EVODiff: Entropy-aware Variance Optimized Diffusion Inference
Authors: Shigui Li
Wei Chen
Delu Zeng
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion models
Inference-time refinement
Efficient sampling
Conditional entropy
Variance optimization
Primary area: deep_learning
TL;DR: EVODiff is an inference-time refinement method for diffusion models that improves sampling efficiency and generative fidelity by systematically reducing conditional entropy, without relying on reference trajectories.

Abstract: Diffusion models (DMs) excel in image generation but suffer from slow inference and training-inference discrepancies. Although gradient-based solvers for DMs accelerate denoising inference, they often lack theoretical foundations in information transmission efficiency. In this work, we introduce an information-theoretic perspective on the inference processes of DMs, revealing that successful denoising fundamentally reduces conditional entropy in reverse transitions. This principle leads to our key insights into the inference processes: (1) data prediction parameterization outperforms its noise counterpart, and (2) optimizing conditional variance offers *a reference-free way* to minimize both transition and reconstruction errors. Based on these insights, we propose an entropy-aware variance optimized method for the generative process of DMs, called *EVODiff*, which systematically reduces uncertainty by optimizing conditional entropy during denoising. Extensive experiments on DMs validate our insights and demonstrate that our method significantly and consistently outperforms state-of-the-art (SOTA) gradient-based solvers. For example, compared to the DPM-Solver++, EVODiff reduces the reconstruction error by up to *45.5\%* (FID improves from 5.10 to 2.78) at 10 function evaluations (NFE) on CIFAR-10, cuts the NFE cost by *25\%* (from 20  to 15 NFE) for high-quality samples on ImageNet-256, and improves text-to-image generation while reducing artifacts.  Code is available at https://github.com/ShiguiLi/EVODiff.

EVODiff: Entropy-aware Variance Optimized Diffusion Inference
Shigui Li
Wei Chen
School of Mathematics
School of Mathematics
South China University of Technology Guangzhou, China sgl. sgl.shiguili@gma..1 com
South China University of Technology
Guangzhou, China
maweichen@mail.se scut. edu. cn
Delu Zeng
School of Electronic and Information Engineering South China University of Technology, Guangzhou, China; Department of Electrical and Computer Engineering University of Waterloo, Waterloo, Canada dlzengescut..o edw.geesctteu.......
Abstract
Diffusion models (DMs) excel in image generation but suffer from slow inference and training-inference inference discrepancies. Although gradient-based solvers for DMs accelerate denoising inference, they often lack theoretical foundations in informa- tion transmission efficiency. In this work, we introduce information-theorrtic perspective on the inference processes of DMs, revealing that successful denoising fundamentally reduces conditional entropy in reverse transitions. This principle leads to our key insights into the inference processes: (1) data prediction parameter- noise counterpart, and (2) optimizing conditional variance offers a reference-free way to minimize both transition and reconstruction errors. Based on these insights, propose entropy-aware variance optimized method for the generative process of DMs, called EVODiff. which systematically reduces uncertainty by optimizing conditional entropy during denoising. Extensive experi- ments on DMs validate our insights and demonstrate that our method significantly solvers. For
ization outperforms
and consistently outperforms state-of-the-art (SOTA) gradient- example, compared to the DPM-Solvve+++, EVODiff reduces the reconstruction error by up to 45.5 (FID improves from 5. 1.10 to 2.78) at function evaluations (NFE) on CIFAR-10, cuts the NFE cost by 25% (from 20 to 15 NFE) for high- quality samples on ImageNet-256,, and improves text-to-image generation while reducing artifacts. Code is available at https//ggtibbbcoom//ggguLL/////D
Introduction
Diffusion models (DMs) 113 have emerged as powerful generative models, achieving success in
tasks such image synthesis and editing [ 5). text-to-image generation [6]. voice synthesis (7). and vidco generation (8]. DMs generate high- fidelity samples by simulating a denoising process that iteratively refines noisy inputs through diffusion inference guided by trained model. The model is trained via a forward process that corrupts data with Gaussian noise across multiple scales.
Despite their impressive performance, DMs face the challenge of a slow refinement process and discrepancy between training and inference
To address this, training maining-free inference methods
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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2T High
Optimizing Denoising Variance
High Uncertainty
DDIM
High Conditional Entropy
EVODiff
Conditional Entropy
DDIM
Batropy Celluction-Reginnn
Low Conditional Entropy Entropy Low Uncertainty
Low 2
EVODiff
Sampling Steps (t)
Figure I: Illustration of conditional entropy reduction during diffusion model inference. Our EVODiff
(blue) achieves lower conditional entropy in reverse transitions compared to DDIM (gray).
reformulate the denoising process as the solution to an ODE using numerical techniques. Such examples include PNDM (T. EDM [12], DPM-Solver [33. DEIS [14, SciRE-Solver (15), UniPC
[16]. and DPM-Solver-v3 [17. Despite their empirical success, these ODE-focused methods lack an information-theoreti foundation. central limitation is their neglect of information transmission efficiency. This theoretical gap suggests that the principles governing inference remain underexplored.
Our work addresses this gap by developing an centered on conditional entropy dynamics. In this view, the forward diffusion process systematically increases conditional entropy as noise is added, while the reverse process seeks to recover lost information through denoising. Unlike gradient- -haeen ODE solvers that primarily focus on numerical approximation, our framework reveals that effective denoising fundamentally operates by reducing conditional entropy during reverse transitions, principle largely overlooked by existing methods. This insight not only guides algorithm design, but also offers a unified theoretical explanation of the varying inference efficiency across successful strategies, rooted in their entropy reduction efficiency.
framework for diffusion inference
Building on this framework, we propose EVODiff, an entropy- aware method that reduces conditional entropy by optimizing the conditional variance of cach denoising iteration. Our approach provides three key technical advantages: (1) enhances information transmission between successive denoising steps through entropy-reduction optimization; (2) it accelerates convergence by steering samples towards totbbbbbbbbbiityy regions of the data distribution, drawing on principles from statistical physics [18 19: : (3) it minimizes both transition errors and reconstruction errors through reference-free variance optimization (detailed Section 3}. These advantages lead to our main contributions:
We introduce information tetemmttto---horttiic framework for diffusion inference, demonstrating that gradicnt- based methods enhance inference by reducing conditional entropy. Our analysis provides the first theoretical evidence for why data prediction parameterization outperforms its noise counterpart, theoretically grounding previous empirical findings B3 20
Guided by our insights, we propose EVODiff. an entropy-aware variance optimized diffusion inference method. Fundamentally, it differs from existing ODE-based approaches by directly targeting information recovery, not just approximating an ODE trajectory. This approach reduces both transition and reconstruction errors via principled variance optimization.
Extensive experiments validate our insights and demonstrate significant improvements in inference, EVODiff reduces FID by 45.5% on CIFAR-10 (from 5.10 to 2.78 at 10 NFE) and 43.4% on DPM-Solver++ and UniPC, while also mitigating visual artifacts in text-to-image generation.
(from 13.97 to 7.91 at 5 NFE) over strong solvers like
2 Background
Let denote the dimension of the data. The forward process of DMs defines a Markov sequence {æ,, I E [0, 7]), where æ E Rd is the starting state drawn from the data distribution g(ar) 0 2
This sequence pushed forward with the transition kernel: q (æx æa) = - (1x;; ayaoo a I). where ar and or are the noise schedules and af
af/of of is the signal-to-noise ratio (SNR). This transition kemel

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DDIM
DPM-Solver
UniPC
DPM-Solver-v3
EVODiff (Ours)
Gradient-based
Bias term (need io)
X
X
X
X
Variance term
V
Entropy-aware
X
X
X
X
Table I: Strategies employed for optimizing reconstruction enror in different methods.
[3] dx, = f(t) dt + g(f)dwr, 20 q (zo), where Wi denotes a Wiener process, (t) log (t) == dou 2 dog of [21]. the
reformulated as the stochastic differential equation (SDE)
di
denoising inference process, the reverse-tings time SDE of the forward diffusion process takes the form:
dæ, [f())x, - 9 (t) x logg (x+)] dt + g(t)
(1)
where W; represents a Wiener process. The inference generative process based on diffusion (or probability flow) ordinary differential equation (ODE) 3 is governed by dæ. (f(1)az
( log q (xte) ) dt,, where the marginal distribution (are) of æ; is equivalent that of æ. in the SDE of Eq. (T) The model is generally trained by minimizing the mean squared error (MSE) (
Bmg c, [w((() || (ara0o + orE, () - -E12)
where of + of = xo q (2n), e (O,, t ~ (0,, T), and w(t) is weight function w.r.t. t.
Diffusion ODE. Based on the relationship of Eg (æz, t) = -o; V log a (x,) 3 samples can be generated by the diffusion inference process from T to defined diffusion ODE:
dæ;
(t)
f(t)x;
Eo (a,, t), æт N (0, 0).
20,
By æe (x). == (xe oie0 (z), t))/ [21]. the data prediction ODE can be expressed as follows
dæ (f(t) + g (t)
(t) Ty (xc, t). 20
20? a.
at
di
Remark 2.1. A unified solution formula for both ODE formulations in and can be expressed as
pu(t) | de (zw()): (T)) dr.
f(x.) f(xf()====)))))))))){))))
(5)
m(e)
where vi (s(()) = When using noise prediction, have da Eg. f(x;) when using data prediction, we have dg = n.o f(xtt) = St
2t
and x(t)
and n(()) = 22
1.
ae
Gradient- based Inference. Denote hi; == x==((i=)) xl1;) and a
= f f (art.) By
substituting the Taylor expansion of do (2: 1) at Tr, into Eq.
5 we can obtain e ((Ft,
Ex= (k+1)) (K) (II,, t2) + O(h where df" (Iw(+)> (T)) = d do (2))))))(r))
as k-th order total derivative of de (Iw(r)) (T)) w.r.t т. When n 1. this iteration is equivalent to DDIM 9
d d
f f(o...)) f(ã,,) + h4 do (4)
(6)
where ã; depends on the type of dg. For gradient-based inference, a widely used technique is the finite + (*))) (xi,1) - (k) (æ., t) + Ohe).
difference (FD) method [23] as follows: (f++1) (wr, = *.
Then, gradient-based diffusion inference can be derived by using the FD method follows:
f = f(ā.,) be, de (ãr, (.) + _ h2 Fo(s,
(7)
where Fo(s;, ()) = (da 81) da (ā.,, (4)) /fv, denotes the forward FD, he, := (==())) m((s).
3 Conditional Entropy Reduction in Diffusion Inference
Conditional Entropy in Information Transfer. During DM inference, each iteration reduces uncertainty in intermediate representations via structured denoising. From tnformation--heorrtiic view, the information gain between successive states is quantified by the mutual information [24]:
Lp( Rtan) Hp(xv) (x (x.)))
(8)
3

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1e4, PD-based22
1o.3, FD-hmed-2
1e-4, D'Dba
1e-4. DD-bned-
1e3, FD 3
1e.3. ED bacd
FD-bared-
1e KE based
1e RE based
FID
1c-,, RE-ased-22
FID
RE-boo8-2
le-3, RE-asedk
1e RE-heeok3
Ie-4, RE-based-3
Le-4, RE-boned-)
3.1g
2,02
12
15
20
50
100
15
20
100
NFE
NFE
(a) CIFAR-I0 (Discrete)
(b) CelehA-64 (Diserete)
Figure 2: Quantitative results of FID J. show that efficient entropy reduction (RE) method consistently improves image quality compared to FD-based method in (7. across various ablation scenarios.
where Hp (x),, is the entropy of state æt, and Ha ( is the conditional entropy of æt, given Btar1 A lower Hp (2t, [xt, results in a higher (2 111 which suggests that a well-designed method effectively utilizes the information from T4,-1 to refine the estimate of Zt.-
Conditional Variance and Conditional Entropy. We denote
Fo) as p( ( and
Var(æt, (Ttyya) ) as the conditional variance Var(ær,, [ En) for brevity. In DMs [2 3, the reverse transition p(a,, [2t, (o) commonly approximated as a Gaussian distribution under the Markov assumption. i.e., p(zt, (Ttz41) R NH,, E,), simplifying both model training and theoretical
analysis 21 25 26 I2]. Accordingly, the conditional entropy ( simplifies to:
Hx ( x C 1/2 log det (Var( ()))))
where C 1/2 d(log 2x + 1). Thus, Hp ( is intrinsically tied to its conditional variance:
(2 (Zt..)) log det (Var(a,, (tt4)))
(10)
This indicates that minimizing conditional variance directly reduces conditional entropy.
Reconstruction Error and Conditional Variance. In the forward process, q(æx,) approaches standard Gaussian as t increases, but (<oz remains structured around scaled versions of æo- Since (2, æo) (x,xx) information about æo persists in the modeled 24, and the inference process seeks to recover it. Let pe = Eg [4t, [ The MSE between the inference states and
its posterior mean is [1 |444 || Tr(Var,(ar; | ))- Leveraging the connection with conditional variance, under the isotropic assumption commonly used in DMs, we obtain:
min H2 ( + min E,[1 - Mu.ttr
(11)
We now decompose the reconstruction error between æt, and 2o using the law of total expectation. Proposition 3.1. Note that at,
27g (art (t.lteen) (Harltees Fo), we have
Eg [||. - =] [|all,
Ey || Mesltar [13 + Ey [IlMatta zp znll
(12)
Variance term
Bias term
where details of this reconstruction error decomposition are provided in Appendix
This decomposition reveals two distinct inference approaches for reducing the reconstruction error: (1) minimizing the conditional variance term directly; and (2) optimizing the bias error term with the prior which is often approximated by deterministic DM sumplers. Although the total reconstruction error includes both the variance and bias terms, optimizing conditional variance becomes the main actionable mechanism, since do not often have access to the desired zo during inference. Thus, optimizing conditional variance is central to inference; various methods are summarized in Table T
Bariance-Driven Conditional Entropy Analysis. We demonstrate how entropy reduction effectively steers samples toward the target distribution, supported by both theory and empirical evidence. Theoretically. DM denoising functions an entropy reduction mechanism grounded in Langevin dynamics 19 and non-equilibrium thermodynamies [8]] This principle dictates that more efficient entropy reduction will accelerate convergence by steering samples toward high-probability regions of the target distribution. Figure Tigure[vvsin visually illustrates this, showing the trajectories of DDIM versus gradient-basse based inference. Further empirical evidence is presented in Figure which details an

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Algorithm EVODiff: Optimizing Denoising Variance Diffusion Model Inference.
Require: initial æT, time schedule {t, [t))==o> model æz.
1: Zen Ir, brg n((t-1) k((t),, MoggNN (i) == log <(t)) (((t))log ((te+1)) log 1((.-1)-log K(T.)"
2: Denote g( =
Su-1 vi; + ote Ne, wg ( 4.). # Euler's or DDIM's iteration. on
3: for i + N to do
4: Dt; t (((:;))))
5: 2t.-1 + g( ) + 04.1 2
Ba (o(t,,,44) 41.
6: Bo(ti) + (1 (1- 44) BB (4) + 2 Be(ti, 4i), where ni is refined by Eq. (5
(1-4)
7: x1,-1 t g(x,,) + o 20
(t,);), where S is refined by Eq. (25).
8: end for
Ensure: 20.
ablation study on CIFAR-10 and CelebA-64 comparing our entropy reduction-focused (RE) method with traditional FD-based gradients.
Our analysis reveals that gradient-based methods excel at driving entropy reduction by optimizing conditional variance, which efficiently guides noisy states toward the desired distribution. For theoretical tractability, we assume independence between estimated noise at different timesteps, in line with DDPM's ' training objective of independent MSE minimization. While neural network parameter- parameter-sharing during training could introduce dependencies, prior works like 9 justify this surrogate by showing that these dependencies have negligible performance impact.
We identify two sources of conditional entropy in the reverse transition of DMs: the inference path uncertainty (e-g., from ODE/SDE solvers) and the model's own intrinsic uncertainty. While simple ODE-solver paths address the former, they may not effectively reduce the entropy contributed by the model. We therefore propose an approach that moves beyond these simple paths to target the total conditional entropy. From this perspective, we first derive Proposition 3.2 (Proof in Appendix C.2)
Proposition 3.2. The gradient-based inference in Eq. efficiently than the first-order inference in Eq. (6) when les E he, <(1, Var(ea(( + Var(eo(@1,..)
(7) can reduce conditional entropy more (ae.
This shows that gradient- based inference can achieve larger reductions in uncertainty when the step
size ratio is appropriately chosen. A practical interval for Proposition 3.2 is discussed in Remark C.I
Furthermore, we identify that solvers like DPM-Solver be understood through conditional entropy reduction, with details in the Appendix C.3
13] and the Heun iterations in EDM 12 can
Proposition 3.3. The acceleration mechanisms of DPM-Solver and the Heun iterations in EDM can be unified and explained as specific implementations of the conditional entropy reduction framework.
Finally, we theoretically establish why denoising iterations using data prediction perform better than those using noise prediction. The proof is provided in Appendix C.5
Theorem 3.4. Data prediction parameterization reduces reconstruction errors more effectively than its noise counterpart. Under independence assumptions, it also reduces conditional entropy.
In summary, we provide variance-driven conditional entropy analysis for diffusion inference, which theoretically explains the superior performance of gradient-based inference and data prediction parameterization. In particular, Theorem 3.4 demonstrates that data parameterization, by directly targeting the data distribution, avoids the error-prone chain of E, + Dz + Zo.
4
Optimizing Diffusion Model Inference via Entropy-aware Variance Control
4.1 Denoising Variance Analysis for Gradient-based Inference
Our focus on multi-step iterations using data parameterization, which has shown superiority through the theoretical result in Theorem 3.4 and prior empirical evidence from [3] 20).
5

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NFE=5
NFE=8
NFE=10
NFB=12
DPM-
Solver
20
EVOD if[
Figure 3: Sample comparison of our method vs. baseline using the pre-trained EDM on CIFAR-10.
Note that ( æt for the ODE for data prediction defined in Eq. 5 ((ttL.)) Formally, the multi-step iteration can written as:
#4-1 24
04-1 94,
1 Bt. FG he, wo (wt,, t.) + hi Bo(ti,t44))
(13)
0
Tt
where (t 1) := (x (2,, t.) xa( (,+1)//h.... denotes the backward FD. In this iteration, the smaller step size |/t,1 compared [/, h4.41 in the single-step case (Appendix C.6 reduces the conditional entropy, offering greater potential to improve the denoising process.
Denote Çi (1 (z). A straightforward improvement for Eq. 13 can be formulated as follows:
1t4-1 It, h11 we (2 (2) 2 h 4)()..4),
1
(14)
0t
Ot:
where ao (x1,;4) 4.) G0oo (#t, ti) + ( 4++1 represents linear interpolation. Similarly, the implicit approach is: #0 2t, = h4 as ( tx) + h?, Bs(s,, t,), where Øz( $4) =
06-1 de;
(oz t4-1) + (2r, t4). Note that these two improvement approaches can be unified as
Ft:
h4, wo (I 1.) + 2 h2, Ci (t:;
1
(15)
T oz
where (o; Bo(si, (,) when u, 8t. and (1 u:) Bo(4i, when u; = 4.
Remark 4.1. The RE-based multi-step iterations in Eq. (15) reduce the conditional entropy of iteration Eq. (3) by leveraging model parameters from law-variance regions.
We provide the convergence guarantees for the RE-based multi-step iteration described in Eq. (15)
Theorem 4.2. If xe (æ,,t) satisfies Assumption D.I the RE-based multi-step iteration constitutes globally convergent second-order iterative algorithm. The proof is provided in Appendix D.2
However, a key question arises: how should & and he, be determined? As x4 (Zt,, t.) is predict the clean data To from the noisy data ot,, we present remark for ideal cases. Remark 4.3. If Var (Io (It,, t()) x of the variance minimization can be achieved by setting G = / +
of,-,) for æn (Is,, ; 8i) and G / + ofnn) for ag (4).
Furthermore, we can improve the iteration of Eq. 13 by incorporating Bo(t,, st) and Bg(s,,
1 Ct It. fe, xe (Ie, tx) + h ((1 Bo t4) + n. Bo(,, 2
(16)
Tt.
where determines the gradient term variance. Therefore, from the lens of conditional entropy reduction, we can establish optimization objective directly minimize this variance.
4.2 Optimizing Denoising Variance with Evolution State Differences
We observe that the conditional variance in gradient- based iterations can composed of two critical components: the variance of the gradient estimation term itself and the variance between the gradient

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ImsweNet
#9180-44444
Hon
Heas
DPM Solver-
UIM Subeer
LnFC
UniPC bai
UniPC 73
6.2
EveDil
BuuDilf
7.
3.0
Droud
5.0
6.0
2.8
5.0
4.0
2.6
4.0
3,3
24
20
25
35
5.0
24
Figure 4: FID ( scores for gradient-based inference methods on ImageNet-64 and FFHQ-64.
term and the first-order term. Specifically, the unified iteration Eq. (15) can be rewritten as:
Ft,-1
Et. hr, ((1= Ce hig De (2 (4) + G he, wy (I Hi)) 2 Myy.
(17)
01: O2
2 hyp
As xy (Z (4) and zo (It,, 1.) are known in multi-step iterative mechanisms, the Var( xt is controlled by the value of G. Inspired by Eq. (17), we observe that Çi balances the variance between the gradient term and the first-order term. By harmonizing their statistical characteristics, we can efficiently reduce the conditional variance. To achieve this, we define G(G.) = (1 - (i) æg (#t,, 4))
Song par 14) to balance the variance between two terms. Moreover, the variance the gradient term itself should be balanced by Tk, should optimize тÿ in Eq. (16).
Retining S with Evolution State Differences. Our goal is to refine & using the available information at current step. As discussed above, we can optimize Çi to control the conditional variance by formulating an objective involving G(G:). On the one hand, we can rewrite the iteration in Eq. (17 as
11 = Ot: Dt.. - or, G(G)).
(18)
0t.-1
Notice that ãt, in Eq. (18) is determined by St. On the other hand, consider æ1,-; starting point and perform an inverse iterative from t to t, to approximate It, as follows:
(((A))
St.
Ft.
By (a)(r)) (T)) dr.
(19)
Tt:
Jn(s 1)
Similar to the Eq. 113) this inverse estimation of Eq.
199 is as follow:
1 h/ Ba (s., St. T2, 0z, 3a t1-1) os,
22.
(20)
0t:-1
Drawing from equations 18)) and (0 we can determine & by minimizing the differences between two estimations. Then, the optimization objective for Ç; is defined as follows:
min (G) ||( x.,) + (#2., x1,) 4.00
(21)
Directly solving this objective is challenging, as the optimal a is unknown. Fortunately, as (Ct)) S || + 22,,, ||2x,,1), we observe that ||2x,,1 || is independent of the G. Then, we can use Tt. to replace 24, as when optimizing C. (G.). Denote P( $2,1, a=:11 Ft.-1 2
di,
for brevity. Then, (G,) can be rewritten as: (G,) [[( oe, G(G))l-
Lemma 4.4. When the constraint on Ça is relaxed, min L?(() possesses the closed-form solution:
Ç = -(vec" (D.)vec (.)) (o,, N, vec vec (D::)vec
(22)
where P;:= := h4 (mi, 44), D₂ == Fg (w., ,44 2o (0e, (2), and vec(.) denotes the vectorization operation. The proof is provided in Appendix D.3
7

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Table Quantitative results of FID and IS f scores for gradient- based methods on ImageNet- 256, FFHQ-64, and CIFAR-10. The results are evaluated 10k and 50k samples for various NFEs. The + is our baseline. Error optimization strategies across methods are shown in Table
5
8
10
12
Model
Method/NFE
Marooy--waare
FID.
1st
F.D4
1st
FID.
1st
FID
IST
CIFAR-10 EDM, 50k
Heun
X
270.75
1.87
52.84
6.93
22.82
8.72
10.74
9.37
DPM-Solver++
X
27.96
7.47
8.40
8.80
5.10
9.14
3.70
9.36
UniPC
X
27.03
7.69
7.67
9.07
3.98
9.40
2.76
9.62
logSNR-tim
EVODiff (our)
17.84
7.89
3.98
9.37
2.78
9.64
2.30
9.80
15
20
Model
Method/NFE
Entropy- -waar??
FID4
1st
FIDL
1St
FID.
1S1
FID.
IST
Heun
X
347.09
2.29
29.92
3.03
9.95
3.19
4.58
3.34
FFHQ-64 EDM, 50k
DPM-Solver++
X
25.08
2.99
6.81
3.27
3.80
3.29
3.00
3.35
edm-time
UniPC
X
28.87
3.20
6.65
3.25
3.40
3.28
2.69
3.37
EVODiff (our)
19.65
3.18
5.31
3.32
3.04
3.35
2.66
3.38
5
10
15
20
Model
Method/NFE
Reference-
FIDL
1St
FID
1st
FID
1S1
FID
1ST
ImageNet-256 ADM, 10k
DPM-Solver++
X
16.62
98.07
8.68
143.59
7.80
152.01
7.51
153.89
UniPC
X
15.37
104.41
8.40
146.95
7.71
152.16
7.47
154.30
DPM-Solver-v3
V
14.92
105.85
8.14
146.82
7.70
153.79
7.42
154.35
uniform-time
EVODiff (our)
X
13.98
110.79
8.14
147.53
7.48
154.78
7.25
157.79
Refining %/ by Balancing Gradient Errors. Our goal is to refine 7/ using the available information at current step. Denote ~ (1 (1:))) (8(8 +))+ B8 (44, 4). F((,,11,4 44) == Bø 4) -r (2)
(1) (æt,, 4)
as gradient error. For balancing this errors, we formulate the following optimization objective:
min C2(m)) II(1 na E() 44) E(4,444)) ME(0.1]
(23)
We can rewrite C2(m) as C2(Tk) = JA x6) (zt,, (4) ||. Denote (2s = |ا Then, (2())) S (2((1))) + |ang (st,, (4) |l. where n. (art, 14) can be regarded as specific constant term independent of the target i/. Thus, the y/s can be obtained by optimizing the tractable (n:)))
Lemma 4.5. When the constraint on q is relaxed, min (m:) possesses the closed-form solution:
41
N (veec" (B, (ee))ee(BBB (Bo 4))) (vec" (B.)vec(B:))
(24)
where B, = Bg(s,, (.) (g 4.). The proof is similar that of Lemma 4.4
From Lemmas 4.4 and 4.5
(G)( and (2,(7)) have closed-form solutions when the constraints are relaxed. Then, the parameters in Algorithm [ are derived by mapping these solutions as follows:
T] = Sigmoid (1n)1), S Sigmoid (-(1C7) p)),
(25)
where Ç and nf are defined Eqs. (22 and (4). respectively, and μí the shift parameter. This optimization-driven approach captures the state differences during iteration while avoiding the computational cost of constrained optimization problems (27). Additional details are provided in Appendix E.7 Ablation studies μ provided in Table 9 of Appendix E.7. 7.
Finally, we prove the global convergence the proposed inference method for data parameterization.
Theorem 4.6. The diffussio inference method in Algorithm exhibits second-order global conver- gence with a local error of (h2.). The proof is provided in Appendix D.4
5 Experiments
We experimentally validate our method on a diverse suite of DMs and datasets, including CIFAR- 10, CelebA-64 -6, FFHQ-64, ImageNet-66 ImageNet-256, and L.SUN-Bedrooms Our evaluation uses standard metrics such as Fréchet Inception Distance (FID) and Inception Score (IS) across a varying number function evaluations (NFEs). We also compare our method on Stable Diffusion v1.4 and v1.5 using CLIP and Aesthetic scores. Our evaluation focuses on the data prediction parameterization on different diffusion-based generative models including pixel-space diffusion and latent-space diffusion. Additional experimental details and results are provided in Appendix E
8

---

NFE=25
NFB=35
DPM-
Solver
++
20
UniPC
EVOD I
Figure 5: Random samples from the Stable- Soole-Diffusio--1..55 model [28 with guidance scale of 7.5, using varying NFEs and the prompt "Giant caterpillar riding bicycle". Even at a low 25 NFE, EVODiff produces high-fidelity, semantically correct images while competing methods fail with severe artifacts, demonstrating superiority of our entropy-aware variance optimized method.
Table 3: Ablation study of our variance- driven gradient-based approach applied to DMs with data parameterization ImageNet-256 and CIFAR-10. The DPM-Solver++ is our baseline method.
NFE
Method
Model
5
6
8
10
12
15
20
Baseline
27.96
16.87
8,40
5.10
3.70
2.83
2.33
RE-based in Eq.
CIFAR-10
21.39
12.14
4.81
2.98
2.44
2.15
2.08
EVODiff
17.83
9.17
3.98
2.78
2.30
2.12
2.06
Baseline
16.62
12.86
9.73
8.68
8.17
7.80
7.51
Eq.
15 with balanced C
15.31
12.14
9.46
8.57
8.07
7.76
7.48
ImageNet-256
Eq.
T6
with refined G
13.80
10.91
8.91
8.23
7.89
7.58
7.36
EVODiff (Flogsnk)
13.98
10.98
8.84
8.16
7.81
7.52
7.32
EVODiff (Fretined)
14.33
11.16
8.95
8.14
7.79
7.48
7.25
To validate the contributions each component of EVODiff, we performed a constructive ablation study (Table 3). which builds upon our initial finding that entropy-reduction (RE) based methods consistently outperform traditional FD-based approaches (Figure 2}. Starting from baseline second- order solver, we incrementally introduced our entropy medrupy-reduction (RE-based) formulation and the final wolltion-state-driin parameter optimization. Each step demonstrates a clear improvement FID, culminating in the full EVODiff algorithm which consistently achieves the best performance. Further detailed ablations on hyperparameters such as the step-size ratio r, and the shift parameter f can be found in Appendix E.7.1 (Tables II and 9 in of Appendix E.7.1). confirming the robustness of our method. Additionally, we demonstrate the generalizability of our core principles by applying them to enhance other frameworks like DPM-Solver-v3, with results shown in Table 14 and Figure
We evaluate our method against advanced gradient-basse solvers, including DPM-Solver++ [20]. DEIS 14. UniPC [16], and DPM-Solver-v3 [7 on CIFAR-10. FFHQ-64, ImageNet-64, and ImageNet-256 datasets. The results Tables and and Figure 4 consistently demonstrate the superior performance of EVODiff. Furthermore, we evaluate our method with available logSNR and EDM noise schedules, further validating consistently robust performance. The results are shown in
Tables 12
19
16
We also evaluate our method the text-to-image generation task, as shown

---

Table FII score (1) and generation time comparison between EVODiff, DPM-Solver++ (DPM++ for short), and UniPC. All methods were evaluated on latent diffusion model 28 trained on the BUUN-Bedrooms dataset with 50k samples.
FID Score (4)
Generation Time (s)
NFE (2m++(2m) (3M++(3)
(nm)C(33) EVODiff
Gain
DPM++ EVODii
Gain
5
21.286
18.611
13,969
7.912
43.4%
3577.6
3488.4
-89.2 (2.5%)
6
10.966
8.519
6,556
4.909
25.1%
3800.6
3719.4
-81.2 (2.1%)
8
5.127
4.148
3,963
3.756
5.2%
4273.3
4046.9
-226.4 (5.3%)
10
3,881
3.607
3.563
3.332
6.5%
4746.7
4699.6
-47.1 (1.0%)
12
3.516
3,429
3.357
3.084
8.1%
4703.8
4678.1
-25.7 (0.5%)
15
3.341
3.284
3.182
2.918
8.3%
5973.1
5913.5
-59.6 (1.0%)
20
3.251
3.167
3.075
2.853
7.2%
7238.4
7154.2
-84.2 (1.2%)
Table 5: Single-batch sample quality comparison for text-to-image generation, measured by CLIP
score (f) and Aesthetic score (+) using Stable Diffusion v1.4 and v1.5 under identical settings.
NFN=10
NFN=25
model
method
CLIP
Aesthetic
CLIP
Aesthetic
Average
maximum
Average maximum Average
maximum
Average maximum
DPM-Solver++
33.07
34.12
5.71
5.83
32.66
36.12
5.74
5.92
sd-v1.4
EVODiff
33.79
35.00
5.77
5.87
32.83
34.50
5.79
5.97
DPM-Solver++
32.98
35.84
5.74
5.99
32.54
34.70
5.79
5.92
sd-v1.5
EVODiff
33.07
36.62
5.75
5.98
32.53
34.72
5.81
5,99
in Table Figures 3] and 5 provide a visual comparison of a generated sample. Finally, Table [5 compares the inference time at various NFEs ImageNet-256 between method and the baseline.
Finally, beyond pixel-space DMs, we evaluate EVODiff against advanced gradient- solvers on dataset (Table EVODiff consistently achieves
popular latent-space DMs. On the the best FID scores across all NEE settings, with particularly sigmificant improvements at low NFE counts (43.4% reduction at 5 NFE, from 13.969 to 7.912 compared to UniPC), Notably, EVODiff also reduces generation time by up to 5.3% while maintaining superior quality. For text-to-image generation using Stable Diffusion v1.4 and v1.5 (Table 5). EVODiff achieves competitive CLIP scores and the best Aesthetic scores compared to the strong DPM-Solver++ baseline, demonstrating its effectiveness in preserving both semantic alignment and visual quality.
Conclusions
In this work, we propose EVODiff, novel inference-ting refinement method based on entropy-aware variance optimization. It significantly improves both efficiency and generative quality without relying on reference trajectories. Specifically, our work first establishes a principled, Information-theored foundation that explains why data-prediction parameterization outperforms its noisy counterpart and demonstrates how optimizing conditional variance reduces transition and reconstruction errors without relying on reference trajectories. Building on these insights, EVODiff systematically reduces uncertainty in each denoising step, thereby accelerating convergence and significantly improving sample quality. Extensive experiments demonstrate EVODiff's effectiveness across diverse settings with SOTA performance: on CIFAR-10, 10, it outperforms the 10 NFE (from 5.10 to 2.78 FID); on ImageNet-2550 it reduces NFE cost by 25% (from 20 15 NFE) while maintaining high-fich fidelity generation; on -LUUN-Beroomss it achieves up to 43.4% FID improvement over UniPC with 5.3% faster generation; and for eet--to-image generation with Stable-Diffusie models, it produces superior visual quality while preserving semantic alignment. Our method achieves SOTA results and establishes reference-free, variance-controlled inference framework, effectively addressing the trade-off between sampling efficiency and generative quality.
baseline by 45.5% at
Limitations and Broader Impacts A limitation of EVODiff is that it currently relies on the data-prediction parameterization for diffusion model inference. Additionally, leveraging information- theoretic principles to enhance inference efficiency and optimize information flow during the sampling process remains promising direction for future research. While EVODiff improves both generation quality and efficiency, we acknowledge its dual-use nature, similar to that of other generative models.
10

---

Acknowledgments
This work was supported part by grants from National Natural Science Foundation of China (52539005), the fundamental research program of Guangdong, China (2023A151501 1281), ), the China Scholarship Council (202306150167), Guangdong Basic and Applied Basic Research Foundation (24202107190000687) Foshan Science and Technology Research Project(2200001118660))
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Appendix
Table of Contents
A List of Notations
24
B Related Work.
B.1 Diffusion Models.
B.2 Training-based Inference for DMs.
B.3 Training-fres Inference for DMs.
26
Analysis and Proofs Variance-Driven Conditional Entropy Reduction
C.1 The Proof of Proposition 3.1
26
C.2 Proof of Proposition 3.2
C.3 The Perspective of Conditional Entropy Reduction for Some Accelerated Iterations
C.4 Difference Analysis of Bradient-based Iterations in Multi-step Framework
29
C.5 Proof of Theorem 3.4 Conditional Entropy Comparison Between Data Prediction and Noise Prediction parameterizations
C.6 Single-step Analysis
30
D Proofs for the EVODiff Optimization Framework in Section 4.2
D.1 Assumption
D.2 Proof of Theorem 4.2
D.3 Proofs of Lemma
4.4 and Lemma 4.5
D.4 Proof of Theorem
4.6
E Experiment Details
E.1 Experimental Computational Resources and Data
E.2 Sampling Schedules
E.3 Parameterization Settings the Sampling Process...
E.4 Evaluating Sampling Efficiency and Image Quality in Generative Models.
E.5 Conditional Sampling in DMs
34
E.6 Single-step Iteration Details.
E.7 Multi-step Iteration Details
E.7.1 Ablation Study
E.8 Comparison of Reference- Free EVODiff and Learning Based Methods with Refer- ence Trajectories
E.9 More Experiments for EVODiff.
40
23

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A
List of Notations
Symbol
Description
t, s,
Time variables and endpoint in the diffusion process, t E
[0, T].
j.j.k X(1)
Indices for discrete time steps.
log-Signal-to-Noise Ratio (log-SNR) time, defined log(as/0t))
Pt. Pt, St
State vector at time (continuous, discrete, or approximate).
Zo, FT ai, 01
Endpoints: Zo q(æu), æy ~ (0, 0²I).
Noise schedule parameters (signal preservation a, noise level 02).
Random noise sampled from (O, I). Forward and reverse Wiener processes. Signal-to-noise ratio: a //?
Wi, W. SNR
Parameters of the neural network model. Network predictions: noise, clean data, unified noise and data.
(o do(-)
())
k-th order derivative of de w.r.t. Forward process distributions, e.g., ((2)) g(ar), (( (aro).
a(-)
p(at, (tte))
Reverse transition distribution.
(),
Gaussian distribution with mean μu, covariance Σ; uniform distribution (a, b].
9(t)
Drift and diffusion coefficients, f(t) d log as g(t) satisfies
(2) = do? 20log o²
Transformation: "b (noise-prediction) or (data- prediction).
f(x)
x(t), (b(T)
Time reparameterization k(t) and its inverse xb()) with ((x())
cloogg .)
Difference term: f( ,) (
he: hes
Step sizes: x(t;-1) - n(tt), auxiliary x<s:) - n(fz))
hx,
Step size in the log-SNR space: X(t;-1) X(tz).
rs, FlogSNRR (1)
Ratio of consecutive step sizes in log-SNR space, used for gradient estimation.
Fg. Bg, Bg
Finite difference terms for gradient approximation (Forward, Backward, and unified).
Var(.I.), H.())))))))))))... Meslees1 E:;
Conditional variance, entropy, mean, and covariance matrix. Optimization parameters (interpolation, complement, bal- ance, shift).
G. Tis 1t
[(0), L1, C2, w(t)
General training objective (e.g., Eq. (3)), specific optimiza- tion objectives (C,, L2), and training weight w(t).
G(S.), P(x
Auxiliary interpolation and optimization terms.
D,, E(; 12), Ag.
Difference, gradient error, and weighted gradient terms. Gradient, expectation.
Va, EE
Tr(-). (r(-),det(.)). de.l.),.1.11 (.c(.)
Trace, determinant, and norm operators. Vectorization operator.
Rd I,O.C
d-dimensional real space. identity matrix, zero vector, Gaus- sian entropy constant C' = dd(oo 2T + 1).
B Related Work
B.1
Diffusion Models
Diffusion Models (DMs) represent powerful class of generative models rooted in stochastic ther- modynamics and statistical physics 229 30]. The foundational work by Sohl-Dickstein et al. [

---

adapted these principles to deep generative modeling through a Markov chain approach based on non-equilibrium thermodynamics. This pioneering research addressed the critical challenge of balancing tractability and flexibility in probabilistic models. The field of diffusion modeling was substantially extended by Song and Ermon [31 32 who introduced score-hased generative models with noise conditional score networks. Their methodology enabled efficient estimation of score functions Vx log Por (x) across multiple noise scales, coupled with annealed Langevin dynamics for sample generation. Subsequently, Ho et al. [2] proposed Denoising Diffusion Probabilistic Models (DDPMs), which offered a significant methodological refinement by parameterizing the reverse process. Their contribution included a well-formulated training objective: L = Ex,elce ca ( t)l?? where E represents the original noise and Eg denotes the predicted noise by the model at time step t. This formulation facilitated high-quality sample generation with remarkable stability. In landmark contribution, Song et al. [3] established a comprehensive theoretical unification by formulating score- based models and DMs within continuous- stochastic differential equation (SDE) framework. This theoretical advancement provided a unified mathematical foundation that elegantly bridged previously disparate approaches in diffusion modeling research.
Building upon this theoretical framework, DMs have demonstrated exceptional capabilities across a wide range of domains. In image synthesis, they have achieved SOTA performance 4 and established new benchmarks in photorealism [12]. Their success has extended to multimodal generation tasks, including text-to-image synthesis 66 (3), speech generation [7], video synthesis [8], 3D content generation (34. Moreover, DMs have also advanced related areas, including doep Gaussian processes [35), diffusion bridges, and density ratio estimation [36] 37.. Furthermore. DMs have shown remarkable capabilities in controllable generation tasks [38] such as image editing, style transfer, and inpainting [5] 39]. In theory, Despite these significant advances, DMs continue to face a critical challenge: the inherently slow sequential generation process, which limits their real-time applicability in certain domains.
B.2 Training-based Inference for DMs
Training-based models methods accelerate DMs through novel training strategies and architectures. Knowledge distillation techniques, such as Progressive Distillation [40], enable efficient sampling by allowing student models learn compressed sampling processes from teacher models. Recent advances have further explored innovative approaches to diffusion model inference. Meng et al. 41 investigated knowledge distillation in guided DMs, addressing model efficiency challenges. Complementing these efforts, Karras et al. [42 43 conducted a comprehensive analysis of training dynamics for DMs, offering critical insights into the underlying mechanisms of model performance and generation quality. Consistency-based methods, exemplified by Consistency Models [44 and Latent Consistency Models [47. achieve parallel generation by learning score functions through consistency training, grounded in probability flow ODE frameworks. Reflow [48) further optimizes the generation paths by reformulating rectified flow ODEs with paired retraining strategies. Architec- tural innovations have also played an important role in improving efficiency. Latent DMs (28) reduce computational complexity by operating in dimensioonl spaces using an auto-encoder frame- work 44). EDM [12) introduces a parameterrizaiion and principled weighting schemes, allowing fewer sampling steps without compromising the generation quality. Shortcut models (50) and Mean Flows [51) achieve efficiency by step-aware network learning and mean field modeling. In addition, there are some other approaches that exploit architectural characteristics and learning-based solvers to improve the efficiency of DMs [9].
Despite these advancements, training- based methods often require specialized training procedures and a careful balance between quality and speed trade-offs 44 60 50 [1], making them computationally expensive. Additionally, their training data are typically obtained through iterative sampling from pre- trained DMs using deterministic, training-fre free samplers such as DDIM 19] or DPM-Solver [13, 20). which introduces further computational overhead and sampling dependencies. These factors limit their practicality in resource-comee constrainedd environments. Furthermore, while some methods achieve generation through a single neural network pass (referred to as one-step generation), similar to GANs [6]] they often sacrifice the iterative refinement process that is a hallmark of traditional DMs. This process, which involves progressive noise reduction and iterative refinement, a core strength of DMs, and its absence may adversely affect the quality and controllability of the generated outputs.
25

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B.3 Training-free Inference for DMs
In contrast, training-free inference methods focus on denoising strategies for DMs without requiring any additional training, making them more adaptable and practical for use with open-source DMs. Early sampling methods in DMs primarily relied on ancestral sampling (2. Score- models B] used predictor-correcto methods to refine samples and introduced PF ODEs as faster sampling alternative. DDIM [9] advanced sampling methods by introducing a non non-Markovian deterministic process that enables deterministic sampling through a variance-minimizing path, significantly re- ducing the number of required steps. PNDM [T demonstrated the adaptability of ODE solvers to diffusion sampling by effectively utilizing linear multistep methods. EDM 122 explored the design space of DMs with a p-paramcterization linked to the signal-to-nois ratio (SNR), analyzed noise dynamics to optimize time steps, and achieved high-quality samples using the Heun solver.
DPM-Solver [3 introduced an exponential integrator- based (EI) sampling framework that discretizes PF ODEs in the semi-log-SND SNR space with high-order solvers for accelerated sampling. DEIS [14] investigated the effectiveness of EI addressing the stiffness of diffusion ODEs. DPM-Solver++
20 extended DPM-Solver to guided sampling by using lata-based parameterization. Based on DPM- Solver, UniPC 16] designed high-order predictor-correcto schemes within unified framework and demonstrated strong empirical performance. These methods effectively focus on optimizing the variance term of reconstruction error, as formulated in our proposed decomposition Eq. (3.1).
Beyond these solvers, another approach reformulates ODE solvers by treating the solutions at multiple steps as ground truth. Leveraging prior information about the target distribution, this strategy simultancously optimizes both the variance and bias terms in the reconstruction error, as decomposed in Eq. B.I). For instance, DPM-Solver-v3 [7 accelerates the sampling inference in DMs by optimizing the ODE solver using empirical model statistics (EMS), where the EMS coefficients are learned from the sampling results of DPM-Solver++ with 200 function evaluations (NFEs). In addition, other studies have explored discretization techniques and noise schedule tuning 62
74
Although various numerical discretization techniques and approaches have been proposed for training- free methods, the underlying mechanisms driving their acceleration are not adequately understood. Despite their empirical success, these ODE-based methods lack an iiformatiion-thorreli foundation. central limitation is their neglect of information transmission efficiency during the reverse process. This theoretical gap suggests that the principles governing diffusion inference remain underexplored. Our work addresses this limitation by introducing a framework that unifies efficient numerical iterations, such as DPM-Solver and EDM, through the lens of conditional entropy reduction. We demonstrate that by explicitly optimizing for conditional entropy dynamics rather than focusing solely on optimizing the numerical error of ODE solvers, we can achieve better sample quality across inference steps. EVODiff is designed to fill this gap by introducing an explicit and optimizable thoooetii objective.
C Analysis and Proofs of Variance-Driven Conditional Entropy Reduction
C.1 The Proof of Proposition 3.1
Proof. We prove the dccomposition of the reconstruction error using the orthogonality property of conditional expectations in Proposition 3.1 First, we express the squared norm as a sum of components:
[z,
22 |1(x,, Melless ) + (Heglts41 x0)i1
(26)
= |x,, - Malles 112 ||Meslees 20012
+ 2( - Mailes Paltess x00)
(27)
Taking the expectation of both sides, we obtain
E 11 xoll Eyll Me,le 113 ElllMet
-0ol +0ll2++2C + 2C
(28)
26

---

where C' = Eyl(er, - Meiltrr2 1 M4 #it1 - zu)l. Next, we show that the cross-term C' vanishes:
= Ey
Bg (zoe. - Meilten Halltss - xo) Tes+11
(29)
= Bug
(Bng (at, - Mailts Cteeal Seeeil-ealtett Meolest1 moo)
(30)
Since Mellt++ Ex,, Dt.41 by definition, we have Ex Meilects Dt441 = O, and therefore
C = 0. Therefore, we obtain the final decomposition:
Eg
xoll = Byll Mt |t 11
Byll Mellec - 10ll
(31)
Variance lerm
Bias term
The proof is complete.
C.2 Proof of Proposition 3.2
Proof. Denote the Gaussian transition distributions governed by the iterative equations (6) and 77 as pi (f ..) fl (a+))) and p2 [] respectively. Without loss of generality, we the common part f of the two iterative equations the mean of both distributions. The remaining components represent the perturbation terms associated with each transition distribution, respectively. Since the noise prediction model is specifically trained predict the noise, we can interpret 60 (#t, as representing the noise perturbation term. Since the estimated noise by the model at different time steps can be considered mutually independent, the conditional variances of the remaining terms for the two different iterations are, respectively, expressed as follows:
Varp, h2, Var Varp2 hr. (1
he: 2he,
h4
Var(co(@u., t+))
-Var(co (ā.,: s;)).
4h?.
(32)
Denote AH( Hpp (#tt., [+t.)= Hp, (#t... (...) Then, by equations 332 and we have:
he,
h?
h2 Var(Es(@a,,
(p = log
+
(33)
hr:
4h2,
4h2
Var(Eo(@1,, 4:)
Therefore, (D < and only if and note that iu, & ht the proof is complete.
M 48f
+
A?
Varieo
(i1)
I
By solving this inequality bee
by,
(arl)
As the reverse process in DMs aims estimate p( [ z.)
2
25
examine Var(co (ar(ee((e (.,, t)
2o) to capture the model's uncertainty in noise prediction conditioned on the clean data. For brevity, we denote this variance as Var(æ (.t,() ()). Based this consideration, can establish the practical interval for Proposition 3.2 using the prior-like conditional variance from the forward diffusion process.
Remark C.1. In the forward process of DMs, the clean data at each step can be expressed by
2o = æt/a; - at/ a,E If we assume that Var(Ea(@z,)) ()) x of/aî to quantify the extent of deviation
from the clean data. Under this prior-like assumption, we obtain Var(en((e,,
Var(ea(ūi, SNR(s:) Then,
SNR(1.)
the conditional entropy reduction condition in Propostion 3.2 h2.
(N)(s.)
SNR(C SNR(s.)
C.3 The Perspective of Conditional Entropy Reduction for Some Accelerated Iterations
As an application of conditional entropy analysis, we deepen our understanding of the iterations in accelerated denoising diffusion solvers, such DPM-Solver (13) and EDM [22. by elucidating the associated changes in conditional entropy. We then demonstrate that the iterations of both well-known solvers are denoising iterations grounded in conditional entropy reduction and represent two special
cases of RE-based iterations.
Firstly. let revisit the accelerated iteration introduced EDM 12). Formally, the iteration formula of EDM can be written as follows:
f f(....)) f + h, (iz,, t.) + (er,199-11
(34)
27

---

which can be equivalently rewritten as the following gradient estimation iteration:
f( f(tt) ) he, Eo (ir,, (4) h3 ca (it.1, t2-1) - (2
(35)
hr,
As hu, he, in the iteration of EDM described by Eq. (35). based Remark following conclusion:
C.1 obtain the
Remark C.2. The EDM iteration in Eq. (34) can reduce conditional entropy more effectively than the DDIM iteration in Eq. (6). Thus, the iteration of EDM can be interpreted as an iterative scheme for reducing conditional entropy.
Next, we revisit the accelerated iteration framework established by DPM-Solver [13 with exponential integrator. Specifically, the sampling algorithm of DPM-Solver decouples the semi-linear structure of the diffusion ODE, with its iterations formulated by solving the integral driven by half of the log-SNR- The exponentially weighted score integral
can be written as follows:
f(x)) f(x.)) = - e (((()))((T)))))))))))))))))))( dr.
2(1)
(36)
A(e)
where X(t) log o: It follows that Eq.
36) and Eq. 5 can be mutually transformed through
the function relation A(t) log(xst)). Denote hx, A(t+-1) A(ta) and hx. == A(si) A(ts). Formally, the second-order iteration of DPM-Solver can be written as follows:
= ((tt,)
Tti-1 (chsc-11 - co (it, ti) a:--
ob (chse- - Eg (Ta,, 81) - ED (ãt, ts)
2r1
(37)
h2, ba;
where 8; 1 (A(te) + r1h2)). Note that r1
k(t;-1) oi.-1 and h4 <(t=-1) k(tz). As
x(t.) ca> x(b-1)' then 0u-- (chas 1) -he,,- Thus, this second-order iteration can be equivalently rewritten as:
an,-
he, her co (# 81) co (mte: (4)
f(iu.)) ((t1,) + he, (Te, 1.)
(38)
2
hx.
Note that the 8; here in DPM-Solver differs from the one in Eq. (7), due to the variations arising from the function space. Based on conditional analysis, similarly, we have the following conclusion.
Remark C.3. Based on Remark C.1 when Ax; E [1,
SNR(s.) hx; SNN(()) the DPM-Solver's iteration
in Eq. 37 can reduce conditional entropy more effectively than the DDIM iteration in Eq. 6 Note
that in the practical implementation of DPM-Solver. Thus, SNR(s:) (l,)),), the DPM-Solver's iteration be interpreted iterative scheme for reducing conditional entropy.
Finally, we summarize the relationship between these two iterations and RE-based iterations. In fact, the iteration described in Eq. (34) is RE-based iteration within the EDM iteration framework. Clearly, the RE-based iteration within the DPM-Solver iteration framework can be formulated as:
h4 hx, co (Woss si) co (#1, t.)
f(it,.)) = f(x,,) + he, (yee (āe,, st) + (1 E) (Zc,, tc)) +
2
hx:
(39)
Therefore, the iterations in both EDM and DPM-Solver be interpreted specific instances of RE-based denoising iterations from the perspective of the conditional entropy.
This analysis reveals that methods like DPM-Solver and EDM have implicitly leveraged principles of conditional entropy reduction. Our work, EVODiff, makes this process explicit, optimizable, and adaptive for the first time, which is the key its superior performance.
28

---

C.4 Difference Analysis of Gradient-basee Iterations in Multi-step Framework
On one hand, let us revisit the multi-step accelerated framework established by DPM-Solver++ Formally, the second-order iteration of DPM-Solver++ can be written as follows:
Ot (e ha; - -1) 2g (mt,
f ,) = f
06--1
(40)
Me: -1) (ãt,, t.) 29 ( t++1)
044-1 (e
01,-1
2r;
Ma hx.
where To (Zt, ti) denotes the data-prediction prediction model and ri
Since
04-1 he -
04-1 -he in data-prediction prediction models, Eq.
40 can
-4--1 be rewritten as
f( = f(+,,)) + le, (ar,, (2) + he, hx, 2g (Tez (2) wg (Te l241)
(41)
2
ha
On the other hand, we can rewrite the iteration presented Eq. 13 as follows:
I ( ) f(xr.)) + he (ātt, 4)) + D2, Do (T),, (i) Dg (Zesss t4+1)
(42)
he
It has been observed that the differences in the multi-step iterations presented in Eq. (41) and Eq. (42) are still caused by the variations in r.. Therefore, gradient estimation-based iterations, the core characteristic of the DPM-Solver++ iteration is the determination of r; in the half-logarithmic SNR space. For convenience, we will hereafter refer to half-logarithmic SNR simply as 'logSNR'.
Without loss of generality, the core differences between various gradient estimation- iterations can be generalized variations in the determination of r;. Then, natural question arises: how can r; be determined better or systematically? Therefore, a principle for determining r; is of great importance. This inquiry drives our investigation from the perspective of conditional entropy within the context of multi-step iterations.
C.5 Proof of Theorem 3.4 Conditional Entropy Comparison Between Data Prediction and Noise Prediction parameterizations
Before presenting the formal proof, we provide the core intuition. This theorem aims to show that data-prediction parameterization is more efficient because it directly estimates the target æp, thereby minimizing reconstruction eтrоr more directly. In contrast, noise prediction follows more indirect path (x, + Co + zo). which can accumulate more variance. The following steps formalize this variance reduction and its connection to conditional entropy.
Proof. Without loss of generality, we only need to prove that the conditional entropy of the first-order iteration using data-predictto parameterization is lower than that of the first-order iteration using prise-prediction parameterization, Let us revisit both first-order denoising iterations. Clearly, based on Eqs. (6) and (5), the first-order iteration of data-prediction parameterization as follows:
01, Lt, + to
Olt:
Olt:
Ft,_.
Do (i t;),
(43)
Tt:
Tt.1
ot.
Lumi lincar
Nuui non-linear
3s, -o,, cal (t) The first-order iteration of noise-prediction parameterization
where my (mt,, 14)
an
as follows:
at,11 It, + at.-1
0t Tte
co (t).
(44)
Ott
C:t-1 ait
Losit:: lincar
Nusu: noo-lincar
Denote the Gaussian transition kernels governed by the iterative equations Pi (it;-g (20) and p2 (Ze.,, (20). respectively. In both iterations of equations
and as
4 and
44
the

---

randomness of the linear term is solely related to the noise introduced in the iterations preceding time step E,, whereas the randomness of the nonlinear term depends entirely on the noise at the current time step 4, Since the noise introduced at each time step in DMs is independent, under this assumption, the randomness of the linear term is independent of the randomness of the nonlinear term in both iterations. Therefore, we will consider the variances of the linear and nonlinear components separately. Formally, the variances of the linear terms for the two different iterations are, respectively, as follows:
Var (ddta xo)= = of 1 Var (Tt, æo), Var Lnoise xb)) of.
ar Var(ă xo).
(45)
For simplicity, denote Var (ã,, xo) as Var (ãx,) where appropriate. Based on monotonicity, 04-1 < 01,-1 as ar monotonically decreasing with respect to time and or monotonically Ty ax;
increasing with respect time t. Therefore, Var (Lama) < Var (Lnoise). Subsequently, consider the variance the non-linear terms for both iterations. For clarity, we denote c(t,, 4111 = Ck2, 0t.-1"
06t1100 Then,
0ti1
at:-1
0t:
-1 c(ts, ta-);
a:
0t:-1
0t: at, an;
ctt,, t;-1).
(46)
Oz Ot,
Ft:
an
Thus, the variances of the nonlinear terms for the two different iterations are, respectively, as follows: Var(NNoisi)
2(le,li-1)
(-c(li,(4-1)) Var (xs (t).
Var (Ee (āt, Var(NNata)
a?.
oi,
(47)
Note that
it. o2,
Eg ED (Tt,,
o7 Var (E,, Ey (t),
Var (xe (āc,, t4)) Var
(48)
Ote
as Lt. =0, + Er. Then,
Var( (Ndaa) (-c(6,,4-1) (-1))2 of Var (Et, - en (4) 2 ? (1i,li--) Var (Eo (0i, 44) - G..)
oi
at.
(49)
Clearly, since eg (mr,, (2) is designed to predict the injected noise into the clean data at time step 4. and based on Eq. ( the variance Var (es (ā+,, 4.) Ex,) can theoretically approach arbitrarily small values as the accuracy of the model's estimation improves. Therefore, as Var (ea (Zt, f4) <x:) < Var (ea (it, tz)), we have Var(Naa)) < Var (Nniose). Since the randomness the linear term is independent of that of the nonlinear term in both iterations, and given that Var (Laata) Var (Lnoiee)
and Var (Nama) < (ar(onnss)) we have
S (a zo)) Var(Laaaa) + Var(NNaai.
(50)
< Var(Lnoise) + Var(NNnoise) Var(pa mo))
Consequently, based on Eq. distribution, we have Hp. (fi.... x0) Hp2 zo). The proof complete.
which provides the conditional entropy formula for Gaussian
C.6 Single-step Analysis
For single-step iteration, one insight is that the model parameter Eg (ă.,, 8t) can be used further to improve the iteration governed by Eq. (7). without additional model parameters. Formally, we can formulate the iteration as
h2 hr, G(v)) + Fo 44),
1
(51)
where G(7;) = 7;46 (#,,, 8.) + JxEB (Ztt, 44, 7: = 7ss % E (0, 1]. This improved iteration shares the same limit state as the vanilla iteration in Eq. 7 when 8; + 4. For convenience, we refer to the vanilla iteration as the FD-bused single iteration. For clarity, we identify the denoising iteration by reducing conditional entropy as the RE-based single iteration.
30

---

In the analysis of conditional entropy, we can compare the different components of Eq. 7 and Eq. Then, the variance of the key distinct components in each conditional distribution follows:
Varp: n/. Var(, Varpa (7)) 7, iig Var(Eo(@ss 84)) 1,
(52)
Varpi>
where 91 = -1-m. Then, the difference in conditional entropy between two gradient estimation- baseed iterations is
Varpa (7;) log (1 + v(7))
AH log
(53)
Varp1
where v(n)) -2% + 7/ 7 (2,, Due to 74 (0, and SNR(4)) < SNR(s:).
2 Var(ea((E.,.
(p < 0 consistently holds under the assumption that Var(ee(ā,, t)) x of/aî. Therefore, this improved iteration can more efficiently reduce conditional entropy compared the vamilla iteration by using subsequent model parameters in lower- variance regions as guidance. Consequently, based on (p < O, we have the following Remark.
Remark C.4. The RE-based single-step iteration specified in Eq. (51) consistently achieves a more efficient reduction in conditional entropy than the FD-based iteration.
Accordingly, Remark C.4 show that the RE-based iteration can consistently surpass the FD-based iteration in reducing conditional entropy.
D Proofs for the EVODiff Optimization Framework in Section 4.2
D.1 Assumption
Assumption 1:
The total derivative d&/²) (m>)r)), (T()) == d
at dodo( exists and is continuous
necessary, where & determined by the specific context.
Assumption 2: The function do (I vTT)) Lipschitz w.r.t to its first parameter Tv(r))
D.2 Proof of Theorem 4.2
Proof. Denotes é,- f(Ø)) for short. Without loss of generality, the RE-based multi-step iteration described in Eq. (15) can be decomposed into:
h2 Bo Bo ti).
z Zt. he, do (3e, tx)
2
and
Rr C. + 7hv; (do (ā 82) do (#tt. 4)))
Clearly, 2, ãt,- + O(2)) based the Taylor expansion. Since the model dg (ă,,t) satisfies the Lipschitz assumption with respect to Zt. then
| 2ull 11y/he, (do (T.,, 8:) - do (ãt,, 12))
L, Li, |dg (0a,, si) do (Tz,, Li) S Lzhe, (0., Tt,, O (3).
(54)
Subsequently, by the triangle inequality, have
| 2z || T + a.. Ze, S || - 2x |||l 1 S., 00 (55)
where the last equality holds because |/r,1 2 hr.l--
Therefore, we prove that the local error of the RD-based iteration is of the same order as the corresponding Taylor expansion. Consequently, the REE-aased iteration Eq. 5T a second-order convergence algorithm. The proof is complete.
D
31

---

D.3 Proofs of Lemma 444 and Lemma 4.5
Proof. Without loss of generality,
a ||A oh (AF) (1 2)
a (vec (A- - oh (AF1 + (1 - -))F2)) vec (A - oh (AFI +(1 -1)F2))
vec (0(A-OH)) oh (AP + (1 (1- 1)) vec (A oA +(1 A)F2)
07
=2vec' T. (-oh (Fi - F2)) voe (A - oh (AF + +1-
0| (1-N)3)) 0, we have
Let
02
vec
(Fi F2) vec (oh) - F2) - (A ohF2)) 0.
(56)
Therefore,
vec (F) F22 vec (A ohl22)
A oh vec (Fi - F2) Vec (Fi - F2)
(57)
The proof is complete.
D.4 Proof of Theorem 4.6
Let us review the EVODiff iteration, without loss of generality, in Algorithm ! as follows:
1 ( + he, arg (
G
(b444)
2
= f(()) + h4, xg (Tt, t,) + C;
h
Bo(si,t)) tú) + (1- 7)
2 Ba(ti,4i)
2
2
= =f((() + he,
② to (ā),, + (1
y1
(itt bi))
2
112
Ci
1i Ba (8s ti) (1
Be (ttll14))
2
where Bo(t;,te41) 2e (0 .1) - (#o 1141).
heepa
In the following, we now proof the convergence properties of this EVODiff iteration scheme and establish its convergence order.
Proof. Denote &, f(x;) for short. The RE-based iteration in EVODift can be decomposed as:
tr,_, 2r, + 2 Lun + (1 (1- ?7
2,12
2
(0t. +Zur) ) + (1
1/1
(it, xpz)),
2
where
h2
h2 Spp = hr (ār, (i) + (.)o(4)4
1 = h1, wo (Tt, 4:) + Ci Bo (8t, li).
2
Let us now consider the case of Ci, + a.. Denote
1.
a 1144 0z Eu Bag St, + hr xe (ār, (2) + Bo Bo t.).
2
122 Ba (si, 4.). Note that E = = + + ((1))) and Ba(si, = (1v..)
Then æ1,,; 1 = p + (Ci - 1) based on the Taylor expansion. Therefore, we have
Si -
| - -2,11 = /21 Rr.
+
11 Ba (85, Bo 4i)
2
S5 - 1
(58)
< - - |
hイ Be Bo tx)
2
= -(h?)) + 11 1(h() = O(h)))
32

---

where L1 is a constant because Çi can be bounded by 1. Denote _2,1--1 = -t, + Byz. Symmetrically, we obtain
| R1,1l || -hh;))
(59)
Now, combining the results, we obtain
- ((,) + (1 (1-
?} (02
[
11.11
(01,1
- d1,4 - 04, + (1 (1- 1 |02,,14 - #2,,1l
1i
(60)
2
= " O(h) + (1-1
? 0 = (h
2
Thus, we have shown that the local error of the RE-based iteration EVODitI 1
O Conse-
quently, the RE-based iteration in WODDifTTacheee second-order global convergence. The proof is complete.
E Experiment Details
In our experiments, we utilize several standard pre-trained models. Specifically, we employ the discrete denoising diffusion probabilistic model (2). the continuous score-based model [], and the uncond EDM model
[12]. all trained on CIFAR-10 (75.. For larger-scale evaluations on high- dimensional data, adopt the pre-trained models trained the ImageNet dataset [76] from the baseline method [4] Additionally, we use the pre-trained I Latent Diffusion Model and Stable Diffusion model (28), where the latter trained the LAION-5B dataset
77 using CLIP [78 text embeddings
as conditioning signals.
Experimental Computational Resources and Data
All experiments were conducted on NVIDIA GPUs. For sigh--imensional datasets like ImageNet, we utilized the NVIDIA GeForce RTX 3090 GPU with 24GB VRAM. For other cases like CIFAR-10. 10, experiments were performed on NVIDIA TITAN X (Pascal) with 12GB VRAM. To ensure fair comparison with prior work, we maintained consistent pre-trained models and experimental settings across both scenarios. We list some of the datasets and codes used in Table
Table 7: Some of the datasets and codes used.
Name
URL
CIFAR10
sttps://wxzz cs toronto.ooo.oooo.. edu/ kriz/cifar html
LSUN--edroom
https://ww.yff.oo//
ImageNet-256x
ScoreSDE
EDM
Latent-Diffusion
Stable-Diffusion
DPM-Solver
APM--olver++
http://gttabbcccc////ogggt///ggp
SciRE-Solver
UniPC
https///giuhb..cc////-hhhha////
DPM-Solver-v3
E.2
Sampling Schedules
Sampling schedules in DMs define how the noise scale evolves during inference and play a crucial role in balancing sample quality and computational efficiency. Several widely used schedules include the Time-uniform schedule 2 3 the LogSNR schedule [33, and the EDM schedule [12]. Although optimized schedules have been proposed 668 69), they typically require significant computational resources for optimization. In our experiments, we follow the default schedule of the baseline methods.
33

---

E.3 Parameterization Settings of the Sampling Process
In the sampling process of DMs, various parameterization settings are used define the target prediction at each iteration step. Below, we list the adopted parameterizations:
Noise prediction parameterization [2]
This parameterization directly predicts the noise injected during the forward diffusion process. The connection to the score function is formalized as:
Eo (Zt, = - V log q (z)),
(61)
where Ve log (xx) denotes the score function [3]
Data prediction parameterization [21]: This parameterization estimates the clean data Do from the noisy input æt at given time step The predicted data satisfies:
To xo(2,, Dt o,E0 (,E0(11,1)
(62)
Although these parameterizations have practical predictive value, they may insufficiently minimize discretization errors. Building upon earlier DPM-Solver versions [13 20) DPM-Solver-v3 [I7 extends the parameterization strategy by incorporating empirical model statistics (EMS). This is an approach that requires reference solution. Essentially, leveraging prior information about the target distribution optimizes both the variance and bias terms in the reconstruction eπror, as decomposed in Eq. 3.II. Specificallyy, they formulated the continuous- time ODE as follows:
dæ yx
12
aA (axc0 ( >) - 1xmxx,
(63)
a
where À represents the continuous time parameter, and L an optimized prior statistics term.
In our ablation study, we employ the default parameterization of the baseline method in all of our experiments. It is important to note that main baseline is DPM-Solver++ (20), while the other parameterizations are only auxiliary setups intended to validate the effectiveness of the variance-driven optimization.
E.4 Evaluating Sampling Efficiency and Image Quality in Generative Models
The Fréchet Inception Distance (FID) 79 evaluates the quality and diversity generated images by comparing the statistical distributions of generated and real images in a feature space. It uses pre-trained Inception-v3 network to extract features [80), computing the mean μ and covariance for both distributions. Specifically, My and Σg represent the mean and covariance of features from generated images, while jy and Σ, correspond to real images. Specifically, FID is calculated as:
FID
|kg /r + Tr [x + E, 2( E,)1/2)
(64)
Lower FID values indicate higher similarity between generated and real distributions, reflecting better image quality 24
The Number of Function Evaluations (NFE measures computational efficiency by counting neural 13 [2]. Lower NFE values indicate faster sampling.
network function calls during sampling 3 9
26
Balancing FID and NFE is crucial for practical applications where both high-quality outputs and computational efficiency are required. Joint evaluation of these metrics provides a comprehensive perspective: FID assesses distribution fidelity, while NFE evaluates algorithmic efficiency.
In this paper, we adopt the evaluation framework used in prior studies 13 20
combining FID
and NFE to jointly assess the quality of generated images and the computational efficiency of sampling algorithms. This comprehensive approach, validated in several studies 4
3
13
12
offers standardized benchmark for comparing different generative models and sampling methods. Moreover, we adopt CLIP-Score (78), aesthetics score such as PickScore [8] and ImageReward [82
estimate the quality of generated images using method Stable-Diffusion 28
E.5 Conditional Sampling in DMs
Conditional sampling DMs enables controlled generation by incorporating conditioning information (c.g., class labels or text) into the sampling process. This is achieved by modifying the noise predictor
34

---

Eg(2), t, to guide generation toward satisfying condition c. Two main approaches exist: classifier- free guidance [83] and classifier guidance [4] Classifier-free guidance (CFG) combines conditional
and unconditional predictions:
69 CCF (xt,,,c)) := (1 w), 4, - wes (æt, 4, (,0),
(65)
where 9 denotes the unconditional case and w > O is the guidance scale. This method is simple and efficient as it requires additional models.
Classifier guidance (CG) uses an auxiliary classifier pp(c Ft, 1):
c t, c) = () - so Vn, mi log Pa(C Zt. (),
(66)
where s controls guidance strength and o; is the noise level time t. While computationally more expensive, this approach can provide finer control over the conditioning process.
In our experiments, adopt the default guidance approach of the baseline method.
E.6 Single-step Iteration Details
Our goal is validate that variance-driven conditional entropy reduction can improve the denoising diffusion process. Compared to iterations based on traditional truncated Taylor expansions, RE- based iterations achieve better sampling performance, as demonstrated in DPM-Solver. This is because DPM-Solver iterations represent specific instantiation of RE-based iterations, as shown in Proposition 3.3 Nevertheless, through extensive experiments on CIFAR-10 75), CelebA 64
[84, and
ImageNet-256 76, we validated that RE-based iterations can further improve the denoising diffusion process by minimizing conditional variance. In this validation experiment, we adopt 13 as our baseline. Since the single-step iteration mechanism only requires the information from the starting point to the information before the endpoint, RE- RE-based iterations depend on prior variance assumptions lo reduce the conditional variance between iterations. Below, based on the principle of minimizing conditional variance, we demonstrate how to select parameters under the assumption of prior variance.
For clarity, we simplify the RE-based single-step iteration in Eq. (1) as follows:
f( f(+.)) hr, ((7+ (1++)) To 2 co (T.., s:) + 7 7) ca (i)r
(67)
where ri fig To reduce variance of iteration (67) in each step, configure the parameter 7> in accordance with the effective variance reduction interval prescribed in Remark C.4 Based on Remark C.4 since 7/8
SNR(t,) (SN(()))
SNR(#:)) when considering only in
SNR(C)+
isolation, we recommend three specific selections of prior parameter 7/s % SNI(())++SN((s)
SNR(4)
and % 1 Due to ri E 1, SNR(()+SNK((.] based on Remark C.I Based on the proof in 2 when considering only Ta in isolation, ponential optimal value for T is given by SNR(C. SNRR(s)) We recommend three specific selections of prior parameter P;: Yy and
4 SNR(ss)
(R()))
2 SNR(())
Based on empirical performance, recommend the combinations
(SNR())
2 SNR(s;) (r. 1, 4) or T1 SNI(())+SN((ss) 7t SNI(())) for balanced inference.
SNR(C)
We compare the performance of RE- REsbd iterations against several established solvers, including DDPM (2). Analytic-DDPM 26). DDIM 19, DPM-Solver [33 F-PNDM [I]] and ERA-Solver Remarkably, this consistent
64
The comparative results are presented in Figures 2 and Table g
improvement in the conditional variance enhances image quality across various scenarios, as demon-
Notably, compared to the 3.17 FID achieved by DDPM with 1000 NFES (2] on 110AR-10. our RE-based iteration achieves 3.15 FID with only 84 NFE, establishing a new SOTA FID for this discrete- time pre-trained model while realizing approximately 10x acceleration. visual comparison is shown in Figure 6
strated by the ablation study with 7 1/2 and r in Figures 2
E.7 Multi-step Iteration Details
In this section, we explore the potential of variance-hasee conditional entropy reduction to further enhance the denoising diffusion process. Unlike single-step mechanisms, multi-step iterations can
35

---

Table 8: The performance comparison of sampling methods on CIFAR-10 75 suggests that RE-
based iterations can further improve the denoising diffusion process with enhanced equality.
Discrete
Continuous
Cond. EDM
3.17
2.55
1.79
DDPM
Hybrid PC
EDM
3.26
2.64
1.79
T-PNNM
DPM-Soovvervv
Heuu's 's 2nd
3.15
2.41 RE-based
1.76
RE-based
RE-baaed
DDPM
Analydi-DDPM
DDIM
Analytc-DDIM
DPM-solver-2
ver-
FID
F-PNDM
ERA ERA-Soliver
RE-based-2 (ours)
RE-based-3 (ours)
3.15
12
20
50
100
200
1000
NFE
Figure 6: Comparisons of FID for different iterations on discrete DMs in CIFAR-10.
leverage information from previous steps, providing additional context. Building on this advantage, propose training-free free and efficient denoising iteration framework aimed at improving the denoising diffusion process through variance-driven conditional entropy reduction. Specifically, the framework minimizes conditional variance by reducing the discrepancies between actual states during the denoising iterations.
Challenge. Formulating the optimization objective achieve this goal presents significant challenge, requiring mechanism that can effectively capture subtle state variations across iterations. The key lies in developing algorithm that can identify meaningful features from state differences and transform these insights into signals that improve the denoising process. This involves not only quantifying state differences but also understanding the underlying deep information patterns in these variations, enabling more precise control over the denoising diffusion process.
Our optimization objective is formulated by considering both the discrepancy between the actual data states and the variation in the gradient states. Building upon these foundations, we outline this efficient conditional entropy reduction iteration mechanism driven by variance minimization in EVODiff which offers an effective means to integrate variance-drive conditional entropy reduction into the denoising diffusion process by minimizing actual state differences.
Practical Considerations. Our goal to develop an iterative denoising sampling algorithm for pre-trained DMs that requires neither additional training nor costly optimization procedures. However, in the iterative scheme aimed at minimizing the variance-driven conditional entropy reduction, we need optimize the key parameters Ç; and 7); that control the conditional variance of the denoising iteration. As discussed in the main text, to balance optimality and computational efficiency, we adopt an optimization guidded streamlined approach obtain optimized pariance-reduction control parameters Çi and Specifically. the original optimization problem was standard constrained mathematical programming problem. We observed that the problem possesses a closed-form solution when constraints are removed. Therefore, directly obtain the optimized parameters in one step, we choose to apply nonlinear nonnegative mapping to this closed-form solution, using the mapped non-negative substitute our final parameters. Since this nonnegative substitute solution has
36

---

Table 9: We conducted ablation experiments with different shift parameters in EVODift using the pre-trained model | on ImageNet-256x44 776 various NFEs and and guidance scales.
We report the FID & evaluated on 10k samples for
NFE
Method
Guidence
Shift Parameter
5
8
10
12
15
20
/ 0.25
13.96
10.97
8.85
8.18
7.82
7.51
7.27
EVODiff
$=2
1 0.50
13.98
10.98
8.84
8.14
7.79
7.48
7.25
1 0.75
14.01
11.00
8.83
8.10
7.80
7.54
7.32
= 0.25
14.43
11.08
8.90
8.30
7.92
7.58
7.53
EVODiff
$=3
= 0.50
14.37
11.04
8.87
8.31
7.89
7.56
7.51
0.75
14.32
10.99
8.85
8.23
7.87
7.56
7.50
0.25
17.80
12.91
9.73
8.75
8.51
8.01
7.92
EVODiff
$=4
0.50
17.57
12.73
9.61
8.66
8.35
8.01
7.93
0.75
17.39
12.57
9.55
8.61
8.41
8.01
7.94
already achieved the objective of quantifying the differences between states, can serve as an effective alternative for parameter optimization, simultaneously ensuring computational efficiency and preserving the capability to capture critical state variations.
E.7.1 Ablation Study
Parameter settings. In our implementation, we primarily employ the sigmoid activation function, which is one of the most prevalent activation functions in neural networks [55] Its mathematical expression is Sigmoid(z) = -te-z- In our experiments, the following improved version often yields
o:ت+1 - 1t), where Ci computed using Eq. 22 and js shift parameter introduced fine-tune the solution space.
better results, particularly for high-dimensional datasets: G = -Sigmoid(-
Conceptually, I serves as a dynamic sensitivity regulator, allowing for nuanced control over the transformation of the activation function. By adjusting μ4, the inflection point of the sigmoid function can be shifted, effectively modulating the model's responsiveness to input variations across different regions of the input space. For nigh-ddmensiioal datasets, this provides principled mechanism for adaptive sensitivity calibration. The shift parameter enables more precise capturing of subtle state variations by expanding contracting the function's most sensitive transformation region.
Empirical results show that this approach achieves judicious balance between computational effi- ciency and the model's ability to discern critical state transitions. systemaittla examines the impact of shift parameters on image generation performance pre-trained DMs, using comprehensive ablation experiments the ImageNet-256 256 dataset. Key findings include:
Global Performance Characteristics: A consistent downward trend is observed in FID scores as NFF increases, indicating a progressive refinement of sample quality. Performance differences among the tested shift parameters μ E (0.25, 0.50, 0.75) remain marginal, reflecting the robustness of the sampling process across configurations.
Shift Parameter Behavior Across NFE Stages: Performance variations exhibit nuanced characteristics:
At lower NFE stages, performance differences between values are more pronounced.
As NFE increases, the performance of different μ values converges.
Different values exhibit unique progression patterns at various guidance scales, despite only marginal differences.
Impact of Guidance Scale: The sensitivity shift parameters varies with guidance scales:
Lower guidance scales (e.g., s=2) slightly more pronounced performance variations with μ. with a change magnitude of 0.05 FID at 20 NFE.
As guidance scale increases (to s=3 and s=4), the influence shift parameters becomes subtler and more stable, with change magnitude of 0.02 0.03 FID 20 NFE.
Beyond pixel-space DMs, we also conducted an ablation study on l within latent-space DMs to verify its efficacy and robustness in more computationally efficient frameworks. Specifically,
37

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Table 10: Ablation Study: Effect of the Shift Parameter on EVODiff Performance for the latent-
space diffuusio model (28] (LSUN-Bedrooms dataset [86].
Method
Model
Dataset
NFE
FID
Relative to μ = 0.5
0.25
+3.5% 7.6328
0.50
7.912
baseline
0.75
8.1845
-3.4%
Latent Diffusion
0.25
3.3357
-0.1%
EVODiff
LSUN- -UUNooeroomss
10
0.50
3.3318
baseline -0.3%
0.75
3.3409
0.25
2.8369
+0.6%
20
0.50
2.8534
baseline
0.75
2.8728
-0.7%
used a latent diffusion model trained on the observations across NFE stages remain largely consistent with the ImageNet findings, but reveal specific trends for latent space:
dataset. As presented in Table 10 the
Low-NFE Sensitivity: At the lowest 5 NFE, the shift parameter μ exhibits the largest influence. μ 0.25 yields the best FID score of 7.6328 (+3.5% relative to the baseline). This supports the notion that μ is most critical during the early, high-variance sampling phase.
Robust Convergence: As NFE increases (from 5 to 10 and 20), performance differences across μ values shrink significantly, confirming the robustness of EVODiff across parameter settings. The = 0.50 baseline performs optimally at NFE=10, while μ = 0.25 is marginally best at 20 NFE, with the total variation across all y being minimal (<s 0.7%).
Conclusion: The LSUN-B Bedrooms results validate the role of μ an effective fine-tuning mechanism that introduces negligible instability, even when applied the complex latent space of high-resolution image generation.
In summary, the extensive ablation studies on both pixel-space (ImageNet-256x556 and latent-space -esu-oBedoooms) diffusion models validate the function of the shift parameter. The results demon- strate the fundamental robustness EVODiff across diverse configurations, showing only marginal performance variations across values in high-NFE scenarios. Critically, μ functions an effective adaptive fine-tuning mechanism, providing the most significant benefit in the low-NFE, high-entropy sampling phase (e.g., / 0.25 leading the performance at 5 NFE in the L.UN-BBedrooe study). This confirms that y introduces negligible instability while offering a refined tool for sensitivity calibration in both high-dimensional pixel and latent spaces. Moreover, the properties of the aforemen- tioned shift parameters collectively ensure the convergence and distinctiveness of our variance-driven conditional entropy reduction iterative scheme during the sampling process. Specifically, although these subtle variations are negligible on ImageNet-2566 2arNNeet-5622555 256, their distinctiveness substantiated through experimental validation on the stable diffusion model, as shown in Figure II
Reducing Conditional Variance with Prior rj- multi-step iterations, we require probing step (an iteration step of Single-step Iteration Framework) to obtain the model value at the next state. Reducing conditional variance is crucial for improving the stability and accuracy of iterative algorithms; thus, we need to balance the conditional variance of the gradient term and the first-order term (see the above Conditional Variance Analysis part). We found that while logSNR typically performs well with larger step sizes, its advantages diminish as the NFE increases, illustrated in Figure and Table II. For clarity, we revisit the logSNR as follows:
log de log 111l
FlogSNR (t)
de+1
(68)
0 log
dc-1 log
This balance concept of loRSNR leads to two potentially useful types of substitutions.
From the perspective of balancing variances, one might consider the following form:
Var
Vare
/
Var, Var
T Tnornarr (t)
(69)
Vare+1
Vare
38

---

where Var, can represent any assumed variance, and satisfies Var, > Var,-1. If Var, < Vary-1- then simply swapping the roles of Var, and Var,-1 in Eq. (69) will suffice. Another substitution idea is to change the function space of the step size, for example, to the arctangent space:
arctan(hs)
Tarctan (1) arctan(he-1)
(70)
where hy denotes the step size from t + 1 to t.
In our experiments, we observed that nonlinear combination of these two substitutions leads to improvements in certain scenarios. We define this nonlinear combination as refined Tú, and the ablation study of both logSNR and refined r; in the context of EVODiff can be found in Table 11 Table TT shows that even when using the same r, as the baseline, our mathematically principled construction of EVODiff consistently outperforms state-of-the-art ODE solvers. Moreover, Table 11 also demonstrates that employing a more effective r; within our EVODiff framework further improves performance. Beyond this, we investigate variance-driven approach that adheres more closely to theoretical principles. Specifically, function of the cosine similarity between Ba (t;, L4) and Tt, at time step t,. This strategy demonstrates strong performance on both CIFAR-10 10 and ImageNet-256, As shown in Table 2 improvements on CIFAR-10.
= FlogSNK (t) * Weon fidence, where Weonnfiennee is
it yields significant
In summary, the ablation study on the shift parameter ja demonstrates the robustness of EVODiff. While different l values show minor performance variations in specific NFE ranges, the overall results are consistently parameter and can achieve excellent performance with a default setting (e.g., / 0.5).
indicating that our method is not highly sensitive to this
E.8 Comparison of Reference-Free EVODiff and Learning- Methods with Reference Trajectories
significant advantage of EVODiff is its reference-free nature, enabling it to achieve state-of-the performance without the overhead required by methods that rely on pre-computed or learned reference trajectories. This section substantiates this claim by presenting comprehensive comparisons against major classes of reference- based methods, followed by analyses of computational efficiency and the generalizability of our core principles.
Superiority over Reference-Basee Solvers and Learning- Methods. Our advantage is particularly pronounced when benchmarked against advanced ODE solvers that explicitly incorporate reference information. As noted in our main results in Table 2 DPM-Solver-v3 leverages Empirical Model Statistics (EMS), which is a technique requiring prior knowledge from a high-NFE reference solution to optimize its steps. This essentially provides the solver with "cheat sheet" the data distribution. Despite this additional optimization information, EVODiff, with its on-the-fly adaptive strategy, consistently demonstrates superior performance. On CIFAR-10 10. it achieves a remarkable FID of 3.98 at 8 NFE and 2.78 at 10 NFE, decisively outperforming DPM-Solver-v3's scores of 4.95 and 3.52, respectively. This trend is not limited to low-dimensional data; on ImageNet-256. EVODiff also maintains competitive edge, further underscoring the robustness of our approach (Table
Furthermore, EVODiff also excels when compared to another class of reference-baseee techniques: learning-based methods that distill knowledge from prior trajectories. As shown in Table [3] while specialized methods like UniPC [LD3, [59] are highly effective, EVODiff surpasses them 10 NFE with a leading FID of 2.74. It is crucial to note that this result achieved without the need for an expensive offline distillation training phase, highlighting a significant practical advantage in terms of flexibility and resource efficiency. Collectively, these results furnish compelling evidence that the reference-free paradigm of EVODiff is not a compromise but fundamental strength.
Computational Efficiency. A critical consideration is whether these performance gains come at the expense of computational efficiency. The end-to-end generation time comparison in Table and and Table confirms that EVODiff introduces negligible or even reduced computational overhead/cost compared to the highly optimized DPM-Solver++ baseline. This because our algorithm a second-order method, yet the adaptive optimization of parameters Ç, and 1/: relies on closed-form solutions involving lightweight vector operations (as shown in Lemmas 4.4 (.5). Consequently, these steps add minimal latency relative to the computationally intensive forward pass of the neural
39

---

Table 11: We conducted ablation experiments under different guidance scales and different random seeds. Quantitative results of the gradient estimation-baxe denoising iterations using the pre-trained model 4 ImageNet-256xx
[76]. We report the FID for 10k samples evaluated under various Bold values indicate the best FID each iteration step column, while italicized values
NFEs.
represent the second best.
NFE
Method
Model
6
8
10
12
15
20
DPM-Solver+++2
16.39
12.77
9.92
8.88
8.31
8.03
7.76
DPM-Solver++-3
15.64
11.64
9.21
8.51
8.12
7.97
7.69
UniPC-2
15.15
11.79
9.41
8.63
8.16
7.93
7.71
UniPC-3
14.93
11.22
9.21
8.55
8.19
7.98
7.70
-PM-
(s=2, seed=1234)
14.88
11.21
9.17
8.51
8.12
7.90
7.67
DPM-Solver-v3-3
15.62
11.73
9.57
8.89
8.37
8.01
7.65
EVODiff (MogSNK)
13.94
10.96
9.02
8.38
8.01
7.83
7.54
EVODiff (Frefined)
14.21
11.21
9.05
8.34
7.97
7.80
7.48
2
16.62
12.86
9.73
8.68
8.17
7.80
7.51
DPM-Solver+++33
15.69
11.65
9.06
8.29
7.94
7.70
7.48
UniPC-2
15.37
11.78
9.22
8.40
8.01
7.71
7.47
UniPC-3
Guided-Diffusion
15.05
11.30
9.07
8.36
8.01
7.72
7.47
DPM-Solver-v3-2
(s=2, seed=3407)
14.92
11.13
8.98
8.14
7.93
7.70
7.42
3PM--olver-v3-3
15.51
11.77
9.37
8.67
8.18
7.73
7.52
EVODiff (flggNRR
13.98
10.98
8.84
8.16
7.81
7.52
7.32
EVODiff (Fretined)
14.33
11.16
8.95
8.14
7.79
7.48
7.25
2
16.27
12.40
9.55
8.66
8.18
7.84
7.61
D
15.93
11.49
8.98
8.39
8.11
7.74
7.63
UniPC-2
15.44
11.64
9.11
8.46
8.17
7.75
7.62
UniPC-3
Guided-Diffusie
16.11
11.88
9.25
8.58
8.14
7.77
7.72
DPM-Solver-v3-22
(s=3, seed=3407)
17.97
12.04
9.17
8.40
8.11
7.76
7.67
DPM--Solve-v3--3
20.87
14.94
10.68
9.29
8.57
7.92
7.77
EVODiff (MogsNR)
14.37
11.04
8.87
8.37
7.89
7.56
7.51
EVODiff (Frefined)
15.93
11.94
9.21
8.31
7.89
7.58
7.54
network. In many low-NFE scenarios, our method is even marginally faster. This finding is crucial, it establishes that EVODiff offers Pareto improvement, achieving superior sample quality at no additional computational cost.
Generalizability of the Core Principles. Finally, to demonstrate the fundamental nature of our proposed principles, we tested whether our variance-control concept could enhance other state-of-the- art frameworks. We integrated our entropy-aware approach into the EMS- EMS-parameter parameterized structure of DPM-Solver-v3. As evidenced in Table (labeled "RE-based") and Figure 8] this hybrid method surpasses the already formidable performance of the original DPM-Solver-v3 (e.g., achieving 10.61 FID vs. 12.21 NFE on EDM). This result provides the strongest validation, elevating entropy- aware variance optimization from mere algorithmic heuristic powerful and universal principle for diffusion model inference. Moreover, it suggests that the improvements from our entropy- aware optimization and the Mase- approach may be orthogonal, opening promising avenues for future work in combining these principles for even greater performance gains.
E.9 More Experiments for EVODIIT
conducted additional experiments to assess the robustness and versatility of EVODiff. These tests span various pre-trained models, datasets, noise schedules, and complex conditional generation tasks, aiming to demonstrate that EVODiff's superior performance arises from entropy-aware, reference-free free design, rather than being limited to specific conditions.
40

---

Table 12: Comparison of FID scores for different sampling methods on CIFAR-I0 with the uncondi- tional EDM model.
NFE
Method Model Reference-bassee Entropy-aware?
5
10
12
15
DPM--Slver++
X
X
27.96
16,87
8,40
5.10
3.70
2.83
UniPC DPM-Solver-v3 EDM
X
X
27.03
17.32
7.67
3.97
2.76
2.23
X
11.60
8.22
4.94
3.52
2.81
2.40
EVODiff
X
17.84
9.17
3.98
2.78
2.30
2.12
Table 13:
Comparison on CIFARIO with recent learning-basee and learning-free methods under 10 NFEs. Bold indicates the best FID in each column, italicized indicates the second best.
NFE
Dataset
Method Type
Method
8
10
UniPC (3M)
13.12
4.41
3.16
GITS (70) (UniPC prior)
I1.19
5.67
3.70
CIFARIO Leamnin-basse based with prior trajectories
LD3 (59] (UniPC prior)
5.92
3,42
2.87
Leamiin-frree and reference-free
EVODiff (2m)
9.07
3.88
2.74
A key indicator of a sampler's utility is its consistent performance across diverse settings. We first demonstrate this quantitative consistency standard benchmarks. As shown in Table CIFAR- 10, EVODiff excels with both the ScoreSDE and EDM pre-trained models, achicving a state-of-the-art FID of 10.61 just NFE EDM. Furthermore, its superiority is maintained under different noise schedules; Tables 16 through 19 show that EVODiff consistently secures the leading FID scores high-resolution datasets like FFHQ-64 and ImageNet-64, regardless of whether "logSNR" or "EDM" schedule is employed. This consistent dominance across varied models and schedules strongly indicates that EVODiff's performance gains intrinsic to its algorithmic design rather than an artifact of a specific setup.
Moving beyond numerical metrics, we evaluated EVODiff's qualitative performance on the highly demanding task of text-to-image synthesis with Stable Diffusion v1.4 and v1.5. This setting tests a sampler's s ability to handle complex semantic guidance and generate coherent, high-fidelity images. As visualized in Figures 10 and 12 our method produces images with significantly fewer artifacts and greater structural integrity. Most compellingly, Figure 13 highlights crucial advantage in semantic consistency: for the prompt "an astronaut riding a horse," competing methods generated an anatomically incorrect horse with five legs, common failure mode in DMs. In contrast, EVODiff correctly rendered four-legged animal, demonstrating its superior ability to preserve semantic and anatomical plausibility. This suggests that our entropy-aware optimization leads to more stable and accurate information flow from text prompt to pixel space.
A critical consideration for any practical sampler is whether performance gains achieved at the expense of computational efficiency. We explicitly address this by comparing the end-to-end generation time of EVODiff with the highly optimized DPM-Solver++ baseline. The results, presented in Table 15 confirm that EVODiff introduces negligible computational overhead. In many low-NFE scenarios, it is marginally faster. This finding is crucial, as it establishes that EVODiff offers Pareto improvement: superior sample quality at no additional computational cost. This makes it a highly practical reference-free solution for real-world deployment.
Finally, to demonstrate the fundamental and generalizable nature of our proposed principles, we tested whether our variance-control concept could enhance other SOTA frameworks. We integrated our entropy-awar approach into the EMS EMS--paaamen parameterized structure of DPM-Solver-v3, As evidenced in Table [4aand and Figure 8 this hybrid method (labeled "RE-based") surpasses the already formidable performance the original DPM-Solver-v3. This result provides the strongest validation that entropy- aware variance optimization is not merely set of heuristies for single algorithm, but powerful, universal principle for improving diffusion model inference.
41

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Table 14:
Quantitative results of FID + scores for gradient-based methods CIFAR-IO. The results are evaluated on 50k samples for various NFEs, some results are borrowed from the original papers. The "RE-based (our)" demonstrates the application of entropy-awand principles to the DPM-Solver-v3 framework.
NFE
Method
Model
8
10
12
15
20
DEIS [4]
15.37
4.17
3.37
2.86
DPM-Solver++ 20
28.53
13.48
5.34
4.01
4.04
3.32
2.90
UniPC [66]
ScoreSDE
23.71
10.41
5.16
3.93
3.88
3.05
2.73
DPM-Solver-v3
12.76
7.40
3.94
3.40
3.24
2.91
2.71
RE-based (our)
13.54
8.56
4.11
3.38
3.22
2.76
2.42
Heun's 2nd [2
320.80
103.86
39.66
16.57
7.59
4.76
2.51
DPM-Solver++ 20
24.54
11.85
4.36
2.91
2.45
2.17
2.05
UniPC [6]
EDM
23.52
11.10
3.86
2.85
2.38
2.08
2.01
DPM-Solver-v3 17
12.21
8.56
3.50
2.51
2.24
2.10
2.02
RE-based (our)
10.61
8.22
3.37
2.43
2.21
2.07
2.01
Table 15: Comparison of Computational Overhead between EVODiff and DPM-Solver++ on ImageNet-256 with 10k samples 3090
GPU.
NFE DPM-Solver-2m
Total Time
EVODiff
Total Time
9.56s/it
1:03:24 (hemss
9.45s/it
1:02:15 (h:m:s)
10
18.40s/it
2:02:39 (h:m:)
18.39s/it
2:00:39 (h:m:s)
15
27.24s/it
3:01:34 (h:m:)
27.25s/it
3:01:47 (b:m:s)
20
36.07s/it
4:00:28 (h:m:s)
36.11s/it
4:00:48 (h:m:s)
Table 16: Comparison of FID scores for different sampling methods on FFHQ-64x64 using the
logSNR schedule.
NFE
Method
Model
logSNR schedule
5
10
15
20
25
Heun +
342.28
45.46
7.60
3.25
2.71
28.96
6.87
4.07
3.29
2.97
UniPC bhl
FFHQ-64, EDM
35.78
4.00
2.81
2.60
2.52
UniPC bh2
27.00
5.44
3.38
2.87
2.67
EVODiff
20.04
3.93
2.72
2.55
2.46
Table 17: Comparison of FID scores for different sampling methods on FFHQ-64x64 using the EDM schedule.
NFE
Method
Model
EDM schedule
5
10
15
20
25
35
Heun
347.09
29.92
9.95
4.58
3.41
2.71
+ UniPC_bii FFHQ-64, EDM
25.08
6.81
3.80
3.00
2.75
2.59
28.87
6.66
3.40
2.69
2.58
2.50
UniPC_ bh2
24.09
6.17
3.35
2.73
2.58
2.50
EVODiff
19.65
5.31
3.02
2.64
2.56
2.48
42

---

Table 18: Comparison of FID scores for different sampling methods ImageNet 64x64 using the
logSNR schedule.
NFE
Method
Model
logSNR schedule
10
15
20
25
35
Heun
231.754
23.269
7.413
3.597
3.020
2.513
DPM-Solver++ UniPC_bhi UniPC_bh2
32.529
7.052
3.922
3.077
2.738
2.465
ImageNet 64, EDM
41.366
5,453
3.041
2.578
2.415
2.286
31.063
5.795
3.312
2.714
2.490
2.319
EVODiff
23,979
4.289
2.688
2.384
2.242
2.143
Table 19: Comparison of FID scores for different sampling methods ImageNet 64x64 using the edm schedule.
NFE
Method
Model
EDM schedule
TO
15
20
25
35
Heun DPM-Solver++ UniPC_bii
248.402
15.129
5.301
3.136
2.739
2.424
27.243
5.785
3.480
2.866
2.606
2.393
ImageNet 64, EDM
39.158
5.649
3.456
2.701
2.424
2.268
UniPC_bli bh2
26.354
5,042
3.118
2.639
2.434
2.284
EVODiff
21.894
4.734
3.316
2.594
2.308
2.167
DPM-Solver+
UniPC
DPM-Solver-v3
EVODiff (our)
5
NFE
(a) UniPC, 25 NFE.
(b) DPM-Solver-v3, 25 NFE.
(c) EVODiff (our), 25 NEE,
28 with a classifier free guidance scale 7.5, using the text prompt "environment living room interior, mid century modern, indoor garden with fountain, retro, m vintage, designer furniture made of wood and plastic, concrete table, wood walls, indoor potted tree, large window, outdoor forest landscape, beautiful sunset, cinematic, concept art, sunstainable architecture, octane render, utopia, ethereal, cinematic light". Our EVODiff method demonstrates consistent improvements actoss both low (5 NFE) and higher (25 NFE) inference steps.
Figure 7: Random samples from
43

---

(a) 3PM-Solver-v3. 12.22 FID.
(b) RE-based, 10.61 FID.
Figure 8: Random samples of EDM 12 on the CIFAR-10 dataset with only 5 NFEs. Within the paaaeearaetee iterative framework provided by DPM-Solver-vá, the RE-based iterative approach improves FID by explicitly balancing the conditional variance of the gradient term itself.
DPM-Solver++ 7.76 FID
UniPC
DPM-Solver-v3 7.67 FID
EVODiff 7.48 FID
7.71 FID
Figure 9: Random samples from the pretrained Guided- -ifffu--if model 4 with 20 NFE the ImageNet-256 dataset (76). Our EVODiff method reduces reconstruction error without relying on reference solution, while can retain the sample quality benefits of methods like which relies on EMSSbased statistics optimized using a reference solution.
DPM-Solver++
DPM-Solver-V3
EVODiff
Figure 10: Random samples from Stable-Diffusion-vl.. 288 with classifier-free guidance scale 7.5, using 10 NFE and the prompt "A beautiful castle beside a waterfall in the woods, by Josef Thoma, matte painting, trending on artstation HQ" Images generated by our EVODiff method exhibit greater clarity and naturalness, along with more coherent and complete structural content.
44

---

(a) EVODiff D (c=0.75).
(b) EVODiff [ (==05))
(c) EVODifI 0=0.1).
Figure 11: Random samples from Stable-Diftusionvl..4 288 with CFG scale 7.5, different shift parameters, using 10 NFE and the text prompt "A beautiful castle beside a waterfall in the woods, by Josef Thoma, matte painting, trending artstation HQ". Our EVODiff inference consistently generates clear and complete content across different shift parameters.
45

---

NFE=5
NFE=10
NFE=15
NFE=25
DPM-
Solver 20
UniPC 16
DPM- Solver- v3
EVOD- iff M
Figure 12: Random samples from Stable Diffusson 28 with guidance scale 7.5, using varying NFEs and the prompt "tree house in the forest, atmospheric, hyper realistic, epic composition, cinematic, landscape vista photography by Carr Clifton & Galen Rowell, 16K resolution, Landscape veduta photo by Dustin Lefevre & tdraw, detailed landscape painting by Ivan Shishkin, DeviantArt, Flickr, rendered in Enscape, Miyazaki, Nausicaa Ghibli, Breath of The Wild, 4k detailed post processing, artstation, unreal engine". Our EVODiff inference improves content clarity, coherence, and overall completeness. In contrast, other methods exhibit partial content collapse at 25 NFEs, whereas ours preserves structural integrity.
46

---

DPM- Solver++ 20
UniPC 16
EVODiff
.4 28 with a classifier-fie free guidance scale 7.5, using NFE and the text prompt "a photograph of an astronaut riding horse". the images at position (2,2), other methods produced anatomically incorrect horses with five legs, whereas our EVODiff inference correctly generated anatomically accurate horses with four legs.
Figure 13: Random samples from
47

---

NFE=10
NFE=25
DPM-
Solver++ 20
UniPC 16
EVODiff I
Figure 14; Random samples from Sibble-Diffusion-11.55 with guidance scale 7.5, using varying NFES and the prompt "A robot chef cooking a meal in a futuristic kitchen, with glowing utensils and holographic recipe book, highly detailed, sci-fi atmosphere". Our EVODiff inference improves content continuity and consistency, and effectively reduces visual artifacts.
48


Reviews:
[{"review_id": "h0Lg9u5LL7", "reviewer": "Reviewer_3tew", "summary": "This paper proposes a new gradient-based solver called EvoDiff for diffusion inference.\nThe idea behind EvoDiff is to use conditional entropy minimization as a guiding principle during denoising.\nExperimentally, this method shows an improvement over the state of the art (UniPC, DPM++, DPM-v3)\n\nDisclaimer: the maths in the paper are outside of my expertise, put simply I didn't follow at all despite my best effort and having multiple reading passes. And unfortunately for me, most of the paper is maths.", "questions": "1. About weaknesses (1), why did you not report the improvement compared to the best baselines from Table 2 for your abstract and conclusion?\n2. Is my assessment correct that the larger the image, the lower the gains offered by EvoDiff? What are the reasons behind this?", "limitations": "Yes", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "I apologize if the review is a bit shallow, I focused on the results and their significance and had to assume the maths were true.\n\nStrengths\n1. Improves state of the art for diffusion sampling as measured experimentally by FID and NFEs gains.\n\nWeaknesses\n1. In the abstract, it is claimed that EvoDiff improves FID from 5.10 to 2.78 for CIFAR10. However in Table 2, it appears that 5.10 was the worst baseline while the best one is 3.52. I believe the ethical choice would be to compare against the best baseline, not the worse one. That is unless the two better baselines are somehow not comparable or unfair to compare to?\n2. Scalability: it seems the higher dimension the images are, the lower the gains become. The best gains are observed on CIFAR10, while they become marginal for ImageNet256 and StableDiffusion. This reduces the significance of the proposed method since large images is typically the area where practitioners want to have the most gains.\n3. Presentation: it seems there are a lot of notations being introduced along the way in the mathematical parts of the paper. I found this to impact clarity, making it similar to understanding code that keeps calling functions within functions within functions. Granted the maths are outside my expertise, I did multiple passes of reading but ultimately couldn't follow despite my best efforts.", "was_revised": true, "final_justification": "The paper is more significant than I originally thought, it's a drop-in replacement at no additional cost for existing samplers.\nWhat prevents from raising my score even further is the clarity aspect which, while addressed in the rebuttal, is hard to visualize without seeing the final draft.", "rebuttal": "*Dear Reviewer 3tew:*\n\nWe sincerely appreciate your thoughtful and insightful feedback on our paper. Your focused attention on the experimental results and their significance, despite the mathematical sections being outside your core expertise, is highly valuable to us. We are especially grateful that you recognized how EvoDiff achieves substantial experimental improvements over state-of-the-art baselines while maintaining simplicity and efficiency, which represents the core contribution of our work.\n\nWe understand your disclaimers regarding the mathematical aspects, and we acknowledge the challenge this posed. Your comments will guide our efforts to enhance the clarity and accessibility of our theoretical framework in the revision. Thank you very much. \n\n**We address each of your concerns below:**\n\n**W1&Q1:**  *In the abstract, it is claimed that EvoDiff improves FID from 5.10 to 2.78 for CIFAR10. However in Table 2, it appears that 5.10 was the worst baseline while the best one is 3.52. I believe the ethical choice would be to compare against the best baseline, not the worse one. That is unless the two better baselines are somehow not comparable or unfair to compare to?*\n\n**Response to W1&Q1:** \n\nThank you for raising this important point regarding our baseline comparison in the abstract. We understand why this might appear misleading, and we want to assure you that our intention was always to provide a fair and accurate comparison. We appreciate the opportunity to clarify.\n\nEvoDiff is developed to enhance *gradient-based inference algorithms*. For fairness in comparison, our method is built upon the framework of DPM-Solver++, as it is currently the most efficient, broadly adopted, and representative gradient-based algorithm. Our approach does not aim to build upon the more complex UniPC or DPM-Solver-v3, nor does it seek to investigate their structures or directly improve their frameworks.\n\nThese methods can actually be divided into two categories:\n\n- Methods using only current model information: DPM-Solver++ (FID=5.10) and our EvoDiff (FID=2.78)\n- Methods requiring preprocessing or additional information: UniPC (FID=3.98) and DPM-Solver-v3 (FID=3.52)\n\nSpecifically, UniPC employs a more intricate multi-step predictor-corrector framework. This framework not only utilizes information from previously computed steps but also estimates subsequent steps, thereby requiring more trajectory information for its optimization and operation. DPM-Solver-v3, on the other hand, requires using existing solvers like DPM-Solver or DPM-Solver++ with 200 NFE to pre-generate final samples, then uses these pre-generated samples as reference solutions to optimize their solver, which is essentially a post-processing optimization method, as summarized in *Table 1 on page 4*.\n\nWe appreciate the reviewer's reminder. While we have already explained our baseline selection in line 58 of the introduction, we recognize that this could be stated more clearly in the abstract. In the revised version, we will update the abstract to be more precise: \"EvoDiff reduces FID from 5.10 to 2.78 compared to DPM-Solver++, the most widely adopted gradient-based baseline\" or \"among gradient-based methods without preprocessing requirements.\" \n\n\n**W2&Q2:**  *Scalability: it seems the higher dimension the images are, the lower the gains become. The best gains are observed on CIFAR10, while they become marginal for ImageNet256 and StableDiffusion. This reduces the significance of the proposed method since large images is typically the area where practitioners want to have the most gains.*\n\n**Response to W2&Q2:**   \n\nThank you for this important observation about diminishing gains at higher resolutions. We acknowledge this is a critical concern for practitioners.\n\n*Key Finding*: While absolute FID improvements decrease with resolution, EvoDiff achieves the best performance without any computational overhead, outperforming even methods that require extensive preprocessing.\n\n*Comparative Analysis*:\n- *DPM-Solver-v3* (with 200 NFE preprocessing): 7.70 → 7.80 FID (1.3% improvement)\n- *EvoDiff* (zero preprocessing): 7.48 → 7.80 FID (4.1% improvement)\n\nThis demonstrates that even at high resolutions where all methods face inherent challenges, EvoDiff provides superior results without additional computational costs.\n\n*Comprehensive Results across NFE Settings* (ImageNet-256):\n\n| NFE | EvoDiff | DPM-Solver-v3 | UniPC | Improvement |\n|-----|---------|---------------|--------|-------------|\n| 5   | **13.98** | 14.92 | 15.37 | Best by 6.3% |\n| 10  | **8.14** | 8.14 | 8.40 | Matches best |\n| 20  | **7.25** | 7.42 | 7.47 | Best by 2.3% |\n\n*Additional Evidence from EDM2 on ImageNet-64*: \n\n| Steps | EDM sampler |         | EvoDiff |         | Improvement |         |\n|:----:|:-----------:|:-------:|:-------:|:-------:|:-----------:|:-------:|\n|      | FID | FD_dinov2 | FID | FD_dinov2 | FID↓ | FD_dinov2↓ |\n| 5 | 28.6718 | 396.113 | **5.31845** | **161.606** | 81.5% | 59.2% |\n| 6 | 10.7555 | 252.302 | **3.99538** | **134.352** | 62.9% | 46.7% |\n| 8 | 3.5605 | 153.608 | **2.8873** | **115.641** | 18.9% | 24.7% |\n| 10 | 2.3892 | 123.77 | **2.2280** | **106.912** | 6.7% | 13.6% |\n| 15 | 1.8366 | 103.687 | **1.7455** | **96.9382** | 5.0% | 6.5% |\n| 25 | 1.7003 | 96.5694 | **1.6496** | **92.8184** | 3.0% | 3.9% |\n\nOur experiments reveal that relative gains naturally diminish with NFE across all resolutions:\n- 5 steps: 81.5% FID reduction\n- 25 steps: 3.0% FID reduction\n\nThis pattern suggests the diminishing improvements are inherent to diffusion model acceleration rather than a limitation of our method.\n\n*Practical Implications*: \n\nFor practitioners, EvoDiff offers the best cost-benefit ratio: consistent improvements across all resolutions with zero additional computational overhead. Even \"marginal\" improvements at high resolutions are valuable when they come at no extra cost, especially considering that IS improvements remain substantial (13% at NFE=5), indicating better sample diversity crucial for practical applications.\n\nWe will add this comparative analysis to our revision to provide clearer insights about the fundamental challenges all acceleration methods face at high resolutions. \n \n**Response to W3:**  \n\nThank you for your valuable feedback on the mathematical clarity of our paper. We sincerely appreciate your diligent efforts to understand the content and recognize that presenting complex theoretical concepts in an accessible manner is a significant challenge.\n\nWhile our paper is indeed theory-driven, and mathematical rigor is essential for establishing convergence guarantees and justifying why evolutionary directions provide acceleration, we wholeheartedly agree that clarity for a broad audience is equally important. We have already structured the paper to be accessible at multiple levels (e.g., Section 2 provides intuition, Algorithm 1 gives a simple 10-line implementation, and Section 4 demonstrates clear empirical gains).\n\nHowever, your feedback highlights that there is still room for improvement in making the mathematical sections more approachable. In the revised version, we will specifically address this by:\n\n- Ensuring notations are introduced incrementally with clear and immediate definitions.\n- Adding more descriptive inline explanations or informal summaries for complex equations and relationships to bridge the gap between intuition and formal derivation.\n- Carefully reviewing the overall flow of the mathematical sections to improve readability and avoid the \"function within functions\" impression.\n- If feasible and beneficial, we will consider providing a concise notation table at the beginning of the appendix section for quick reference.\n\nOur goal is to ensure that the solid theoretical foundations of our method are accessible to as wide an audience as possible, without compromising on rigor. We appreciate you bringing this crucial point to our attention.\n\nWe believe these clarifications address the reviewer's concerns and demonstrate that EvoDiff provides significant value through its simplicity, efficiency, and consistent improvements across all settings.\n\nThank you again for your constructive review. We are committed to improving our paper based on your valuable feedback and believe these revisions will significantly enhance its clarity and impact."}, {"review_id": "RzOqMVm5PY", "reviewer": "Reviewer_uyET", "summary": "This paper proposes EvoDiff, a novel entropy-aware variance optimization method for diffusion inference. The authors introduce an information-theoretic framework interpreting denoising in diffusion models as a conditional entropy reduction process. Based on this principle, EvoDiff systematically optimizes the conditional variance during each denoising iteration, which theoretically and empirically improves the convergence speed and sample quality. Extensive experiments on CIFAR-10, ImageNet, FFHQ, and text-to-image generation tasks demonstrate the effectiveness of EvoDiff comapred to existing training-free baselines.", "questions": "See the weaknesses section. I would be willing to increase my rating if authors could address my concerns.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n+ The proposed EvoDiff method introduces a principled way to optimize diffusion inference from entropy perspective.\n\n+ The paper offers detailed proofs supporting its claims, with reasonable theoretical assumptions and convergence guarantees.\n\n+ The experimental results achieve strong performance compared to training-free baselines.\n\n**Weaknesses:**\n+ Presentation clarity could be improved, especially in Section 4. For example, in Eq. (14) the symbol $l_i$ suddenly appears as the time stamp of a virtual sample $x_{l_i}$ but is never defined; readers must infer after the fact that $l_i \\in (t_{i+1},t_i)$ and that $x_\\theta(x_{l_i},l_i)$ is just a linear interpolation of neighbouring predictions. Besides, the intermediate sampling point $s_i$ has different definitions in different contexts (e.g., line 937 vs. Eq. (15) ). \n\n+ Some necessary discussion on prior works on DMs from an entropy perspective is missing [1,2,3].\n\n+ Experimental results could be more comprehensive. \n    + The shift parameter $\\mu$ is only ablated on ImageNet-256 using ADM; similar ablation studies on both pixel-space and latent-space diffusion models would help clarify the method’s generality.\n    + Since this paper focuses on continuous-time diffusion ODEs, it would be interesting to evaluate the proposed method on flow-based models used in recent works [4], such as Instaflow [5], to demonstrate broader applicability.\n\n[1] Zheng, Guangcong, et al. Entropy-driven sampling and training scheme for conditional diffusion generation. ECCV 2022.\n\n[2] Wang, Renzhi, Jing Li, and Piji Li. InfoDiffusion: Information Entropy Aware Diffusion Process for Non-Autoregressive Text Generation. EMNLP 2023.\n\n[3] Lou, Aaron, Chenlin Meng, and Stefano Ermon. Discrete diffusion modeling by estimating the ratios of the data distribution. ICML 2024.\n\n[4] Tong, Vinh, et al. Learning to discretize denoising diffusion odes. ICLR 2025.\n\n[5] Liu, Xingchao, et al. Instaflow: One step is enough for high-quality diffusion-based text-to-image generation. ICLR 2023.", "was_revised": true, "final_justification": "The clarifications and additional experiments address my concerns well. I believe the authors can revise the paper for improved clarity and completeness. I’m raising my score to Accept.", "rebuttal": "*Dear Reviewer uyET:*\n\nWe sincerely thank you for your thorough review and highly constructive feedback on our paper. We are especially grateful for your positive assessment, recognizing the novelty of our information-theoretic perspective, the reasonableness of our theoretical results, and the strong empirical performance of EvoDiff compared to training-free baselines. We truly appreciate the valuable suggestions that will undoubtedly help us significantly improve the clarity and comprehensiveness of our manuscript. \n\nWe have carefully considered all your comments and are fully committed to addressing them in the revised version.\n\nHere are our responses to your specific points:\n\n**Weaknesses and Questions:**\n\n1.  *Presentation clarity could be improved, especially in Section 4. For example, in Eq. (14) the symbol $l_i$ suddenly appears as the time stamp of a virtual sample $x_i$ but is never defined; readers must infer after the fact that $l_i \\in (t_{i+1}, t_i)$ and that $x_\\theta(x_i, l_i)$ is just a linear interpolation of neighbouring predictions. Besides, the intermediate sampling point $s_i$ has different definitions in different contexts (e.g., line 937 vs. Eq. (15)).*\n\n    We deeply appreciate this meticulous feedback. We agree that clearer definitions and consistent notation are crucial for readability.\n    * *For $l_i$ in Eq. (14):* You are correct that its definition appears slightly after its first use. We will revise the text to *introduce and define $l_i$ (as a linear interpolation point within $(t_{i+1}, t_i)$) immediately before Eq. (14) (specifically, at line 157)*. This will ensure $l_i$ is clearly understood before it is used in the equation.\n    * *For $s_i$ in Eq. (15) and Line 937:* We acknowledge the potential for confusion due to the dual use of the symbol $s_i$. While these are used in different contexts (our main algorithm vs. DPM-Solver++'s definition in the appendix), to avoid any ambiguity, *we will replace the symbol $s_i$ used in Line 937 (Appendix E.2) with a distinct, unused time symbol* to prevent any misinterpretation.\n    * More broadly, we commit to a *systematic review of Section 4's notation* to ensure all symbols are clearly defined upon first use and used consistently throughout. We will also enhance the prose around derivations to provide more intuition.\n\n2.  *Some necessary discussion on prior works on DMs from an entropy perspective is missing [1,2,3].*\n\n    Thank you for pointing out these important related works. We agree that a discussion connecting our work to these prior explorations of entropy in diffusion models would enrich the paper. Our approach specifically focuses on entropy-aware variance optimization within ODE-based inference, distinguishing it from [1]'s training schemes, [2]'s non-autoregressive text generation, and [3]'s discrete diffusion models. Nevertheless, these works collectively demonstrate the value of entropy-based perspectives. We will add a dedicated paragraph in Appendix A (Related Work) to discuss these connections in detail. \n\n3.  *Experimental results could be more comprehensive. The shift parameter $\\mu$ is only ablated on ImageNet-256 using ADM; similar ablation studies on both pixel-space and latent-space diffusion models would help clarify the method’s generality. Since this paper focuses on continuous-time diffusion ODEs, it would be interesting to evaluate the proposed method on flow-based models used in recent works [4], such as Instaflow [5], to demonstrate broader applicability.*   \n    We appreciate your excellent suggestions for more comprehensive experiments, which will indeed strengthen the generality of our method. We have conducted additional experiments:\n    * *$\\mu$ Shift Parameter Ablation:* We have performed an ablation study for the μ parameter on a latent-space diffusion model. \n  \n        *Ablation Study: Effect of $\\mu$ on EvoDiff Performance*\n        | NFE | $\\mu$ | FID Score | Relative to μ=0.5 |\n        |:---:|:--------:|:---------:|:-----------------:|\n        | 5   | 0.25     | **7.6328**     | +3.5%             |\n        | 5   | 0.50 | 7.912 | baseline          |\n        | 5   | 0.75     |8.1845  |          -3.4%      |\n        | 10  | 0.25     | 3.3357    |  -0.1%         |\n        | 10  | 0.50 | **3.3318**| baseline          |\n        | 10  | 0.75     | 3.3409 |-0.3%        |\n        | 20  | 0.25     |**2.8369** |+0.6%           |\n        | 20  | 0.50     | 2.8534  | baseline              |\n        | 20  | 0.75     | 2.8728  | -0.7%                |\n\n        *(This table will be added to the revised manuscript.)*\n\n    * Latent space diffusion (lsun_beds-256) on stable-diffusion\n \n        *FID Score Comparison*\n        | NFE | DPM-Solver++ (2m) | DPM-Solver++ (3m) | UniPC (3m) | EvoDiff    | Improvement |\n        |:---:|:-----------------:|:-----------------:|:----------:|:----------:|:-----------:|\n        | 5   | 21.286            | 18.611            | 13.969     | **7.912**  | 43.4%       |\n        | 6   | 10.966            | 8.519             | 6.556      | **4.909**  | 25.1%       |\n        | 8   | 5.127             | 4.148             | 3.963      | **3.756**  | 5.2%        |\n        | 10  | 3.881             | 3.607             | 3.563      | **3.332**  | 6.5%        |\n        | 12  | 3.516             | 3.429             | 3.357      | **3.084**  | 8.1%        |\n        | 15  | 3.341             | 3.284             | 3.182      | **2.918**  | 8.3%        |\n        | 20  | 3.251             | 3.167             | 3.075      | **2.853**  | 7.2%        |\n\n        *Time Comparison: DPM-Solver++  vs EvoDiff*\n        | NFE | DPM-Solver++  | EvoDiff     | Time Difference | Faster Method |\n        |:---:|:-----------------:|:-----------:|:---------------:|:-------------:|\n        | 5   | 3577.60s          | **3488.36s**| -89.24s (2.5%)  | EvoDiff       |\n        | 6   | 3800.61s          | **3719.40s**| -81.21s (2.1%)  | EvoDiff       |\n        | 8   | 4273.31s          | **4046.90s**| -226.41s (5.3%) | EvoDiff       |\n        | 10  | 4746.74s          | **4699.64s**| -47.10s (1.0%)  | EvoDiff       |\n        | 12  | 4703.76s          | **4678.07s**| -25.69s (0.5%)  | EvoDiff       |\n        | 15  | 5973.14s          | **5913.50s**| -59.64s (1.0%)  | EvoDiff       |\n        | 20  | 7238.42s          | **7154.19s**| -84.23s (1.2%)  | EvoDiff       |\n        | 25  | 8523.72s          | **8394.67s**| -129.05s (1.5%) | EvoDiff       |\n \n    * *Evaluation on Flow-based Models (e.g., Instaflow):*  Thank you for this excellent suggestion. We agree that comparison with flow-based models would be valuable. As an initial step, we present comparisons with recent learned acceleration methods, with comprehensive flow-based evaluation planned as future work:\n\n        *Comparison with Recent Learning-Based Methods*\n        | NFE | Uni_PC[GITS] (Learned)| Uni_PC[LD3] (Learned, [4]) | EvoDiff (2m, Learning-free) |\n        |:---:|:-----------------:|:---------------:|:---------------:|\n        | 6   | 11.19 | 5.92 | 9.07 |\n        | 8   | 5.67 | 3.42 | 3.88 |\n        | 10  | 3.70 | 2.87 | **2.74** |\n\n*Paper Formatting Concerns (e.g., parentheses instead of brackets):* \n\nThank you for pointing out this formatting issue. We use *\\bibliographystyle{ieeetr}*, ​ which should display citations in brackets. We will correct any instances where parentheses appear instead of brackets in the revised version.\n\n---\n\nWe believe these comprehensive responses and planned revisions address all your valuable concerns and will significantly enhance the quality and clarity of our manuscript. We are confident that these improvements, combined with the recognized novelty and strong performance of EvoDiff, will further strengthen the paper's contribution to the field.  \n\nThank you once again for your constructive feedback and for your time in reviewing our work."}, {"review_id": "llQ7X7Z6Yw", "reviewer": "Reviewer_RCMQ", "summary": "This paper introduces an information-theoretic framework for diffusion inference, showing that a successful sampling processs basically reduced the conditional entropy effectively during the denoising process. Based on this framework, the authors propose EvoDiff, a novel entropy-aware gradient-based method that directly targets information recovery by optimizing the denoising variance. Unlike prior ODE-based solvers, EvoDiff emphasizes principled variance control to reduce both transition and reconstruction errors. The method is validated through extensive experiments, showing substantial improvements in sample quality and efficiency.", "questions": "Please refer to the Strengths And Weaknesses section.", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n1. Overall, I think this is an interesting paper. The information-theory perspective to view the sampling process appears novel.\n2. The theoretical results are reasonable and the motivated method based on the theoretical insights has very convincing performance. \n\n## Weaknesses\nI do not have critical concerns about the paper's results. However, I think the presentation of the work should be improved. In particular,\n\n1. While the general ideas are provided at the end of the introduction. The connections between Sec 3 and Sec 4 should be better articulated when deriving the proposed sampling algorithm. Currently, the authors present lots of technical details in the main text but do not provide much intuition. For example, why linear interpolation can give improvements at L158?  In addition, I think section 4 has notations overloaded. Authors should consider simplify them. \n\n2. The statements in Prop 3.3 and Thm 3.4 hide too many details which make it difficult to understand why this is the case. In the revision, I think authors should consider containing more details in statement and the following dicussion why there are the cases.", "was_revised": true, "final_justification": "Thank you very much for the responses. All of my concerns have been addressed, and I expect the authors will incorporate the proposed changes in the revised version.\n\nAdditionally, I believe the work provides novel insights into diffusion inference from a distinctive information-theoretic perspective. Accordingly, I will maintain my current score.", "rebuttal": "*Dear Reviewer RCMQ:*\n\nWe sincerely thank you for recognizing our information-theoretic perspective as novel and our results as convincing. We are particularly grateful for the **Accept** rating and the valuable constructive feedback provided. We greatly appreciate your constructive suggestions and will address all presentation concerns:\n\n**Response to W1:**\n\n*1. Clarifying Sec 3–4 Connections*  \nWe will add a bridging paragraph before Section 4.1:  \n> *\"Section 3 establishes that minimizing conditional entropy requires reducing conditional variance (Eq.(10)). Section 4 operationalizes this: the gradient operator (Eq.(16)) optimizes variance control, while the interpolation scheme (Eq.(15)) leverages lower-variance regions to reduce uncertainty. The intuition before formal optimization is provided in Eq.(17).\"*\n\n*2. Explaining Linear Interpolation (line 158)*  \nWe will clarify after Eq. (15):  \n> *\"Linear interpolation $x_θ(x_{l_i}, l_i) := \\zeta_i x_θ(x_{t_i}, t_i) + (1-\\zeta_i)x_θ(x_{t_{i+1}}, t_{i+1})$ reduces conditional variance by leveraging model parameters from lower-noise regions, as proven in Eq. (52). This directly minimizes conditional entropy per our theoretical framework.\"*\n\n*3. Simplifying Section 4 Notation*\n \nWe acknowledge the notation complexity. We will: \n- Create a comprehensive notation reference table in the appendix with all symbols and their precise meanings\n- Reduce subscript complexity\n- Move technical derivations to appendix while keeping intuitive explanations in main text\n\n\n**Response to W2:**\n\n- *Prop 3.3*: We will expand the statement to include the specific mechanism: *\"DPM-Solver and EDM's Heun iterations achieve conditional entropy reduction by incorporating gradient corrections $\\frac{h^2_{t_i}}{2}B_θ(t_i)$ that reduce $\\text{Var}(x_{t_i}|x_{t_{i+1}})$ compared to first-order methods (Eq.6), as shown by our variance analysis in Eq.(32)-(33).\"*\n\n- *Thm 3.4*: We will add quantitative details: *\"Under independence assumption, data prediction parameterization achieves lower conditional entropy than noise prediction by a factor of $\\frac{\\sigma^2_t}{\\alpha^2_t}\\text{Var}(\\epsilon_θ - \\epsilon_t)$ (Eq.(49)), because it directly estimates $x_0$ avoiding the error-prone transformation chain $\\epsilon_t \\mapsto x_t \\mapsto x_0$.\"*\n\nThese revisions will significantly improve clarity while preserving all technical contributions. Thank you for helping us enhance our paper's accessibility.\n\nWe once again thank the reviewer for their insightful comments, which will undoubtedly help us produce a higher quality submission."}, {"review_id": "T7hLqRLzRv", "reviewer": "Reviewer_yvrP", "summary": "The authors propose to accelerate diffusion inference by simulating reverse diffusion with updates that minimize conditional entropy between two successive reverse diffusion states. The proposed algorithm shows competitive performance on a variety of benchmark datasets, such as CIFAR-10, FFHQ-64, and ImageNet-256.", "questions": "I am willing to raise the score to *borderline accept* if the authors provide explanations to address concerns in **Clarity** and **Quality**.", "limitations": "The authors discuss limitations in the conclusion.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths and weaknesses are denoted [S#] and [W#], respectively.\n\n**Quality**\n\n- [W1] Several notations / equations are undefined or imprecise, so I had to put in some effort to fill in the gaps. For instance,\n    - Line 76 : what do the authors mean by \"when $d_\\theta$ represents the noise prediction model\"? Does it mean $d_\\theta = \\epsilon_\\theta$?\n    - Line 96 : \"we denote $p(x_{t_i}|x_{t_{i+1}},x_0)$ as $p(x_{t_i}|x_{t_{i+1}})$.\" Those two distributions **are not equal in general**. The authors should not use those two interchangeably.\n    - Line 158 : What is the relation between $l\\_i$, $\\zeta_i$, $t_i$, and $t_{i+1}$? Is it $l_i = \\zeta_i t_i + (1-\\zeta_i) t_{i+1}$?\n\n**Clarity**\n\n- [W1] There is a misalignment between the main idea and the implemented approach. The main idea of this paper is to minimize $H_p(x_{t_i}|x_{t_{i+1}})$, where $x_{t_i}$ and $x_{t_{i+1}}$ are two reverse diffusion iterates, to accelerate generation. However, **the authors consider probability-flow ODE iterations, not reverse-SDE iterations**, such that $x_{t_i}$ is a deterministic function of $x_{t_{i+1}}$. So, $H_p(x_{t_i}|x_{t_{i+1}})$ is always zero. I do not understand how the authors were able to derive non-trivial theoretical results such as Proposition 3.2 under the ODE setting.\n\n- [W2] I also find the result of Theorem 3.4 very weird. It is well known what diffusion inference with data prediction parametrization and noise prediction parametrization are equivalent. Specifically, $s_\\theta(x,\\sigma) = - \\epsilon_\\theta(x,\\sigma)/\\sigma = -(x - D_\\theta(x,\\sigma)) / \\sigma^2$, where $s_\\theta$ is the score function, $\\epsilon_\\theta$ is the noise prediction model, and $D_\\theta$ is the data prediction model [1,2], so there is no difference between diffusion inference with data prediction and noise prediction parametrizations.\n\n- [W3] Several statements are written colloquially or ambiguously, making the logic unclear. For instance,\n    - Proposition 3.2, \"the gradient-based inference in Eq. (7) tends to reduce conditional entropy more efficiently than the first-order inference when...\". What do the authors mean by the phrase \"tends to\"? Does it mean that the statement holds in an asymptotic sense? Or in a probabilistic sense?\n    - Theorem 3.4, \"... diffusion inference using data prediction parametrization is more effective at reducing conditional entropy ...\". Again, what do the authors mean by the phrase \"is more effective\"?\n    - Remark 4.1, \"the RE-based multi-step iteration in Eq. (15) reduce the conditional entropy of iteration in Eq. (13) by leveraging model parameters from low-variance regions\". What do the authors mean by \"leveraging model parameters from low-variance regions\"? I assumed that diffusion model parameter was fixed to be $\\theta$. Also, variance of which random variable?\n\n**Significance**\n- [S1] The proposed method shows strong performance, when compared to state-of-the-art fast diffusion samplers.\n\n**Originality**\n- [S1] Accelerating diffusion inference through the lens of conditional entropy reduction is original and interesting.\n\n[1] Generative Modeling by Estimating Gradients of the Data Distribution, NeurIPS, 2019\n\n[2] Score-Based Generative Modeling through Stochastic Differential Equations, ICLR, 2021", "was_revised": true, "final_justification": "While the rebuttal has addressed my concerns, I still feel the paper should undergo major revisions to improve the presentation, e.g., abused notations, undefined notations, clarify the fact that the proposed framework reduces model entropy, not transition entropy, etc. Hence, I have raised the score to **borderline accept**.", "rebuttal": "*Dear Reviewer yvrP:*\n\nWe sincerely thank you for your thorough and insightful review. We are greatly encouraged that you found our core idea of accelerating diffusion via conditional entropy reduction perspective to be **original and interesting** and our method's performance **strong** compared to SOTA samplers. We are equally grateful for your critical feedback on the paper's clarity. Your identification of the tension between conditional entropy and deterministic ODEs, in particular, was an invaluable observation that prompted us to substantially clarify our theoretical framework, which we believe makes the novelty you pointed out even more apparent. \n\n**Quality W1:**\n\nWe address each of your concerns below and will incorporate these clarifications into our revision.\n\n*1. On the notation for the prediction model (Line 76), Does it mean $d_θ = \\epsilon_θ$?:*\n- Yes, $d_θ = \\epsilon_θ$ when representing the noise prediction model. We use $d_θ$ as a general notation in Section 2 to unify both data prediction ($x_θ$) and noise prediction ($\\epsilon_θ$) parameterizations. We'll clarify this in the revision.  \n\n*2. On the notation for conditional probability (Line 96):*\n\n- Thank you for pointing out that $p(x_{t_i}∣x_{t_{i+1}},x_0)$ and $p(x_{t_i}∣x_{t_{i+1}})$ are mathematically distinct distributions and should not be used interchangeably. But in our manuscript, our intention of *\" denote\"* was to use $p(x_{t_i}∣x_{t_{i+1}})$ as **a notational shorthand** for $p(x_{t_i}∣x_{t_{i+1}},x_0)$ within contexts where the conditioning on $x_0$ is implicitly assumed throughout the derivation, following conventions in DDPM and DDIM.\n\n*3. On the relationship between l_i, zeta_i, and the timesteps (Line 158):*\n\n- $l_i$ is **not** an interpolated time step. The relationship you proposed, $l_i=\\zeta_it_i+(1−\\zeta_i)t_{i+1}$ does not hold. Instead, $l_i$ is merely a symbolic placeholder.  \nAs defined in line 158 (page 5), $x_\\theta(x_{l_i},l_i) := \\zeta_i x_\\theta(x_{t_i},t_i)+ (1-\\zeta_i)x_{\\theta}(x_{t_{i+1}},t_{i+1})$ is a linear combination of model predictions at two distinct time steps.  We use this notation as a shorthand for the \"interpolated synthetic model prediction\" to distinguish it from actual model evaluations at $t_i$ and $t_{i+1}$. Thus, $l_i$ has no physical meaning as a time point.\n \n\n**Clarity W1:**\n\nOur implemented approach follows our theoretical analysis, and our use of the term \"conditional entropy\" should be understood from a broader perspective than ODE and SDE theory. Our framework does not analyze the stochasticity of the ODE path itself, but rather the **uncertainty inherent in the model's predictions**. We elaborate on this below:\n\n*1. Theoretical Foundation: The SDE Framework and Model Prediction Uncertainty*\n   - Our information-theoretic framework is inspired by and rooted in the more general perspective of reverse Stochastic Differential Equations (SDEs). In the SDE setting, the reverse transition $p(x_t​∣x_{t+1}​)$ is a stochastic process with non-negligible variance, making its conditional entropy $H(x_t​∣x_{t+1}​)$ a meaningful, non-zero quantity to optimize.  \nThe critical step in our paper, detailed in Section 3, is linking this **conditional entropy** to **conditional variance**. As shown in Equations (9) and (10), under the common Gaussian transition assumption, $H_p​(x_{t_i}​​∣x_{t_{i+1}}​​)$ is directly proportional to the logarithm of the conditional variance $\\text{Var}(x_{t_i}​​∣x_{t_{i+1}}​​)$. Therefore, *minimizing conditional entropy is equivalent to minimizing conditional variance*. This variance is non-zero even in an ODE setting because it measures the *uncertainty of model's prediction*, not path stochasticity.\n\n*2. In the ODE Setting, Conditional Entropy Analyzes Model Prediction Variance*\n - While the ODE sampling path is deterministic, the vector field that drives this path is provided at each step by our neural network, $d_θ​$ (e.g., $ϵ_θ​$ or $x_θ$​). This prediction is inherently uncertain. For instance, the model's noise prediction $ϵ_θ​(x_{t_i}​​,t_i​)$ is not perfect, and its variance, $\\text{Var}(ϵ_θ​(x_{t_i}​​,t_i​))$, precisely quantifies the model's uncertainty at that state.\nOur core thesis is that a superior ODE solver iteration should more effectively utilize the model's predictions to construct an update step with **smaller variance**, thereby achieving more efficient information transfer. This \"smaller variance\" is what we equate to a more \"effective reduction in conditional entropy.\"\n\n*3. Proposition 3.2 is Derived from This Principle*\n  - The proof does not analyze the stochasticity of the ODE path. Instead, it compares the *variances of the perturbation terms* resulting from two different iteration schemes: the first-order method  Eq. (6) and our gradient-based method  Eq. (7).\n  - As shown in Eq. (32) of the appendix, we explicitly calculate the conditional variances of the update terms,  $\\text{Var}_{p1}$​ and  $\\text{Var}_{p2}$​. These variances are determined entirely by the *variance of the model's noise prediction, $\\text{Var}(ϵ_θ​)$​*.\n  - The proposition concludes that, under certain conditions, our proposed gradient-based method Eq. (7) constructs an update step with a lower variance than the first-order method  Eq. (6). Based on the direct relationship between entropy and variance  Eq. (10) , we interpret this as a more efficient reduction of conditional entropy.\n \n---\n\n*Theoretical Framework Summary:*\n- *SDE Motivation:* The principle of conditional entropy reduction is derived from stochastic reverse-SDEs, where minimizing $H_\\theta(x_t \\| x_{t+1})$ corresponds to reducing path variance (Eq.  (1)).\n- *Variance-Entropy Proxy:* We establish that $H_\\theta(x_t \\| x_{t+1}) \\propto \\log \\text{Var}(x_t \\| x_{t+1})$ under Gaussian transitions (Eq. 9-10). This allows using model prediction variance $\\text{Var}(\\epsilon_\\theta(x_t, t))$ as a practical surrogate for conditional entropy in both SDE and ODE settings.\n- *ODE Implementation:* In deterministic ODEs, entropy reduction manifests through the minimization of the model's prediction variance $\\text{Var}(\\epsilon_\\theta(x_t, t))$. Although the ODE path is deterministic, $\\text{Var}(\\epsilon_\\theta) > 0$ due to imperfect predictions (Eq. (2)).\n- *Theoretical Justification:*  Proposition 3.2 establishes sufficient conditions for gradient-based methods to maximize entropy reduction by achieving greater decrease in $\\text{Var}(\\epsilon_\\theta)$ than first-order methods.  \n\nThis clarification reveals that our method is the first to provide a unified information-theoretic framework for both stochastic and deterministic diffusion sampling, explaining why certain numerical schemes outperform others, a longstanding open question in the field.\n\n**Clarity W2:**\n\nIt is well known that the equivalence of diffusion inference with data prediction parameterization and noise prediction parameterization only holds under ideal theoretical conditions. While different parameterizations can indeed be converted to each other through the relations you mentioned, they lead to different diffusion ODE formulations, which in turn result in different numerical discretization schemes.  Specifically:\n\n1. *Different ODE formulations*: As shown in lines 74-76 on page 2, different parameterizations produce different ODE structures:\n   - Noise parameterization->Noise prediction ODE in Eq. (3)\n   - Data parameterization->Data prediction ODE in Eq. (4)\n\n2. *Different discretization schemes*: When we discretize these ODEs, we obtain different numerical iteration formulas. Even though the continuous-time limits are equivalent, the discrete-time approximations have different truncation errors and numerical properties.\n\n3. *Variance analysis*: Theorem 3.4 demonstrates that the discretization scheme derived from data parameterization more effectively reduces the denoising variance compared to the scheme from noise parameterization. This is a property of the numerical schemes, not the continuous-time equivalence.\n\nThe key point is that mathematical equivalence in continuous time does not imply equal performance under discretization. Different discretization schemes can have vastly different numerical stability, accuracy, and variance properties. For instance, in diffusion models, the variance exploding (VE) and variance preserving (VP) formulations are mathematically equivalent through appropriate change of variables, yet their numerical behaviors and optimal discretization strategies differ significantly in practice, as exemplified by Score-based DMs [2] versus DDPM.\n\n**Clarity W3:**\n\nThank you for identifying these ambiguous statements.  We clarify each below and will revise for mathematical precision:\n- *On \"tends to\" in Proposition 3.2:* This is neither asymptotic nor probabilistic, it's a deterministic inequality under our analysis framework. We will change \"tends to reduce\" to \"reduces\" for clarity.\n- *On \"more effective\" in Theorem 3.4:* We prove that the discretized iteration governed by data prediction ODE (Eq. (4)) achieves strictly lower conditional entropy than the discretized iteration governed by noise prediction ODE (Eq. (3)). We will replace \"more effective\" with \"achieves lower conditional entropy\" and state the precise inequality.\n- *On \"model parameters from low-variance regions\":* We meant \"model predictions at timesteps with lower noise levels.\" For example, when $s<t$ in the reverse process,  $\\text{Var}(x_θ​(x_s​,s)∣x_0​)<\\text{Var}(x_θ​(x_t​,t)∣x_0​)$.\n\n---\n\nWe appreciate your constructive feedback and willingness to reconsider. By clarifying that we analyze model prediction uncertainty rather than path stochasticity, our theoretical framework is now more coherent and our contribution more apparent. Our principled approach is empirically powerful, achieving a 45.5% FID reduction on CIFAR-10 at 10 NFE without extra training. We are confident these clarifications and strong results satisfy the acceptance criteria."}]

Author rebuttal:

Meta-review: This paper proposes EvoDiff, an entropy-aware variance optimization method for diffusion inference. By framing denoising as conditional entropy reduction, the authors provide an information-theoretic justification for why data prediction parameterization outperforms noise prediction and how optimizing conditional variance reduces transition and reconstruction errors. Building on these insights, EvoDiff systematically reduces uncertainty during denoising, thereby accelerating convergence and improving sample quality. Extensive experiments on CIFAR-10, ImageNet, FFHQ, and text-to-image tasks demonstrate consistent improvements over state-of-the-art training-free solvers, including up to 45.5% lower reconstruction error and 25% fewer function evaluations for high-quality generation.

The reviewers find this work interesting, novel, and well-executed. The information-theoretic perspective on diffusion inference is original and insightful, supported by solid theoretical analysis with reasonable assumptions and convergence guarantees. The proposed method is well motivated by the theory and demonstrates strong empirical gains across diverse benchmarks. Overall, the work is regarded as both theoretically sound and practically impactful.
Decision: Accept (poster)