 for the estimation of the error bounds of the terms W2(TT,,((Y,)) W2(C(Y,,); ((Y,,) and
W2(C Y (,), (()))))) The error bound for (( is derived along the same lines of Bruno
ct al. 2025
Proof of Thoorem 10). Putting these four estimates together leads 27 and 28
Modified Half-Normal Distribution
In this section, we recall the probability density function of the modified half-normal distribution, see e.g.- Sun al. (2023). used in Section 3.4.1 and defined as
2 exp (-Ex² + y/r)
g(x)
x 2 0,
(69)
V
where U.E> 0, ab E IR, and the normalizing constant
00
( + 2
2
V
[(n) n!
We point out that the half-normal distribution, truncated normal distribution, gamma distribution, and square root the gamma distribution are all special cases of the modified Half-Normal distribution
is the Fox- Fox-Wright function Fox
1928
Wright
1935
69 The distribution 29 follows by taking the symmetric
extension of 69 ie. g((=l)/2, and choosing and v -1.
E
Table of Constants
Table displays full expressions for constants which appear in Theorem 19 and Theorem
21
26

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Published in Transactions on Machine Learning Research (09/2025)
Table 3: Explicit expressions for the constants in Theorem 19 and Theorem 21
CONSTANT
DEPENDENCY
FeLa EXPRESSION
C1
O(vā)
21
B11X0|11 + va)
C2
o(va)
v2
/111X/1 va
O(e (4) J 21 ke at
265 (1)
K. de
c)(
(T).
(MMerdatipae Tp2al (ALL
(3
16Ke
2 2/0" P)
7700 207)
c 2 +125
(T),
1024K
4)2)
O(de 220t p cAul)
2(C + ""
- ""
Ca(T,e)
Olvde platl olv paatt1
EMusc 2Kg (, 13
CEMon.4
Old.
8(CmM, + K Romn (1024CrM 8192(1 BI(0)+M) 144d
+++ +4kg -7-- -
K3s- 17 CEM 1d(1 SK3 +
+ 20 kQ( + 8(CAL + 10 (0)
(,(T,
gloti 22/1)
+ 46 4d1 d(1
sKG (1 + 27
x [(1 + i + (ee..a77))
+ 32k (1 + ) + 2ē + 210"1²²1
2(11 skğ(1 + 47 C 2kı + SEAL + 810 Pyya1))
x |dv2(1 skŽ(1 + 1/22
8//66( B[|xo| va)²)
Oblained solving using Proposition
In(ule" 27 1) (K/# ()/
11)+1)
Ts
(n( 12 vV)/88
gos.k. ktK//++-
(6 20 orT 014 J 008.K.
28N,5
(8<C/16)(T c) 20110-00-0-0 20111//
76
x (C (1 2Ks + +2 (EA
(8/(AV 1/ - e) - - hagp://
min
(1 1 Ce 4d/1 SKŽ (1 4772))
+ + k + 8 00 (3)
74
+ 46 4c- a(1 + 8K + 47
x [(1 16K T (eM,
+ 32Kk (1 + 720) [1 + 25 + 2
+2((1 sKŽ + 17 22 )) 122 0//2 2K, + SEAL 810 1327
-1/(20)
x [dv2(1 + SKg 1)22
27

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Published in Transactions on Machine Learning Research (09/2025)
Table 4
provides the main notation used throughout this work and indicates where each symbol is introduced.
Table 4: List of the main notation.
SYMBOL
REFERENCE IN THE TEXT
Equalion
Assumption
EAL
Assumption
aU
Definition
K
Assumption
Assumption
ii1
a
Assumption
ESN
Assumption
Definition
fL
Equation
33
Equation
B(,,0,,;K))
Equation
28


Reviews:
[{"review_id": "9zkTDyGABd", "reviewer": "Reviewer_o7dK", "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\nThe paper is a theoretical paper, which provides detailed proof to support its claim. The studied problem: Wasserstein convergence for SGMs, is interesting and important.\n\nWeaknesses:\n\n1. There is some notation abuse. For example, in equation (11) $\\partial U(x)$ is defined to be a set in equation (1). However, it is used as a vector and takes the inner product with another vector. It should be $\\nabla U$ given that $U$ is differentiable or something else. Moreover, I find that Definition 6 directly relates to [1] without good reference or comparison (if they are different).\n\n2. In Remark 10, I do not quite understand why you mention the specific function class, and why “the gap may vanish”? What do you mean?\n\n3. The presentation of the main results (Theorems 13,15) is very ambiguous. It follows the writing of [2], replacing $L_{MO}$ with an integral of $\\beta$. The presentation and remarks on those results puzzled me. First, the choice of $T$ arises from solving a complex equation. At first glance, it appears that $T$ could depend on $\\log d$. Since many of the “constants” in the bounds involve terms like $e^{T^{2\\alpha +1}}$, this could introduce additional dependence on $d$, casting doubt on the claim of “optimality” on the dimension. Second, it's unclear what notion of “optimal” is intended here. In prior work, optimality typically refers to the dependence on $d$ in terms of sample complexity or iteration complexity. However, your discussion only focuses on the rate term preceding the step $\\gamma$. Moreover, optimality usually corresponds to a matching lower bound. Third, “resulting from numerical techniques introduced in Kumar & Sabanis (2019) and employed in the proof of Theorem 15 to achieve the optimal convergence rate of order $\\alpha \\in [1/2, 1]$”, appears to be directly copy-pasted from [2], and it's unclear what is actually being conveyed here.  \n\n4. As I have pointed out in 3, this paper highly resembles [2], except for a slightly weaker concave assumption. I don’t quite understand why this improvement is essential for this setting.\n\n5. The paper makes several assumptions on the estimated score function. While similar assumptions have appeared in previous works, I believe some of them are too strong and less suitable compared to what is commonly adopted in the literature. In Assumption 3.1, it is unclear why the \"Lipschitz constant\" for the $\\theta$ and $x$ terms is assumed to depend only on $t$, and not on 𝜃$θ$ or 𝑥$x$ themselves. This seems overly restrictive. Moreover, the parameter $\\alpha \\in [1/2, 1]$ appears somewhat artificial. In Assumption 4, the expectation is taken over an auxiliary process. However, this seems less appropriate than taking the expectation over the forward process, which is directly related to the training procedure. Since all training data and objectives are derived from the forward process, using the auxiliary process introduces a mismatch. \n\n---\n\n[1]  Gentiloni-Silveri & Ocello 2025, Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance\n\n[2] Bruno et al., 2025, On diffusion-based generative models and their error bounds: The log-concave case with full convergence estimates", "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 work provides Wasserstein-2 convergence guarantees for score-based generative models (SGMs) targeting semiconvex data distributions with potentially non-smooth potentials. The theoretical analysis shows that it can achieve $\\sqrt{d}$ dimension dependence rate in the stepsize term. It generalizes previous analysis to more complex non-smooth settings.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please add some introductory paragraphs before each theorem and lemma. For example, clearly explain the purpose and significance of Lemma 7. For Proposition 8, instead of jumping directly into the content, first motivate the result: explain why this proposition is needed, what problem it addresses, and how it contributes to the main theory.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "bFuZpsyhNx", "reviewer": "Reviewer_qYo2", "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**:  The strongest part of the paper is the theoretical contribution. It broadens the scope of SGM convergence guarantees to more realistic data distributions, which are often non-smooth in practice. The proofs are rigorous, and the results are cleanly stated with explicit constants.\n\n**Weaknesses**\n- The weaknesses are mostly around accessibility and empirical support. Some technical constructions (like the weak convexity profile and function `f_L`) could use more intuition or explanation. Readers not deeply familiar with convex analysis may find some sections hard to parse. \n- Also, while this is a theory paper, a minimal empirical illustration (e.g., on synthetic 1D or 2D distributions) would help ground the results.\n\n- Another potential issue is that while Assumptions 2 and 3 are weaker than those in prior work, they still require conditions that may not be easy to verify when using neural networks as score approximators.", "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 proves non-asymptotic convergence guarantees in Wasserstein-2 distance for score-based generative models (SGMs) under relaxed assumptions. Specifically, the authors assume the data distribution has a semiconvex potential and allow for discontinuous gradients, removing the need for smoothness or strong log-concavity. They derive explicit bounds with optimal dimension dependence (`O(√d)`) and convergence rates (`O(γ^α)`), covering practical examples like Gaussian mixtures, double-well potentials, and elastic net.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "* Briefly explain the core assumptions (e.g., what semiconvexity and strong convexity at infinity mean) in more intuitive terms.\n* Include a toy numerical example that shows convergence under non-smooth or multimodal targets.\n* Consider adding a table of notation and constants, as there are many parameters across theorems.", "broader_impact_concerns": "There are no major concerns. The work is theoretical, with no direct applications that would raise ethical or safety risks.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "EwBcgbfKSn", "reviewer": "Reviewer_Zoog", "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- The contribution is important in understanding score-based generative models in the context of nonsmooth, non-log-concave data distributions. Previous works have to assume either log of the data distribution is concave or twice continuously differentiable. \n\n- The paper is generally well-written;  The motivation discussion and the contributions are pretty clear -- a detailed comparison with previous works is presented in Table 1.\n\n# Weaknesses:\n\n- The paper focuses solely on the Ornstein-Uhlenbeck process. I wondered what the difficulties would be when extending to other probability paths, or, at least, to more general SDEs.\n\n- Assumption 3.b. (needed for Theorem 15) looks strong to me: the bound needs to hold uniformly for $\\theta$ which is the neural network's parameters. I do not think it would be the case in practice. I wondered if one can relax this assumption by also allowing some growth for $\\theta$ in the bound.\n\n- Some mathematical details need a more careful revision; some recent references are missing -- See Requested Changes.", "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 establishes bounds on the Wasserstein-2 distance in Score-based Generative Models—specifically, between the data distribution and the final step of the backward process -- under the mild assumption that the negative log of the data distribution is weakly convex (and possibly non-differentiable). The bound presented in Theorem 13 is optimal with respect to the data dimension, scaling as $O(\\sqrt{d})$. Additionally, Theorem 15 provides an improved bound in terms of the discretization step size $\\gamma$, assuming Lipschitz smoothness of the score approximation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Page 2: The sentence \"...our estimates in Theorems 13 and Theorem 15 are derived without imposing any restrictions\non the stepsize of the generative algorithm\" deserves some comments on which restrictions on the stepsizes in the previous works are. Maybe you can also include this information in Table 1.\n\n- Page 3: About the mild assumptions on the target distribution that admits the backward process (3): Please specify what the assumptions are and how they effect your main assumptions. Because this paper is all about \"mild assumptions\", I do not think we can ignore even this assumption.\n\n- Page 5: about the selector of the Frechet subdifferential, $h(x) \\in \\partial U(x)$: It seems we also need $h$ to be regular enough, i.e., Borel-measurable. Under which conditions such a measurable selector exists?\n\n- Assumption 2: It seems to me that the R's in Assumption 2(ii) and (iii) need to agree, i.e., same R. If yes, please add this extra detail to the assumption.\n\n- Remark 2: $\\nabla p_t(x)$ is continuous for $t$ and $x$. Do you mean it is continuous w.r.t. x for fixed t, or it is continuous w.r.t. both variables?\n\n- Assumption 3.a.: It is worth discussing the condition that $\\alpha \\in [1/2,1]$. I noticed you used the same condition as in [Bruno et al. 2025] but it is more complete if there are some comments in this paper.\n\n- Definition 6 and Proposition 8: appropriate citations are needed to show the equivalence between two definitions of weak convexity. Also, in Proposition 8, (i) can you give a comment on why $\\tilde{\\mu} > 0$ and (ii) make sure the $R$ in item 2 is the same as the R in item 1 -- it is not obvious to me, especially for the case $\\mu > L$. \n\n- Corollary 9: I think we have to exclude $0$ in $[0,T]$ since the gradient of $\\log p_0$ does not exist.\n\n- Remark 10: I have difficulties understanding this one. (1) The statement \"This gap may vanish for different functions...\" needs citations; (2) If the gap can vanish, why don't you pick that function from the beginning? Because when you replace $L$ with $K+\\mu$, the inequality (19) no longer holds (the inequality sign is flipped). Please provide more explanations in this regard and discuss how your results are still valid.\n\n- About the critical time $t^*$ that the distribution evolves from non-log-concave to log-concave, I think a numerical illustration is very helpful for the readers' understanding. \n\n- Related works: I found the following work very relevant to your work: Wang, Xixian, and Zhongjian Wang. \"Wasserstein Bounds for generative diffusion models with Gaussian tail targets.\" arXiv preprint arXiv:2412.11251 (2024).\n\n# Minor:\n\n- Proposition 8: You used $L$ without giving its connection to Assumption 2, i.e., in Assumption 2, there are only $\\mu, K, R$.\n\n- Theorems 13 and 15: you are using $K$ as an index, viz., $Y_K^{EM}$. Note that $K$ is already used as the weak convexity modulus throughout the paper. I suggest using another character.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)