A Sharp KL Convergence Analysis for Diffusion Models under Minimal Assumptions
Authors: Nishant Jain
Tong Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: diffusion models
probability flow ODEs
score based generative models
convergence analysis
Primary area: learning theory

Abstract: Diffusion-based generative models have emerged as highly effective methods for synthesizing high-quality samples. Recent works have focused on analyzing the convergence of their generation process with minimal assumptions, either through reverse SDEs or probability flow ODEs. The best known guarantees, without any smoothness assumptions, for the KL divergence so far achieve a linear dependence on the data dimension $d$ and an inverse quadratic dependence on accuracy level $\varepsilon$. In this work, we present a refined analysis for the standard Exponential Integrator discretization that improves the dependence on $\varepsilon$, at the same time maintaining the linear dependence on $d$. Following recent works on higher order/randomized midpoint discretizations, we model the generation process as a composition of two steps: a reverse ODE step followed by a smaller noising step, which leads to better dependence on step size. We then provide a novel analysis which achieves linear dependence on $d$ for the ODE discretization error without any smoothness assumptions.
    Specifically, we introduce a general  ODE-based counterpart of the stochastic localization argument from Benton et al and develop new proof techniques to bound second-order spatial derivatives of the score function -- terms that do not arise in previous diffusion analyses and cannot be handled by existing techniques.  Leveraging this framework, we prove that $\tilde{O}\left(\tfrac{d \log^{3/2}(1/\delta)}{\varepsilon}\right)$ steps suffice to approximate the target distribution—corrupted by Gaussian noise of variance $\delta$—to within $O(\varepsilon^2)$ in KL divergence, improving upon the previous best result requiring $\tilde{O}\left(\tfrac{d \log^2(1/\delta)}{\varepsilon^2}\right)$ steps.

Published as a conference paper at ICLR 2026
A SHARP KL CONVERGENCE ANALYSIS FOR DIFFU- SION MODELS UNDER MINIMAL ASSUMPTIONS
Nishant Jain & Tong Zhang University of Illinois Kirana-Chaappagg {nj27, tozhang) .ite..111 illliinnooi edu
ABSTRACT
Diffusion-basee generative models have emerged as highly effective methods for synthesizing high-quality samples. Recent works have focused analyzing the convergence of their generation process with minimal assumptions, either through reverse SDEs probability flow ODEs. The best known guarantees, without any smoothness assumptions, for the KL divergence so far achieve a linear dependence on the data dimension d and an inverse quadratic dependence on accuracy level In this work, we present refined analysis for the standard Exponential Integrator discretization that improves the dependence on &, at the same time maintaining the linear dependence on d. Following recent works on higher order/randomized midpoint discretizations, we model the generation process as a composition two steps: reverse ODE step followed by a smaller noising step, which leads better dependence on step size. We then provide a novel analysis which achieves linear dependence on d for the ODE discretization error without any smoothness assumptions. Specifically, we introduce general ODE-based counterpart of the stochastic localization argument from Benton al. (2023) and develop
proof
techniques to bound second- -oder spatial derivatives of the score function terms that do not arise in previous diffusion analyses and cannot be handled by existing techniques. Leveraging this framework, prove that Ô (dlog//2(1/8)) steps suf- fice to approximate the target distribution distribttionccorupped by Gaussian noise vari- ance 5-t0 within (ss2) in KI. divergence, improving upon the previous best steps.
result requiring Ô dllog"
INTRODUCTION
Recently, diffusion based models have picked up momentum for various use-cases involving generative modelling. They are widely used for image generation (Song Ermon, 2019; Croitoru et al., 2023; Song et al., 2020a; Nichol et al., 2021: Song et al., 2021; Ho et al., 2020), video generation (Epstein et al., 2023; Chen et al., 2023d), semantic editing (Lugmayr et al., 2022), generating text (Li al., 2022) audio signals (Liu et al., 2023), protein design (Gruver et al., 2023; Guo et al., 2024), and many other areas. The success of these diffusion models largely stems from their ability to generate high-quality samples using denoising mechanism. This is achieved by defining a forward noising process that gradually perturbs data from the target distribution, and learning a score function using these noisy observations. New samples are then generated by iteratively simulating the reverse of this process, guided by the learned score function. The forward process can be modelled as stochastic differential equation (SDE) (Song al., 2020b), and consequently the generation can be carried out by simulating its reverse-time SDE through discretization. Corresponding to this reverse-time SDE, there also exists probability flow ordinary differential equation (ODE) Song et al. (2020b), which shares the same marginal distributions all times. Consequently, two main approaches have emerged for sample generation: simulating the reverse SDE (Song et al., 2020b; Chen et al., 2023c) and simulating this probability flow ODE (Chen et al., 2023b; Lu et al., 2022).
Several works (Chen et al., 2023c;a; Lee et al., 2022: Benton et al., 2024; Li Yan, 2024) have targeted the theoretical underpinnings behind the working of these diffusion models, under various

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assumptions. These studies established polynomial convergence rates with respect to the data dimen- sion d, assuming accurate score estimation together with regularity conditions such as smoothness of the score function or bounded support of the data distribution. More recent efforts (Chen et al., 2023a; Li & Yan, 2024; Benton et al., 2024) aim to minimize such assumptions and obtain guar- antees just using accuracy assumption for the estimated score. The best existing result (Li & Yan, 2024) shows O(d/T) convergence rate in the total varation (TV) distance for the Denoising Dif- fusion Probabilistic Model (DDPM) (Ho al., 2020). recent work (Benton et al., 2024) also achieved linear dependence on the data dimension for convergence in KL divergence, requiring ( steps to achieve KL divergence within =² with respect to Gaussian perturbation of the true data distribution. Since TV is bounded by square root of the KL divergence, it is an important issue to investigate whether a better convergence rate is achievable in the KI. Kl.-divereence. While the linear dependence on d seems satisfactory, the quadratic dependence on may not be optimal. In this work, we are interested in investigating the following question:
Can we improve the dependence of the Kl.-divergence on & while maintaining linear dependence on the dimension, thereby establishing stronger convergence guarantees for diffusion models?
To achieve this goal, we explore the line of works (Chen et al., 2023b; Gao & Zhu, 2025; Li et al., 2024b;a) which investigate the probability flow ODE for generation. This perspective is motivated by the observation that we can have better discretization dependence for each interval when analyzing the Wasserstein type error directly using the ODEs (Chen et al., 2023b). However, aggregating and bounding the error across all the intervals just based on this ODE requires additional assumptions either related to the error in divergence (Li et al., 2024b) or the Jacobian (Li & Yan, 2024) of the approximated score. Therefore, Chen et al. (2023b) instead considers smoothness of true and approximate score function at all times to bound the Wasserstein error in each interval using the reverse ODE and adds a noising step utilizing Langevin dynamics to then convert the Wasserstein error to TV. However, this noising via Langevin finally results a suboptimal dependency on e. Given the improved dependence on step size the probability flow ODE can offer, we also consider using it but instead of additional assumptions or the Langevin dynamics, we just consider taking a smaller step in the forward (noising) direction. This way are able to consider the error due to discretization on the reverse ODE and then convert it into KL error via the noise addition with better dependence be aggregated across all the intervals. The combination of the ODE step and a smaller noise step can be interpreted as an alternative simulation of the reverse SDE based generation process. This idea of noise addition along the forward process to convert the Wasserstein type error achieved via ODE-based deterministic step KL has been used in works targetting second order discretization (Li & Cai, 2024), randomized midpoint analysis under smoothness (Li & Jiao, 2024) and also for convergence analysis for the ODE-based consistency model framework (Jain al., 2025).
step size for each interval, which can then
Unlike Chen et al. (2023b), we work in the minimal assumptions scenario similar Chen et al. (2023a); Benton et al. (2024) and just consider the accurate score estimation assumption. A straightforward way then is to consider the analysis of Li Cai (2024) and adapt the standard DDPM sampler. This improves the dependence on e but worsens d- dedependence leading to
complexity of Ô ////2) This is discussed further in section 4.1. Therefore, achieving the desired
linear convergence rate for the KL-divergence based on these prior works is non-trivial.
To achieve the linear dependence on d in this setup for our considered ODE step followed by nois- ing path, we take inspiration from Benton et al. (2024) (which considers the reverse SDE and by establishing equivalence to stochastic localization directly picks up known result from the litera- ture) and investigate additional relations between the score function and its derivative. As discussed in the paper, our analysis along this ODE based path introduces additional challenges: it involves terms containing Laplacian of the score function along with the terms containing both score func- tion and its Jacobian, making it more complicated than the SDE counterpart. By establishing the required novel relations between the score function and its higher order gradient terms, to achieve the linear dependence on data dimension d for the ODE, matching the result of Benton et al. (2024). Due our improved dependence on discretization step size, this translates into new
able
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state-of-the guarantee for the KL convergence: requiring Ödd/e) iterations to achieve =2-KL divergence improving up the previous best result of Ō (Benton al., 2024). Also, since the TV distance is upper bounded by the square root of KL.-divergence, this becomes the state-of-the-art convergence guarantee for diffusion models against the TV convergence guarantee provided in Li & Yan (2024).
1.1 RELATED WORK
Here, we provide a review of the recent works targetting diffusion based generation broadly catego- rized into whether they consider the reverse SDE or the probability flow ODE.
SDE-Based generation. The effectiveness of this forward noising and the corresponding de- noising process for generation was first majorly advocated by the Diffusion Probabilistic Models (DDPM) framework introduced by Ho et al. (2020), which utilized Gaussian transition kernels for noising and estimated the parameters of the corresponding Gaussian denoising kernels using denois- ing score matching during training. Going further it was shown that this forward noising in DDPMs can be seen as an SDE (Song et al., 2020b) and the generation process then corresponds to the re- verse SDE. Since then there have been various works (Chen al., 2023c; Li et al., 2023; 2025; Lee et al., 2022) targetting the convergence the real world, some recent works (Chen et al., 2023a; Benton et al., 2024; Li & Yan, 2024) have also targeted setups for the SDE based generation methods requiring minimal assumptions (just the bound on score estimation during training via denoising score matching) and have achieved state- of-the-art convergence guarantees. Specifically, Benton et al. (2024) shows only (ddea) steps are required be ec-close in KI. w.r.t a Gaussian perturbation of the target distribution. On the other hand, Li & Yan (2024) considers the TV-distance and shows O(d/e) steps are required to achieve e-close TV of the perturbed data distribution.
this generation process. To advocate for the usability in
ODE based generation. Song et al. (2020b) highlighted that corresponding the forward noising process for this diffusion model setup, there also exists a probability flow ODE along side the reverse SDE which shares the same marginal distribution at all times. also advocated that this Probability Flow ODE can lead to faster sampling using the ODE solvers. Taking inspiration, Song et al. (2020a) then proposed a deterministic counterpart of the DDPM sampler and since then various works have attempted to investigate the convergence of these deterministic samplers (Li et al., 2024a; Gao & Zhu, 2025; Huang al., 2025; Li et al., 2023; 2024b) under various additional assumptions. The current best result (Li et al., 2024a) achieves TV distance of 8 (w.r.t. perturbation of the true data distribution) ( steps under score estimation and an additional assumption on the Jacobian of the estimated score. Another work (Li et al., 2024b) requires weaker assumption the divergence of the estimated score but achieves sub-optimal results. These works have also argued that under just the score estimation assumption, the TV-distance for these deterministic samplers is lower bounded unlike SDE and thus, such additional assumptions are required. Another line of work is based on the predictor-correct sampling (Song et al., 2020b; Chen et al., 2023b) which uses an ODE step and addition of small noise using Langevin dynamics for smoothening the trajectory to avoid the error blow-up due ODE, For this scenario, the convergence can be achieved (Chen al., 2023b) under standard assumptions on score estimation and the smoothness of the true score as well as the approx- imated score function, which can be further improved using the randomized midpoint discretization in the predictor step (Gupta et al., 2025). Instead of this langevin step, some recent works (Li & Cai, 2024; Li & Jiao, 2024) have considered the ODE step with second order or randomized midpoint (under smoothness) discretizations and then directly adding noise along the forward process, thereby improving the dependence on e. In this work, we also take a similar route but for the DDPM sampler and achieve state-of the--arttttttttt KL convergence rate under just the score estimation assumption.
PRELIMINARIES AND SETUP
We now discuss the formulation behind diffusion models in detail, including both ODE and SDE- based generation. Following this, we discuss the assumptions used to achieve the results provided in the next section.
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SDE considered and its discretization. As discussed previously, diffusion models are based forward noising process and the corresponding reverse generation process. The forward process for d- di-mmnnssinal setup can be seen as taking the given samples and gradually corrupting them using the SDE of the following form (Song et al., 2020b):
dx(()) -(((()), t)dt + g(t)du;
where z(0) = y ~ Pdata. x(t) E R. and correspond to the drift and diffusion coefficients, wy is the diddimeniional Brownian motion. Following the popular choice of the OU process, we consider the following SDE:
dx(t) = ==((ddt + 22dw:
The corresponding OU process would be:
(I) where Pt denotes the law at time 1, .r(t) Pt and y Pdota- Also, the joint distribution of the random variables generated via this process at time-stamps corresponding to sequence [f1.-., fx } (Ie Ttg) is denoted as Pt.....t. The resulting reverse SDE (Song al., 2020b) for generation will be:
x(t)= = c + v1 e e(t),
e(t) ~ ~N(,
(t = - 2V In (x)(t)d v2dw;
(2)
where In (xx(()) is referred as the score function and Wi is again the Brownian motion. forward process is run from time T then initializing from pr and going along this reverse process for a time T will result in the marginal Pt- For the corresponding probability flow ODE (Song et al., 2020b), we have the following equation:
dx(() x(t)dt s(t, ((())
where we denote st., = log Using the Exponential Integrator discretization (Chen et al., 2023a) where divide the overall generation time into small intervals and fix the input to the score function for each interval to be the value at the start (from the reverse direction), leads to the following ODE for the interval [fx-1, tk]:
dx(t) = -i(tdd s(tk, 2g
Empirical Counterpart. Practically, we do not have the true score function s(t,) .) and instead during training is approximated via denoising score matching (Song et al., 2020a). Denoting that approximated score function (l,,-), we have the following empirical version (discretized and using the approximate score) of the true ODE:
di(t) =(t)dt ([(t,.tx)0r
(4)
where denote the law of this empirical process time as pr. For any particular discretiza- tion {&x N of the reverse process, this process usually initialized using a normal distribution fta ~ ~ (O,Ta) and denote the joint distribution for the true (Zez, -.. Ftx reverse ...,,..... Fex)) processes as Pt Px......x. Also the conditional distribution at 44-1 conditioned on t& is denoted as Pie-ilte> Pta-1 12 for the true and empirical processes respectively. We now discuss the assumptions used in our theoretical framework.
.....
Assumptions. As discussed in the introduction, for our theoretical analysis, we take inspiration from the line of works operating under minimal assumptions (Benton al., 2024; Chen et al., 2023a; Li & Yan, 2024), and just use the following standard assumptions:
Assumption 2.1. For the discretization sequence { discussed in the next section (and used in the Inference Algorithm 1), the score function estimate {3(t, DiSIST satisfies:
K+1
1
M
Egpppy [[8((2, x)) - s(12, |?3] S 2cooee
(5)
T
where h2 t4 tw-1 corresponds to the step size of the discretization.
Assumption 2.2. The data distribution Pdata has finite second order moment Eryoggtn [[|
m2 < 20.
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2.1 NOTATIONS
As discussed above, corresponds to the data distribution Pduta. rrt) (with its law denoted by and score function as s(-) corresponds to the forward OU process and x11) (with its law denoted by ge and score function s-)) is the variance exploding counterpart of the forward process discussed in Appendix. (t)) corresponds to the discretized version of the true reverse process. H denotes the sequence of random variables generated by our algorithm for discretization sequence {tx} and their law is denoted by Pi.- The step size ty - is denoted as fig. 2g corresponds to the ran- dom variables generated by the forward process for this time sequence. Bx-1. - corresponds to random variables generated by running our proposed scheme (with true, empirical probability flow ODE respectively) for single interval [ [x] starting from de. l2. 5..0.5. f1-0055 corresponds to the random variable generated by taking two steps along the discretized (Empirical, True respec- steps of true probability flow ODE from Fg. Vs(t, z) denotes the Jacobian of the score and å corresponds to the partial derivative w.n.t. time t. We further define 0; as the partial derivative w.r.t. 7th coordinate of the spatial variable x (or z discussed in appendix). It can also be interpreted as Oz, 0z./O... We also
tively) probability flow ODE in reverse direction starting from Tg- - denotes
define I Laplacian operator A Σ Ei= 0, The jt element of the score vector s(-) is denoted by s(.);- (0, Ia) denotes the d-dimensional standard Normal distribution. For two terms P, P S Q means there exist an absolute constant C₁ such that P S CQQ.
3 MAIN RESULTS
As discussed above, using previous works (Chen et al., 2023b) the probability flow ODE, we can directly analyze the Wasserstein-type error under smoothness conditions by using the Young's 's and Grönwall's inequality. However, the aggregation leads to blow-up in the error and thus, noise is added via Langevin dynamics based corrector after a small ODE step to instead convert the error into TV distance, which can then be aggregated. Here, instead take more simplistic perspective for the diffusion use-case where we consider first taking a step along the reverse ODE to bound the Wasserstein-type error (without any smoothness conditions) and then taking partial step in the noising (forward) direction to convert this error to KL. As discussed in the introduction, this inspired from recent works (Jain et al., 2025; Li & Jiao, 2024; Li & Cai, 2024) which target the consistency model setup or randomized midpoint/second order discretization schemes.
We first define a discretization sequence 0 < 8 to t1 t2 < <... < tK tk+1 = T for the generation process, where T denotes the total time, initializing from the standard normal distribution (0,, Ia). Denoting h4 = 4 f4-1, we provide the inference procedure in Algorithm 1. It is based on the Exponential Integrator discretization of the empirical probability flow ODE (Eq. 4) in the step 4 followed by noise addition along the forward process (Eq. 1) in step 6. The step along the ODE can used to control the Wasserstein-type error and then the noise addition convert this into KL. This is discussed further in the next section and in detail (along with the technical lemmas) in Appendix A.1. We denote the generated sequence from our algorithm by random variables &e. (corresponding to time (ς) and their law by Pt, and the joint distribution for the complete sequence as Pts.....x..kk. Similarly, corresponding to the sequence generated by the forward process along these time-stamps, we will have the joint distribution as Pt (x0+1 Figure I shows this aggoritmm/generatin process.
Algorithm Inference Algorithm for Diffusion Models
1: Given: Discretizing sequence (to, l1,.,, LK, 1K+1). by = -1 - 1k-1, Pexee as the normal dis- tribution //00, Id)
2: Sample d'x +1 ~ P1x+1
3: for k K 1,K,..., do
4: 8/-0.5 = huthi 18/ + (c 1)8(te, 2%)
5: Sample 7h ~ NN(0
6: P/-1 - -- 0-5 + v1 c- 2-2
7: end for
8: Output #1
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Po R. Piduto
NO
PK N0, Ia)
Time
4
Uk
b
tu
tx-11
1x+1
8(-)
8(-)
Figure 1: Demonstrating the updates: (a) along the generation process using 8(-) and (b) the forward noising process (A())), of our proposed scheme.
We now provide the guarantee for the distribution generated by Algorithm in terms of KL diver- gence w.r.t. perturbation of the true data distribution (law of the forward process at 41): Pt. This corresponds to early stopping (with i > > O), similar Benton et al. (2024); Chen al. (2023a), we also avoid smoothness assumptions the data distribution.
Theorem 3.1. For T 2 1 and K > d(log()) + T), under Assumptions 2.1 and 2.2, consider the generation process in Algorithm with discretization times 0 defined by the step size rule h2g tx tk-1 = c min (1, tx} for some constant > 0. Then, denoting Pt, as the marginal distribution ty for this algorithm and Pt the distribution of the forward process (Eq. 1) at time we have:
lg < 1 < [K+1 T
m2) KL (pt. (pp.) S (d m2)e T de & + T'E
score
(6)
where x E y means there exists an absolute constant C such that x S Cy.
We provide the proof of this theorem in the Appendix (A.5) and discuss a sketch of the complete proof in the next section. The first term corresponds to the error due to initializing the algorithm from (0 Ia), second term corresponds to the error due to discretization and the third term is due to error in score estimation (Assumption 2). From the definition of c, it can be observed that (discussed in the proof as well) e² should be O (log 1 ++7)") Also we can observe that T required to just have K3
logarithmic dependence d and thus, the second term corresponding to the discretization error will be R2
d²
We formalize this in the following corollary discussing the iteration complexity.
Corollary 3.2. Under assumptions 2, 2.2 running Algorithm for the SDE based generation via diffusion models for a total time = log
d+-m2
with an exponentially decaying step size sequence
Excore
achieves a KL-divergence error of log( 1oR(+)+T)
ba = 1u 4--1 cminn42,11 1} where c +
K
dl d(log( )//2)
Ō(E2cara) with an iteration complexity K = (Benton al., 2024)
improving upon the previous best
Secore
complexity O ddlog Seevee
4 PROOF SKETCH
We now provide brief sketch of the proof for Theorem 3.1 and the complete details are pro- vided in the Appendix. We begin by first discussing the decomposition of KL divergence into the Wasserstein-type error aggregated dence the discretization step size (Chen et al., 2023b), this serves the main motivation of our Algorithm 1. Then, we discuss bounding the discretization error along this ODE path in the non- smooth scenario. Finally, we discuss on how the optimal dependence d can be achieved for this non-smooth setup, leading to state of the art convergence guarantee for the KL divergence.
each interval. Since the ODE can result in a better depen-
KL control for diffusion via Wasserstein-type error. between the generation process and the forward process at by KL (pt, (.,)) using the data process-
We can first decompose the KL between

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ing inequality and chain rule follows (Lemma A.2):
K+1
KL (pr, (l.)) & KL (Price: ((xx+.) + E Ep
Σ KL (pt. 11th (-12x) Pr.. alen (ITa))
where Pt..al denotes the conditional distribution of the true process f--1 given Iy at tx and similarly pt,- lt. for the generation process. The first term on the RHS is just the initialization error (the error by using the standard normal distribution for initialization as against the distribution of the forward process after time 7) and can be bounded following previous works (Chen et al., 2023c;a) as (d++mg)e
The second term denotes the summation of the KL error aggregated in each interval [f when the true and the generation process start from the same point (IR).. Now, to calculate this term, we will consider the following update for the interval [ empirical ODE (Eq. 4) and noise:
44] starting from ry using the
k 0.5 =e hethe-l x + (e 1)8((e, T.k
-K-1 =e - -h--1 -.S 0.3 V1 e-2hx-1ER ENe
Ek (O,
(8)
Based on this, the KL (1te (+12x) Pen- l4 (:122x) term can be written (Lemma A.1);
KL (Pt, alth (+1xx) pec 11t4 (-12%)) e--2h, 201 x 0.5 - 2.5-
2(1 2hx-1)
where 2k-0.5 denotes the reverse process time Eq h4 h This Wasserstein-type error
to KL conversion and then aggregation is inspired from the recent works (Jain et al., 2025; Li & Cai, 2024; Li Jiao, 2024). We now diseuss on how the expected value of the |zg-0.5 0 0.5 l term in the RHS of the last equation can be bounded to finally bound the expression obtained after applying the chain rule.
4.1 BOUNDING B||[ - -x-0.6
For error control of this term, we define additional process for the interval [f (starting from Øg and governed by Exponential Integrator discretization of the true probability flow ODE in
Eq. 3): Tz:
k = ththe 20 + (e he the ()s((A.
(9)
- =c ha- ix 0. + v1 - - 2-2h- Ck N(O,T).
(10)
Now, decompose the target term corresponding our scheme E - xk-0.5 for each interval as follows:
Elll==-0.. i2 0.5 S Blllax 0.5 - 5x 0.5 12
B[ 0.5 o.s-0.5 (11)
where T. is the error due to using the approximate score function á and Ta is the error due to the discretization true process. The score estimation error term can be written as (chuth,, - 1)* zx)) s(ta, xx)111 (Lemma A.3) and aggregated across all the intervals can be bounded as (hhse2core) using Assumption 2.1 (further discussed in the proof Theorem 3.1 in Section A.5).
To bound the discretization error, we first define rescaled version of the original process 2(t) = eta(t) (Section A.3) with the law at time denoted by gt, score function denoted as s,,(f, =(t)). This is done to simplify the calculations in the analysis. Now, using the ODE path of Eq. 14 and the Integral Remainder form of the Taylor Expansion, we bound this discretization error (Lemma A.4):
E = 0.5 - E o......l13 S (he + h2
"F || (t. x(t)) de
t2-2
where the derivative of the score sf,t, can be calculated as st, x(t) = - (t,)t,((t)) = (t.z).))
(t.2) dz(t)
This is different from previous works (Benton et al,, Chen et al.,
dz
===(t)

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2023a) which instead consider the reverse SDE in Eq. 2 and thereby incur the discretization error contribution for each interval in the overall KL divergence as: Je E [[ls(ta, 2x) s(t, x(t)) dt, bounded using the Jacobian of the score. This results in sub-optimal dependence on h& (O(h))) Specifically, the best bound is achieved in Benton et al. (2024) which directly expresses the deriva- tive of E [[|s(ta, =x) stt, [+² term w.r.t. in terms of Ep. [|] =)1 and then integrates the bound on this, resulting in the optimal linear dependence d and sub-optimal (h)) depen- dence on the step size.
Now, we discuss in detail on how to bound li E [|s, (t. 2(t)) dt following Eq 14. Using the trick proposed in Chen et al. (2023a), thereby rewriting score function time t: sr sr, as Eyl= T- (Lemma A.5). A straightfor- ward option is to then calculate the Jacobian (f,,(t, 2), partial derivative w.r.t. time (0,s, z)) and then substitute in the expression of st (t, =t)) to get an expression for E ||s (t, 20))
write 2(t) as Gaussian perturbation in y Pdatas
This can then be upper bounded using the fact that discussed in Lemma A.8). This is just the adaptation of the calculations proposed in Li & Cai
-= ~ Ia) (Chen et al., 2023a) (also VE--I
(2024) for the considered DDPM sampler and leads to a better dependence on h4 than Benton et al. dt and thus,
(2024) but will lead di dependence of our target term (J
a d322 dependence for KT. (Pt, (p,,)) which is worse than the d-dependence achieved for KL in Benton et al. (2024).
4.1.1 ACHIEVING THE OPTIMAL d-DEPENDENCE FOR ODE
Lemma A.8 shows that E (t,, 2) can be bounded O O(za_T) against ( for
d
E ([, Therefore have the linear d-dependence, take inspiration from Benton et al. (2024) which first establishes the equivalence of reverse SDE based to Stochastic Localization and then exploits well known result from the Stochastic Localization literature (Lemma in the paper). Since we are considering the ODE path, instead of directly utilising such result, we begin by first establishing 1., [||>, (1, ]]] = -2e VI 8, (1, 1){] Lemma A. Given our target term for the discretization error (Iit Eg. d Eg, [|)s, 2] can improve the d² contribution from this term to d. of the Jacobian term, using zi This serves as the motivation for the remaining sketch.
dt depends on the integral (w.r.t. time)
As discussed above, since Benton et al. (2024) involves the reverse SDE, just requires bounding Eg, V (t, =) term. However for our considered probability flow ODE path we need to bound
the overall derivative term: E || (t, ]]] which includes partial derivative w.r.t. time (t, making the analysis much more complicated which discussed next.
We first convert the time-derivative to spatial derivatives using the the Fokker-Planck equation coun- terpart for the score function (Lemma A.9):
(t,, 2t As, (sr, S (t,
where recall from Section 2.1 that A denotes the Laplacian of the score sr. This, results in the overall derivative term being represented only in terms of spatial derivative as follows (Lemma A.10):
Eg (1, =)]1"] = =e = [| =)], + || (1, 8, (1, =1)
T + Ey [(As T (Vs,(t, 2) (t,
(12)
Since this overall derivative involves a term containing both score, its Jacobian and term contain- ing the Laplacian of the score, bounding this involves more complex analysis compared to the
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SDE scenario in Benton al. (2024), Now, based the motivation discussed above achieve d
optimal d dependence by expressing dt gg. ( (t, 2)]l
= -2 E [l 1) we further es- tablish similar relations of the RHS terms in Eq. 12. For the term comprising both s, and ss, in Eq. 12, we provide the generalized version of Lemma A. which considers general power m in d at Ryy. (t, 212) and term of the form E. (1, =) Vs, (t, =) (Lemma A.12):
e e- d g (t, (t, m.g. q. [| 2) - 2 sr (t, 2) 1
dt
m(m m - 2) Egg || (t, 2)
[
We then utilise this equation first write the second term in the RHS of main Eq. 12 in terms of d B. (t, )- To target the first term, we establish another novel relation by start- at
ing with the term de
d gg s,, =)1 and expressing it in terms of f (=)] Vs, (t,,(t)))) I),
Eg As, =)1 and Eg [| =)
Then, we rearrange and express E. II (1, =)
in terms of f (=) (t, dt,, Eg [I| [t,,,, 1) and a. (t, []]]] (Lemma 1.) We bound the term f (=)] (t. dl as follows (Lemma A.15, Ca present in the lemma statement is ((1) since C. S 12 for 2 10):
/ V=( =)]l $ (e - 1)3
d
2-d
F. 11,,, (1,
(e2t 1) dt
leading to an overall bound on the E,g, (s, 2|1 as (Lemma A.1
d2
de-2t 2t
Eg. (t, E (c2t 1)3
B B [t, (t, 2)1
(621- - 1) dt
d
By [IVs, (t, ||= + d
-e 21
dt Bg Bg.
[Iss,(t,))]]]
dt
Finally, this leads to the following bound on By 111s (1. 2 (Lemma A.17):
de
add d d
B (t,, $ (e
Eg ||| 2)
1)3 (e21 1) dt
2 Em [1 =) 2) + di Em || (1, 2) =)]141
di
Integrating this and summing up across all the intervals, choosing he = emin 1} following the previous works (Chen al., 2023a; Benton et al., 2024) and scaling back to a(l) along with accounting for the score estimation error and the initialization error leads to the following final expression for KL (pr. ( (section A.5, refer to the analysis there for more details):
KL (p..) S (d + ma)e TT + de TT
log( +)T which results in K
where due to the exponentially decaying step size e x 1/ ( to achieve Ö(=²) KL (pr. ((.,,) error.
K
5 CONCLUSION
In this work we provided an improved analysis for generation process of the diffusion models under just the -a-accurate score estimation and finite second moment of the data distribution assump- tion. We showed that by modelling the SDE based generation process as an ODE step followed by noising and thereby targetting the discretization error along this ODE path can lead better depen- dence on the step size. We also introduced a novel analysis framework for this ODE path which expresses the overall derivative of the score funetion in terms of spatial derivatives and establishes relations between the score and its first, second order spatial derivatives. This resulted in achieving linear dependence on for the considered ODE path, leading to a new state-of-the-ar convergence guarantee for KL divergence. Since KL upper bounds the square of the TV-distance by Pinsker's inequality, our result also provides a stronger guarantee than the best existing rate for the TV con- vergence achieved in Li & Yan (2024). An interesting future direction can be to investigate if the dependence on the step size can be improved further when considering this ODE step followed by noising framework, thereby enhancing the dependence on and achieving faster convergence.

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ACKNOWLEDGMENTS
This material based upon work supported partially by NSF under Grant No. 2416897, Grant No. 2505932, and by ORN under Grant No. N000142512318. Any opinions, findings, and conclusions or recommendations expressed in this materi al are those of the author(s) and do not necessarily reflect the views of the funding agencies.
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Nishant Jain, Xunpeng Huang, Yian Ma, and Tong Zhang. Multi-step consistency models: Fast generation with theoretical guarantees. arXiv preprint arXiv:2505.01049, 2025.
Holden Lee, Jianfeng Lu, and Yixin Tan. Convergence for score-based generative modeling with polynomial complexity. Advances in Neural Information Processing Systems, 35:22870-22882, 2022.
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A PROOF OF THEOREM 3.1
A.1 BOUNDING KL (Pt.llet)) AS AGGREGATION OF B -.. INTERVAL
FOR EACII
We now discuss two lemmas: a) The first one converts Wasserstein type error between empirical and true process to KL for each interval and the second one aggregates the KL across all the intervals.
Lemma A.1. Denoting Pta-s]tx It be the conditional probability of it- given Tt and let Pte-slte be the conditional probability of xk-1 given Dy. using steps of ODE and one step of noise similar to our algorithm. Then, for the updates in Eq. 7, Eq. 8 (the updates of our Algorithm 1 for each interval given the same starting point for both the true process at tg: Ikh, we have:
KL (Ptz llth. (-1ax)) Pre. alte (-1z)) = -2h2-1 1 0.5 - 0.5
2(1 - e 2h2-1)
where we recall that hu ix tk-1 denotes the step size, T.50.5 corresponds to two steps of Probability Flow ODE from Ty and thus, the law is same as of the forward process at time L5 - his
hk-1-
Proof. For this, we know that from Algorithm that the conditional Pra-1 142 (((x)) for the generation process is the following Gaussian:
Pu Iie (+1xx) N -o--h-1- (1 2ha-1) la)
where La is the d-dimensional identity matrix. Similarly, for the true process we can just write:
Ph- lte (+1zs)) N (e (e-ht-tzz (1 e 2hx-1) 1a)
Now, since the covariance matrices are same for both, we can just use the following formulae for calculating KI. between two Gaussians with different means but same variance:
1 (m - 12) x 'mn 122)
KL. (Pte- (+1=x) P ( (+I=x))
2
where μi, #2 corresponds to the mean of the two distributions and Σ corresponds to their covariance. For this case, we have:
f1 hx-1 -
#2 = é hy - U.5
Σ = (1 -- 2hh 1)
Id
Merely substituting these values in the KL formulae will lead the desired term.
For KL-agggegation, have the following lemma:
Lemma A.2. For the discretization sequence $1...,,4 and the law corresponding the gen- eration process in Algorithm 1, we will have (where Pte denotes the law of true process at time tz):
KL (pt, (12,,) SKL (pt, Ie Pa....
K+1
-KL (pex.u (ex.))) + Eppe 4811
KL (Pen-1 (-1xx)]l@e--11e (f12x))
Proof. The first inequality is just the data processing incquality and second equation is the chain rule for KI..
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A.2 ANALYSING B|||==-0.5- 2..u.sll
We begin by first decomposing the term into a discretization error component and the error due to using the estimated instead of true score.
Lemma A.3. For the sequence uk- generated by Eq. 7, we have:
U.- 8.-
0k-0.5 2(chethaa 1) mx) 8(te, 24) xx)
where we recall 2.-0.3 corresponds to two steps of true probability flow ODE from Ey and Øg uu-
corresponds to the discretized true process update defined Eq. 9. The first term in the RHS corresponds to the discretization error (Ta) and the second term is the score estimation error (T.).
Proof. We can just bound the LHS as follows:
v [B|||==-0.. 0.5 8E o.- S P Fh o.s
Blll 0.5 8x 0.5
Ta
T.
where, discussed in the lemma, 0k-0.5 0.5 is defined in Eq. 9. Squaring both sides and using 2ab S a² + 12, we will have:
= - o.s < 2R|l| - T #x-0.5 2B BB]lix--0. 8x 2.--0.5l)))
Bounding Tes. Now, utilizing the Eq. Eq. 9, we have:
# 0 [12] (e - (8(12; $(12, rxk) 113
thhthhh- 1)² - 3(1x, xx
We discuss the analysis (and eventually bounding it) of the discretization error term Ta in the sub- sequent subsections.
A.3 ANALYSING THE DISCRETIZATION ERROR ALONG THE ODE PATII
Considering rescaled process. 1) as z(t) = etx(t), leading to:
We consider rescaled version of the original OU process (Eq.
21)) = y Ve 1-7: 77
(o
(13)
where y corresponds to the data distribution: 2o xo = y ~ Pdata with the corresponding forward
SDE being:
dz(t) x(t)de' + +''z((t) =(t)e' +t (-z(t)dt + V2dw.) = 22 dw,
We denote the law of this process time t by q:(:) where de is just pushforward of pt(-). Also, we denote the score function of this rescaled process as s, where we will have 8, =(t))
The probability flow ODE becomes:
d=(t) = -t (1, (t))
(14)
Using the Exponential Integrator discretization for a given interval [t here also, we define Ek-0.5 0.5 for this interval when starting from 24:
1 20x- (62hxx 1)s, (1k: =k) - = =k + z
(15)
where tx-2 = to hR h Now, we have the following lemma for bounding E. [1|5 E Ex-n.sl13] where za 0.5 is steps of true probability flow ODE Eq. 14 from 2k-
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Lemma A.4. For the -05 defined in Eq. 15, we have:
(hrx + hw- )3 f" e [1()(((,+))))1]] de
E [1 0.. S 2
hw-2
where ((t) is the derivative of s, 2(t) w.r.t and can be calculated using:
s,(t, = (t,2)
(t,(t,2) dz(t)
+
01
Oz di
===(()
Proof. Since, we have used the Exponential Integrator discretization, the ODEs for the interval [f corresponding to 2k: žn (given Ex = 2ạ) are:
dz(t)= = -e st, (t,,)
dž(t) = -e2 =²"ss, 2x 2x)
Therefore, we have:
fk d(ž(t) z(t)) Smm th 22 (sp(tk, 2k) (t))) dt
2k5 2k-0.3 -
sp(4k: + S. s,(uu, za) du
20 sy(tks 2k)
de
Taylor's Integral Remainder
where in the last step have just used the Taylor's Integral Remainder form for the score function. Using this, we will have:
1. 1x-2 S e" e, ((u,)((())))))
Bill 005 - 2.- 12]
E
(hx + hx-1) f" dt 11 E 2" (, 2 x (u,
1'h (Ix (Ix-1) 1'h 11 ((,))) dudl
(hx + hx-1)
14-2
pu
(4x- (ldl 1.M E "[ x(1)) du
(hg + 1h-11))
le-2
AuE ||5, ((,,((1)))1) du f 20-2
= =(hs + 11--1)
(1x - l)dt
16-2
(h+ h2-1))
th
A"E
[|s, 2(4)))
du
2
We calculate and bound the spatial gradient since the partial gradient time be written in terms of the spatial gradient using the Fokker Planck Equation (FPE).
Calculating the Jacobian Vs, 2(t)) for this rescaled process: We can now observe the fol- lowing for this rescaled process:
(x)2) = f (zly)))aaaa (y)dy x
/
||a-y1?
2122 11 Pipaa (y)dy
(16)
which takes us the following formulation of the score function:
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Lemma A.5. For the rescaled process in Eq. 13, we have:
(t, Byl= (e2t 1)
Proof. As discussed in Eq. 16, we can just write the score function as:
1z-v12 2(- Pdata (V)
1s-u12 Pdaat (y)dy
s,(t,2) = V=og = Vgr(z) V g((z)
y=z (1= e.
2602t
ll x112 2(e21
1a 1e
Se
Pdata (y)dy
ʃe
2(c21 Pdata (y)
/ P(ulz) 11 --2
dy
(e2t 1 1)
2
By = (E 1)
1s-u12 fly.)ddy and P(y, 2) =e 20 1) Panta (y)-
where P(y|z)
P(y.a)
Lemma A.6. Jacobian of score. We have the following expression for the Jacobian of the score Vs, (1,2)) for the rescaled process z(1);
Ia
(s, Var yar
2
e²t
where Var denotes the covariance matrix.
Proof. We begin by calculating (the gradient of P is w.r.t. second variable a in this Lemma), P((l2)) VP(,, Ply z)dy P(y. 2) PVy,
(ʃ P(2, v)d))
From the calculations in last Lemma (A.5), have:
PP(y,)) = y 22f P(y.2)
and therefore:
y
2(y,2)
P(y,2)
J 2-
y=2 P(y, dy
22 P(y, z)dy
J P(y, z)d
P(y, z)dy
P(yl=)
222 - Egll
リ-
リ 2
Thus, we can calculate the V (1, 2) as follows:
(s,(t,2)= = /
e22 P(yl2)) dy
2/ z
La
/
P(ylz)dy
22
/ P(yl2)
2
1/
Ia
Eyl=
dy
222
222
2t
2t
1/
1) 222
y
Ia
Eyl=
Eyy
e2t
2tt
1
21
1
1/
Id
Vary|=
2t
e2
Bounding Eg, s, (t, |),, Egg [/ (t, [? and other spatial gradient terms. Since, we NO, Ia). we first provide a helper lemma to bound the moment of the multivariate Gaussian distribution. Then using that and the formulae for score function, Jacobian provided in Lemma A.5, bound E. []| =).lP EE s1, (1, =)]]]] for a general p.
)-2(() know that v-l
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Lemma A.7. Gaussian Moment. We have the following result for the Gaussian random variable ~ Ia)
E 11mm T [|%]] E [1m]]" S (d + 2p)
Proof. We will have:
Imm T || Tr ((m (T((m=)) )) = Tr (m (pmmmm) "=mmmm=> = "7'"(T((mmm (n)
Thus, we have l!vm T = (n (n' 1)) ||ml22 Since ~ N Ia) and thus, the vector has i..dd
normal entries, thereby:
Σ TH (d)
-| = [[())) where X x2))
where (d) denotes the chi-squared distribution with d degrees of freedom. Now, can just use the formulae for moments of X (d), leading us to:
[||]| 2r
E[X = 2"
T(p + 4) (a) T
S 2"
2
d = 2p))
+D)
where I' denotes the gamma function and for inequality (a), we have just used the gamma function bound.
Lemma A.8. We have:
d
(d+ p)p/2 (e2r 1)p/2
By [1))(((.((())]]]]
2 T By (1,=((1((00)1))))) <
2d 6d
Eg. [t,ss
(e 1)
Proof. Similar to the Chen et al. (2023a), here we also utilize the fact that
v-x(0
Gaussian.
Vi-
Using Lemma A.5 we have:
Eq [11s, =)]
(6
E
Bgnqe
Bylz
2
2r
2
(c2t
1
Eqg Eyle
Ve
(0.1a) [|]]||]]
2t
1)
d
222
For a general p 2 2, it becomes:
y
Eg.
(e2t 1)p/2 Eyg
Byla(t)
(e 1)
11 y-
Es [y))(())
(e2t 1)p/2
Ve -1
E En.-i(o. [|m]|P]
(elt 1)p/2
(d + pp)p/2
(Lemma A.7)
20 1)P/2
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Going similarly, naive bound Ey.
[|[V s, (t, 2)
will be:
Egngq V| (t, 2)]1]
Benge
Var
2
La
(Lemma A.6)
2
e22
1
P
y
y/
2d
Bylz
e
1
(e 1)2
y
y
2d
21 2.e Bulz
2
22
(e 1)
2d
B Bynorrco.aa [1lv11]] + (e.2 1)2
(e21 1)2
2d + 6d
(e2 1)
A.3.1 EXPRESSING THE Ep, (()))11 IN TERMS OF SPATIAL DERIVATIVES
Since we also need to bound 0t,, 2) to bound the sy (f, =). we will utilise the Fokker Plank equa- tion associated with the forward/reverse processes which relates the partial derivative w.r.t. with the spatial derivative. The Fokker-Plank equation corresponding to the rescaled process z(1) (Eq.13) would be:
d
0zq((2)= = Σ 0.
= 22 (z)
(17)
Since score function is just [q.(s(t)) cess to relate the Öts,(t, 2) with spatial derivative in the lemma below:
9 we provide the corresponding score-fpe for the rescaled pro-
Lemma A.9. We have the following counterpart of the Fokker-Planck equation for the score function of the rescaled process defined in Eq. 13:
at 2) e2 (t, 2 [ e² As,rrt,,
2e"' 2e st, (t, (18)
Proof. To arrive at the equation for the score function, we first derive an equation for O log by considering the following term:
d
V (z)
(22)).V9((()) (.4.(2) Vz)
20
O V log (x)) = 2t
Σ
0
20
(x(2)
pē(2)
1=1
(.)))())) 9e(2)
26
22 / log
which results in:
0 log (=)= = (.)(2)) (=)
d
a (.Vaa(2)
2t
20 Σ
0 (V log) + 2 1 log (z)
(x(2)
1=1
1=1
Now, again taking spatial gradient:
v0, log (z) = 20
vd, (V (z) + "V (=)l²
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Interchanging the operators result in the score Lokker-Planck equation for the forward process (on the reverse it would be negative):
(t, = e² (s,(t, + + V 2)1 = (t,) 2e V)r(t,)) st,,t)
Now based on this lemma, we express the overall derivative in terms of spatial derivative in the following lemma.
Lemma A.10. We have the following relation,for the overall score derivative st, z) and (t, for the rescaled process following the reverse ODE (Eq. 14):
Bg (t, 2)
Bgg. |ts (t, + ett (t, sy st, 2)112] + Bug [2e (t, 2)) (Vs, "T_ (t,
(19)
where (recall from Section 2.1) A denotes the Laplacian of a vector.
Proof. Now, utilising the Fokker- Plankkk equation (FPE) for the score function, we will have:
Eg ((t))
Be (t, (s, [t,
(a.s.(1.2) (1,
(
+
dz
di
(a) Eyg
(1,,,(1;2) - " [(, st(t, (Tx) (1) (Vs1(,))))) sx(t,
[10,s, 2)1 + /
V3,
(t,
22 (t,, (t,,(t st st((,)))
E
l1ea lle As, (t, 20 st, Vs,
(ts st,t, 2)11
- -2 (t2 Vsr ( (V
(Vs, To(t.2))
= -Eg
1 (411sss, 2)12 + (8, (, =) 26 26 (As,(11))) (V5r(12) st(1, st(1,))) x))
where in (a) we have used reverse ODE Eq. 14 and in (b) we have used the FPE for score from Lemma A.9.
A.4 BOUNDING THE REQUIRED SPATIAL DERIVATIVE TERMS
Now to bound the terms Ey (t, =). E (t, Vs, 2)1 appearing in Eq. 19, we analyze the relationship of these spatial gradient terms with d Egg [t)s,(t, 2)] Since the naive bound on Ey. [t, [|] proportional to d as against d2 in case of Ey [|, 2) )) this can lead to improved d d-dependence of the discretization eπror upon integrating this for given time interval. We first discuss two lemmas: the first one establishes relation between di E.g (t) and E.g V (t, ). Then extending this lemma for general power m in di Ey (1, 2) m*] leads to the terms comprising both Vs, and 8, from which we can bound the (t, Then, utilising these lemmas and bounding terms comprising &g. and E [|s, (t. [?]. we finally bound E,, [[]As, (t, [|] by applying Integration
d
d
term E [|ls,
Vs, in terms of d de
By Parts.
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A.4.1 ESTABLISHING RELATION BETWEEN SCORE AND ITS FIRST ORDER SPATIAL GRADIENT TERMS
We first analyze the term ae Bg [11s, =)[[] and manipulate it to relate it with Egg V/ =) leading to the following lemma.
Lemma A.II. We have:
d B. (t, 2]]1 -22 B. (t,z)))
dt
Proof. We begin by analysing the LHS term, taking the derivative inside the integral and utilise Fokker-Planck equation (FPE) (for qe and 8, (Eq. 17, Eq. 18) for the rescaled process to convert to spatial derivative and finally utilise Integration By Parts (IBP):
d dt d
R. (t,
ʃ (tt))lls,tt
dt / (.q dz + 2 (tt))s,tt,)) (ts,
(
/ ( d= +2 (1)))),(,)))
(1), dz (=))){(()2)) (As, V ||,, (1, =) dz
1
" / Vol(:)
)]1 -2
( 20 / Va((2) (t, 2) 2) + -222!
(t)),,(t,2) (As,(t, 2)) dz
2¹ / Vo((2) (t,,, 2) 12 dz 2e
((t)) St st, 2)2
jointly for IBP
(d) 2 /Va(tt)
(Vultt) (t, Txr(1 z)dz
d== 262t
20 / ((, (1, 20 f [a(=)\Vs,
where in (a) we have used FPE Eq. 17, 18, in (b) just use IBP, in (c) we use (1,)) (2)(2), in (d) we again use IBP and in (e) we use [tls, 2) = 2 (t,s,(t,2)) s, so the first and second terms cancel out.
We now generalize this lemma by establishing relation between di Egg [||s, (t, 2)]l for gen- eral m > 2 and E,, [I,V,, =)] For m the RHS should have terms comprising both Vs, (1, =), (r, 2) and thus the result can be used to bound the second part of the RHS in Eq. 19.
Lemma A.12. We have the following general result for the score function s, 22) of the rescaled process 24 defined in Eq. 13, holding for any m 2:
2)113 2 (t, 21 Ey (1, 2)1 = mm g. s, dt
m (m - 2) Eng. [|st (1,
-- =
(1, 111)]]]
Proof. Here also, we start with analyzing the LHS similar to previous lemma (as discussed Sec- tion 2.1, 2 corresponds partial derivative w.r.t. 1th coordinate of A Σ, 0,0, is the Laplacian,
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sr()); corresponds to tth element of 8, which implies (t,) Ei= 2)) (all the vari-
ables under Σ range from 1 to not mentioned):
d e
2r
(t, 2)
dt
(
21
(r9e 2) + m f (tt st,(xx)) 3o( (ts,(t, (,,,,)))
(t
m / (<==)]lsr((,2)13
(r((, (10 dz
d
m
2),z
2
1,
d Σ
/
(0,0,s, + 0, z),) dz (t, 1 (1)
+ m
i m
Σ (x))]ls,(t a),,s, (t, 2),
m
Σ
(() =),z
()g((2) ||s| (t, 0is 2),d
Σ
(1()))
2); 2)
m Σ
pc()) (1s, 2)1 (t) 2)
1 -m(m - 2) Σ
(c(2) 1|s, 2) s, (t, (t, Ots,(t, 2),d2
k.j.k
(((2) (t, (t,s,
m(m 2)
Eyg ||s,, (t,,))lm= m- [[ |
mlg [|]s) 2)11 m - Vs, (t,
where in (a) we have used 0el1s, (1,
2 m||s, (4, =)|km= s- (1, 2) 0, O,s, (b) implies the use of FPEs Fq. 17, Lemma A.9 (c) is the application of Integration By Parts on the first term, (1, then subtract it from the second part of the second term and use at, 2), = 2s,(t, 2t) z),, (e) implies again using Integration By Parts on the second term where one term is jointly considered as I1 and the other remaining.
(d) uses (,g((2)
(f) is derived using 0.g = (t,)),, 2) on second term, cancelling the first two terms and writing 0, s (1, >) (m 2) (t, => (t, 2),
Now, discussed before, consequence lemma of this lemma is that bound the second term comprising both 8, and Ss, in our main Eq. 19 (since the RHS contains these terms for a general m). This is stated as lemma below.
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Lemma A.13. Defining Xm [e(())l (1, =>]l"=="||0")
S [q((2)lssr(e (0d,,))[[livsssss 2)? X
|| (1, 1112 we can bound it as follows for any > 2:
Xm S _ --- 21 d Be (1, =)] dt
X
26 E. [11 =)11
m(m 2)
dt
Proof. For this, considering the Lemma A.12, we have:
-1 e e-2 d F. (t,111s,(t, 2)1 11 s 2)11 m Vor 2)1
m di
Xm
(m
2)
Ey.
[1sr 2)11 3>1l--11(v (s, 2) 12
X
When > 2. will have Xm, X 2 0. thus, Xm S --e L. (t, =) and
n m(m 2) ge [||ss, (t,=))]))))) =)]
A.4.2 BOUNDING THE SECOND ORDER SPATIAL GRADIENT SCORE TERM
For this, we first provide lemmas to bound terms comprising second order spatial derivative of the law qi and first order spatial derivative of the score 8r.
Lemma A.14. For the rescaled process z(t) in Eq. 13, we have:
(q)(2) (2)
-d
lly 1 (e2c 1
+ Eyl=
(e2
Proof. Here also similar to the score function calculation in Lemma A.5, we have:
1|=-112
(q:(2) V (q( (a)
2
Pdata (
90(2)
(2)
I1s v12 f
2tc2
Pdata (y)dy
Pdata (v)dy + S (ve Hlaat
(---17)
Pdata (y)
( dy
l|s=yl12 Le 26ete-1
Pdata (y)dy
1
Zt
Pdtta (y)dy + +fe 2ie
y-z Pdaaa (v)
w-z
dy
11
(122-11)
(02t (c2e-
11 fe
2(025-1) Pdata (v)dy
(e20 - 1) + / P(ulz)
-d
y-2 2
y 2
(e21 11 dy
(020 -
-d
11 =1/
( - 1) + Ryyl
(e2t
1)2
where in (a) we have taken the expression also used in Lemma A.5 and as discussed before y Pdota-
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Lemma A.15. For the rescaled process z(t), defined in Eq. 13, we have the following bound for the term involving the Laplacian of margin and Jacobian of score:
I Aai(t)IVs,, (t, V
Cad2
de
2(1 + log d ) 1) Ey [||s,, 2)
(e2t-1)
(e2e - 1)3
1
where Co (1-21og 5) (1+log d)
Proof. We start by writing Laplacian as E, 0,0; and decomposing the term as follows:
/ (ss
/ Σ
/ Σ 0:0:ge (t) Vs,
/
c2t 22 Bug [ (t, =)1
/ (x)) V >)1
Cm V (t,, 2) 2/10 dz
d
2- B [tvs,(t,))]
(6) cod Vst + e
m ʃ (<t))lVVs,tt>>)) =11
1
d E [s, 21)
22t
where J((:) = Σ, (.):g(())
(2) 2 1 I and m are constants where 1 > 1, 1/1 + 1/m = and in (b), have just used ab S ta + m
1." with a (1) b Cm. Now, we utilise Lemma A.6 for
the spatial gradient of score, Lemma A.14 to write J₁(2) 21 + +yyl I-= (e 1) Also, for the
second term, can just use Lemma A.1 11, leading to
Cm / (1())
lly
213 1) 1)2
1
Ia
E
Var, y/=
dz
y
(E2t
2
1
22t
F
e 21
d
d
'
B.g.
[1| [1, =]
2
2 1
dt
Cm Eeqe
Ryyl
11 - =
E
Byle
(c2t 1)2 ) ||| 2))2
|ly
=11
d
(e2t
1)2
(c2t 1)2
e 2t
d
d
cm
E
2
21
dt
Cm
Eq Eylz
[ly - 2112
14-2
d
C
-l
By Byl=
[ly [ -111==11 2 1
(1ee 1)33
1
(e21
1)²
(e21 -1-2
1
e 221
d
d
E [t,, 21,
(Jensen's Inequality)
2
m
2t
dt
where in the last step we first multiplied the terms in the first part and then used Jensen's 's inequality for each. Now, utilising the fact that w-z NN(O, and using the Gaussian moment bound
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from Lemma A.7, the last term can be further rewritten and bounded as:
Cm
E ((
|Inlla
21+4
d||n112
2 =1)
d
d
11, (t,2)
1(e2t 1)1t2
2
m
di
(d+2) 21 + +4) 2
cm
d
d
Eg [|s,, 2)1
1(e 1)++
2
e
dt
d (c"l
m
2t
(e
11
d
(d+ 21 + 4)
E (1,s,(t,
1(e21 1)44
2 1
dt
(cm
[e 111//m
8 + 21/d + 4/d) 142
2t
d
d
d
Ey.
1(c21 1) -1/m
2
m(e21 1)
2 -
dt
d²
(1 logd
G G d+3
222
d
d
d
Ey (t,,,()))))
(e2t 1)3
(1 log d)
2
( (1
(c (c21-1)
E -11- dt
logd
Ca
(1 1 log d, m
+ log
Cad
e 21 de
d
<
Eg (t,
(d = e)
(e21
1)3
2(1 + log
(e22
1) dt
where we have used Cm (1tm-1) 1 1 + log d,
logd which results in Ca
(1+2 logd + 4) 4+3
(1+logd)
A.4.3 BOUND LAPLACIAN OF SCORE
Now, using the previous two lemmas and the Lemma A.13, we bound the second order spatial derivative term of the score function in the following lemma.
Lemma A.16. We have the following bound for the second order score derivative term in Eg. 19:
32Cad²
2 d
8
Eg. (t,2)l)
d) B
2)11
Ey. [/[ 2) >){
13(e2t
1133
(e2t 1)
13
20e-2t d
Ey [t|s,(t,)))
13 dt
where Ca is defined in Lemma A.15.
Proof. The proof is just careful utilization of the integration by parts, Fokker-Planck equa- tion (FPE) and the reverse ODE Eq. 14. The proof starts with manipulating the term: d E. [ I?? to break it down in the target term and remaining terms from the previous two lemmas and Lemma A.13. Then the target term term is expressed via this term and the remain- ing terms where replace the bounds for the remaining terms from the mentioned lemmas. It is
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as follows (again we use O for the derivative w.r.. 8th coordinate and the Laplacian by E, 0,0,):
di / [[=)]][ss
=f [
13 (=)00, Σ
(1,
in
= ʃ E Σ
qi(2) 2 Σ
058,(1, (500(,,)),00000(11, dz
+===1
(Eq. 17 for first term)
(a)
/ 22 [=)
22 / a((2) Σ (t)s
Σ (n0ns)((12)) + Σ 0,s)(t,2)2 dz
Ad
(b)
25
(,0.,n(3]1VV - 2e
Σ / (0) 0,s,,tt,)) dz
ijjk
20
(/Σ (, (t,
(c)
21
Σ / (((2) Σ (r,(t,2),0ss((tt,)) (0:0 +0,, de
2 Σ a1(2) Σ 0,0 2). Σ (OnOss, + + (t, dz
where (a) implies use of Lemma A.9 for the second term, (b) implies using Integration By Parts for the second term where we consider the term 4 as one part and the remaining as other, (c) implies using 02 (2)ss, z), and then Ô st, 2), (t, 2), in the second term. Now, we consider the terms except first, treating E, (t, 2)5 = 2 E3 Sr (t, (), 2)} = by and E, (t, a, these terms can be written
Σ b, (a: + b.) 2a,(a, + +b) Σ -2a b² -3a,b; - Σ -2a
b3 +
3 a 6b}
8
which leads us to:
2
2t d dt
/I Σ (1
13
Σ / (e(z)
By
s
2t,2)
dz
8
To
Ti (target term)
2
20
Σ / (0(2)
(t,
dz
1
T2
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Denoting the first term in the RHS Tu, second or the target tern 7i and third term T2, have the following expression for our target term Ti (rewriting E, a,O, as Laplacian operator):
13 Ee [|As, (t, 2
To + 2072 - d
Fgg | (1, 21;
dt
2t d
To + 5gg.
[ (t, 2) 2)1
E [| 2)]{]
(rewriting Tb)
dt
(0)
4d
2t de
d
$
g. [1| sr (t,
2e dt Ey [t,,,(t,))] 2)
20
(e 1)3
2(1 + log ( - 1) dt
2t d E Vs, 2)
dt
where in step (a) we have just used Lemma A. for To term and the observation that the third term (obtained by rewriting T2) just the X in Lemma A.13 for which we bound using the Lemma A. Now since d > 1, we have approximated the value 13(1+ mgaa negative, can be seen from Lemma A.13) leading to the final bound.
4e
< (since d क En [1/ss, 2) |
A.4.4 BOUNDING THE DISCRETIZATION ERROR FOR EACH INTERVAL
Utilising the lemmas discussed above for bounding the spatial derivaitve terms in Eq. 19, we now
provide lemma which using these bounds provides final aggregated bound for E. 1/ (t, 1)]
Lemma A.17. For the rescaled function, we have the following bound for (1,
3c21 de d
10
2t d Eg. ||Vs, 2) 2)
gg (t. 2)1 S 13(c2t - 1)3 4(E2 - 1) dt E (ts,, 2)1
13 dt
d Ey (t,
3e
dt
where Ca is taken from Lemma A.15.
Proof.
gg (1,s =]
Egg
ee ||s 2)11 Vt, (t, 2) 2) 20 (ts, (Vs,(t, 2)T s, 2))
$ Eg
5
41 ( (t, 2)
5c [t, 2) s, (t, 2)
+ 4||b12 for a Rd) ||all (2a (2a-
1
5
5
EE, |1|Ar, (1, =)113]
+ "E,, || s,, (1, x) =
4
4
5
5 21 d
"EEy [1]ss,(t (t, =)]
E
(1)
4
8
dt
20 et
2t d
10
2 d
Ey. |||s (t, (t,
By (t,
13 - -1)3
2( 1) dt
13 dt
265 2 d
Ey (t,
104 dt
where the first equality uses Lemma A0 the second last inequality uses X (m 4) bound from Lemma A.13 and the last inequality uses Lemma A.16 for the first term. Now since
di E [t, 2) [+] will be negative from Lemma A.13, then here we can use 104 <
265
leading to
the final bound.
Now, we have the following Lemma for bounding the discretization error 2(t): E. [[|= Ex-0.5
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Lemma A.18. The discretization enror for each interval E [||2k-0.5 0.5 -k- [12] A.3 can be bounded as (where R/g- = hig + ha-1 1 and recall L2 = la- hx 11-
discussed Lemma
(2Ca + +10dd + 24d) (11b) (chi - 1)
26 2B [|=
Ex-0.5
[2] S
(1
2ta-2) 2)3
(h2))² 3
edtt
10
Bag
(t, + 3.m Ills, (t, =2)111) ] tN
2
13
5(hL))
Ee (t.2))))t))]])
8(1 - E 2))-))
Proof. Using Lemma A.4 and Lemma A.17, it can be bounded as
e 2E [1120-0.5 Ex
00
fr
<ea2 21x = (h+ + 13
E
E
|| (t, (t,2()))
dt
4--2
(hx + 1/-1 1) cother
Ex
(t,,(tt,(())) dt
2
1k-2
(h1- + -2211) )33
40
5e
clothen
B IIs (t, 2)1A1) dt
13(E21 - 1)3
4(e2t 1] dt
2
(h+ + -/--1)3 Nuithe
tu
10 d En [IVs, _ IIsss ((2201))))) de
Eat
dl
1x-2
200 (hs + h2 3 hn + he tha
13(1 e 201-233
5(hx + h4- 3
1" 2e²" 111 2)12]at
[0 Be (1 2)11311
8(1 - c 2-x 2)
(hy- + h4- )3 bother
10
to Vi, 2)1l + 3., (1, z)
Eng
2
13
24-2
10
+ (hx + h4 1
20
Egg
[| (t, 2) 2)1 + +3Eg. (t, 2)11
dt
13
1x-2
( 20Cad (hx + h4 (E 13(1- - e 21x 2)3
1)
10(h: to (1--1)3eemm )ze/uthhe-dd for 621 c dt
+
8(1 - E 2))
1x-2
20( + 6d)
+ 6ef + 12d
+ (hu + hx-1)² (e
dt
13(e21 - 1)2
(e20 - 1)²
by
(hx + hx-1)) g Nather
([ eat
10
Vr (1, 2) ]]] + 3E [11s (1, ((,,)))))))) n
)
En
2
13
5(hx + hi 1het
[e Bee |l| (t, 2) || =
8(1 -(x-2)
((2aa + 10) + 24d) (hse + ha 11-11)1113
3 (els-th-i- -1)
(1-6-212-2)
(hi + Iys 133 Mathers 11
10
[1, + +3 ((,s,((,))11) )r "
Bq
2
13
5(h2 + 1x- )sebethie-dd df
[e [|| (1, (1,,))122 =)]
8(1 2112-2)
where (c) uses f
at " dt S --
d dt. fite 2e dt = log ( log(1+ +
a:) S for the first term and implies applying the Integration By Parts to the second and third terms,
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(t, 2) as one term and use the max where in the second term, we have considered e² Ey. (t, value of the remaining term since know from Lemma A.11 that e2x di B, (t,,, 2)]l I in step (d)we recollect the integral terms and since they have positive contribution, just replace the term inside the integral with the upper bound from Lemma A.8. In the last step, similar to step (c),
have used _ - dt log ( for l
e" tef-Ijrdt - Ji j dt
24k-
and finally log(1 + x) S I.
A.5 PROVING THEOREM 3.1
We first discuss a lemma based on standard calculus which would be utilized in the theorem proof.
Lemma A.19, For some fixed с E (0, 5 and denoting a == 2 - c). b = 2 - c. For x E (0,1) define
(2 c) r eh euv
(2 c)3 x33 c.o
f(x)
g(x)
(ear 1) (1 st
Then f and 9 are increasing on (0, 1) and there exist absolute constants Cj, Cg > 0 (independent of and x) such that for all (0,1).
0 f(x) - ((1-o0)) < Cy
0 05)(2) - 1(1--((1)))) - -c)22) s Cgg
x"
1 e
Proof. Using de log(en*
ax 1)
3 ae
3
agar
a²
(log ()())
(log ())))))())))))))>))))) b
2r
e
For t >> 0 we have the elementary bound
J I f: hence
2 2c
00
S a +
1
Therefore, for x E (0, 1).
(log f)(z)((x))))
2
3 x
b
(a
1
1
+
2
> 0.
and
(logg)(()) 2
3
+ a
+
+
2 > 0.
x
1
Hence f and g are increasing on (0, 1). Using ear x E 10
1 >
ar and
(2 c) < 8, 6 S 2, we get for
(2 3)3 de
8x 62
f(x)
Se x²
Consequently,
f(x)
86 2
g(x)
22
1 1- - 4
1 -
27
From above, we have:
fxxx = (log < f(z) (3 3 b) < 20 (3 +2) < 40c²x.
Similarly,
g'(x) = g(x) (z)) s s(x)( {z)g))())>{((({ +b) s Sc²
x²
3
2 s 40e
e
x
x
E
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Now for y :- (1- ( -1= (1- (2c- (2c-c))x= s 2cr. By the mean value theorem, for some E (v, C (0,1),
[(x) - [(y) = (5) (a - u) S 40c² & (2cx) < 80e² ex²,
which yields f(z) - f((1 - x) s Cy cx² where Cf = 80c² is an absolute constant. Likewise, for some 17 (y,z)
g(x) - gly) = 5'))) (s y) s 40e²
"
(2er) S 80e2
x e- 27 2
c و
since
et is increasing and 17 S x. This gives
g(z) g - x) s Cg
x
e)
where Cg = 80e2 is an absolute constant.
Proof of Theorem 3.1. Now, using the Lemmas discussed above, we provide the proof for Theo- rem 3.1.
Proof. We first bound the discretization error term for the rescaled process in Lemma A.18 aggre- gated across all the intervals. For this, using 4/g hg + hg-1, we bound it as following:
K+1
21x-2
E [[|2 2k-5- - 2x-0.5
2hp-1
-1
K+1
((20s + 10)d² + 24d) (11)) (E - 1)
Σ
(62hx-1 1)(1 2-1x-2)
km
K+1
10
42-2
Egg.
[1Vs,
+E [IIss (t, =) =)1(4)
2(c2hrx1 -1)
13
K+1
2t
2
Byg (1,
2(c2he-1 - 1) 4(1 E 20-=)
2=2
K+1
(22 + 10) + 24d) (M1))%M4 M/ (e/ -
(1/)8&42 R((o)
Σ
(e2ha-1 11)(1 e -222-23
62h1
(1/)3 32
+ 2h2 -1 R(tt)+ + (e2h: - 1)(1 e E2200)
(iot + (62h2 - 11) e-2(1)
R1(t1)
(k242)
3 N3 (11
(11x+1)3
(h'x)3
(e2ha-1 11) R((x)) (62h -1) R((x+1)=
2ehx-i-1 R((x)
e2hx+1
(1(+2) 324
(11)882
(e2tr 1 1 1)(1 c-2tu) 1- - -222 2) R1((t)
11(k42)3e" bk-
(11z) )
(e2hxc 1)(1 C 2 Ri (1((x+1)= - (e2hk-) - 1) e- 26x
Ri R1x)
where R(t)
Le (10 g [| (1, (1,=) + 3R (1))(1))))))11))
2
0.
Rí R1 =
5d 26
E,, (t, 2)] 2. 0.
8
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Selecting the step size. Now for the mentioned choice of the step size h2 tkk -x-11 emin h. we will have tk-1 = (1 - c)ta, hx-1 = (1 - hx when to $ 1 and hy = c for
remaining. Since to = 6, we will have:
ô = (1 c) (1-c)ar T-1 c(K + - M)
for some M S + with Lm 1. Thus, will have c S log (+)+ and will have very small value for the mentioned condition K 2 d(} 5 + T). Also, for the coefficients of terms containing R,
K
for L. S 1 we will have 1x-1 = (1 - c) 11-1 = (1 - c)ht,, hx-2 2 = (1 - c)a and thus, we will
have:
(ha42 thet M 1 1i441 Ah++l
hhu-thh-1)* 2
1
(2- (2-c) Nz (2- (2-0)/242 (2- (2-c) 1 h2gel--- for 211 c/h2-2-1
2211 ojh
k when tx+2 I 1 and 0 for the rest. This can be written as (1x+42 - f(xx) where f(x) =
(2 would be an increasing function w.r.t. x for x < 1 in the small region (c < 0.5). -
el
For this, we will also have f(x) f(11-))3) S cx² (Lemma 1.19). Similarly for the R1,
(2 (2-0) 00-0)'>0(00000
have to consider: g(x))
(e201 0/ 1) 1-c 201-0"x and it will also be increasing on (0, 1) for
small e (c < 0.5) and g(x) g((1 2)2x) S
20-0 x2 from Lemma A. Since ha (20) is
an increasing sequence, we can use the upper bound for R((), Ra (1) using Lemma A.8 as:
R(t) I 4d + l1d
5d
R1(t) S 8(1 22)
(1 e 21)2
Also, the term Cd from Lemma A.15 is Ca = (1+2 +9) 9)oogd+
12 for d 2 10 and thus we will have Ca as O(1). Since R(t),R((t) > 0, the negative terms corresponding to R(Ik), R()x+1), (1)),, ((( can be dropped and we will finally have:
(1+logd)
K+1
2h
E |1=x-0.5 - -050
2hu
K+1
((20a + 10)d² + 24d) ee (e -1)
(62hx-1 1)(1 E-
-21
2
(
(
(h2) 3
R(to)= + 22h2 - R(ti) +
Rt(
2ah1
(eth1 - 1)(1 e-220)
(e4)3
K-1
(h/ (b2-2) 3e
(11
e-211) R1(l
+
Σ
18((x)
(22h2
1)(1
che11 -1
(e 1 -1]
2=
(hb42) 2ehke12
(M1))eMM
(t(tt)
(e2he+1
1)(1 -224)
(e2he-1 1)(1 ---21-22
k=2
K+1
d2
+ +1ī (R(to) + R2(6
+11 (R(())
(14)
+
11 c 24 2)3
-
e-2-2 -20
1 c e- 211
k=
M
R((tx)
K-1
(2
+
Σ
chi
R((a)) + 1- 26u
Σ
- 222a-2
2=
k= M+1
M
K+1
Bo
222
2
Σ
+
e
21
(1 e 24 20e-23
(1 -
c 20o
(1 - e
2 )
k=2
k=
K+1
Σ de
2
where tj < 1 for k < M. Now using Lemma A.2, Lemma A. Lemma A.3 and the scaling back the
above bound on the aggregated error for the rescaled process غ, we will have (using < e - < 2u
29

---

Published as a conference paper at ICLR 2026
for (0, 1)):
KL (pt, (e,))
K+1
< KL (pexti Pexti) + E Ep
Σ
KL (pin-alte (+2xx) (l (-12x))
K+1
2hg
KL (pexts [[pext))
+
E 2 0x50.512
2h2-1
2=
+ e --
2-21 i :
[) TR)) 8(tz, 3x.11
K+1
K+1
S KT (pexts [[ee).))
dec
hg (1s, a(t,,, [|
k=
The last term can be just bounded using Assumption 2.1 and the first term is the initialization error discussed below.
Initialization Error. The term KL (Ptx-. ( ) is the error due initializing the generation using the Normal distribution and can bounded via convergence of the forward OU process after the total time T (Chen et al., 2023c):
KL (Pex+) [[Prc..) S (d + m²)e
where my B|| [1.] Thus, have the final expression as:
KL (pe, (ppe.) S (d + m2)e + d K + +T
30


Reviews:
[{"review_id": "skFRJ7UwXo", "reviewer": "Reviewer_mkej", "summary": "This paper investigates the convergence rate of DDPM in terms of KL divergence and establishes an iteration complexity of $O(d/\\varepsilon)$, improving upon previous results in Benton et al. (2023). In particular, the authors first quantify the discretization error via the corresponding ODE and then translate this error into a KL divergence bound by injecting noise into the current sample.", "questions": "Please refer to the Weakness part.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "The theoretical result provides an improved convergence rate in KL divergence compared with Benton et al. (2023).", "weaknesses": "The main contribution of this paper lies in extending the previously established total variation bound to a KL divergence bound. However, this extension appears to be of limited significance, as it does not improve the dependence on either the dimensionality or the target accuracy.\n\nMore importantly, the authors claim to introduce a novel technique that **controls the KL divergence for the probability flow ODE with injected noise through the squared $l_2$ discretization error of the probability flow ODE (as stated in Lemma A.1)**. Nevertheless, this observation has been widely recognized and utilized in prior works, which the authors fail to acknowledge. \nA lot of researches has focused on reducing the KL divergence by designing ODE-based sampling methods aimed at improving the squared $l_2$ discretization error of the probability flow ODE. For example, [1] investigated a second-order strategy, [2,3] analyzed the randomized midpoint method, and they improve the convergence rate of diffusion models. While it is fair to note that this paper is among the first to apply such a technique to the first-order method (i.e., DDPM), the claimed methodological novelty is overstated.\n\nIn summary, given the limited theoretical advancement and the overstatement of technical novelty, I find it difficult to provide a positive assessment.\n\n[1]. Gen Li, and Changxiao Cai. Provable acceleration for diffusion models under minimal assumptions. arXiv preprint arXiv:2410.23285, 2024.\n\n[2]. Shivam Gupta, Linda Cai, and Sitan Chen. Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallel. arXiv preprintarXiv:2406.00924, 2024.\n\n[3] Gen Li, and Yuchen Jiao. Improved convergence rate for diffusion probabilistic models. The Thirteenth International Conference on Learning Representations, 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "tuY1AUr20Y", "reviewer": "Reviewer_LSV4", "summary": "The paper shows that a hybrid sampling scheme consisting of a two time step backward ODE followed by one time step of forward SDE leads to a KL error that can be bound proportional to $1/\\varepsilon_{\\text{score}}$, improving on Benton et al (2023/4) that establishes a bound in $1/\\varepsilon^2_{\\text{score}}$. This error bound only concerns the distribution at some stopping time $t_1$.", "questions": "The proposed scheme is:\nTwo time step backward EI ODE followed by one time step of forward SDE:\n\n$$\n\\hat{x}'_{k-0.5} = \\exp(h\\_{k}+h\\_{k-1}) \\hat{x}'\\_{k} + (\\exp (h\\_{k}+h\\_{k-1})-1)s(t_k,\\hat{x}'\\_{k})\n$$\n\n$$\n\\hat{x}'\\_{k-1} = \\exp (-h\\_{k-1}) \\hat{x}'\\_{k-0.5} + \\sqrt{1-e^{-2 h\\_{k-1}}}\\eta_k.\n$$\nThis can be written as a one step scheme:\n$$\n\\hat{x}'\\_{k-1} = \\exp (h\\_{k}) \\hat{x}'\\_{k} + (\\exp (h\\_{k})- \\exp (-h\\_{k-1}))s(t\\_k,\\hat{x}'\\_{k}) + \\sqrt{1-e^{-2 h\\_{k-1}}}\\eta_k.\n$$\n\nThis looks close to a convex combination of a backward EI step for the SDE and a backward EI step for the ODE (although not exactly). My question is: Can you interpret the proposed scheme as the discretization of a backward SDE?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "* The proposed sampling scheme is simple, with a single score approximation by iteration. However, it does not correspond to the dynamic of a continuous equation (see Question).\n* The convergence result improves on Benton et al. regarding dependency on $\\varepsilon_{\\text{score}}$.", "weaknesses": "The paper presentation should be improved.\n\nAlgorithm 1 lacks of clarity: \n* line 2: Why sample from $t_K$ not $T = t_{K+1}$ (line 236) as a start using the standard normal distribution (line 238)?\n* For the last step $k=1$, what is used to define $h_0 = t_{0}-t_{-1}$?\n* What is the purpose of lines 9 and 10 recalling notation of distributions within the algorithm?\n\nAlso Theorem 3.1 is badly stated regarding stepsize/time discretization. $t_0<t_1<...$ are first given as a fixed discretization sequence, but then $h_k$ is given thus fixing the $t_k$s by a fixed rule.\n\nl. 180: Empirical counterpart: Should initialization be mentioned here? Also confusing notation $t_{k}$ and $p_{t-1|t}$ in the same line.\n\n\nMinor remarks:\n* l. 076: dependence or dependency?\n* l. 111: DDPM \"showed that the corresponding denoising kernels are also Gaussian\" It is my understanding that the kernels are chosen to be Gaussian by design. \n* l. 161: The dependcy in $t$ should also appear in $\\varepsilon$, or speak only of the marginal of the process not the OU process itself.\n* l. 180: donot\n* l. 209: Say that $z(t)$ is the variance exploding counterpart\n* l. 214: space issue\n* l. 248: Algorithm For Diffusion -> for\n* l. 316: +1 is missing in first term of RHS.\n* l. 354: $T_{est}$ police issue for est\n* Several references are ArXiv of published papers, eg Benton et al 2023 -> ICLR 2024, Song et al 2023 -> ICML 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "vmshdFTBH9", "reviewer": "Reviewer_EBD6", "summary": "The paper theoretically proves that $O\\left(\\frac{d}{\\varepsilon}\\right)$ diffusion step is sufficient to reach KL divergence $O\\left(\\varepsilon^{2}\\right)$ to a blurred target distribution by taking a probability-flow ODE step then adding a small forward-noising step. It improves on prior $O\\left(\\tfrac{d}{\\varepsilon^{2}}\\right)$ iteration complexity rates under similar assumptions on time averaged score-estimation error. This is based on the observation that the discretization error contribution to KL divergence drops like $O(\\frac{1}{K^{2}})$ as the number of steps $K$ increases.", "questions": "Experimentally showing that the discretization error contribution to KL divergence drops like $O(\\frac{1}{K^{2}})$ would put the paper on much stronger footing.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper presents strong theoretical analysis with genuinely new ideas. The framework and bounds are carefully developed: assumptions are explicit. It is a well written paper that improves on previously known iteration complexity under minimal score estimation assumption.", "weaknesses": "There is no experiment validating whether the improved bounds translate to any practical improvements.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)