What does guidance do? A fine-grained analysis in a simple setting
Authors: Muthu Chidambaram
Khashayar Gatmiry
Sitan Chen
Holden Lee
Jianfeng Lu
Venue: NeurIPS 2024 poster (2024)
Keywords: diffusion
guidance
sampling
probability flow ode
Primary area: diffusion_based_models
TL;DR: Diffusion with guidance does not do what its motivation suggests; in linearly separable settings it samples from the edges of each class

Abstract: The use of guidance in diffusion models was originally motivated by the premise that the guidance-modified score is that of the data distribution tilted by a conditional likelihood raised to some power. In this work we clarify this misconception by rigorously proving that guidance fails to sample from the intended tilted distribution. Our main result is to give a fine-grained characterization of the dynamics of guidance in two cases, (1) mixtures of compactly supported distributions and (2) mixtures of Gaussians, which reflect salient properties of guidance that manifest on real-world data. In both cases, we prove that as the guidance parameter increases, the guided model samples more heavily from the boundary of the support of the conditional distribution. We also prove that for any nonzero level of score estimation error, sufficiently large guidance will result in sampling away from the support, theoretically justifying the empirical finding that large guidance results in distorted generations. In addition to verifying these results empirically in synthetic settings, we also show how our theoretical insights can offer useful prescriptions for practical deployment.

What does guidance do? A fine-grained analysis in a simple setting
Muthu Chidambaram
Khashayar Gatmiry
Sitan Chen
Duke University
MIT
Harvard University
muthuêcs. duke. edu
gatmiryemit edu
sitanêseas www.aaardd...... edu
Holden Leef
Jianfeng Lu Duke University
Johns Hopkins University hlee2830jhu. edu
i iianfengmmath duke, edu
Abstract
The use of guidance in diffusion models was originally motivated by the premise that the guidance-modee tional likelihood raised to some power. In this work we clarify this misconception by rigorously proving that guidance fails to sample from the intended tilted distri- bution. Our main result is to give fine-grained characterization of the dynamics of guidance in two cases, (1) mixtures of compactly supported distributions and (2) mixtures of Gaussians, which reflect salient properties of guidance that manifest on real-world data. both cases, we prove that as the guidance parameter in- creases, the guided model samples more heavily from the boundary of the support the conditional distribution. We also prove that for any nonzero level of score estimation error, sufficiently large guidance will result in sampling away from the support, theoretically justifying the empirical finding that large guidance results distorted generations. In addition to verifying these results empirically in synthetic settings, we also show how our theoretical insights can offer useful prescriptions for practical deployment.
score is that of the data distribution tilted by condi-
Introduction
With diffusion models having emerged as the leading approach to generative modeling in domains like image, video, and audio [1 34 18 12
29 26), there is pressing need to develop principled methods for modulating their output. For example, how do we impose certain constraints on generated samples, or control their "'emperature"" To formalize this, suppose one has access to an unconditional diffusion model approximating the data distribution p, as well as a conditional diffusion model approximating the conditional distribution p(- =) for various choices of class labels or prompts 2 Given z and parameter w, one might wish to sample from the distribution p tilted by the conditional likelihood. i.c. the distribution p²." with density
33
35
36
pa 2. (x) x p(x) :p( x=)+**
(Throughout,
lower-case letters denote densities, and capital letters to denote distributions.)
By varying w, we can naturally trade off between diversity and quality: If w then =* the unconditional distribution, if w = 0 then p² is the conditional distribution p(- 2) by Bayes' rule, and as 30, 2- converges to being supported on the maximizers of the conditional likelihood.
"Lead authors, equal contribution
1 Equal contribution
"In some settings, instead of having access to p(- | 2). one has access to some model for the conditional likelihood p(= -). The distinction between the "classifier-free" setting [19 and the "classifier" setting is immaterial to this paper, and theory applies to both.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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34
34
45
(a) Ground Truth
(h) Conditional Score (w 0)
(c) Guided Score (w 3)
Figure 1: We consider sampling from the positive class of 2D mixture of uniforms (a) using the probability flow ODE with the conditional score (b) and the guided score (c). As can be seen, increasing the guidance weight w clearly biases the distribution of samples to concentrate towards points far away from the other class support.
In practice, the standard approach to try to sample from the tilted distribution is to use diffusion guidance [3 19). The idea for this is roughly as follows. To sample from given distribution using diffusion generative modeling, one numerically solves certain ODE or SDE whose drift depends on the score function, i.e. gradient of the log-density, of convolved with various levels of Gaussian noise. By definition, the score function of p satisfies
v logg".. = V logp + (1 + w) logp(z |)) -V logp + (1+ w) V p( 2).
(I)
Assuming we have access to approximations of both terms on the right, we can run the corresponding ODE or SDE whose drift can be computed using the above to approximately sample from tromp.... p²:".
There is however one fundamental snag in the reasoning above. The aforementioned ODE SDE involves the score function at different noise levels, i.e. we would need access to V log 2. where the notation q: to denote the distribution given by running certain noising process (see Section 2.1p for time t starting from a distribution q. Here p. means we tilt before adding noise, that is, take q = p²1 and then apply noise q. Unfortunately, Eq. no longer holds, i.e.
as >> the analogue of
V Poggpt 3. - log Pt + (1 w) W log 2).
In other words, the operation of applying noise to p and the operation of tilting the direction of the conditional likelihood do not commute. Nevertheless, in practice it is standard to use the Sampling using this approximation is called
right-hand side of Eq. (2) as an approximation 13 19
diffusion guidance.
Intriguingly, for appropriate choices of w, this heuristic results in generations with high perceptual quality. Yet despite the popularity and empirical success of guidance, our theoretical understanding of this approach is lacking. this work, we ask:
What distribution is diffusion guidance actually frmpllingfrom:
A motivating example. To see clearly that diffusion guidance is not simply sampling from the tilted distribution, consider the following simple setting. Suppose that there are only two classes -1 and +1, and that the corresponding conditional distributions p(- 2 -1) and p(- +1) have disjoint supports , Ω. In this case, the conditional likelihood is simply given by p(2 = ( x [[x E ..). In particular, the conditional likelihood is binary-valued, which implies that for any > 0, the tilted distribution - is exactly the same! On the other hand, as Figure shows, increasing w changes the distribution of generated samples to concentrate towards the edge of the support of the guided class.
This simple example already reflects two properties of diffusion guidance that manifest on real-world data as the guidance parameter w increases:
Drop in diversity: The entropy of the distribution over generated samples resulting from diffusion guidance tends towards zero.
Divergence towards "archetypes": The generated outputs drift more and more to the extreme points in the support of the conditional distribution p(- i) to which the diffusion model is

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being guided. Note that these extreme points do not necessarily coincide with the set of maximizers of the conditional likelihood, our example shows.
These phenomena are surprising because we have shown that they can occur even when the conditional likelihood contains no geometric information about the data distribution: in our example, all points E S have the same conditional likelihood under p(= +1 a:), so it is not at all clear why the sampling process should end up preferring points in S2 which are furthest from points in Q_
In this work, we hone in on simple settings in which we can precisely characterize the dynamics of diffusion guidance and explain this counterintuitive behavior. Our main result is to execute such an analysis for mixtures of compactly supported distributions:
Theorem (Compactly supported setting, informal - see Theorem 4j. Consider a data distribution
P - () ) + 20( 1) where y(4) p(-1) are 3-bounded and supported on disjoint intervals [ax o22] and [-xx, - respectively (see Assumption Suppose that one runs the probability flow ODE
with guidance parameter w which is larger than some absolute constant. Then with probability
1 e- -e-((w), the resulting sample lies in the interval
(oc(t-OL/VVmw)).o)
(3)
where the O()) notation hides constants depending on (11 02,0
We also conduct this analysis in a setting where the conditional distributions p()) and p(-1) do not have compact support by proving an analogous result for mixtures of Gaussians. The proofs in this setting turn out to be somewhat simpler:
Theorem 2 (Gaussian setting). Consider the data distribution p 1) (N,,1, Suppose
that one runs the probability flow ODE with guidance parameter w which is larger than some absolute constant. Then if the resulting sample is denoted by z(1), we have
(( 2 0) > 1 1 e- ()w)))
P(ū(1) 2 Vw) 2 - e - D(w)
Theorems and 2 illustrate that diffusion guidance results in strong bias towards points in the support of one conditional distribution which are far from points in the support of the other.
These results apply even when the unconditional and conditional diffusion models in question incur zero score estimation error. One shortcoming however is that they fail to corroborate a third commonly observed behavior of guidance in practice:
Degradation when guidance is too large: In practice, even ignoring issues of diversity, there is typically a "sweet spot" for the choice of w such that past that point, the quality of the generated output begins degrade.
Next, we show how to leverage ideas in the proof of Theorem to explain this degradation. Concretely, we give a simple example where a small perturbation to the score estimate at the tails of the data distribution is enough to take the sampling trajectory given by diffusion guidance far away from the trajectory predicted by Theorem [
Theorem 3. Given 0 < 6 < 1, assume 2 52( /lg log (1/()))),, where the hidden constant factor is sufficiently large. There exist densities ,
(p1 satisfying the assumptions of Theorem well as
functions s (1) St satisfying
Voogp. (1) + 111 2 Sc and
V Pt + sell Sc
Late!")
such that, if one runs the probability flow ODE with guidance parameter but with V log (1) and 4 log, Pt replaced by (1) and st respectively, then with probability at least 1- e Q(1)) the resulting sample lies outside of the domain of
In other words, for any level of score estimation error, if one takes the guidance parameter w to be too large, the sampler will end up going off the support of the data distribution Roughly speaking, the idea derives from the proof of Theorem]] T As we will see, one key feature of the guided ODE in the setting of Theorem is that the trajectory first swings past the edges of the support of p and into
3

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the tails of the noised data distribution pt before returning. As result, errors in score estimation at these tails can move the sampling process away from the intended trajectory and thus prevent the trajectory from ever returning to the support of p. leading to corrupted outputs. We stress that this phenomenon is not an issue of numerical precision: Theorem 3 applies even if one runs diffusion guidance with infinite precision.
Taking inspiration from our theory, we posit that the optimal choice of guidance (from the perspective of sample quality) for compactly supported distributions that approximately satisfy the assumptions of our theory is the largest possible w for which the resulting trajectory does not exhibit this behavior of swinging away from the support of the data distribution and returning. Specifically, we propose a rule of thumb for selecting the guidance strength based on looking at certain monotonicity property of the trajectory and experimentally validate this rule of thumb in both synthetic settings and on image classification datasets. Additionally, for compactly supported distributions that fall outside the scope of Theorem 1 we propose an alternative heuristic based on the ideas of Theorem 3 we should choose the guidance strength as large as possible while still ensuring that final samples are contained within the distribution support. See Section 3 for details.
1.1 Related work
has been observed previously [5 20 that the score of the tilted distribution convolved with noise is different from what is used in diffusion guidance, i.c. Eq. (2). These works conclude informally that as a result, diffusion guidance should not be sampling from the tilted distribution. In contrast, work gives rigorous justification for this and provides fine- teeeggained analysis of the behavior of diffusion guidance on simple toy examples, shedding new light on several key features of the dynamics of guidance.
To our knowledge, only two prior works have sought to theoretically characterize the behavior of guidance, one by Wu et al. (37) and one by Bradley and Nakkiran [3]. Here we discuss the connection to these works in detail and briefly summarize some other relevant results.
Comparison to Wu et al. (371. This previous work studied the effect of the guidance parameter when sampling Gaussian mixture models. They considered two summary statistics: the "classification confidence" and the "diversity" of the generated output.
The former refers to the conditional likelihood p(> 2),, where 2 is the index of the component of the Gaussian mixture model to which the sampler is being guided, and æ is the generated output. They prove a comparison result showing that the classification confidence of the output of the guided sampler is at least as high as that of the unguided sampler, and they give some quantitative bounds on how much the former exceeds the latter. In particular, they prove that as w - oo, the classification confidence tends to 1.
As for diversity, they show that the differential entropy of the output distribution of the guided sampler is at most that of the unguided sampler, though they do not provide quantitative bounds on the extent to which the entropy decreases with w.
Instead of studying summary statistics of the generated output, we instead give a fine- tine-grained analysis of where exactly the trajectory ends up at different times in the reverse process. While we do not directly study classification confidence, note that in the setting of our main result, Theorem [ for mixtures of compactly supported product distributions, the statement that classification confidence increases is uninformative because, as mentioned previously, the conditional likelihood for any point in the support of the target class is 1. The dynamics that we elucidate in our results can be thought of as more geometric notion of classification confidence. As for diversity, implicit in our Theorems and 2 are quantitative bounds on how the diversity decreases as w increases.
Additionally, the analysis of how score estimation error impacts diffusion guidance, as well as our empirical findings on real data, are unique to our work.
Comparison to Bradley and Nakkiran (3. During the preparation of this manuscript, a very recent theoretical work 33 also studied the extent to which diffusion guidance fails to sample from the tilted distribution. They provided a simple example where the conditional likelihood p(z x) is Gaussian (so that the tilted distribution is also Gaussian) and the probability flow ODE with guidance provably does not sample from the correct tilted distribution. Interestingly, they also study the reverse

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SDE with guidance and show that it behaves differently under this example than under the probability flow ODE with guidance.
They also observed that diffusion guidance is equivalent to a predictor-correctt scheme where the predictor makes step according to the conditional distribution p(æ: =), and the corrector makes distribution.
step using Langevin dynamics with respect to the moised-then-tinn
In addition, they considered the mixture of two Gaussians example that we study here. They provide numerical, but non-rigorous evidence that diffusion guidance results in a very different distribution than the tilted one. In contrast, we provide rigorous analysis of the dynamics proving that this is the case. On the other hand, to our knowledge, the example of a mixture of distributions with compact support has not been considered previously, and the qualitative difference in the behavior of guidance in this setting versus under the mixture of Gaussians setting has not been reported in the literature.
Sampling guarantees for diffusion models. Most of the theoretical literature on diffusion models has focused on unconditional sampling, e.g., proving that SDE diffusion models can efficiently sample from essentially arbitrary data distributions assuming L'-accurate score estimation [2] S 4
I 9
Note the notion of L eтror matches the objective function used in practice, Similar results hold for the ODE under additional smoothness constraints or by using a corrector step 23
We also mention various recent works on understanding other aspects of diffusion models using mixture models, including provable score estimation 30 10
g 17 and feature emergence 25
Finally, an unrelated work that touches upon guidance and conditional generation is that of [6]. They give representational bounds on how well conditional score functions can be approximated by ReLU networks nonparametric settings, which translate to sample complexity bounds for conditional score estimation. In their work, "classifier-fre guidance" does not refer to the sampling process that we focus on (indeed, they take guidance parameter w = so that the tilted distribution simply the conditional distribution p(r 2)). Instead, it refers to the training of neural network that simultaneously parametrizes the unconditional and conditional scores.
2
Preliminaries and proof overview
2.1 Technical preliminaries
Mixture models. We focus data distributions p which are uniform mixtures of two constituent distributions, taking the form
1 2000 2 (-)
(1)
p
Throughout, we freely conflate probability measures with their densities. Here p0)) and p(-1) are meant to represent class-conditional distributions, and p is meant to represent the unconditional data distribution.
We will denote sample from p by the pair (x., where = E {+1} specifies the class (+:1 with probability 4). Given class z, the conditional distribution on x given by p(z).
Probability flow ODE. Here we briefly review some basics on diffusion models, specifically the probability flow ODE, tailored to the mixture model setting outlined above. Throughout, let be time variable which varies from ( to some terminal time T, such that the output of the sampling algorithm is the iterate time
To formally introduce the probability flow ODE, we define the parameters a₂ = e -T+ be ai Let p-(-) be the distribution of (a,r + + 2) where x ~ p is sampled from the mixture and denotes its class, and & (0, bž) is Gaussian noise corresponding to time of the backward process. Hence, the marginal (Ir) is the convolution of the target p scaled by a, and (0, b2). Denote by a.p the distribution of a.X where ~ p. Then the distribution at time given = given by pe(-Iz = (1) * N (0, bZ) and pr(-1z 11) ac.p(-1) (0, 62), respectively.
The probability flow ODE with respect to the component p()) is given by x' (t) = =(t)= + v (logp(z(())2= 1),
(5)
5

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and analogously for the other component. This ODE has the property that if zz) is distributed as sample from ar.p(1) (0, ), then xrt) is sample from acr- (cr-.).p()) NN(, -)
Guidance. Our goal is understand the effect of introducing guidance into the probability flow ODE (5. Given guidance parameter w, the resulting guided ODE is given by
(t) srt) V
+ (w 1) log ()gr(((((((()
(6)
= xx) + (we + 1)V po (=(())))) - wV pe
(7)
where in the second step used Bayes' rule. We will sometimes refer informally the position of x<t) (or tim-rrpparmettrizations thereof) as a particle.
O, this is identical to the vanilla probability flow ODE in Eq. for 2 = 1. When w = 11, then this is identical to the probability flow ODE for the unconditional distribution p. Our goal in this work is to understand the behavior of the guided ODE for general w, especially large w. In particular, as noted at the outset, the "guided score" term (wc + 1)V log pa (x(+))=²) - w log pt (xx(t) in Eq. 7 does not correspond to the score function of the tilted distribution convolved with noise, so it is not priori clear what the distribution over the final iterate x (T') actually is.
Note that when
2.2 Intuition for our characterization of the dynamics of guidance
Having formalized the probability flow ODE with guidance. we now provide high-level overview of our proofs by presenting general intuition for the effect of guidance. The behavior of the guided ODE in the setting of mixtures of compactly supported distributions is the richest, so we focus on illustrating the proof of Theorem [ In that setting, roughly speaking. we will show that there are three distinct regimes for the evolution of the guided ODE, depending on how the posterior probabilites pe(z = - [[)(()) and (= = 1()(()) relate to each other.
First, when the posterior probability pa(= - (()) is much larger than pz(= = 1 (())) then the score function of the convolved mixture model is dominated by the score of p(-1) convolved with the appropriate Gaussian; in particular, the guided score term in Eq. ( is almost
(w + V log Pt (x(t)]2) wV log Pr (z))) = (2w + 1) log (=((tt))=)
i.. zt) gets pushed toward the pt)) component with maximum velocity.
The second case is when the posterior probabilities p.(2 1[x(t)) and pt(2 -1]x(t)) are approximately equal. In this case, the score of the convolved mixture is almost zero since the influences from p!)) and p(-1) cancel each other out. Hence, the guided score term in roughly becomes
(w + V log (x((t)2) wV log (x(t)) = (w + 1)V log
We can see that here xt)) will still converge to the right component with high velocity proportional to + in this regime.
Finally the third regime is when pr(z = 1|x(t)) > pa(= = ((x( i.c. artt is "closer" to p())
Then the RHS roughly becomes
(w + 1)V log (x(((())=) wV lo s V (logp((((()a))
In this case (t) converges to the right component with minimum speed, i.c. proportional to constant independent of w.
Overall, observe the behavior that when the particle is close the wrong component, guidance adds more biasing on it to repel it from that component toward the correct one, whereas when the particle closer to the correct component, it decreases in velocity. This intuitively means that guidance somehow biases the distribution of the correct component to points that are "farther" from the other component, intuition that rigorize in Appendices A and where we prove formal versions of Theorems [ndd] 2
Experiments
Here we empirically verify the guidance dynamics predicted by Theorems 1 and 2 in this section were conducted on a single A5000 GPU. We use Jax 2 for the experiments in Section 3.1 and PyTorch [27] for all other experiments.
All experiments

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3.1 Synthetic experiments
We first revisit the distribution used in Figure (mixture of uniforms), and then consider the case of mixture of Gaussians Appendix C.1 The distribution in Figure 1 is constructed by taking p(*) to be Uniform([-1/2, 1/2 @ [- 1 shifted by 2z in the a-coordinate. Note that although
(a) this is a 2-D distribution, the distributions p(2) and p can be written as pj 69 q with q shared, which does not affect the dynamics of Theorem [ We demonstrated in Figure I how sampling with larger guidance parameter yields a distribution of samples that more concentrated than the true conditional distribution of the data. We now examine the ODE dynamics that produced these samples and compare them to the dynamics predicted by our theory.
We generate samples using guidance by numerically solving the guided probability flow ODE (7) using the Dormand-Pring evaluation steps and take T = 10, which we is sufficiently large based on the stipulations of Theorem [ For obtaining the unconditional and conditional scores necessary for the ODE, it is straightforward to write down exact expressions for this case (which consist of integrals that we can numerically approximate). However, we estimate the scores using more general approach that can be effectively applied to any mixture distribution for which we can sample both conditionally and unconditionally from.
method [14 as implemented in JAX (2. For solving. we use 1000
For brevity, let Ar follow the distribution of ar X, where de is defined as before and X ~ (the mixture distribution). Similarly, let Ar, follow the conditional distribution a,X z. Lastly, letting & denote the density of εt. we have the following expressions for the scores:
E., cp A:)
V (o
(8)
EA, [po.e x0 - Ar]l
EA, 17 p Ac) - An)
(9)
EA [oo. (xc Ail
EAI, /pb.e ( - Ac,z)l
V log 2)
(10)
EA... (06,e(1 Ar,:)]
Both (8) and (1O) follow from rewriting the convolutions as expectations and then using dominated convergence pass the gradient into the expectations. Using the above, we can compute the scores by standard Monte-Carlo.
We use this ODE solving procedure to generate 500 samples from the conditional distribution p(x] +1) with varying levels of guidance. For each generated sample, we project the computed ODE trajectory on to the z-coordinate (as this is the coordinate handled by our theory in this case).
Theorem D suggests that as increase the guidance parameter w, the ODE dynamics will push samples farther and farther in the direction of the guided class support before ultimately pulling them back to the support if necessary (i.e. is large). As we show Thcorem 3 this behavior can be undesirable if the sampling trajectory moves too far away from the desired class support, as it can amplify score estimation errors and lead to issues in the fidelity of the final produced samples.
We can thus intuitively think of increasing the guidance parameter as not only trading off diversity and sample quality, but also trading off stability with sample concentration. This observation indicates that there should be some range of guidance values that allow for the sampling concentration effect while not entering the unstable regime; i.e. those guidance values that do not lead to sampling trajectories that move far away from the guided class support.
To verify this, we plot the mean of the projected ODE trajectories for increasing guidance parameter values alongside the final produced samples from each trajectory in Figure 2] Since large choices of the guidance parameter lead to some trajectories diverging due to numerical instability/score approximation errors, we visualize only the samples and trajectories that were "good" in that they produced final samples constrained within the guided class support (and we indicate this proportion on the plots). The results show that the projected trajectories do indeed follow the predicted dynamics, with larger choices of leading to pronounced pullback towards the end of the trajectories.
Furthermore, as suggested earlier, the qualitatively best choices of we appear to correspond to trajectories that do not (significantly) exhibit this pullback effect. In our case, w exhibits the
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Guidance
Guidance
Guidance
04
Good: 0.82
Good: 0.70
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Guidance
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Good:
Good:
02
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-0.4
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250
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750
1030
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(1)
(a) Samples
(b) Trajectories
Figure 2: Final samples and mean sampling trajectories produced from solving the probability flow ODE guided towards y = +1 in the distribution of Figure The proportion of good samples (i.e. those that were correctly in the class support) is shown with each sample plot, and a standard deviation band is shown around each mean trajectory.
sharpest sample concentration while having significantly better sample fidelity when compared to higher guidance values.
3.2 Approximately separable image data
While the synthetic experiments serve to verify theory, they obviously do not constitute practical setting in which guidance is used. The most popular use case for guidance in the literature is sampling from image data, and indeed this is what motivated our investigation of guidance for distributions with compact support in the first place.
However, typical image datasets used in the diffusion literature such as ImageNet [1] are known to not be linearly separable, and therefore cannot fall under the exact conditions of Theorem [ That being said, simpler image datasets are known to be close linearly separable in particular, MNIST.
We thus consider using the classifier-free guidance formulation (which corresponds the second equality in (T) of [19] to conditionally sample from MNIST with guidance. We use the open-source classifier-free guidance implementation of 28 designed for MNIST.
Although MNIST is perhaps the simplest generative image modeling testbed, it still presents a significant increase in complexity from the synthetic setting. Firstly, compared the experiments of Section 3.J and the setting of our theory, we are no longer considering only two classes. Furthermore, there is guarantee that the class supports are well-separated, or even disjoint. Even we do not have access to approximations of the true score functions of the conditional distributions that are guarantced to be close in (8) and (10); we have to instead learn a model-based score.
worrying,
We address the multi-class issue by using the standard one-vs-all reduction. In particular, we fix single class the positive class y +1, and then let the union of all other classes represent the negative class = -1 We note that after this reduction, the distribution is close to linearly separable, and an appropriate basis.
thus at least close in spirit maintaining the separation from Theorem under
To obtain projection direction for the sampling dynamics analogous to what was done in Section 3.1 we generate 100 samples from the positive and negative classes using a guidance of w = 0, to approximate sampling from the conditional distributions. We then let the projection direction be the difference between the two sample means. For sampling, we use DDPM [8] with 400 time steps and linear noise schedule, and we found that training the guidance model of (28 for 40 epochs was sufficient to generate high quality samples.
Figure 3 shows the mean projected sampling trajectories alongside the final produced samples for the same choices of guidance parameters used Figure[[and the positive class fixed to be the digit O. We observe the same phenomenon before: after the guidance parameter w is taken to be sufficiently large, there pullback effect in the projected sampling dynamics. Furthermore, once again as before note that the qualitatively best choice of w (again 3) is the largest choice for which

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Guidance
Guidance
10-
Guidance w =
Guidance w
5
300
203
300
400
300
700
300
400
Harratio
(a) Samples
(b) Trajectories
Figure 3: Final samples and mean projected trajectories produced from sampling using the classifier- free guidance model (28). For the positive class, we fix the digit to be O, and the negative class corresponds to all other digits. Each row of samples from top to bottom corresponds to increasing guidance values.
we can preserve monotonicity of the projected sampling dynamics. These results are not sensitive to the choice of positive class; show similar plots for every other choice of positive class (i.e. all the non-zero digits) in Appendix[C.] [C.2] Interestingly, for almost any choice of positive class used in the reduction, the qualitatively optimal choice of guidance amongst the values consider remains roughly the same.
3.3 ImageNet experiments
Although we previously mentioned that experiments on more complicated datasets such as ImageNet are outside the scope of Theorem [ we show this section that it is still possible to make qualitative guidance recommendations in such settings based on the ideas of Theorem [ The idea is that we scale the guidance parameter w to be large, we start to obtain samples that are no longer within the original data distribution support due to amplification of score/precision errors.
To conduct experiments on ImageNet, we use the classifier-guided ImageNet models available from [13. This is due to the fact that there are no classifier-free free guidance models available from [19]. The classifier-uuidance formulation corresponds to the first equality in (T) To be consistent with the notation in [13] and to also clearly distinguish the classifier cerffieer-uieee guided setting from the classifier-free setting of Section 3.2, we will use
throughout the experiments in this section.
First, we illustrate that the behavior exhibited in Figure 3 no longer holds when running diffusion with guidance ImageNet, at least using the same experimental setup before. To parallel the experiments of Section 3.2 we use the 256 x 256 conditional diffusion model released by [33 to generate samples from a fixed ImageNet class (corresponding to y = +1 before), and then use the same model to generate samples from all other classes (corresponding to 50 samples from the positive and negative classes (due the cost of sampling at this resolution and the overhead of storing the entire sampling trajectories), and then compute the normalized direction between the two sample means as before [ For sampling, we use DDIM [32 with 25 steps, once again because storing the entire sampling trajectories using DDPM with a large number of steps is prohibitive.
1)). We generate
For sampling with guidance, we use the unconditional diffusion model of 13 with the 256 x 256 ImageNet classifier also released by [13. Note here that [3] combined diffusion guidance with their conditional model for their best results, but this does not fall in to the formulation of (I) and so we use the unconditional model. We use DDIM with 25 steps for the guidance samples as well.
Shows shows the final produced samples alongside the mean projected trajectories for an arbitrarily fixed positive class in the experiments of Section 3.2 We see that even for extreme guidance scales s 25 the previously observed non-monotonicity phenomenon in the projected trajectories no longer
"We should note that, in contrast to the MNIST experiments, this choice of direction is very noisy in the ImageNet setting. However, we use the same setup as before both for consistency and to show that Theorem be applied even in this imperfect experimental setting.

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Erron: 0.00E+00
Emoo:: 0,00E+00
RIB
Meen Projetton
Guidance 5 10.0
Guidance s 25.0
500
Error 4.26E-06
Erro:: 6.71E-03
50
250
520
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3000
0
250
500
750
1050
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(2)
(a) Samples
(b) Trajectories
Figure 4: Final samples and mean projected trajectories produced from sampling using the classifier- guided ImageNet diffusion model of [3]. The positive class here taken to be 292 (tiger). As before, each row of samples from top to bottom corresponds to increasing guidance values.
longer close to separated, and as a result the direction corresponding to the difference in sample means is no expect the dynamics of Theorem T (in fact, we can expect that
occurs. We suspect this can largely be attributed the fact that the class supports
longer direction for which there is no such direction along which these dynamics occur since the data is not linearly separable even after reducing to two classes). However, we note that as increase the guidance strength, the final sample correlation along this mean difference direction continues to increase, more akin to the result of Thcorem 2
In tandem with this increasing correlation, we also observe an increase the mean "support error" of the final samples, which is overlain to the trajectory plots in Figure This error is computed by taking the mean absolute deviation of every dimension the final produced samples from the range of valid RGB values (0, 255]; dimensions that are outside of this range truncated so to form valid images. We find that, at least qualitatively, the largest guidance value (s = 5) for which we have support
perform the best, taking guidance values larger seems introduce various visual idiosyncrasies and taking guidance small leads to insufficient concentration on the correct class (as we are guiding an unconditional diffusion model).
We verify that these observations hold for a number of different choices of the positive class; these experiments, along with further discussion of limitations of our experimental setup, are available in Appendix [C] We emphasize again that this is merely a minimal demonstration a possibly useful heuristic, and once again point out that this setting is outside the scope of theory. Still, an interesting direction for future work could be to run more comprehensive experiments regarding this heuristic (and other heuristics in this section) such experiments outside the scope of our available compute resources.
Conclusion
In this work we gave the first fine-gratn fine-grained analysis of the dynamics of the probability flow ODE with guidance, focusing on two toy settings involving mixture models in one dimension. Our key finding was that not only does the guided ODE fail to sample from the tilted distribution that originally motivated the formulation of guidance, but in fact the guided ODE implicitly leverages geometric information about the data distribution even such information absent the classifier being used for guidance.
Our results open up number of interesting follow-up directions. For example, our guarantees are restricted to oie-dimensional settings, and it would be useful to obtain analogous guarantees for non-trivial high-dimensional settings such as mixtures of bounded densities over disjoint convex bodies. Additionally, we have made no effort to optimize the choice of w in our theoretical guarantees, and it would be interesting to see how small one can take w in theory while still obtain the behavior in our results.
10

---

Acknowledgments
SC thanks David Ding and Yilun Du for illuminating conversations on diffusion guidance. The authors thank Adil Salim for insightful discussions at an earlier stage of this project.
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12

---

A Mixtures of compactly supported distributions
Consider the case when p(1) and p( are supported on [01, Ozg] and [-02, -01], respectively, for dealing with two points, i.e., when a1 = 02. However, this case is trivial as adding guidance has no effect: the final distribution with any guidance parameter w remains point mass 022-
parameters 0 < o1 0g- The simplest case is when
In contrast, the behavior unclear the length of the intervals is nonzero, which the focus of our results below.
We make the following assumption on p()) and p!
Assumption 1. We assume the densities p()) 1111) have 3-bounded densities with respect to each other for some 3 2 Namely Vx1 E (-a2, 01), 12 E (2).
1 p (m1) 3.3.
3
(12)
Under Assumption I we show that by rumning the guided ODE starting from Gaussian initialization, we sample from distribution concentrated at the edge of p()) In particular, we prove the following result:
Theorem (Convergence to the edge of the support). Assume Iu(w) 16
In (Z)
In lnl
?
1024
Then running the guided ODE with T 2 2log 2 and parameter w for the mix-
ture model defined in 4 under Assumption with probability at least 1 - exp( resulting sample lies in the interval
wai 512 ) the
012
1
32 vin(w)
.
(11)
The formal proof of Theorem 4 Which combines all the pieces that we present in this section comes at the end of the section. At a high level, show that the particle goes through major phases: first, we have the condition that the initial point satisfies z=0)= 2 O ( then it goes towards the support of p(1) and swings past it, ending at position ( Then, in the second phase, it comes back to the support of p( but by the time it reaches the rightmost edge of the support, it is already close to the end of the process; in particular, we show that it cannot move too much the support of
p()) once it arrives there. Hence the particle gets stuck near the edge of the support. We formalize this intuition below.
A.1 Reformulating the ODE
In the setting of this section, the score can be written as
I-a.a a, 2 (2b2) da
V Vlogp((>) (a) b
(a e (x-aaaa)) (2b da
and
z-raa V Vogg +1) Sac(o1,02)
e - (x-aja) / (2b)) da
bf
Saejan (x=aaaa)) (22))
To simplify the notation, for all t [0, T). denote the random variable aur + & by X, (observe that X, 2). Then by Tweedie's 's formula, the score at time t can be obtained from the posterior mean of Xo given X₃:
Vlog(p((v)) B[x, vl y).
(12)
V (og( = =1)) E[ y|X, = y, = =1]
(13)
We denote E[Xo|| = y] in short by F[(oly)] and [FXooXx, = v. +1] short by B[X, ly. 2
+1.. Then, the probability flow ODE can be written
w +
d' z((1)= = x(t) +
1] xx))
(a.E(Xo ((t))
b3
13

---

Next, we consider the change of variable s(t) _ =' and its corresponding inverse function
t(s) = is) + T. and ø(s) x(t(s)). where s varies in the interval [sp. 1] for sn = e TT Define
the following conditional probabilities of the two components of the mixture, conditioned on X₁:
qi +1 P(2 = 1|X; xxt)),
( (-1) = P - =tt)).
qi
Before proceeding with the proof of Theorem 4 we present a key algebraic manipulation of the probability flow ODE in 7 which enables to analyze it more easily:
Lemma (Alternative view on probability flow ODE). The probabiliiy flow ODE can be written as
x' d'(t)) = (a.E(Xolz(t), = 1] - af)((t)
1 by
w (-1) (a,E(Xolz(t),, = 1] = = -1),
b3
(14)
or with respect to i(s) in variable i' ('(s)
1 (E((o)ã(s)) 1] a.)(s)) + ,7ti (E])o) 2
(-1)
b
(15)
Proof. Note that combining 7 and (33 we can write
x' (1) - x(()==
w (0) = 1] x())
b2
w
a (a.E|Xol=(()) = 1] xxt)) + -1) (a. = -1] - -x(1))
w + 1
(a.B(Xula(()) = 1| x(t)) -
b3 (q, (()), x(t))
03
69
w
qi
(-1)
((o,E(Xolx((t)<) = 1| x(t)) -
-1] - x(()))
(a,
(,))) xt)) + पे (c. -,, =-1)),
b7
which using at + li2 = implies
x' (t(t)==
(a, == =1] by
1
w (-1) a)]()(((t))+++ -1]
6? b2.0 (a. = - -1]
(16)
Now changing variable from l s() - or, taking the derivative we get x' x'(1) ds 5' =0 (8) Plugging this into the above, we obtain its equivalent form in variable s:
z' (E(Xolā(s), 1] a,)()) b
oz 4:
(-1) (E[Xolā(s), 1] E[ ((s) 2 1]) C
A.2 Analyzing the guided ODE
We begin by sketching our argument in greater detail. First note that the second term in 14 namely
w (-1) (B(Xolæ(t), 2 = - (t) = -1))
b7 q
(17)
= 0(1ww), -01/w)) which happens only when ztt) (.))())) First, we show that (1) starting from high probability region for z0)), the particle will get to xtt) 2 ((I[()) at some time t, using the dominance of the second term (Lemma 3). Then, in the case where x=(n)) = ?(Inw)) for some time lg. (2) we show lower bound on the time that takes for (1) to get back to the proximity of the origin and then an upper bound on how much it can move inside the support (Lemma 5) Finally in Lemma we show that a:() does converge to the interval, provided the event of Lemma holds. Building upon these Lemmas, we prove the final result in Theorem
a positive dominant term (compared to the first term) unless we have -
We now proceed with the formal proof. We start with a Lemma showing that as long as xxf)) is less than the threshold log(1e) then x' (t) is least of order 12(ww); hence the particle has a strong ag
push toward the positive direction.
14

---

Lemma 2 (Positive push toward right). If xx(1) I
Tog(
21v
02
log(w] and given that
16ag
1 a1 nivu Φ 2 log(w), then for times t s.t. a; S 2 we have
0,01 V/w
z'(t) 2
2
Proof. From the assumption s² a. _ 2' note that we get - and at s² s b2. Hence, for any two points 21 E (~ -0,01) and 22 E (0,0), 0,02)) we have
e (=(())=z))/(ab) (20))
(=(()=21)(/(2())
e (2(1))))))))) / -000))))))
- (x((t) (1)))))a)))
sazz(0)
(oo(lor(w)/(1600)))))) Vw.
Therefore,
Ja p ()) (z))e (m(t)) / dx
f [ p(-1) (x)) (x(t) -x) 126 dx S B/W,
which gives
4 (+1)
P(z = 1 1 = x(())
P(: = 1 X, = xx()) S $.yw.
(-1)
41
Therefore, using 8,w w 2 1,
(-1)
4
+ // 2 2
Therefore, for the second term in Equation
14
we have
D7 4t (-1) (aEE(x))))) = 1] o, E =
-1)
2
b 9t
W (-1)
2
ag00 /
Bb3
On the other hand, for the first term we have
((), az;x(t) 2 x()
by
Plugging this into Equation 14] and using the fact that sr())
and a, S 1, we get
29
x'(t) 2 4:01 /w
2,36
Next, we show that starting from (ss) 2 (VV the particle ã(s) reaches at least ((I((w) before time 1. This results from the strong acceleration force toward the right in this phase, coming from the dominance the aforementioned second term in (14.
Lemma 3 (First phase). Assuming given that (()(0)) 16,8
In(w) < VVaal T 2 2ln,, under the conditions of Lemma
an 3
there exists time Su such that
In(w)
F(so) 2 16ag
where recall (0) -T is the initial time for the ODE 15
15

---

Proof.
Note that as long as i(s) < In(w)/((66))) Lemma gives
{((s x' ds((tt)
Vwai
/wa1
2
2,3
which means starting from (()(0)) 2
Vaiia 163
we definitely reach
/woo by time 8 = 2 if we continue
163
with the same speed. (Note that the condition at S 1/2 of Lemma 2 is satisfied up to time 2.) But In(w]
since In(w]
L/an 1602 4/3
<
then we definitely have to pass the point 16ng
Next, show that once the particle passes the threshold ((s) In(w)/ then by time 1 it
cannot reach any point much to the left of a2 the edge (right end-point) of the interval. To show this, need the following helper lemma showing that near the end of the reverse process, the particle is quite close to its conditional denoising.
Lemma 4 (Conditional Gaussian estimate). Given
S
S 1 for constant e₁ 2 1 with
v Vn[w)
In(ar) In [ 2 16c]
[22] [i
2 we have
[[Xo]((s)) 2 = 1] m(+) S 2 b..
π
Proof. We can observe the quantity [[Xa| (s)), = 1] -(s) as the expectation of a gaussian variable, g(t), which is conditioned on the union of intervals ( (-az. -a1) U (a1, a2), i..
E Jar 1 el ((a)))/((0)))))) V2 S (a V2nb, el= ((5)) + S 1 c(z-)(s))) / (26 (x - dz
i(s))dz
22
[(-0p-or)V(nn02)) V2nb.
(2))
(18)
For the first integral, we estimate the numerator by half of the integral of absolute value of centered Gaussian with variance bj, denoted by E., and the denominator by Gaussian tail bound:
02
ela-#(0)) (268)) {s))ddz S Elg
2 S
by-
(19)
V2rb,
π
For the denominator,
(aar-anuutacco V2rb,
c (00=(())2 dx [
e(m= ((2b)) (22))
(20)
(a1,az) 2rrb,
((20)) dx
2 - 2
(21)
V2Tbe
But using integration by parts:
V2 e x² (22b)) dx
bb x² (2b)))
b
x² (2(())) dx
e
+
xV xV nr
21102
2
x2 /2T
b
(1itas)
) 1(2b/)
b²
1
x2 ((2b)) dx,
a1 /2x
2
1 11ln 122 V2Tb2
which implies
C31
1
e ((s))(()))(())) dr <
b
( )²/ )/((6)))
V2
1 4b2 (a) + o2)
2 t 2
2h.
(08/()07)).
< 1 4b/ /
02 2
2m
Therefore, combining with Equation 21
|( (1)001)))((()00))) 22
(x (20))
do
>
2hp
1
(a3/(8071)
2h
2
16

---

completes the Now using the assumption on by.
[
V2Tb, ((===(s))) {(x)) /(6)) dr 2 1
4c1
(22)
(()to.a)) 00)
2 In(w)
2'
where the last inequality is due to the fact that In(w) 2 16c.. Combining this with Equation can bound the first term in Equation (88)
19 we
V2 fle b()) 1266) (s))
a1
b4-
(23)
J(aa,aan)U(aaaa)) 22Tb,
gl=- /(66))
π
On the other hand, for the second term in Equation 18 we can upper bound the numerator
a1
c0 (-- (0-2(0))) d( & 20²
e l0 ((o) /(2))) dr.
(24)
V2Tb,
22Tb,
But
/ 2
1 c= ((66) dx S
a1
(x- /(2)))/(222)) dx
-00
V2
1
2² 1(6/)) dr.
2
Using similar integration by part for the tail and the assumption on by, we get
1 [(oe-aa) V2Tb.
1 (a ((0)) dr S
h. 20:1
1
(2a7/(03)
V2T
1
2011
bt
20?1(c9a3)
0.1
222T
by
22 (17(3)
w
2
b
2 In(w)
16 2
Plugging this back into (24)
(x- ((8)) (x (())
b
2
16.2
Combining this with Equation Equation (8):
22
we obtain the following upper bound for the second term in
Soa V2nb.
an
(x== =(s)))
-
b
<
f(-aa--o) (() V2nb. 2()
8/2m
Finally combining this with our upper bound for the first part Equation proof.
With this helper lemma in hand, we are ready to prove that near the end of the reverse process, the particle does not move much the left of a-2.
Lemma (Second phase). For some time 8q E [0) 1}, suppose (ss(0)) 2
In(w] 160g
oi S In(w), and
In(ar) In (m) 2 1024
02 [qA1
Then for all s E [sv. 1). we have
i(s) 2 a2 1
v In(ww
17

---

Proof. From Equation
13
as long i(s) 2 O, we have
ar0(s)
Cx5(s)
5(s)
('(s)
2 by
[ 2 = 1| a,}(ss))
2
by
8² 1 1
which implies
(m()s))) (m() 2 In s) In(1 su).
(25)
If we denote by $1 the first time 8 2 Su when z(s) = Or(s) = s02, then is certainly larger
than the first time that sr()) = 02 as for all I, a, - 1. Let denote the first time 2 80 such that x(s) Qg- From Equation (25).
1 - 52 <
x(22)
80
[(so)
which implies
02
16a2
81 2 82 2
In ln(w)/(16e))
In(w)
But reach the interval (4,00,00412) we lower bound a'(s) using Lemma In partieular,
note that b2 ((s)) = - of
32a2 Hence, we can use Lemma with 8, and given that In(w)
In(wr) In (n()) 2 1024 (1A1)
the conditions of Lemma 4 are satisfied, so we get
2 bt.
[B])ol = 1] - =(s)) 2
Therefore, for 22 81,
[E[Xo]
( (3, 2 11 as)(s))
z
[E[ 2 11 =(s))
2 1
x b
4
7 52)
4
x² s)
On the other hand, note that from our definition of $1:
160
i(s1) = 0231 2 022 1
In(w)
Therefore, using of < In(w),
x(1) 2 2(31)
T2 8)
16(1
31)
-(s1)
16a3
a
16
In(wo)
2
In
32a2
02
Vin(w)
Next, we show that when the process gets close to the end, if the particle is on the right side of both of the intervals, then the first term Equation (14) will dominate the second term and the particle most likely to converge some point in the interval (a1, a2).
18

---

Lemma 6 (Convergence to the support). Assume 02 a0 2 en an for all E [su, 11. Then, for any E [$1, 1] where
16 vluw
For 2 suppose i(s) 2
2
16(1 s0)
1
$1
50,
(1 2(11 -80)) (x((80) ox2)
have
3(8) S 02 ( - s0) (1+ a
))
at
Proof. First note that for every E (-0,02; -0 and x2 E (0x00, ax02) when 2 2
(xa())==))/((21
(a(1)) (1)0)_=))
(a((t) m1) -(2 (120))
But since x²t) 2 a t02, we have
(a(1)= x (22 x1)
(a:00 0,02 2x2)(222 x11
by
2b
(12 11
(ara) 22)((22 21)
21
26
4 ?
2b²
Therefore
6 (=2)
2 (6200/(2(1-5)))
e
2 (0)/1(1-0),
Therefore
4 (-1) < <--0\\\(--)
which implies
-a2/4(1-o)
(o.)
1] a, (t))
(2sa2)
s2
a0,/((1--)
5 a2
But integrating this upper bound from sq to 1 by changing variable
(hence dx [1 (1- ds)
of /1(--))
02
ds
02
1-6
O22
-aqee/ di
Ko
No
4azw
a²KO
402W
(1) (1 80)
(26)
of
19

---

On the other hand, for the first term in Equation [5]. as long as i(s) 2 2s0 22 we get
1
02 spā(s) S 02 =(s)
Therefore,
1 [f(xol((s) = 1] ax0((s))
1
(a2 sg)(
1 - 52
1
$
(02 x(s))
2(1 82)
1
(02 x(s))
(27)
4(1
8)
Now let
1
di (s,
(B[Xp) x 02, = 1] ax o2))
(28)
(8,
(29)
4(1 8)
and define the following ODEs for E [s0, 1]:
(x(s) = (8, B(s)), d'(s) Φ2 (8, di
with
(sso) d(s0) i(so) -22
Note that using
27 and (29). we have
(1) S (2(8500)
Define the integral of the first term as
[" أn (s, (s))
A(s1) e
E [[ a, ds -
(30)
First comparing Equations
15 and 33), since the second term in the RHS of the ODE in 15 is
always positive, from ODE comparison theorem we get (ss) 2 a(so) + A(s).
Therefore, assuming i(8) Z 2 02 and letting s 2 be the first time that B(s1) 2=a-I 022
by ODE comparison theorem we have B(s;) B(ss) - d(s1) d(so). Hence
A(s1) < B(s)) - ((su) = B($)) B(su) S d(s))) - d(su).
(31)
But can solve the ODE in
d(s1) d(so)
s1
In
In
Note that inequality 31 is valid more generally up to time sf, when (ss) reaches 250 02 02
1-2(1 3 02g. Now we solve for the value of š when d(5i) reaches T-2(1- 02, which upper
2(1-8a)
2(1-8o)
bounds 81 due to (31)
2(1-8g) I 1-2(1-8 30)
1/4
31
(80)) 02
1 8n
Therefore
16(1 50)5
51 =1
(1 - 2(1 50))² (I(sp) az)
and from definition for this choice of $1 we get from Equation (31
02 02 (ā(su) - o2)) = 2(1 80)
A(51) < d(s1) d(so)) = 2.0 1
- (8(ss) 02). (32)
2s0 1
20

---

On the other hand, note that for any time 2 s before ø(s) reaches 91 (if it ever reaches that value), the first term [5) is always non-positess con-positive. Hence, we have
2(1 - 80) 022 (5((0) 02)) (s52) v
(33)
280
Note that the inequality (33) true when ((s)) I 20 the right hand side is positive and the left hand side is negative in (33) in this case. Hence, overall we showed that for any time 82 2 $1 8g. as long as ă has not reached autt2, we have (33).
Finally combining the upper bounds on the first and second terms in RHS of the ODE in 15 that we derived in Equations (26) and (32
4az
((22) S ((00) + A(82)
(1 80)
of
< ((5)) + 2(1 - -00
02 - (i(s)) - a2)) + 4028
(1 - s0)
28g 1
2
40
00
+ (1 s0)
+
1 2(1 su)
af
1 02 w
02
1 1 - sp)
of
where we used so
This completes the proof.
Next, combining all the pieces, we prove Theorem
Proof Theorem First, note that sample the initial point s=(0) according (0,, 1), hence wa?
with probability at least - 5120
we have
Vwar 163
I(so)
Then, from Lemma 3 we get that for some time S 1,
In(w)
#(so) 2
16az
Let Sp be the minimum such time. Now plugging this into Lemma then implies for all s 2
32
i(s) 2
(34)
/n(ww)
Now take sn 2 so V. Then we use Lemma 6 with so because 02
1
32
+02
Vln(w)
so its condition is satisfied from 34 then, Lemma iiplics that there exists a time $1 such that for all E [$1, 1]:
((s) S 02 1 4(1 4(1 - so) (1+ O2w ))
(35)
af
Note that šg can be picked any value in the interval (su V 1' 1). Therefore, picking šo - 1,
wu?
Equations 34 and (35) show that with probability at least 1
s32 .is)) converges to the interval
1
B Mixtures of Gaussians
In this section we adapt our analysis to the setting of mixtures of two equal-variance Gaussians. Given mean parameter and variance of, consider the random variables
z Uniform((#1))
x: ~ No22,, oi)
21

---

The random variable z is distributed according to a mixture of two Gaussians. Let Po denote its density.
Whereas in the case of mixtures of compactly supported distributions, we found that the guided ODE converges the edge of the support of one of the components, in the case of mixtures of Gaussians we find that as the guidance parameter increases, the resulting sample moves further and further towards infinity
rate that scales with (vw). This formalized in the following main result of this section. For convenience, we take μ = 1 and øo 1, but our analysis naturally extends to other choices for these parameters.
Theorem 5. For larger than some absolute constant (see Lemmas Aand R). ODE with parameter w for the mixture + results sample @ (1) for which
running the guided
(( 2 0) - 1 -
P()(1) Vw 1/4) 2
(?(w)
The first bound implies that an overwhelming fraction of the probability mass in the output distribution is given by positive values. In fact, our proof says more. Based on where the particle is initialized, there a bifurcation in where the guided ODE sends the particle: points to the left of -2w do not result in positive samples, whereas points to the right of -2w (In w:) result in positive samples.
The second bound in Theorem [ gives qualitatively stronger guarantee at the cost of weaker high-probability bound: not only is the overwhelming majority of the output distribution concentrated on positive values, but most of that mass concentrated on values least 52 (V)w).
B.1 Reformulating the ODE
We begin by performing some preliminary calculations to simplify the guided ODE, culminating in the simplified expression in Equation (39 below.
Recall that (-))) denotes the distribution of (a.æ + Et, where x ~ p sampled from the mixture and z denotes its class, and & ~ NO0, b,) for a, = e T++ and b, = / - a.. We will often omit the subscript when refereneing a and 6, when the context is clear. Hence for & ~ N (0,
(y ay-ap)² 2(n2aa) + b2) exp 2(a2a) 12)
ap)
(v( exp
Let o; a²oz + 6². Then a straightforward calculation shows that
(-y + ap tanh
apy
v logp:(u) = o?
o,
We also have
20pa
exp
2(a
exp
(((=]))=
exp
[g=-a)) 2a
+ exp
-(y+gu)" 2 ))
exp
aug of
+ exp
only
V Vlogp((2lg)
aj
tanh
auy of
o?
Then
V llogp( + (w + V) log
(w 1)zau y - wap tanh
apt
o7
oo
As sanity check, for w 0, this gives J/4 (zapu y). which is the score for N (2ap, of).
The probability flow ODE for the guided model is then
aspar(t) of
<'(1)= = x(() oā
(w 1)201)u xx(1) - wayu tanh
()p((() of
= xxt)) +
(2a.h x(t)) + warf tanh
(36)
of
22

---

where z00) ~ (0. I) and varies in [0, T.. For simplicity we assume μ 50 = 1. Then, writing the guidance equation for this mixture in the form 7
z'(t== x() - (w (z((t) az) + w(x(() az tanh(a,z(t))).
Now changing variables to s T+, F(s) =<In(s) + T), we can rewrite the ODE in terms of a(s): (Note that we have 8z' = =1)
SF (s) = #ss) - (w + 1) 5) + w (ā(s) tanh(s2(s)))
= s(w + 1) - sw (tanh(s((s)))
which boils down to
(s() (w 1) w tathhnn(s0 for s [so, 10 = =T with initial condition Ø(so) = zz()= x.
(37)
Now sending T - 0o, the interval [so; 1] converges to [0. 1], and define the corresponding ODE in variable (s)):
x(0) 3; 5' (s) (w 1) (anh())())),
(38)
(39)
where x ~ (0,, Below, we study the behavior of the ODE in (39) for different initial conditions (()) I.
B.2
Analyzing the guided ODE
The proof of Theorem is based on two key results which break down the behavior of the guided ODE dynamics into two cases depending on where the initialization lies.
First, Lemma 7 below controls how far a(1) moves to the right when the initialization x(0) x is in the interval 2w + O(n w). 0]. This gives rise to the first bound in Theorem 5) Lemma g below handles the case when the initialization is in the interval [-O(v /w), Oj, giving rise to the second bound in Theorem In the first case, show that the movement of z(t) can be as large as ((w)
while in the second Temma we show movement of order ((vw).
Lemma 7 (Characterizing when the particle moves to the positive side). Suppose 2 100. Then provided that z(0) > 22 + 26 inww,, we have i(1) 2 0.
Before proceeding with the proof, we note that up to the O w) term, this analysis nearly tight in the regime of large w. The reason is that if a()) < -2w - 1, then because the velocity in Equation (39) is upper bounded by 2w + all times, the particle will remain negative at all times.
Proof. First, note that the right-hand side of Equation then a(1) 2 (0)) + w 2 0. More generally, this implies that as soon as the particle becomes positive, it continues move to the right rate lower bounded by 1.
39 is lower bounded by 1, so if a(0) > 0,
Next, we handle the case of (0)) < O. Observe that w 2 100 implies for Z(s) < O we have E' (s) 2 w + 1. i()) > ww, then we are done as this implies that the particle moves at rate least w + to the right until it becomes positive, which point remains positive by the argument in the previous paragraph.
18 In(w) + Moreover,
In the rest of the proof, we assume
ww. Define In(w). Now for the first E (0,c/(w + +1]
window of time, using the fact that -1 S (anh(()(s)) (x)) < 0, we get
s(c/(w + +1) 2 x + (w + 1)
=
x. +
w +
c
Z(c/(w (w + 1)) - x + (2w + 1) -
< Ir- + 2c,
where the last inequality is due to the definition of x and the fact that w 2 18c + 2 2 6c - 1. From this, we see that for time su = c/ (w + 1),
(50) I -c.
23

---

Now let 51 2 So be the first time (if any) where
818(81) = cc
In the following we upper bound 81. Defining by bye/w = 1 + tanh(-c). we see from the definition of ODE (39) that for all so I 81:
f'(s) 2 2w + -1166 c.
Therefore, we get for all times so 81:
0 2 x(6) 2 T(c/(w + 1)) + (2w + 1 1-6) ()
c
w +
(40)
c
2 x c (2w e)
8
2 x c (2w + 1 - -))8.
w +
This means from the definition of $1
(x + (2w + 1 c)81)81 -c
Therefore, if s* is the larger zero of the following quadratic function (in variable s),
(x c + (2w + < +)8) + e
then 81 S 5* Now we estimate s by completing the square:
xx + c
c
+
= 0,
2w + 1
2w 1 -
which implies
8 -
)
)²
x + C
)²
c
- +
C
2
$
2(2 -1
2(2w + c)
2w + 1 -
2(2w + 1c
- +
(41)
This means
+
81 $1 8* S
(42)
2w + 1
+
Using this bound on 81 we want to show that x(1) 2 0. Indeed, it is enough to show that for time s* which is upper bound on 81, for the remainder of the time (1 s1), the movement of 2(s*).
which is at least (1 s")(w + 1), is at least -ī(s*). First note that
2w + 1 - 1
C (w + 1),
(1 (w + 1) N
2w + 1--
-x +
and using the inequalities 2n + S 2w + 2 and -I + C. S
2w we get
(1 (1-s")(w + 1) 2w + - E + c c
(43)
2
2
On the other hand, note that from 337 In particular, from the definition of 81,
5' ('(s) < (2w + 1), we have ((s) S a(50) (2w 1)(s 5o).
81 (ā(so) + (2w + (8 - -50)) 2 -c.
(44)
In particular, consider the following quadratic
(ss) (2w + 1)s² + s (ā(s)) (2w + 1)so) + c= 0,
(45)
and let its roots be s < si First note that Q(0) c > and Q(so) = syā(su) + c S 0. hence Q has root in the interval [O, sp] and s S sp. On the other hand, 81 > Sp, so implies that $
should greater or equal to the larger root si. Now we lower bound si. Defining G
#(ap) (2w++))e
2(2w++1)
completing the square for (5) gives
2c
2
G = G
(46)
2w +
(ī(s0) (2w + 1)30)
24

---

Note that for the last inequality in (46 to hold, we need to show
2c
G [(su) + (2w + 1).00)
or equivalently
(- + (2w + 1)80)² 2 Ac((w + 1).
But from < ww, get (=i(so) (2w + 1) s)))) (w-2c)², which follows from the assumption
w 2 18c+ 2. Now based on (46) and using 2(50) _ w + and so = c/(w + 1) S c, we have for the larger root si:
2c
81
8 2 -2G +
(8n)
(2w + 1)80
2c
2 -2 +
2c
2c + 80
2c
2 22
W
4c
2c
2u
+ 1
1 c
But note that
W
1
2w + 1
w - c 2
C
2
4
2
4
Hence
- 12c
8*
2w +
Therefore, from 440)
|i(s") - s| + - (2w + 1 - -)8
|z| + (2w c) - 12c
-I-12c 12c
13c +c
I 13c + c. (47)
2w +1
2w +1
where the last inequality follows from < 2w. Combining Equations show the following to prove (1) > 0:
43 and
47 it is enough to
2w + 1 1 |r| c
+ 13c
2
Using x 2 2w 26c it suffices to show e S 1 But unwrapping the definition e, this equivalent to showing
2
2 +1 2
From the definition c= In(w), this inequality holds for w 2 100.
Lemma 8 (Moving the mass to @(v Assume w 2 0. Then for z(0) = x 2 -V + I/2 we
have
2(1) 2 v w + 1/4.
Proof. Below, we will use the fact that w 2 implies 1
vー- S 1.
As in the previous proof, note that for ā(s) < 0 we have
F'(s) + 1.
Hence so is the first time that (ss) reaches zero, we have 8n S 2/wFT Note that for ((s) 2 0 we have (s) I w w++. so for s I 50 2v we have ((s) S V + 1/2. But since tanh < 1/2. we get that if s S so + 1 I VetI then (anh(sū(s) 1. Therefore, we get that for all
5 50 + 2/wFI'
x' (s) 2 w
W
2
2
25

---

As result, for 80 + 2vw+1 S 1, we see that
f(s1) ((so) + (51 su) w
+
+
4
which completes the proof, as [(1 {($)) because the particle continues to move the right as long as it is positive.
We can now conclude the proof of the main result of this section:
Proof of Theorem 5 The first inequality follows from Lemma as for the standard normal we have
P(x -2w + In(w)) I e- The second inequality follows from Lemma 8 P(x
-V w + 1/4) c (()).
C
Additional experiments
C.1 Gaussian experiments
Here we consider a 2-D version of the mixture distribution of M'TTheore 2 i.. p()) &
and pf=1) () 6 (0, 1). We follow the exact experimental setup Section 3.1 and generate 500 samples from the conditional distribution p( = +1) with varying levels of guidance.
We once again plot the mean of the probability flow ODE trajectories (projected to the z-coordinate) for increasing guidance parameter values alongside the final produced samples from each trajectory in Figure The figure is analogous Figure 2 except for larger choices of the guidance parameter w and the fact that the proportion of "good" samples here is the proportion of samples that did not result in NaNs (since
longer in the compact support setting).
We use larger w values to better illustrate the behavior predicted in Theorem 2 from Figure 5 (a), we produce more samples with larger (positive) a-coordinates we increase However, we also get significantly more numerical instability, and as a result the mean trajectory plot in Figure (b) is much less meaningful than it was in Figure
As can be seen
C.2 MNIST experiments
In Figures 14 we collect the MNIST experiments considering every other possible one-vs-all reduction. As mentioned in Section 3.2 they have near-identical behavior to the experiments of Figure
C.3 ImageNet experiments
Figures 15 to 18 show the results of repeating the experimental setup of Section 3.3 for different
choices of the positive class, and also illustrate limitations of this experimental setup in the context of ImageNet.
approximately the same behavior as in Namely, guidance values for which we have support error lead to distorted samples. Similar to Figure the choice & = 5 works well in Figure 15 However, for Figures 16 and 17,
In Figures 13 to 17
that have non-zero support error even for s = 5. In these latter two cases, we expect the qualitatively best choice of guidance to be somewhere between 1 and 5.
Figure 18 demonstrates some limitations of our experimental setup for classes which have high levels of noise/variance. Indeed, we see that for the "basketball" class the support error is not even monotonically increasing with the guidance parameter, and that sample quality poor across guidance levels. Furthermore, the projected trajectories become progressively more negative as opposed to positive, indicating that the direction between the means the positive class samples and the negative class samples is likely useless in this case. Despite these various issues, there appears to still be some positive correlation between support error and sample distortion.
26

---

Guidance w
Guidance w
3
Good
Good:
Guidance
Guidance 127
Good: 0.93
Good: 0.89
-2.5
0.0
25
5.0
7.5
30.0
-25
0.0
25
50
75
10.0
(a) Samples
Guidance
Guidance
Mean Coordinate
25
Guidance w
Guidance 127
100
75
50
25
250
500
Iteration (1)
(b) Trajectories
Figure 5: Mixture of Gaussians analogue to Figure 2 to non-diverged samples. Some trajectories explode due to numerical instability, leading to less meaningful mean projected trajectory plots.
Proportion of good samples corresponds
Guidance
Guidance
/
Vean
300
200
300
400
300
200
300
400
(a) Samples
(b) Trajectories
Figure 6: Experiments of Figure but with the positive class fixed to be 1.
27

---

222 2 2
Guidance
W
2222
Guidance m =
Guidance w
15
2 2222
2 2223 2 2
330
220
300
Banabion
(a) Samples
(b) Trajectories
Figure 7: Experiments of Figure but with the positive class fixed to be 2.
3 332 333 3
33 3 3
Guidance m =
Guidance
3333
3333 3
300
200
300
400
300
220
300
Iterodin
(a) Samples
(b) Trajectories
Figure 8: Experiments of Figure 3 but with the positive class fixed to be 3.
4 4
4 4 4 4
Moan Prodton
Guidance
Guidance 15
44
300
200
300
Iterabion
(a) Samples
(b) Trajectories
Figure 9: Experiments of Figure B but with the positive class fixed to be 4.
5 5 555 5
Guidance
Guidance
5 555
5550
5
5
Guidance =
Guidence 15
5 55555
5
5 5 5 5
300
300
300
Hecration
(a) Samples
(b) Trajectories
Figure 10: Experiments of Figure but with the positive class fixed to be 5.
28

---

7
Guidance
W
9
6
Guidance m =
Guidance w 15
6
6
6
6
330
400
220
300
Banabion
(a) Samples
(b) Trajectories
Figure 11: Experiments of Figure 3 but with the positive class fixed to be 6.
7 7 7
771 7
7
Guidance
Guidance
7777
7 77777 7
300
200
4000
0
300
200
300
Iterotion
(a) Samples
(b) Trajectories
Figure 12: Experiments of Figure but with the positive class fixed to be 7.
8
8 8
8
8
8
8
8
Guidance m =
Guidance w 15
8
8
8
8
300
400
220
300
Iterabio
(a) Samples
(b) Trajectories
Figure 13: Experiments of Figure 3 but with the positive class fixed to be 8.
9
9
9
Guidance
Guidance
9
9
9
9
Guidance
Guidence 15
9
9
9
9
9
9
9
9
300
300
Iecration
(a) Samples
(b) Trajectories
Figure 14: Experiments of Figure but with the positive class fixed to be 9.
29

---

Guidance = 1.0
Guidance 5.0
0.00E+00
Error: 0.00E+00
Moan Proecton
Guidance
Guidance
Error: 4.77E 05
Erron: 9.75E 03
(a) Samples
(b) Trajectories
Figure 15: Experiments Figure except with the positive class taken to be 84 (peacock).
Erro: 0.00E+00
Eror: 3.07E-05
Mean Projettonn
Guidance 6 10.0
Guidance 25.0
Error: 5.78E-04
Eroo: 1.41E-02
200
100
250
3000
250
500
750
2000
leeation (1)
(a) Samples
(b) Trajectories
Figure 16: Experiments of Figure except with the positive class taken be 148 (killer whale).
Guidance
Guidance
Error: D. 0.00E+000
Error: 10110EF4 -77
50
Mean Prjecton
Error: 6.43E-04
7.26E-03
500
3000
250
500
750
1000
(a) Samples
(b) Trajectories
Figure 17: Experiments of Figure 4 except with the positive class taken to be 334 (porcupine).
30

---

0:00E+00
3.25F
Moan Popectin
Guidance = 10.0
Guidance 25.0
+00000000
Erre:: 94E-02
-40
250
750
250
500
750
2000
Iteaaiion (2)
(a) Samples
(b) Trajectories
Figure 18: Experiments of Figure except with the positive class taken be 430 (baskethall),
31

---

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33

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38


Reviews:
[{"review_id": "7mw99qreWD", "reviewer": "Reviewer_5E9W", "summary": "This paper explores the mathematical basis for the principle of \"guidance\" in generative models built out of dynamical transport of measure and provides a mathematical analysis on why certain effects are empirically observed. \n\nThey provide this theoretical analysis for a mixture distribution and then test if these results hold in the case of images for classifier and classifier-free guidance.", "questions": "Can you please clarify what you mean by this notion of noising a distribution *p* (which doesn't really make sense to me, I think you mean noising the samples e.g. convolving *p* with a Gaussian) and tilting not commuting? I don't fully understand the point still. There is a valid transport equation for *p_t* and therefore also an associated probability flow, so it's a bit unclear to me still by what you mean. See the above paper.", "limitations": "The authors have not addressed many limitations of this work, though the reviewer does sympathize with not having easy access to compute or the image models necessary to really do some good experimentation.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "This paper motivates well what its aims are. In addition, the authors provide a suite of experiments ranging from simple synthetic examples to support the main theoretical claims about the evolution of the probability flow ODE under different guidance scales.", "weaknesses": "The reviewer finds the paper pretty disorganized, to the point that it is hard to follow the validity of some of the theoretical statements as well as their implication. In particular, I'd like to draw the following comments to the authors in hopes that they can improve these aspects of the paper:\n\n- There are some statements early on that I found confusing, and without theoretical justification. For example, the statement: \"In other words, the operation of applying noise to p and the operation of tilting it in the direction of the conditional likelihood do not commute,\" confuses the reviewer, in the sense that it is not clear why the relation they refer to breaks down. There are proofs in other papers that show that there is a valid transport equation for the classifier guidance setting, e.g. the appendix C in [1]. The question the reviewer thinks the authors should be trying to ask is how to interpret this tilted density (their first unmarked equation). \n- The organization of the theorems in the paper makes them a bit hard to follow. The authors introduce Theorems 1 and 2 early on in the motivation of the paper, but don't provide a preliminaries section until 2 pages later that try to introduce some of the distributions under consideration. Following this, there is a section on numerical experiments to motivate these theorems, but then a return to results on the mixture of uniform distributions relevant for the theorems. This organization needs serious work to be compelling. It's pretty hard to follow which aspects of the theoretical results one should be keeping track of to see if the experiments really support them. The paper then ends with a high level sketch of this last proof.\n- Certain equations are introduced with no clarification of notation, for example the probability flow ODE (eq 4), nor is it clear where this equation comes from unless you know the literature. It's also unclear why a different formulation of it is included in Lemma 1.\n- The reviewer appreciates the efforts of the authors to include results on image generation, however the experimentation is a bit thin and heuristic, only relying on this pullback effect. For example, on the MNIST experiments, how do you quantify this as outside the support of the density?\n\n\n\n[1] Ma et. al. (2024) https://arxiv.org/pdf/2401.08740", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for taking the time to review our paper and for the helpful feedback. We apologize for some of the organizational issues, and hope to address the main weaknesses below.\n\n**Weaknesses**\n\n> There are some statements early on that I found confusing, and without theoretical justification. For example, the statement: \"In other words, the operation of applying noise to p and the operation of tilting it in the direction of the conditional likelihood do not commute,\" confuses the reviewer, in the sense that it is not clear why the relation they refer to breaks down. \n\nRegarding tilting and noising, please see the joint/global response where we have further clarified what we mean by noising and tilting.\n\n> There are proofs in other papers that show that there is a valid transport equation for the classifier guidance setting, e.g. the appendix C in [1]. The question the reviewer thinks the authors should be trying to ask is how to interpret this tilted density (their first unmarked equation).\n\nIt appears to us that Appendix C of Ma et al. (2024) only shows that the heuristic derivation of the probability flow ODE extends to the more general flow models that they consider. In fact, the only formal claim made in Appendix C is that the guided drift used at time $t$ is the score of what we refer to in the joint response as the noised-then-tilted distribution, and what they refer to as the “tempered distribution.” In light of the argument we provide in the joint response and in our paper, however, the noised-then-tilted distribution is different from the tilted-then-noised distribution, and **guidance provably does not sample from the intended tilted distribution**. In our original submission, we also gave a very simple example in Lines 47-54 supporting this claim.\n\n> The organization of the theorems in the paper makes them a bit hard to follow. The authors introduce Theorems 1 and 2 early on in the motivation of the paper, but don't provide a preliminaries section until 2 pages later that try to introduce some of the distributions under consideration. Following this, there is a section on numerical experiments to motivate these theorems, but then a return to results on the mixture of uniform distributions relevant for the theorems. This organization needs serious work to be compelling. It's pretty hard to follow which aspects of the theoretical results one should be keeping track of to see if the experiments really support them. The paper then ends with a high level sketch of this last proof.\n\nWe thank the reviewer for their organizational suggestions. Please see our joint response for a discussion on this, and in particular the changes we plan to make to the organization.\n\n> Certain equations are introduced with no clarification of notation, for example the probability flow ODE (eq 4), nor is it clear where this equation comes from unless you know the literature. It's also unclear why a different formulation of it is included in Lemma 1.\n\nLemma 1 was useful in the proof in the supplement as it gives a more explicit form for the drift of the ODE in the two-component Gaussian mixture setting that we consider. However, in our reorganization (see joint response), we will omit it in favor of a more clear exposition of the probability flow ODE in the introduction and defer the details of Lemma 1 to the supplement.\n\n> The reviewer appreciates the efforts of the authors to include results on image generation, however the experimentation is a bit thin and heuristic, only relying on this pullback effect. For example, on the MNIST experiments, how do you quantify this as outside the support of the density?\n\nFor the MNIST experiments, we use the notion of class support more loosely than in the case of the synthetic experiments, since we do not have a precise definition to work with. We more so meant to draw attention to the fact that increasing the guidance parameter leads to sampling trajectories that move further and further along the separating direction between a fixed class and all other classes (visualized in Figure 3), and that this correlates with qualitatively worse samples. When we revise, we will add plots of the intermediate samples corresponding to when the trajectory has moved very far out to provide better intuition for the undesirable effects of large w in these cases.\n\n**Questions**\n\nThe main question is addressed above in the first two points under weaknesses.\n\nWe hope the above addresses the main concerns of the review, and we are happy to engage further and answer any additional questions you may have.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6bB8UNuXP4", "reviewer": "Reviewer_xts9", "summary": "Previous authors show that tilting the score at any given noisey time t corresponds to the score of a tilted-at-that-time-t distribution, and they use this to motivate conditional sampling algorithms, but it is shown here that this is not the score of the noised version of the titlted-at-time-0 distribution (which is the one intended to sample from)\n\nAs such, there's no reason to believe priori that the guided samplers are sampling from the intended titled conditionals.\nThey make rigorous their observation that when you want to sample X|A but the current particle during sampling is currently close to fulfilling event B != A, the particle gets repelled at maximum velocity (under some constraints) away from set B and towards set A, and due to some weird dynamics, particles end up getting stuck on the edges of the support of X|A.\n\nTo analyze what's going on the authors consider some simple low dimensional examples as well as MNIST + Imagenet, and then provide theorems that make rigorous the above phenomena. \n\nThe authors connect the theory well to previous works on this topic (Wu et al) and in general this is an insightful read + carefully executed theory wise and experimentally.", "questions": "Nothing notable for now, but I will add additional comments when some questions come up during the discussion period.", "limitations": "Nothing notable.", "rating": 9, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "See above summary for strengths. In short, I'd like to add that too much diffusion + generative models literature focuses too much on showing that one particular setup is able to achieve a good result on a dataset.  On the other hand, this work adds much needed questioning about what common empirical choices are doing (at best, at optimum, in any situation, etc)", "weaknesses": "Nothing notable.\n\nminor:\n- explain second equality in (1) to the reader (that the coef. gets normalized out + dropped due to grads)\n- broken ref right before (2)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their positive feedback and for finding our work to be an insightful read with carefully executed theory and experiments. We are encouraged that they agree it is important to question what common empirical choices in the practice of diffusions are actually doing. We will also incorporate the small fixes that were suggested. We are also happy to answer any additional questions that may come up during the discussion period.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SJXhvIiwdA", "reviewer": "Reviewer_gYu8", "summary": "This paper discusses the impact of diffusion guidance, especially when noting that the guided score function does not correspond to that of tilted distributions. The authors theoretically justify that a large guidance scale can lead to low-entropy and \"extreme\" samples. The authors further discuss score estimation in the real world and propose that a sufficient large guidance scale is more likely to lead to a sample outside the distribution domain. The authors finally introduce experiments to discuss the optimal choice of guidance scale and show that a guidance scale that is too large introduces swinging away from the support of the data distribution.", "questions": "1. **How applying noise and tilting the distribution do not commute?**\n   In Line 37-39, the authors mention that applying noise and tilting the distribution do not commute. Since this is a crucial point for the paper, could the authors provide more details or intuitions to explain this?\n\n2. **How experiments on ImageNet are related to the Gaussian setting?**\n   In Line 260-262, the authors discuss that the dynamics of \"farther movement\" resemble to those of the Gaussian setting, instead of that of MNIST. Seemingly the authors do not provide details of the experiments of Gaussian setting in the paper. Could the authors provide more details on this?", "limitations": "The limitation and broader impact are not discussed.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The paper focuses on a frontier research field, diffusion guidance, and provides systematic analysis.\n2. Theories are well justified, make sense, and well explained.\n3. Theoretical analysis is combined with experiments, which makes the paper more convincing.", "weaknesses": "1. **The introduction of the paper could be better organized.**\nThe authors organize the paper in a way that the introduction is a bit unclear. The current introduction consists of the background, the main results, and the related works. It would be better to separate them in individual sections. Also, the authors could provide a summary of contributions in the introduction.\n2. **There seems to be uncomplete or missing parts in the paper.**\n   - The conclusion section is missing.\n   - The limitation and broader impact are not discussed.\n   - Line 245: the authors mention the choice of positive labels but the appendix does not provide the details.\n\n**Minors:**\n   - Figure 1-4: graphics are not vectorized.\n   - Line 35: missing reference.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for taking the time to review our paper and the helpful feedback. We hope to address the main concerns below.\n\n**Weaknesses**\n\n> **The introduction of the paper could be better organized.** The authors organize the paper in a way that the introduction is a bit unclear. The current introduction consists of the background, the main results, and the related works. It would be better to separate them in individual sections. Also, the authors could provide a summary of contributions in the introduction.\n\nWe thank the reviewer for their helpful organizational suggestions. We will separate the background (lines 14 to 46), motivating example and main results (lines 47 to 110), and related works (Lines 111 to 148) in Section 1 into three separate subsections. For the second of these three subsections where we describe main results, we will add a concluding section where we summarize our main contributions, namely 1) two toy settings which illustrate that diffusion guidance fails to sample from the tilted density in two different ways, 2) theory illustrating the impact of score estimation error on the behavior of guidance, and 3) synthetic and real data experiments validating our theory and offering practical prescriptions.\n\n> **There seems to be uncomplete or missing parts in the paper.**\n\nThank you for pointing out the missing MNIST experiments in the appendix and we sincerely apologize for the appendix being cut off in the submission - please see the joint response that includes the majority of the cut off MNIST figures (some left out due to single page limit) which affirm line 245. We also address the issue of the conclusion section in our joint response on organizational changes we will make in the final manuscript.\n\n**Questions**\n\n> How applying noise and tilting the distribution do not commute? In Line 37-39, the authors mention that applying noise and tilting the distribution do not commute. Since this is a crucial point for the paper, could the authors provide more details or intuitions to explain this?\n\nPlease see the joint response for more details.\n\n> How experiments on ImageNet are related to the Gaussian setting? In Line 260-262, the authors discuss that the dynamics of \"farther movement\" resemble to those of the Gaussian setting, instead of that of MNIST. Seemingly the authors do not provide details of the experiments of Gaussian setting in the paper. Could the authors provide more details on this?\n\nIn the Gaussian setting, we show in Theorem 2 that as we take the guidance parameter $w$ to be large, points move towards $\\pm \\infty$ depending on the guided class. In this case there is no pullback effect like we show in the compactly supported setting. What we observe in lines 260-262 is that this same kind of behavior appears in the ImageNet experiments, where the sampled points move further and further along the mean-separating direction. \n\nWe originally excluded the Gaussian experiments simply for space constraints in the main body; an empirical verification of Theorem 2 is also available in the joint response PDF in addition to the MNIST experiments. We apologize for excluding these originally - thank you for pointing out that it would be useful to have them.\n\nWe hope the above discussion addresses your main concerns, and we are happy to answer any further questions you may have.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "XPXS7NpOWp", "reviewer": "Reviewer_CopY", "summary": "The paper offers a theoretical investigation of the use of guidance in diffusion models. Through two stylized models, the paper fully characterizes the behavior of using guidance, which violates the commonly adopted intuition.", "questions": "As mentioned in the \"Weaknesses\" section, I wonder:\n1. How general are the phenomena revealed in the stylized examples? Would it be possible to investigate it at least through simulations?\n2. Would it be possible to provide a concrete implementation of the use of guidance with theoretical guarantees (even under the stylized models)?", "limitations": "The authors have addressed the limitations.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper focuses on an important question, i.e., using guidance in diffusion models, revealing an overlooked phenomenon through rigorous theoretical treatment. The illustrated phenomenon is likely to have large effects on practice, which I think is a major contribution.", "weaknesses": "1. The phenomenon revealed in this paper is restricted to highly stylized models, and it is unclear whether it is generally applicable.\n2. The main message of this paper is the potential failure of using guidance in diffusion models. However there are no rigorous recommendations for implementing the method (e.g., the recommended choice of w is heuristic).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for taking the time to review our paper, as well as the positive encouragement and helpful feedback. We hope to address the main weaknesses/questions below.\n\n**Weaknesses/Questions**\n\n> How general are the phenomena revealed in the stylized examples? Would it be possible to investigate it at least through simulations?\n\nThe phenomena we reveal are quite general. The stylized experiments in the paper are meant to conform to our theoretical setting, but similar behavior can be observed even when we relax constraints. For example, even though Theorem 1 concerns compact distributions with separated support (at least in the coordinate of interest), the empirical results of Section 3.1 hold true even when we allow the supports to overlap. Furthermore, as Theorem 1 suggests, we can consider significantly more complicated distributions than uniform over an interval; we tried experiments where we sampled from various convex bodies with and without overlap and this behavior of moving to edges of the distribution still appears. When we revise, we will include at least a subset of these experiments in the appendix.\n\n> Would it be possible to provide a concrete implementation of the use of guidance with theoretical guarantees (even under the stylized models)?\n\nUnfortunately this is a big open question even if the tilt is Gaussian, which corresponds to the heavily studied setting of posterior sampling for linear inverse problems. For the compactly supported setting we consider, note that it is trivial to sample from the tilted distribution: it is simply the distribution over the compact support selected by the classifier, and this is independent of the choice of guidance parameter.\n\nWe hope the above provides more clarity, and we are happy to answer any further questions you may have.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vAx4VRykxb", "reviewer": "Reviewer_4NtD", "summary": "The paper characterizes the distribution from which diffusion guidance samples. It proves that guided diffusion sampling tends towards the edges of the supports of the class-conditional distributions in scenarios involving mixtures of uniform or Gaussian distributions.", "questions": "Please see weaknesses.", "limitations": "The consideration of only two classes for the condition and uniform and gaussian distribution is kind of simplistic.", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "* The paper clearly proposes the focused question, to characterize the distribution diffusion guidance sampling from.\n\n* The paper clearly introduces the relationship with prior works on diffusion guidance and conditional diffusion models.", "weaknesses": "* The paper's structure needs improvement. The experiments should be settled at the end of the paper, rather than in Section 3. The motivation for theories could be shorter. The current placement interrupts the coherence of the theoretical discussion. \n\n* It is not clear which main results are explained in Section 4. While it discusses the convergence speed towards $p^{(1)}$ on different cases​, it lacks solid support.\n\n* The role of score estimation error isn't described clearly. Do Theorems 1 and 2 require ground truth scores?\n\n* The presentation of theoretical results is not coherent. For example, there is no definition of $\\tilde{x}(1)$ when it first appears in Theorem 1.\n\n* Assumptions 1 and 2 are the same, but Theorem 4 cites Assumption 2, which is in the appendix, causing confusion.\n\n* Typo: Line 35, reference missing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for taking the time to review our paper as well as the helpful feedback. We hope to address the main concerns below.\n\n**Weaknesses/Questions**\n\n> The paper's structure needs improvement. The experiments should be settled at the end of the paper, rather than in Section 3. The motivation for theories could be shorter. The current placement interrupts the coherence of the theoretical discussion.\n\nWe thank the reviewer for their organizational suggestions. Please see our joint response for a discussion on this, and in particular the changes we plan to make to the organization.\n\n> It is not clear which main results are explained in Section 4. While it discusses the convergence speed towards $p(1)$ on different cases​, it lacks solid support.\n\nThe rigorous support for our theory is provided in the appendix. Due to space constraints and the technically intricate nature of our proofs of Theorems 1 and 2, we chose to opt for a higher-level discussion. Our reasoning was that the role of guidance is of broad interest to the diffusions community and thus merits a more conceptual approach to exposition which we will build upon in our forthcoming re-organization. We also thank the reviewer for their helpful suggestion to clarify which parts of the discussion in Section 4 correspond to which Theorems.\n\n> The role of score estimation error isn't described clearly. Do Theorems 1 and 2 require ground truth scores?\n\nTheorems 1 and 2 assume ground truth scores, and we quantify the effect of score estimation error in Theorem 3, which we originally omitted from the discussion in the main body due to space constraints. Given the cuts we plan to make to Section 5 as outlined in our joint response on organizational changes, this will free up space to include additional discussion about Theorem 3 in the main body.\n\n> The presentation of theoretical results is not coherent. For example, there is no definition of $\\tilde{x}(1)_1$ when it first appears in Theorem 1.\n\nWe thank the reviewer for pointing this out and will omit the notation of $\\tilde{x}(1)_1$ in the theorem statements in Section 1. This notation is just meant to denote the first coordinate of the final sample, which is already clear from the rest of the text in Theorems 1 and 2 and thus unnecessary.\n\n> Assumptions 1 and 2 are the same, but Theorem 4 cites Assumption 2, which is in the appendix, causing confusion.\n\nWe apologize for this oversight. This was caused by an improper compilation of the main body + supplement within the same TeX file, and will be addressed in our revision.\n\n> Typo: Line 35, reference missing.\n\nThank you for catching this.\n\nWe hope the above addresses your main concerns and we are happy to engage further to address any additional questions you may have.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We would like to thank all of the reviewers for taking the time to review our paper and for all of the helpful feedback. We hope to address some common points of discussion below.

**Tilting and noising do not commute.** 

We use the term “noising” a distribution $p$ to mean the distribution obtained by taking a sample $X\sim p$ from the original distribution and then putting it through the noise process, in our case, $X\_t = a\_t X + \xi\_t$, where $\xi\_t\sim N(0,b\_t I)$ is a Gaussian. We will clarify this.

To formalize the fact that tilting and convolving do not commute, we define the tilted distribution (with parameter $w$) by 

$$p^{z,w}(x) \propto p(x) p(z|x)^{1+w}$$

where $(z, x)$ is drawn from a joint distribution and z is the label. The tilted-then-noised distribution is

$$p^{z,w}\_t = ((a_t)\_*p^{z,w}) * \gamma\_{b\_t}\quad (1)$$

where $\gamma\_t$ is the density of the Gaussian $N(0,b\_t I)$ and we use $a\_*p$ to denote the distribution of $aX$ where $X\sim p$. I.e., it is the distribution of $X\_t = a\_t X + \xi\_t$ where $X\sim p^{z,w}$ and $\xi\_t \sim N(0,b_t I)$. The probability flow ODE that would result in the distribution $p^{z,w}(x)$ would use the score function of $p^{z,w}\_t$, i.e., $\nabla \log p^{z,w}\_t$.

However, this is not the score function actually used in diffusion guidance, which uses the noised-then-tilted distribution

$$(p\_t)^{z,w}(x\_t) = p\_t(x\_t) p\_t(z | x\_t)^{1+w}.\quad (2)$$

Here, $p\_t = ((a\_t)\_*p) * \gamma\_t$, and the conditional density $p\_t(z | x\_t)$ is interpreted as the conditional distribution of $z$ given $x\_t$, where $x\_t$ is produced by taking $x|z$ and then letting $x\_t =a\_t x + \xi$ where $\xi\sim N(0,b\_tI)$. In words, it is the classifier for the noisy sample.

(1) and (2) are not the same in general for $w\ne 0$, because if they were, then diffusion guidance (which uses (2)), would produce the tilted distribution $p^{z,w}$; however, our analysis for compactly supported distributions shows that diffusion guidance results in samples that are not even in the support of $p^{z,w}$.

**Re-organization of results in the paper.**

We thank the reviewers for their feedback on improving the organization of the paper and believe the following changes will improve the flow and render our key message more transparent:

1) We will move the experiments section and figures to the end of the main body in order to not break the flow of the theoretical discussion.

2) We will define the probability flow ODE immediately after Line 32 instead of in the Preliminaries so that the algorithm we consider is clear to the reader early on.

3) Theorems 1 and 2 are meant to be self-contained and readable independently of the Preliminaries, hence the ordering in the submission. Indeed, in the theoretical computer science literature, it is standard to have informal theorem statements in the Introduction prior to the section introducing technical preliminaries. Nevertheless, to make Theorems 1 and 2 even more parseable, we will remove the $\tilde{x}(1)_1$ notation, and we will make it clearer that Assumption 2 is in reference to something later in the document.

4) The proof of our main result is quite technically intricate, yet we believe our contribution to ultimately be largely of a conceptual nature that is of broad interest to the practical and theoretical communities working on diffusions. With this in mind, we will significantly rework Section 5 as follows. First, we will integrate the main assumption and theorem in Section 5 into Section 4 to add supporting technical detail to the high-level discussion already present in Section 4. We will then remove the material in Lines 303 to 325 in order to make room for a conclusion section (see below) and for a lengthier discussion in the Introduction. In particular, based on certain confusions among some of the reviews regarding our main conceptual point about the fact that tilting and noising do not commute, we will allocate an extra two paragraphs in the Introduction to make this point even clearer conceptually. 

5) We will include a conclusion section to provide a more thorough discussion of limitations that raise the possibility of future exploration. For instance, we will mention that while we show interesting behavior in simple toy models, it remains an important challenge to characterize the behavior of guidance in higher-dimensional, non-separable settings.

**Experiments.**

Some experiments were cut-off from the Appendix and we sincerely apologize for that, and thank the reviewers for catching that. They are included in the attached PDF, along with experiments in the synthetic Gaussian setting to complement the mixture of uniforms setting.

Meta-review: This paper studies how guidance influences the generated mixture distributions. From my reading and reviewers' assessments, the theoretical results are new and nontrivial compared to the related earlier results by Wu et al.

There are one negative review and four positive review of the paper. The biggest criticism of the current version is the organization (Reviewers 5E9W, gYu8, 4NtD). The authors proposes a reorganization of the paper in the general rebuttal and it was appreciated by one of the reviewer. I believe such revision is necessary and can be a bit heavy for very short period. However, I am willing to trust the authors for making these changes possible in the final version (I would also encourage the authors to carefully proofread the manuscript at least to clear broken links, typos and grammatical issues). Other weaknesses of the paper are a lack of implications to practitioners and the theory being stated for simplified models. These charges remain valid, yet do not outweigh the contributions of the paper.

Therefore, I am recommending acceptance of the paper.
Decision: Accept (poster)