Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
Authors: Gen Li
Yuchen Jiao
Venue: ICML 2025 poster (2025)
Keywords: score-based generative model
diffusion model
denoising diffusion probabilistic model
guidance
Primary area: theory->learning_theory

Abstract: Diffusion models have emerged as a powerful framework for generative modeling, with guidance techniques playing a crucial role in enhancing sample quality. Despite their empirical success, a comprehensive theoretical understanding of the guidance effect remains limited. Existing studies only focus on case studies, where the distribution conditioned on each class is either isotropic Gaussian or supported on a one-dimensional interval with some extra conditions. How to analyze the guidance effect beyond these case studies remains an open question. Towards closing this gap, we make an attempt to analyze diffusion guidance under general data distributions. Rather than demonstrating uniform sample quality improvement, which does not hold in some distributions, we prove that guidance can improve the whole sample quality, in the sense that the ratio of bad samples (measured by the classifier probability) decreases in the presence of guidance. This aligns with the motivation of introducing guidance.

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
Gen Li
Yuchen Jiao
Abstract
generates samples. The reverse process typically involves approximating the score function-defined as the gradi- ent of the log-likelihood of noisy distributions-al various scales by training neural network (Hyvärinen, 2005; Ho et al., 2020; Hyvärinen, 2007; Vincent, 2011: Song & Er- mon, 2019; Pang et al., 2020), followed by solving reverse stochastic differential equation (SDE) associated with the forward process. Recent studies have rigorously established the convergence of diffusion models, demonstrating that the generated sample distribution approximates the target dis- tribution (Lee et al., 2022: 2023; Chen et al., 2022; Benton et al., 2023; Chen al., 2023; Li et al.. 2024b; Gupta et al., 2024; Chen et al., 2024; Li al., 2024a; Li & Yan, 2024; Li & Jiao, 2024; Li & Cai, 2024; Huang et al., 2024; Cai & Li, 2025; Li et al., 2025a).
Diffusion models have emerged as a powerful framework for generative modeling, with guid- ance techniques playing a crucial role in enhanc- ing sample quality. Despite their empirical suc- cess, comprehensive theoretical understanding of the guidance effect remains limited. Exist- ing studies only focus on case studies, where the distribution conditioned on each class ei- ther isotropic Gaussian or supported on one- dimensional interval with some extra conditions. How to analyze the guidance effect beyond these case studies remains an open question. Towards closing this gap, we make an attempt to analyze diffusion guidance under general data distribu- tions. Rather than demonstrating uniform sam- ple quality improvement, which does not hold in some distributions, we prove that guidance can improve the whole sample quality, in the sense that the ratio of bad samples (measured by the classifier probability) decreases in the presence of guidance. This aligns with the motivation of introducing guidance.
As diffusion models become a dominant paradigm for gener- ative modeling in domains such as image, video, and audio, the need for principled methods to modulate their output has grown significantly. For instance, when the data comprises multiple classes, one may seek to generate samples specific to a desired class. In practice, the standard approach is to use diffusion guidance (Dhariwal & Nichol, 2021; Ho & Salimans, 2021), a technique that enhances sample quality by incorporating an auxiliary conditional score function. This method combines the model's score estimate with the gradient of the log-probability of samples conditioned on the desired class through a weighted sum, enabling the gen- eration of outputs with high perceptual quality when an appropriate guidance weight is applied. Reference (Karras et al., 2024) proposed to use a bad version of the model for guiding diffusion models.
1. Introduction
Score-based diffusion models have recently emerged as an expressive and flexible class of generative models, demon- strating competitive performance on image and audio syn- thesis tasks (Sohl-Dickstein
al., 2015; Song Ermon, 2019; Ho al., 2020; Song al., 2021b;a: Croitoru al., 2023; Ramesh al., 2022; Rombach et al., 2022; Saharia et al., 2022). These models operate through forward pro- cess, which progressively transforms data from the target distribution into Gaussian noise, and a reverse process that
1.1. Motivation
Despite the empirical success and widespread adoption of guidance methods, their theoretical foundations remain un- explored. key question persists: why does guidance im- prove the quality of samples generated by diffusion mod- els? Existing literature offers partial insights through case studies, analyzing guidance dynamics in limited scenar- ios such as mixtures of compactly supported distributions, isotropic Gaussian distributions, or linear guidance term (Chidambaram et al., 2024; Wu et al., 2024; Bradley &
Equal contribution DDepartment of Statistics, The Chinese University of Hong Kong, Hong Kong; Email: {genli, yuchenjiaac aeaneeaaaaaaeeeoooeeoeeeoeee cuukk. odu.hk. Correspondence to: Gen Li <genli @ www....u.kkk...
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada, PMLR 267, 2025. Copyright 2025 by the author(s).

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
Nakkiran, 2024; Li et al., 2025b). However, the effect of guidance across general data distributions remains unknown, and we discover that the uniform improvement does not hold even for Gaussian mixture distributions (see Figure I), which highlights& a significant gap in our understanding.
2.1. Diffusion Models
Diffusion models are based on forward process that pro- gressively transforms data from a target distribution into sequence of increasingly noisy representations. Starting
from Xo E I drawn from the target distribution Pdata. the forward process evolves as follows:
1.2. Our Contributions
Xg ~ Pdata:
(Ia)
Motivated by the above discoveries, this paper investigates the improvement on the average of the reciprocal of clas- sifier probabilities under general data distributions. We demonstrate that guidance preferentially enhances the gen- eration of samples associated with higher classifier probabil- ities, which aligns with the primary motivation for adding guidance. Specifically, we prove that the expectation of the reciprocal of classifier probabilities decreases with guid- ance. This metric bears resemblance to the commonly used Inception Score (IS), a standard measure of sample quality (Salimans et al., 2016), which also considers the expecta- tion of the (logarithmic)) function of classifier probabilities. Furthermore, we extend our analysis to practical implemen- tations, with discretee-rrron and score estimation errors. We prove that the discrete-time processes approximate their continuous-time counterparts, ensuring the applicability of our theoretical results in practical settings.
X, = V Pn- 2n n n= I, N, (lb)
where 0 < 3n < 1 is the step-size, (Za) hignSN L.d. (0,, Ia) is a sequence of independent Gaussian noise vec- tors. This process gradually converts the original distribu- tion into standard Gaussian noise as n increases.
An essential component of score-based diffusion models is the score function, defined as the gradient of the log- probability of the intermediate distributions in the forward process:
(x) == llogxx,))), 1 ns
Assuming access to good approximations of the score func- tions, denoted s, (z) R (r),, one can utilize them to re- verse the forward process and generate samples resembling the target distribution. The reverse process is governed by:
YN ~ NN0 (a,T,)
(2a)
Comparison with prior works when restricted to specific distributions: Existing works focus mainly on specific classes of distributions like GMMs, while our work pro- vides a more general theoretical analysis. Here we compare our findings with prior works when restricted to specific distributions. In Wu al. (2024), the authors demonstrate that Pa (1 2 Pe]x, (1 holds under specific con- ditions, while we show that this inequality does not always hold. In addition, Chidambaram et al. (2024) argues that guidance can degrade the performance of diffusion models, as it may introduce mean overshoot and variance shrinkage. In contrast, our result shows that guidance can improve sam- ple quality by generating more samples of high quality. Fur- thermore, Bradley & Nakkiran (2024) shows that classifier guidance can not generate samples from p(zlc)"ppax) "(x))> for GMMs and establishes its connection to an alternative approach, i.e., the single-step predictor-corrector method, whose effectiveness in this specific setting remains unclear. In contrast, we directly analyze and demonstrate the effec- tiveness of CFG.
1
Y VT- Bn
(Yn + Bn))((Y)) + VBnzn.
(2b)
for n = ..... .......... 2, where Z, Lid. N( Ia) denotes another sequence of independent Gaussian noise vectors. This re- verse process has been shown to gradually remove noise and guide the system back toward the target distribution, in the sense that the generated Yn has distribution close to that of X,, in (1).
2.2. Guidance
Conditional diffusion models are designed to sample from the conditional distributions p(-lc), where e represents specific class label. This can be achieved by generalizing the unconditional diffusion model defined in (2), replacing sne (Yn) with sn (Ynlc), as shown below:
YN ~ NO Ta),
(3a)
Y VI 3n (Y,, + (yx)(( + VBnZn, (3b)
2. Background
for n = N.... 2, where 8n (x/c) are good estimates of the gradient of the log-density function Px.. given the con- dition c. That is, sn (2|c) ~ s (alc) = V log c).
In this section, we review basics about diffusion models, guidance, and their continuous limit. Throughout this paper, we shall use n = 1,... NN and 0 < t t 1 to denote the discrete and continuous time steps, respectively.
The noise terms Zn i.d. independent Gaussian noise vectors.
~ NN Ia) represent a sequence of
2

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
To further enhance the quality of conditional sampling, re- searchers introduced guidance techniques. These methods aim to increase the posterior probability Pe (c| by modifying the reverse process as follows:
The above result can be found in Song al. (2021b). When extending this framework to conditional sampling with guid- ance in (5), the reverse SDE becomes
dy (- + + w) Vog Ic)
1
- 2-1
VI B (X + ( lc)
-V log (x,1
_ dt
1 - dd
(9)
w logPel x. (c ))) 23 (4)
where the guidance scale w controls the strength of the modification. Furthermore, reverse process (4) can be ap- proximated as
The continuous-time framework provides powerful per- spective for understanding and analyzing score- based diffu- sion models.
Y n-
v1 Bn (Y + Ba((1 + w)sn(V," (c)
3. Main Results
ws (Y + v B,
(5)
In this section, we shall present main theorem and its proof. For the reverse process with guidance (9), we prove that after introducing a non-zero guidance into the diffusion process, the expectation of a specific decreasing function of the classifier probability will decrease as increases. This is formally stated in the following theorem.
This approximation is derived from the observation that v llggPe Pe| x, (c) z)) = s (z
(2|cc c) - s (z), which is referred to as classifier free guidance (Ho & Salimans, 2021).
2.3. Continuous Time Limit
Theorem 3.1. Let
The discrete-time diffusion process described in Section 2.1 exhibits natural correspondence its continuous-time counterpart. Specifically, the forward process corresponds to the following stochastic differential equation (SDE):
(.(1) == Pel Delxi_.(( (e_(x__))l))) (cl
(10)
which is a decreasing map of Pel (clV). It can be shown that for any < <t1,,
dx, 2(1
X,dt +
VI-
dBe
(6a)
de (e - de (ita)) Y = - Pol (c
with No ~ Pdata: for 0 1
V VlogPx...le (V;"" (c) Vlogxx, (Y:") |/a (11)
where B, denotes the standard Brownian motion, and can be arbitrarily small. It transforms the data distribu- tion into standard Gaussian distribution as t - 1. Simi- larly, the reverse process in (3) corresponds to the following continuous- cartinuous-timee SDE:
where Y: is defined in (9).
The above result reveals that the average reciprocal of clas- sifier probability Pe (cl decreases when we add non-zero guidance. When compared with the case without guidance, that is w O, the total expected improvement over the diffusion process is given by:
dy, (2++ V llgg Pog df 1 dB. for 1151.
Vt
f
W 7 Pel (cl
This reverse SDE effectively transforms the noise dis- tribution back toward the target distribution condi- tioned on c, guided by the conditional score function V lo (Y, c). If the initialization Ys- ~ Pxi-s c it is well-known that Y, has the same distribution with the re- verse process of Xt, which is stated in the following lemma: Lemma 2.1. can be shown that for S Srsssl- 5,
V Vlogxx. 1oog V loggx._.(Yx") (f (12)
This result reflects an improvement in sample quality, as samples with higher classifier probabilities are favored.
The choice of Pa Pa|x1-,cl y) in our analysis is primar- ily for technical considerations. rewards more on the decrease of bad samples with small P. x (c (v). which means it places greater emphasis on reducing the probabil- ity of generating low-quality or misclassified samples. This aligns with the initial motivation of introducing guidance. In practice, Inception Score (IS) is commonly employed (8) to measure sample quality, which related the average
)
X, N
and Ys ~ Pxi-ale then
(Y:) d (X1-t). g
3

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
logarithm of the classifier probability EllogPe p. (c) v)]- This is conceptually aligned with the metric in our analysis, with the difference being that IS adopts Pe x,_ (c y) as
Proof of Theorem 3.1. The relation (13) in the above lemma gives
the weight, while we use Pe| X1 (c] y) but both aim to increase the ratio of high-quality samples (measured by the classifier probability). In addition, to address potential con- cerns, we note that although some practical limitations of IS have been identified (Barratt & Sharma, 2018), it remains a commonly used metric for evaluating sample quality in the study of diffusion guidance (Dhariwal & Nichol, 2021; Ho & Salimans, 2021). Moreover, in our theoretical analysis, we use the true conditional probability, which addresses the estimation issues discussed in Barratt& & Sharma (2018).
0= E Palx1 + (c
- (c)))) IY( = =vl=
Op (c)3))
1 Tr Tr (2'pe X xi_ (c|2t)) (1))))))) -1)
y=yi + 2t
0t
(G-m (( + v Vlogpx. e) 71
+ (c]
+ +
(14)
where the second relation established in Section 4.3. Here, we let 8 > 0 be some small quantity, which depends only yr, and the property of Xo. Similarly, we have
Theorem 3.1 states that guidance improves the averaged re- ciprocal of the classifier probability rather than the classifier probability of each individual sample. This suggests that while guidance improves overall sample quality, it may lead to decline in quality for small subset of samples. This insight encourages the development of adaptive guidance methods that address this issue and achieve more uniform performance gains, which is a potential practical application of our theory.
1 δ
[pcc x(45)=1
- -Pol (clx)) x(t) Yi" = =l]
8 (cl))
1 Tr Tr(V' (p'pel X (clv)))
1== +
0t
2t
((2
+ VP X1 (cl ve) 1-1 (2 + (1 + w) V |c lc)
Our main result is established through the following key observation, whose proof can be found in Section 4.1.
- w loggoxx (w1) 1) + 0(5).
(15)
Lemma 3.2. For any E > 0 and S T S 1 E, we have
Comparing the above two relations leads to
= =E (oolox//Pooopeo [ol| x. (c|2z)) =Ea Xx, x)1 (13a)
E E[0L+s (Y [.m - (Y,²)
Pe|
= 8 w
V loggPx
V (X7")...((7"))
I
or equivalently, for any s I15L
Vpclx1 (c 1 0
( Pel X1 (c (1-)
V Voggxx--llll
Pe| Pc|xi_.(cl2) = EyuNY Y x (clve)) 11))_ = y)
(13b)
-Voog (Y,")
+ 0 (16)
where, Xt and Y, are defined in (6).
where the second relation holds since
With Lemma 3.2 in hand, we are ready to prove our main the- orem. Before diving into the proof details, we would like to first explain the main analysis idea: First, this result comes from the key observation that the function of reverse process, Pa|x, (c) -1 forms a martingale, as stated in Lemma 3.2,
Veel XI (cl - -=
(v (V (cylc) loggx... (v)) (17)
Then we can conclude the proof here.
which is established through a careful decomposition of
Pelx, and Px, Next, the guidance term sy (x(cc) - s<(x) in classifier-free guidance (CFG) aligns with the direction of (=)], which makes us expect that adding the guidance at time can decrease Ex,~x, [pqxx (c|x,)-'|X,
3.1. Numerical Validation
-Vpexx (c/z) = Pe|x, (clz) [[s((x(c)
In this section, we present experimental results on the Gaussian Mixture Model (GMM) and ImageNet dataset to demonstrate that guidance does not uniformly enhance the quality of all samples. Instead, it improves overall sample quality by reducing the average reciprocal of the classifier probability. This observation empirically validate our theo- retical findings.
xr| for all S t. Finally, to achieve the desired result, particular care must be taken in handling first- and second-order differential terms with respect to t for the process (c|,")) due to its randomness nature, which is completed in the following based on the technique of Ito's 's formula.
4

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
-1
(1]xx.(1)x)) PPxx (1(x)))
-1.2
Palx (1
-1.4
-1.6
0.9
-1.8
-2
-2.2
0
4
6
8
10
0
2
4
6
8
10
Figure left: Ratio of samples with improved classifier probabilities for different guidance scales w; right: Expectation -P- (1 (Ya")= for varying w.
0.9
P(pex, (e(lG") 2 Pecx, (clxP))
0.85
Blpax, (c)x?))
-50
0.8
-100
0.75
0.7
-150
6
0
8 10
10
Figure 2. Experimental results ImageNet dataset. left: Ratio of samples with improved classifier probabilities for different guidance scales wt right: Expectation of -p:1 [(") for varying w.
Gaussian Mixture Model: Let consider distribution with two classes e 0, 1. each with equal prior probability (0) = (1) = 0.5, in a one-dimensional data space (d = 1). The data distribution defined as follows:
Here, we focus the conditional class 1. The score functions V log a, |e(2 1). V VlogPx,-T, (z), and the | are provided Ap- pendix (cf. (44), (45), and (46)). To empirically validate our theoretical findings, we simulate the DDPM framework under different guidance scales w. Specifically, we fix N 4000, vary from 0.01 to 10, and perform 10a trials for each w. We compute Y by implementing the reverse pro- cess in (19), and its counterpart Y without guidance. For each trial, we evaluate classifier probability Pel xo Y(") and Pe (1 Y and compute the empirical probability of P(pe) x. (I)))) 2 Pe (1 YP)). In addition, we also calculate the average of -Pc| _ (1 (1 for various w. The results shown in Figure
classifier probability Pe|X,- (1
Xo cc = 1)
1 1 } + N(-1,1).
Xolc=1
According to the DDPM framework with guidance (5), the reverse process adopts the following update rule. Starting from Y ~ NO0, 1), the process evolves for N,
YW
Van (Y,"" + (1 (a-aa)]--- wV log (Y
+ (1 + w) Vlogxx wa 1o(Y [c)]) /1 an Zn:
(19)
ImageNet dataset: We conduct a numerical experi- ment on the ImageNet dataset. Specifically, we gener- ate samples using pre- trained diffusion model (Rom- bach al., 2021) with varying values of the guidance
where Zn Lid. (0, 1) is a sequence of independent Gaus- sian random variables.
5

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
level w, and evaluate the classifier probabilities using the Inception v3 classifier (Szegedy et al., 2016). We com- pute two statistics: P(pe xx (1 Y 2 Pe (1 Y))) and
with the averaged & score estimation error
N
-Elpe1 x, (1 [x"') averaged over 20000 random trials 20 trials for each of the 1000 ImageNet categories. The experimental results are presented in Figure 2.
80 (Y |c)
n=
log PX1
1 (
|c)112] S Scorei (22a)
observed that the empirical probability P(pe x. Y!") 2 X. (Y))) is less than 1 for any w 10, which indicates the guidance does not achieve uniform improvement in classifier probabilities. However, the average of -Pc| x (1 "1"))= increases with w. which explains why guidance effectively enhances sample quality, as predicted by Theorem 3.1. Moreover, we remark that the performance of diffusion models is commonly evaluated by two metries in practice: diversity and sample quality. This study primarily focuses the sample quality measured in a similar way the Inception Score, which increases with w. However, prior work Ho & Salimans (2021) has demonstrated that large values of w can significantly reduce sample diversity, leading to unsatisfactory performance in real-world applications.
N
(
N
n=1
V -an (yan
[|]] S 6 score
(22b)
We further assume that the sample Y, Y, the score func- tion V Floopxx. px,_, (Y, (Y, and the conditional score func- tion V log Px,,, c (Y, cecet |e) have bounded second-order moment, which is stated in the following lemma.
Assumption 3.4. There exists some quantity R, such that the sum of the second order moment of the following three random vectors are bounded by R², that is,
E y mlyy.com .ee.eooot 1/ + V logpx... (Y, (Y,e,cont)) 112
3.2. Discretization and Robustness Analysis
+ V log (ot 10]1[]] S R²2
(23)
Consider that practical algorithms operate in discrete time and are subject to score estimation errors, we provide a supplementary analysis of the discretization error and es- timation error for completeness. Specifically,
In addition, we consider the case with smooth score func- tions in this paper, which is stated below.
aim to show the discrete-time process in (5) closely approximates the continuous-timg time process in (9), thereby validating the observation from Theorem 3. in practical settings. Since our primary focus is on the efficiency of diffusion guidance rather than establishing a convergence theory, the bounds and conditions derived here may not be tight.
Assumption 3.5. Assume that V loog (x) are Lipschitz for all < 1 < such that
V logpx,
- logpx, S L||21 - x2ll22 (24)
With the above assumptions, We could establish that the discrete-time process converges the continuous-time pro- measured by the KL divergence. The proof is post- poned to Appendix A.
In the following, we shall use y. denote the con- tinuous process of (9) in order to distinguish with (5), and let
Theorem 3.6. Suppose that Assumptions 3.3, 3.4, and 3.5 (20)
an II akk 1=1
with ag := 1- Bk
hold true. Then the sampling process (5) with the learning rate schedule (21) satisfies
satisfying
KL(aaa,.cont ,Tm S SC (1 c((( w²) L N
1
aN = Noo
(21a)
N
+ (1
L2 R2 log* N
0g = on + cian - σx) log N
+ (1 + w² w²)= log N) (25)
(21b)
N
N
where Go and Cj are constants.
for some constant C > 0 large enough, where Y w,con and Y1" are defined in (9) and (5), respectively.
Before presenting the analysis result, we make the following assumptions. The first assumption states that faithful esti- mates of the score functions (-) and (-cc) are available for all intermediate steps as follows:
This theorem proves that, after sufficiently large number of iterations N. the sample distribution of the discrete-ting process Y." converges to that the continuous-time process wwrcon The latter corresponds to data contaminated by ai
Assumption 3.3. We assume access to estimates
8n (Y= (cont and 8, (Yar, Tr
(Yw..ont (c) for each s (Y (y,,.cont) and
noise with variance 1 Q1 According to Theorem 3.6, the
6

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
sampling process (5) with the learning rate schedule (21)
4.2. Preliminary Analysis of Xi
satisfies
We begin by establishing some key properties of Pel X, to support the proofs of our main results. Let R < 2o be some quantity such that
E|p(c
& x1 cont) + B[(p(c|V)")-- - (p(c)))))) r)),
where 7 is defined as the largest value satisfying
X < R) > and [11xool2<<110 1
1
(27)
TV(Y -11) < -(p(c)Xx")-- r))
Then there exists some quantity CLA,RR 0 depending only on t, k, R. such that the following bounds hold:
This further implies the following relative influence from discretization, the ratio between the improvements of r and you over XTx, y.ooon obeys
V X1 (xpp
(28a)
El m )
- -1 E
2² Pel x, x,,, (cly)
S Sexp(Cex.
(28b)
E [[p(c ( y -1 E Elp(c L contt
O
T.
(cclv)- S ex
2 1 B[(((c y 1)1l
+)
k
(28c)
(26)
01
Epp(c a y
E[p(cl>at -1]
where p
1 denotes the k-th order gradient with respect y of function Pet Pe]xi_.(c) y)
x (cl
Appendix A.2 presents numerical results for the relative "10"x)""))=""
error E -Bpppe [-1] evaluated under vary- ing total variation distance thresholds 7 on the ImageNet dataset.
In the following, we focus primarily on the gradient Vpc [ (cl (cl))- as the other bounds can be derived us- ing similar techniques. Notice that Vpe|Xi (cly) fies the following decomposition:
satis-
4. Analysis
(c]
In this section, we shall provide details in the proof of main results.
= Pcllx 2
= -PPelx_, 1 log X1 (c|y)
4.1. Proof of Lemma 3.2
Pe (c logpx, (()) log- Moggxx.-11 loggxx.. (29)
According to the equivalence between X, and Y, (see (8) in Lemma 2.1), it is sufficient to focus on the first relation.
Recalling Lemma 2.1 again tells us
In addition, it can be shown later that
Ex,xx, /. px, Px, [x, c)pc (cl xz)) -
=r]
Pe| (clv)) s 2pp(c) exp (Ilxllaa + V(R22
2(1 t)
(30a)
= -f_ Px. olxr (2(o0)2)- exp( 1x-az- l1 20²
and
Px, (x,1c(x(e)
Px, (x7)
||ul|2 + VIR
d
Px, Px-Ic(IT ())p(c) de,
Voggx,, (1))11 <
(30b)
1-
v1
Sa, Px, (1:) (2TT²) -d/2 exp( ||=-ax, )dx-
V Vogpx <
||al|2 + VtR
d
(30c)
1-1
VT-7
Px, Px,1a(< c))p(c)
(x)
where f S&S g implies that there exists universal constant C > 0 such that < Cg. By inserting (30a) and (30b)
(x (c) (c|xx(c|xx)]
into (29), the gradient (Pe]xi-(cclu) can be controlled directly.
where we let let = and o o Here, the first line is just the definition of conditional expectation; the second line comes from the Bayes rule and the relation (7); and the last line can be derived by applying the Bayes rule and the relation (7) again.
Proof of Claim (30a)- - (30c). We begin with establishing (30a). First, according to Lemma 2.1, random variable X1- follows Gaussian distribution (VViX,
7

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
Thus we have
For the first term, we have
(V) = 220 Pxo (/x (u)
Junckepo ED
(2oly)) ly viroll
D
Pxol X2
1-t dro S
v1
=/o pxo (20 (t) d/2 exp (- [IIl~v viroll2 dro 2(1 -t) t)
For the second term, noticing that
[ 20
pxo (20)ddo $ (2T(11 - -))- (2 -))-
1 PX, (2) 2 2 (2TT(1 100- +)) 4/2 exp
+)/-
(- (1lv + VER)2
(31)
2(1 - t)
Moreover, recalling the definition of R in (27), we have
we have
Px...Ic(Ulc) c) 2 I
px. (x)
Pxo (x)(20)((2)) (2x()) -1))=
=P(I|xoll2 < R [
c, N
< R)
J200| V-vieo 1,20
4/2 exp (- 1ly - v /
2 2 =0:1|=o||=<<R
inf
(2 [1 t))
(-22) 2(1
exp (- 11 - v /Bool1 2
11 - VEwo dzo
2(1 t)
1 1- t
2
(32)
(lull2 + viRR23
2 2 (2T(1 - -))
-d/22/22/ exp (lllvlla + VER)2)
( 2(1- -) t)
(33)
S exx
2(1 - t)
10:
x- v-x/ vi-
|1
Pxo (xo)
(- 1ly - vixool2
y/ - v /t:00
where Pl |a cv c) denotes the joint probability density of Xi and the binary random variable indicating ||Xoll2 < R or not, and P ( c) probability density of Xi-t conditioned on the class label c and [ooll <
exp
(-1 2(1 1)
dxo
1 -
2
denotes the
2
(Ilv)l12 + VER)2
cD2 cd
V1 exp
2(1 -11) 1)
R. Combining (31) and (33), we have
px._. (v)
where e a universal constant.
Pe| (e]x)_(cclv)) 1((c)xx-.lo(// cc)
By choosing
(Ilvvlla + VIR)² 2(1 - t)
< 2p)( -1 exp
D=C [ulla [Illl++ vi-L +
+ ViR
Next, we shall prove (30b). For < 1, recalling that the random variable Xo follows Gaussian distribution (VVixo, (1 - t)),, the score function has the following expression
for some constant C > 0 large enough, we have
< vI |1vl12 + VRR
2D
V Woog
1-t
-1-
(v)
Similarly, we could derive that
-PX
Px, (1o) -0 t)) d/2 -d/2
llulla + ViR
d
exp (- I1y - vixol11 y - vixo dzo
V logpx S
1-1 +
vI
2(1 -t) t)
1-t
Pxo X._. (FO (v) 11 vizo drro
(34)
4.3. Proof of Claim (14)
t
We provide a detailed proof of Claim (14) by analyzing the decomposition of the expectation. We start by decomposing the expectation as follows:
Moreover, noticing that for any D > 0,
| loog loggpxx,.(vll
vizo
pxol N2 (2ol8)
dæy
E Pe| X1 x1-xx-1-- (c) (ce/Y++s)= -1 Pa 121xx-100 (.) -1 Y, ys]
200||| v-vian 1,50
= E [e X1- (cl
Pe X1- V1) -1 Vr = y/]
+ I|xx ((v)-1
| >-Vinn ||,>D
/
Pxo (1)) 4/2
(I)
+ E |Pe] Xi- (c)
Po| - (c Y(4s) - yv
.exp (-
lly T 12
y / frv dxp
In the following, we shall analyze these two terms sepa- rately.
2(1 - t)
1-t
8

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
Analysis of the first term.
Applying Ito's formula gives
Analysis of the second term. expressed as:
The second term can be
us
Pe|x, X2 (cll(4o)
- Pol X2 (cl))))
E Pel x (Po PePl (cl (c(Y..)) PY, - vi]
146
V2 (c) (c| ( ds
= E
Os Pelxi, (cl Ye4s) ds Yr= = yr
28
VellXi (cl (c Yz)-1 (2r,
2
Similar to the analysis of the first term, we notice that
+ Vlogpx c)) ds + Ve dB }
1
0s Pel Xi_ (clv)
i π. (c
(35)
= f.
02
3 Pel (cl y) dr,
We further decompose the first term by using Ito's formula again as
and according to (28h),
Tr
V Pel xx (cl1))) - Tr (c (V((
8
E
r Po x P (c|Y y/ x0.
L 2r
Tr Tr 82 Pel (c. (c)
dr
Thus we have
+ VTr V2 Pel X (cl (c(xx)--) (6/
- -Pe| E Pepe| (ele -Pcpcc/csss--Ppep______p (c(14)) y/
5
+ V Vlogpx.-. e)) dr -
+ Vr dB.) }
1
(36)
a Po (cl (cl
06)).
=
r
According to bound (28a), we have
Combining the above two relations, we could get our desired result.
E
Tr
T V (c | Vi
<
E exp(Cr,,,F + Cr Y. Va < oo (37)
5. Discussion
and
In this paper, we present a theoretical analysis of the impact of guidance in diffusion models under general data distri- butions. Specifically, we demonstrate that guidance in the continuous-time process can enhance the sampling process by generating more high-quality samples those associ- ated with higher classifier probabilities in the average sense. Additionally, we prove that the practical discrete-time process converges to the above analyzed continuous-time process, as the number of iterations goes to infinity. These results provide a theoretical foundation for the empirical success of guidance methods.
VTr V2 (c. ((>
EE
V Vlogpx, ((r o0) Y. = 3/1
< 0o.
(38)
Inserting (37) and (38) into (36), we have fort s S t 8,
Tr Pel (c) (c| =
11) + 0(8).
(39)
Similarly, we could get that for t 5t
E Vpelx. (o) -1 ((2rs Y.
In this paper, the convergence analysis in Theorem 3.6 is included primarily for completeness. The dependencies on d, L and may not be optimal, and the smoothness condi- tion might not be necessary. Future research could focus on establishing tighter bounds or analyzing under more general bounds, to broaden the applicability and improve the conver- gence rate. In addition, we are interested in extending these results to the concept of Inception Score (IS), demonstrating similar findings when the weights used in IS are applied.
+ V Floggx llog e)) IY= = V/
VP (cl 2i0e
+ +Vlogxx c)) + O(8).
(40)
Inserting (39) and (40) into (35), we have
1
E |e Pel (cl Yo-s) Pe|x (c) 1i))--iY vel
Acknowledgements
1
Tr V2 (cl + (c((x)
Gen Li is supported in part by the Chinese University of Hong Kong Direct Grant for Research and the Hong Kong Research Grants Council ECS 2191363.
21
-66m + <Vogpx...1<000 10) 1 +06))

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
Impact Statement
Ho, J., Jain, A., and Abbeel, Denoising diffusion proba- bilistic models. Advances in Neural Information Process- ing Systems, 33:6840-6851, 2020.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Huang. Z., Wei, Y., and Chen, Y. Denoising diffusion probabilistic models are optimally adaptive unknown low dimensionality. arXiv preprint arXhv:2410.18784 2024.
References
Barratt, S. and Sharma, R. A note the inception score. arXiv preprint arXiv: 180101970 2018.
Hyvärinen, A. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6(4), 2005.
Benton, J., De Bortoli, V., Doucet, A., and Deligiannidis, G. Nearly d-linear convergence bounds for diffusion models via stochastic localization. In The Twelfth International Conference on Learning Representations, 2023.
Hyvärinen, A. Some extensions of score matching. Com- pulational statistics & data analysis, 51(5):249--212. 2007.
Bradley, A. and Nakkiran, P. Classifier-free guidance is predictor-corrector. arXiv preprint arXiv:2408.0900 09000, 2024.
Karras, T., Aittala, M., Kynkäänniemi, T., Lehtinen, J., Aila, T., and Laine, S. Guiding diffusion model with bad version of itself. Advances in Neural Information Processing Systems, 37:52996-53021, 2024.
Cai, C. and Li, G. Minimax optimality of the proba- bility flow ode for diffusion models. arXiv preprini 3XXv:25003.05583 2025.
Lee, H., Lu, J., and Tan, Convergence for score-based generative modeling with polynomial complexity. In Ad- vances in Neural Information Processing Systems, 2022.
Chen, H., Lee, H., and Lu, J. Improved analysis of score- based generative modeling: User-friendly bounds under minimal smoothness assumptions. In International Con- ference on Machine Learning, pp. 4735-4763. PMLR, 2023.
Lee, H., Lu, J., and Tan, Y. Convergence of score-based generative modeling for general data distributions. In In- ternational Conference on Algorithmic Learning Theory, pp. 946-985 985, PMLR, 2023.
Chen, S., Chewi, S., Li, J., Li, Y., Salim, A., and Zhang, A. R. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. arXiv preprint arXiv:2209.1 1215, 2022.
Li, G. and Cai, C. Provable acceleration for diffusion models under minimal assumptions. arXiv preprint arXiv:2410.2322 2024.
Li, G. and Jiao, Y. Improved convergence rate for diffusion probabilistic models, arXiv preprint arXiv:2410.13738, 2024.
Chen, S., Chewi, S., Lee, H., Li, Y., Lu, J., and Salim, A.
The probability flow ode is provably fast. Advances in
Neural Information Processing Systems, 36, 2024.
Li, G. and Yan, Y. Adapting to unknown low-dimensional structures in based diffusion models. arXiv preprint arXiv:2405. 00..1.5..1666. 14861, 2024.
Chidambaram, M., Gatmiry, K., Chen, S., Lee, H., and Lu,
What does guidance do? a fine-grained analysis in a
simple setting, arXiv preprint arXiv:2409. 3074, 2024.
Li, G., Huang, Y., Efimov, T., Wei, Y., Chi, Y., and Chen, Y. Accelerating convergence of score-based diffusion models, provably. In forty-first International Conference on Machine Learning, 2024a.
Croitoru, F-A.,, Hondru, V., Ionescu, R. T., and Shah, M. Diffuusion models vision: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2023.
Dhariwal, P. and Nichol, A. Diffusion models beat GANs on image synthesis. Advances in Neural Information Processing Systems, 34:8780 8794, 2021.
Li, G., Wei, Y., Chi, Y., and Chen, Y. A sharp convergence theory for the probability flow odes of diffusion models. arXiv preprint arXiv:2408, 02320, 2024b.
Gupta, S., Cai, I.., and Chen, S. Faster diffusion-based sampling with randomized midpoints: Sequential and parallel. arXiv preprint arXiv:2406. 00924, 000:22446.00044, 2024.
Li, G., Cai, C., and Wei, Y. Dimension- convergence diffusion models for approximate gaussian mixtures. arXiv preprint arXiv:2504.0 053000, 2025a.
Ho, J. and Salimans, T. Classifier-fres free diffusion guidance. In NewrIPS 2021 Workshop on Deep Generative Models and Downstream Applications, 2021.
Li, X., Wang, R., and Qu, Q. Towards understanding the mechanisms of classifier free guidance. arXiv preprint arXiv:2005. /9210, 10, 2025b.
10

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
Pang, T., Xu, K., Li, C., Song, Y., Ermon, S., and Zhu, Ef- ficient learning of generative models via finite-difference score matching. Advances in Neural Information Process- ing Systems, 11119175-19888 2020.
Wu, Y., Chen, M., Li, Z., Wang. M., and Wei, Y. Theoretical insights for diffusion guidance: case study for gaussian mixture models. arXiv preprint arXiv:2403.01633) 2024.
Ramesh, A., Dhariwal, P., Nichol, A., Chu, C.,, and Chen, M. Hierarchical text-conditional image generation with CLIP latents. arXiv preprint arXiv:2204.066 00025,, 2022.
Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with la- tent diffusion models, 2021, URL https: htttps://githb.. com/ http///oossssaaetttttc diitttt-ifffiioooo
Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, High-resolution image synthesis with latent diffusion models. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 10684-10695, 2022.
Saharia, C., Chan, W., Saxena, S., Li, L., Whang, J., Denton, E. I.., Ghasemipour, K., Gontijo Lopes, R., Karagol Ayan, B., Salimans, T., al. Photorealistic text-to-image dif- fusion models with deep language understanding. Ad- vances in Neural Information Processing Systems, 35: -3494, 2022.
Salimans, T., Goodfellow, I., Zaremba, W., Cheung, V., Radford, A., and Chen, X. Improved techniques for training gans. Advances in neural information processing systems, 29, 2016.
Sohl-Dickstein, Weiss, E., Maheswaranathan, N., and Ganguli, S. Deep unsupervised learning using nonequi- librium thermodynamics. In International Conference Machine Learning, pp. 2256-2265, 2015.
Song. Y. and Ermon, S. Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems, 32, 2019.
Song, Y., Durkan, C., Murray, I., and Ermon, S. Maximum likelihood training of score-based diffusion models. Ad- vances in Neural Information Processing Systems, 34: 1415-1428, 2021a.
Song, Y., Dink-Dickt,,in, J., Kingma, D. P., Kumar, A., Er- mon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. International Conference Learning Representations, 2021b.
Szegedy, C., Vanhoucke, V., loffe, S., Shlens, J., and Wojna, Z. Rethinking the inception architecture for computer vi- sion. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2818-2826, 2016.
Vincent, A connection between score matching and de- noising autoencoders. Neural computation, 23(7):1661- 1674, 2011.

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
A. Discretization and Robust Analysis
A.1. Proof of Theorem 3.6
Here, we provide a brief sketch for this result. With similar analysis as Chen et al. (2022, Section 5),
KL Y1
N
(11 + (1++))(s comt c)
(Yn
loggxi
(Cut cont
c)]
2
Vlogxx. (Xtt a11 df
w| 18n (m
(Vu.cont)
+ KL V
(41)
Then it can be shown that
| ( (Mr
V log (Y, contt 1
Gn-1
2d
1.2B
y
Y
yruiconn
112
Ra-1
cont
dr
dB,
2 dt
<
L
+ (1 +
- w) log logpx, Ia (c) w log Lo
Jun an) + d2,
2
< [21 2
R²
Inserting the above relation, Assumption 3.3, and Assumption 3.4 into (41) leads to our desired result.
A.2. Numerical Validation
For different values of TV(Y"". ",, we empirically validate the aforementioned result on the ImageNet dataset. Specifically, we generate 2 10t samples Y!" under various guidance level and their counterparts Yo without guidance by using a pre-trained diffusion model (Rombach et al., 2021), and evaluate the classifier probability p(c[Y[") and p(c| by using the Inception v3 classifier (Szegedy et al., 2016). Finally, we evaluate the relative eгror in (26). Here we use [[p(c|{))) -+1 E(p(c)(19))) as an estimate of [[p(c]) O..ont - Elp(c (eptt ) and calculate the ratio of empirical average
E -1)
E - -ac_]-EEpp(Y E(p(c(Y
The results are presented in the following table for various values of the TV distance and w, which indicate that the relative error remains small, particularly for practical choices of w 2 1.
Table I. Empirical values "((((e) ()(e(()))" 1rr]
under different w and TV.
E I " 1
TV
w = 0.2
0.4
0.6
0.8
1
2
3
0.30
0.447
0.196
0.115
0.085
0.029
0.006
0.006
0.002
0.10
0.440
0,194
0.114
0.085
0.029
0.006
0.005
0.002
B. Basis Calculations of GMM
Consider GMM defined as:
Xo
T (pu, 1),
(42)
12

---

Provable Efficiency of Guidance in Diffusion Models for General Data Distribution
where Ty is the mixing coefficient of the k-th component, and P4 is its mean. By Lemma 2.1, have
X1
N N (Vante:))
k=1
K
(- (x - vanllx)
(x)
=
Th (2x) 1/2 exp
2
k=1
The gradient of the log-density log px1 1xg -nn (z)
can be computed as:
V log Px (x) Vpx ma (2)
Th (x vanH)) + Va,
Th Mk:
(43)
Px1- (I)
k=1
k=1
where
(a- (o-van))l)
Th exp
2
Σi exp
[a- Un
2
Using this setup for specific cases (K = 2, 3) leads to
v logPx; cl2 1) x + vān - 2xpp-22vapp 2)))
(44)
1 1 exp( 10pp-22varx) 2v)
Va exp( -2 -2
logpx, na (x)
(45)
exp( vT + 2 exx
[1
(T
2
Additionally, the classifier probability Pel X (1 (t) is given by
PI (1|x)= PXI-T, c c)p(()
1+ expp-2vix a,
(46)
Px--x (x)
1 (ap(-22onn)) -expp
2 vanz)
13


Reviews:
[{"review_id": "SeTg4v5JEt", "reviewer": "Reviewer_EKNn", "summary": "In this paper, the authors analyze the effect of diffusion guidance under general data distributions. Their study reveals that guidance does not necessarily improve sample quality in all cases, but it enhances overall sample quality. Specifically, they prove that under the influence of guidance, the proportion of low-quality samples (measured by classifier probabilities) decreases. A toy experiment on the Gaussian Mixture Model provides empirical support for their theoretical analysis.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks a lot for the reviewer's helpful comments and valuable feedback. \nBelow, we provide a point-by-point response, which has also been incorporated into the revised version of our manuscript.\n\n**Experiments on real dataset.**\nNotice that classifier-free guidance (CFG) was originally validated on the ImageNet dataset [1], where Inception Score is typically computed using the Inception v3 classifier [2]. To further support the practical applicability of our method, we have included an additional numerical experiment using a pre-trained diffusion model [3] on the ImageNet dataset, which we believe provides stronger empirical validation than other datasets.\n\nThe results demonstrate that guidance improves sample quality by decreasing the averaged reciprocal of the classifier probability rather than achieving uniform improvement across all samples, thereby validating our theory. \nWe adopt the guidance scale range in [1] from $0$ to $4$; specifically, in our experiments, we use $w = \\{0.2, 0.4, 0.6, 0.8, 1, 2, 3, 4\\}$. \nThe numerical results are as follows:\n\n\\begin{align*}\n\\begin{array}{c|cccccccccccc}\n\\\\hline\n\\\\hline\nw & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 2 & 3 & 4  \\\\\\\\\n\\\\hline\n{P(p_{c|X_0}(c|Y_1^w) \\geq p_{c|X_0}(c|Y_1^0))} & 0.70 & 0.75 & 0.78 & 0.80 & 0.82 & 0.85 & 0.85 & 0.86 \\\\\\\\\n-\\mathbb{E}[p_{c|X_0}(c|Y_1^w)^{-1}] &-140 & -74 & -47 & -36 & -14 & -3.6 & -3.4 & -2.0 \\\\\\\\\n\\\\hline\n\\end{array}\n\\end{align*}\n\n**Paper structure.**\nWe agree that presenting the numerical experiments as a separate section would enhance readability. \nwe will restructure the manuscript by placing the toy experiment and the newly added results on the ImageNet dataset in a separate section following the main results.\n\n\n**Practical implementation.**\nOne potential application of our theoretical analysis comes from its implication that guidance may reduce sample quality for a small subset of samples. \nThis observation could motivate further research into adaptive guidance mechanisms that ensure a more uniform improvement in sample quality.\nWe will include a remark following Theorem 3.1 to discuss the potential implementation of our theory:\n\n''Theorem 3.1 states that guidance improves the averaged reciprocal of the classifier probability rather than the classifier probability of each individual sample. This suggests that while guidance improves overall sample quality, it may lead to a decline in quality for a small subset of samples. This insight encourages the development of adaptive guidance methods that address this issue and achieve more uniform performance gains, which is a potential practical application of our theory.''\n\n\n**Influence of guidance scale $w$.**\nWe agree that extremely large values of $w$ can degrade the performance of CFG.\nHowever, this phenomenon is consistent with both our theoretical and experimental results.\nIn practice, the performance of a diffusion model is typically evaluated based on two key metrics: diversity and sample quality. CFG attains a trade-off between these two metrics, as noted in [1]. In this paper, our main focus is on the influence of guidance on sample quality, particularly in relation to the Inception Score. \nOur results align with practical observations in the sense that the generated samples are more closely adhere to the conditional distribution with larger $w$.\nIn addition, previous studies have shown that an extremely large $w$ can severely reduce sample diversity and negatively impact realism. While this is an important consideration for practical applications, it is not the main focus of this work.\nWe will add the following remark to avoid confusion:\n\n''In practice, performance of diffusion models is commonly evaluated by two metrics: diversity and sample quality. This study primarily focuses on the sample quality measured in a similar way as the Inception Score, which increases with $w$. However, prior work [1] has demonstrated that large values of $w$ can significantly reduce sample diversity, leading to unsatisfactory performance in real-world applications.''\n\n**Related works.**\nReference [4] proposed to use a bad version of the model for guiding diffusion models. We will cite it properly in our revision.\n\n\n[1] Classifier-free Diffusion Guidance\n\n[2] Rethinking the Inception Architecture for Computer Vision\n\n[3] https://github.com/CompVis/latent-diffusion \n\n[4] Guiding a Diffusion Model with a Bad Version of Itself", "claims_and_evidence": "Each claim in this paper is supported by strong theoretical justification.", "methods_and_evaluation_criteria": "This paper provides a sound theoretical analysis of the effect of guidance across general data distributions, offering valuable insights and contributions to the research community. Its findings are likely to inspire further advancements in the field.", "theoretical_claims": "The theoretical analysis in this paper is sound, with detailed and correct proof processes or relevant literature support.", "experimental_designs_or_analyses": "The experimental results are reliable—I did not find any obvious issues. However, although this paper primarily focuses on theoretical analysis and proofs, its experimental evaluation is relatively limited. The paper provides only a single case study on a Gaussian Mixture Model, which may not sufficiently demonstrate the practical applicability of the proposed method. Including experiments on more general data distributions would help readers better understand the method and provide stronger inspiration for the research community.", "relation_to_broader_scientific_literature": "This paper analyzes the effect of guidance on the sampling process in terms of the reciprocal of classifier probabilities, which, to some extent, is conceptually similar to the Inception Score. Moreover, compared to other works, such as Autoguidance [1], which only provide a qualitative analysis of the classifier-free guidance sampling process, this paper offers a detailed and rigorous theoretical analysis, making a more substantial contribution to the understanding of guidance in diffusion models.\n\n[1]: Guiding a Diffusion Model with a Bad Version of Itself", "essential_references_not_discussed": "The citations and comparisons with related works are comprehensive and well-covered, ensuring a thorough contextualization of the proposed approach within the existing literature.", "other_strengths_and_weaknesses": "**Strengths**\n1. The paper is written in a clear and coherent manner.\n2. The paper provides a sound and comprehensive analysis of the effect of guidance mechanisms on general data distributions. The analysis is thorough and well-structured, offering valuable insights that could significantly inspire further research in the community.\n3. The Toy Example on the Gaussian Mixture Model is a notable highlight of the paper, providing practical support for the theoretical analysis.\n\n**Weaknesses**\n1. Although the theoretical analysis in this paper is detailed and comprehensive, the Toy Experiments on the Gaussian Mixture Model alone feel somewhat limited. Conducting experimental analyses on more general data distributions and larger-scale models would provide stronger empirical validation, making the findings more impactful and accessible for the research community and readers.", "other_comments_or_suggestions": "1. Reorganizing the structure of the paper could enhance readability and comprehension. For example, presenting the Toy Experiments as a separate section rather than embedding them within the theoretical analysis would make it easier for readers to follow the logical flow of the paper.", "questions_for_authors": "1. How can the theoretical analysis and insights from this paper be used to enhance the capability of the guidance mechanism in practical applications?\n2. In my view, $\\omega$ is a crucial hyperparameter in classifier-free guidance. When $\\omega$ is too small, the generated samples do not align well with the conditions, while an excessively large $\\omega$ affects the realism of the generated samples. The experiments conducted on GMM in this paper cover a wide range of $\\omega$, and the evaluation metrics improve as $\\omega$ increases, which seems inconsistent with observations in practical applications. Could you explain this discrepancy?\n3. Could you provide experimental observations on more general benchmarks, such as MNIST or CIFAR-10?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ioZoVDBTS7", "reviewer": "Reviewer_5U8H", "summary": "This paper gives a novel theoretical analysis of classifier guidance. Whereas prior work focused on special cases, e.g. mixtures of Gaussians and compactly supported distributions, this paper establishes a guarantee under minimal distributional assumptions. Specifically, they consider the functional given by the expected *inverse* classifier probability over generated samples, which is correlated with sample quality, and show that this quantity decreases as the guidance parameter increases. The analysis is short but clever: they utilize th", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the constructive feedback! Below, we provide a detailed point-by-point response.\n\n**Explanation of main analysis idea.** In the revised version, we will add the following explanations to better communicate the high-level ideas behind the analysis:\n\n''**A glimpse of the main analysis idea.** First, this result comes from the key observation that the function of reverse process, $p_{c|X_{t}}(c|X_t)^{\\rm -1}$, forms a martingale, as stated in Lemma 3.2, which is established through a careful decomposition of $p_{c|X_{t}}$ and $p_{X_{\\tau}|X_{t}}$. Next, the guidance term $s_t(x|c) - s_t(x)$ in classifier-free guidance (CFG) aligns with the direction of $-\\nabla p_{c| X_t}(c| x)^{-1} = p_{c| X_t}(c| x)^{-1}[s_t(x|c) - s_t(x)]$, which makes us expect that adding the guidance at time $t$ can decrease $\\mathbb{E}\\_{x_{\\tau} \\sim X_{\\tau}}\\big[p_{c| X_{\\tau}}(c| x_{\\tau})^{-1} | X_t = x\\big]$ for all $\\tau \\le t$. Finally, to achieve the desired result, particular care must be taken in handling first- and second-order differential terms with respect to $t$ for the process $p_{c| X_{1-t}}(c| Y_t^w)^{-1}$ due to its randomness nature, which is completed in Section 4.2 based on the technique of Ito's formula.''\n\n**Extension to DDIM.** As pointed out by the reviewer, our analysis is specific to DDPMs. In particular, our analysis relies on the martingale property stated in Lemma 3.2, which relies heavily on the property of $p_{X_{\\tau}|X_{t}}$ in DDPMs and can not be applied for DDIMs. Extending our framework to DDIMs remains an open question due to the absence of this key property. We will add a remark in Section 3.1 of the revised version to explicitly state this limitation.\n\n**Influence of discretization.** We fully agree that our continuous-time analysis for CFG can not be immediately extended to the discrete-time setting with only small KL divergence error. As the reviewer suggests, if $p(c | x)^{-1}$ is uniformly bounded, one could establish the desired result. However, such an assumption is too strong. Instead, we adopt a weaker condition: we assume that $\\mathbb{E}[(p(c | Y_1^w)^{-1}-1)1(p(c | Y_1^w)^{-1} > \\tau)]$ is small for some threshold $\\tau > 0$. We have also verified this assumption numerically on the Imagenet dataset. We will include the following new result in the revised version: \n\n''The sampling process (5) with the learning rate schedule (19) satisfies\n\\begin{align*}\n\\mathbb{E}[p(c | Y_1^w)^{-1}] \\le \\mathbb{E}[p(c | Y_{\\overline\\alpha_1}^{w, \\mathsf{cont}})^{-1}] + \\mathbb{E}[(p(c | Y_{1}^{w})^{-1}-1)1(p(c | Y_{1}^{w})^{-1} > \\tau)],\n\\end{align*}\nwhere $\\tau$ is defined as the largest value satisfying\n\\begin{align*} \n\\mathsf{TV}(Y_{\\overline\\alpha_1}^{w, \\mathsf{cont}}, Y_1^w) \\le \\mathbb{P}(p(c | Y_{1}^{w})^{-1} > \\tau).\n\\end{align*}\nThis further implies the following relative influence from discretization, the ratio between the improvements of $Y_1^w$ and $Y_{\\overline\\alpha_1}^{w, \\mathsf{cont}}$ over $X_{\\overline\\alpha_1} = Y_{\\overline\\alpha_1}^{0, \\mathsf{cont}}$, obeys\n\\begin{align*} \n\\frac{\\mathbb{E}[p(c | Y_{\\overline\\alpha_1}^{0, \\mathsf{cont}})^{-1}] - \\mathbb{E}[p(c | Y_1^w)^{-1}]}{\\mathbb{E}[p(c | Y_{\\overline\\alpha_1}^{0, \\mathsf{cont}})^{-1}] - \\mathbb{E}[p(c | Y_{\\overline\\alpha_1}^{w, \\mathsf{cont}})^{-1}]}\n\\ge 1 - \\frac{\\mathbb{E}[(p(c | Y_{1}^{w})^{-1}-1)1(p(c | Y_1^w)^{-1} > \\tau)]}{\\mathbb{E}[p(c | Y_{\\overline\\alpha_1}^{0, \\mathsf{cont}})^{-1}] - \\mathbb{E}[p(c | Y_{\\overline\\alpha_1}^{w, \\mathsf{cont}})^{-1}]}.\n\\end{align*}\n\nFor different values of $\\mathsf{TV}(Y_{\\overline\\alpha_1}^{w, \\mathsf{cont}}, Y_1^w)$, we empirically validate the aforementioned assumption on the ImageNet dataset. Here we use $\\mathbb{E}[p(c | Y_{1}^{0})^{-1}]-\\mathbb{E}[p(c | Y_{1}^{w})^{-1}]$ as an estimate of $\\mathbb{E}[p(c | Y_{\\overline\\alpha_1}^{0, \\mathsf{cont}})^{-1}] - \\mathbb{E}[p(c | Y_{\\overline\\alpha_1}^{w, \\mathsf{cont}})^{-1}]$. The relative error $\\frac{\\mathbb{E}[(p(c | Y_{1}^{w})^{-1}-1)1(p(c | Y_1^w)^{-1} > \\tau)]}{\\mathbb{E}[p(c | Y_{1}^{0})^{-1}]-\\mathbb{E}[p(c | Y_{1}^{w})^{-1}]}$ is presented in the following table for various values of the TV distance and $w$. The results indicate that the relative error remains small, particularly for practical choices of $w \\ge 1$.\n\n\\begin{align*}\n\\begin{array}\n\\\\hline\n\\\\hline\n\\mathsf{TV} & w=0.2 & 0.4 & 0.6 & 0.8 & 1 & 2 & 3 & 4\\\\\\\\\n\\\\hline\n0.30 & 0.447 & 0.196 & 0.115 & 0.085 & 0.029 & 0.006 & 0.006 & 0.002  \\\\\\\\\n0.10 & 0.440 & 0.194 & 0.114 & 0.085 & 0.029 & 0.006 & 0.005 & 0.002 \\\\\\\\\n\\\\hline\n\\end{array}\n\\end{align*}\n''\n\n**Assessment on real dataset.** We have added experiments on the ImageNet dataset to validate our theory. Please refer to our response to Reviewer EKNn of **Experiments on real dataset**.", "claims_and_evidence": "Yes, please see \"Theoretical Claims\" for details", "methods_and_evaluation_criteria": "N/A as this is a theory paper", "theoretical_claims": "I checked the correctness of the proof of Theorem 3.1, the main calculation for which appears in Section 4, and believe it to be sound. Please see \"Strengths and Weaknesses\" for further thoughts on the theory.", "experimental_designs_or_analyses": "The experiment was a small-scale evaluation for Gaussian mixtures, and the results appear to be valid and consistent with the theoretical findings.", "relation_to_broader_scientific_literature": "This paper fits into the broader literature on establishing rigorous guarantees for diffusion models.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n- This paper's main selling point is that it gives a guarantee for guidance that works for all probability distributions. This is quite exciting, as all prior work focused on specific classes of distributions like Gaussian mixtures, and prior to reading this paper I would not have expected one can prove any interesting, general-purpose result about classifier guidance.\n- The key idea is quite clean: (1) the functional they consider is a martingale along the unguided reverse process, so its infinitesimal expected change (which is zero) can be approximated to first order, using Ito's lemma, by an expression depending only on its derivatives in time/space. (2) Likewise, the infinitesimal expected change for the guided reverse process can be approximated to first order using a very similar expression, with some extra terms arising from the guidance term. These extra terms are precisely what give rise to the decrease in expected inverse classifier probability under guidance.\n\nWeaknesses:\n- The writing does not do a good job of communicating the main idea behind the calculations in an easy-to-understand manner. I am happy to raise my score if the authors can improve the clarity of the writing\n- Unless I'm misunderstanding, the analysis is specific to DDPMs and does not say anything about DDIMs.\n- While the authors show the discrete and continuous-time samplers are close using off-the-shelf methods, it's not clear how this closeness can be combined with the main result on expected inverse classifier probability to say something about how that functional behaves under guidance for discrete-time samplers.", "other_comments_or_suggestions": "Please see \"Weaknesses\" and \"Questions for Authors\"", "questions_for_authors": "- Could you comment on how to combine Theorem 3.1 with Theorem 3.6? It's not clear to me that the KL bound should imply anything about the expected inverse classifier probability. I guess if the inverse classifier probability is bounded, then you can use Pinsker's, but such boundedness doesn't seem to be a reasonable assumption.\n- Have you tried assessing whether expected inverse classifier probability is actually meaningful in real data? I would imagine it's too large in general to be a useful measure of distributional sample quality", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UhnpQSC7fZ", "reviewer": "Reviewer_qwwj", "summary": "This paper presents a theoretical analysis of classifier-free guidance (CFG) in diffusion models, demonstrating that guidance enhances sample quality by reducing the expected ratio of poor samples, as measured by classifier probability. The authors establish a connection between their proposed metric and the Inception Score, a widely used evaluation metric for diffusion model sample quality, thereby justifying the choice of their metric. To validate their theoretical findings, the authors provide a one-dimensional experiment involving Gaussian Mixtures, which serves as a proof-of-concept for their approach.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your valuable questions. Below we provide a detailed point-by-point response.\n\n**Experiments on real dataset.** We have added experiments on the ImageNet dataset to validate our theory. Please refer to our response to Reviewer EKNn of **Experiments on real dataset**.\n\n**Further explanations of the main theorem.**\nWe agree that a more detailed explanation of Theorem 3.1 would enhance clarity. In the revised version, we will add the explanation of main analysis idea (see ''Explanation of main analysis idea'' in our response to Reviewer 5U8H), detailed comparison with prior theory (see ''Comparison with prior works'' in this response), explanation of IS (see ''Issue of Inception Score'' in this reponse), and experiments on Imagenet dataset (see ''Experiments on real dataset'' in our response to Reviewer EKNn) to address this comment.\n\n**Clarification of the toy example.**\nOriginally, we state that *''$P(p_{1|X_0}(1|Y_0^w)\\ge p_{1|X_0}(1 | Y_0^0))$ is less than $1$, which indicates the guidance may not achieve uniform improvement in classifier probabilities''* for the toy example. To make it more accurate, we will revise it to *''$P(p_{1 | X_0}(1| Y_0^w)\\ge p_{1| X_0}(1| Y_0^0)) < 1$ for any $w < 10$, which indicates ...''*, since we have numerically confirmed that this holds true for all $w < 10$, which covers the practical range of $w$ in typical applications.\n\n**Relation to prior works on toy examples.** We agree with the existence of prior analyses of classifier guidance in GMMs. However, our work focuses on different aspects compared to these studies. \nWe will include a detailed comparison in the revised version, as detailed in next bullet.\n\n**Comparison with prior works.** Actually, we have cited and briefly compared with these existing works, which mainly focus on specific classes of distributions like GMMs. In contrast, our main contribution lies in providing a more general theoretical analysis. Nonetheless, we agree that a comparison between our theory--- when applied to specific distributions--- and prior works would further clarify our contributions. We will include a new section in the revised manuscript for comparison:\n\n''Existing works focus mainly on specific classes of distributions like GMMs, while our work provides a more general theoretical analysis. Below, we compare our findings with prior works when restricted to specific distributions.\n\nIn [1], the authors demonstrate that $p_{c | X_0}(c | Y_1^w)\\ge p_{c | X_0}(c | Y_1^0)$ holds under specific conditions, while we show that this inequality does not always hold. In addition, [2] argues that guidance can degrade the performance of diffusion models, as it may introduce mean overshoot and variance shrinkage. In contrast, our result shows that guidance can improve sample quality by generating more samples of high quality. Furthermore, [3] shows that classifier guidance can not generate samples from $p(x | c)^{\\gamma}p(x)^{1-\\gamma}$ for GMMs and establishes its connection to an alternative approach, i.e., the single-step predictor-corrector method, whose effectiveness in this specific setting remains unclear. In contrast, we directly analyze and demonstrate the effectiveness of CFG.''\n\n**Issue of Inception Score (IS).**\nWe offer two clarifications: \n\n1. The reviewer is absolutely correct that IS is not a perfect metric and has known limitations in evaluating sample quality. \n\n2. We disagree that *''IS is outdated''* and argue that IS is still one of the most widely used and informative metrics for assessing sample quality. The precise evaluation of sample quality remains an open problem. Despite the concerns raised in [4], IS continues to be employed as a key metric in recent works on classifier guidance [5] and classifier-free guidance [6] for diffusion models, due to the absence of an alternative metric that is demonstrably superior to IS. Our choice to consider IS aligns with the original studies that introduced diffusion guidance. In addition, most issues outlined in [4] come from the inaccuracy in the empirical estimation of IS, which doesn't apply for the theoretical consideration, as our results are derived using the true conditional probability.\n\t\t\nWe will clarify the reasonability for using IS in the revised manuscript:\n\n''Although some practical limitations of IS have been identified [4], it remains a commonly used metric for evaluating sample quality in the study of diffusion guidance [5,6]. Moreover, in our theoretical analysis, we use the true conditional probability, which addresses the estimation issues discussed in [4].''\n\n**Typos.** We have fixed them in the revised version. \n\n[1] Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models\n\n[2] What does guidance do? A fine-grained analysis in a simple setting\n\n[3] Classifier-Free Guidance is a Predictor-Corrector\n\n[4] A Note on the Inception Score\n\n[5] Diffusion Models Beat GANs on Image Synthesis\n\n[6] Classifier-free diffusion guidance", "claims_and_evidence": "The main result, Theorem 3.1, appears to be theoretically sound and relevant to the current literature on classifier-free guidance (CFG). However, I am concerned that the paper lacks sufficient empirical evidence to support this theoretical contribution. The experimental section is unfortunately very weak, which undermines the overall impact of the paper. To strengthen the manuscript, I would recommend that the authors invest significant effort into improving the experimental evaluation, including more comprehensive and rigorous experiments that demonstrate the practical effectiveness of their approach. Furthermore, giving more detailed insight and explanation regarding the main theorem and its contribution would benefit for the future readers. In its current form, I do not believe that the paper meets the standards expected for an ICML conference paper.", "methods_and_evaluation_criteria": "Unfortunately, the paper falls short in its experimental evaluation. The authors only present a single experiment on a one-dimensional Gaussian Mixture, which is insufficient to demonstrate the effectiveness of their approach. To strengthen the paper, I would recommend that the authors conduct additional experiments on higher-dimensional models, including Gaussian Mixtures, as well as real-world datasets. This would provide a more comprehensive understanding of the proposed method's performance and limitations.\n\nFurthermore, I have concerns regarding the example presented on page 8. The authors claim that P(p_{c|X_0}(1|Y_0^w) \\geq p_{c|X_0}(1|Y_0^0)) is less than one, but the curve suggests that this may not be the case for sufficiently large values of w. Moreover, this specific scenario has been previously analyzed in \"What does guidance do? A fine-grained analysis in a simple setting\" by Chidambaram et al., who demonstrated that classifier-free guidance (CFG) in one dimension can lead to flawed results, including mean overshoot and variance shrinkage. In light of these findings, it is essential that the authors perform experiments in larger dimensions and on real-world datasets to validate their claims and demonstrate the robustness of their approach.", "theoretical_claims": "I did not find an error in the calculations in the paper.", "experimental_designs_or_analyses": "There is only one experiment in one dimension. As mentioned above, even this experiment and its results do not seem completely valid.", "relation_to_broader_scientific_literature": "There is currently an increasing amount of theoretical works that are attempting to explain CFG, so this paper is timely and relevant.", "essential_references_not_discussed": "It is disappointing that the authors have not cited any related work, especially given the substantial body of research currently focused on explaining classifier-free guidance (CFG) from a theoretical perspective. For instance, relevant studies can be found in the following papers: https://arxiv.org/abs/2409.13074, https://arxiv.org/abs/2403.01639, and https://arxiv.org/abs/2408.09000. Including references to these works would provide valuable context and strengthen the paper's contribution to the field.", "other_strengths_and_weaknesses": "I would also strongly advise the authors to move all the proofs to the appendix and to only highlight the main results in the main paper, as well as add related work and stronger experimental results.", "other_comments_or_suggestions": "The authors justify their chosen metric by demonstrating its relationship to the Inception Score. However, it is important to note that the Inception Score is considered outdated and is no longer widely used in recent work on diffusion models. For instance, a well-known study in the field highlights the shortcomings of this metric https://arxiv.org/pdf/1801.01973. I strongly recommend that the authors address this issue by discussing the relevance of their metric in light of these criticisms and by relating the findings of their paper to the proposed metric. This would help clarify the metric's applicability and strengthen the paper's contribution.\n\nThere is a typo on end of page 2, where you have double brackets )) and should have just one: ∇ log pc | Xn (c | x)).\nYour title is \"Title Suppressed Due to Excessive Size\" so please make sure to amend this accordingly. \nPage 2 also mentions multiple times that Z_n represents i.i.d. Gaussian noise, this could be avoided for conciseness and clarity.", "questions_for_authors": "No questions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper has a split among reviewers. Two of the reviewers like the paper, and view it as a solid initial attempt to better understand the behavior of guidance in diffusion models, which one reviewer describes as otherwise not that easy to study particularly in cases when working with general distributions. The remaining reviewer is worried that the experiments are not realistic or comprehensive enough, and thinks that the theory is not validated experimentally in a tight-enough manner.

In discussion, reviewers did not come to an agreement, and instead decided to emphasize these distinct strengths and weaknesses to back up their judgment, thereby illustrating that these differing views come down to a disagreement about what is important for a paper in this area to have. To this end, I think it is also reasonable to have a paper's contributions emphasize more the theoretical aspects - which precisely pin down intuition coming from practice, even if the setup in which this is done is more restrictive - as opposed to empirical aspects. This is in part because some researchers - for instance those in academia rather than industry - may not have sufficient compute access to perform empirical evaluations to the same degree of comprehensiveness, and may be interested in studying complementary aspects to those with an empirical focus.

In light of this balance, my view is to side with the two positive-score reviewers, and I recommend a weak accept.
Decision: Accept (poster)