What does guidance do? A fine-grained analysis in a simple setting

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.

Strengths

* The paper clearly proposes the focused question, to characterize the distribution diffusion guidance sampling from. * 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. * 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. * The role of score estimation error isn't described clearly. Do Theorems 1 and 2 require ground truth scores? * 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. * Assumptions 1 and 2 are the same, but Theorem 4 cites Assumption 2, which is in the appendix, causing confusion. * Typo: Line 35, reference missing.

Questions

Please see weaknesses.

Rating

4

Confidence

3

Soundness

3

Presentation

2

Contribution

2

Limitations

The consideration of only two classes for the condition and uniform and gaussian distribution is kind of simplistic.

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.

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. 2. 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).

Questions

As mentioned in the "Weaknesses" section, I wonder: 1. How general are the phenomena revealed in the stylized examples? Would it be possible to investigate it at least through simulations? 2. Would it be possible to provide a concrete implementation of the use of guidance with theoretical guarantees (even under the stylized models)?

Rating

7

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

The authors have addressed the limitations.

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.

Strengths

1. The paper focuses on a frontier research field, diffusion guidance, and provides systematic analysis. 2. Theories are well justified, make sense, and well explained. 3. Theoretical analysis is combined with experiments, which makes the paper more convincing.

Weaknesses

1. **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. 2. **There seems to be uncomplete or missing parts in the paper.** - The conclusion section is missing. - The limitation and broader impact are not discussed. - Line 245: the authors mention the choice of positive labels but the appendix does not provide the details. **Minors:** - Figure 1-4: graphics are not vectorized. - Line 35: missing reference.

Questions

1. **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? 2. **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?

Rating

5

Confidence

3

Soundness

3

Presentation

2

Contribution

3

Limitations

The limitation and broader impact are not discussed.

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) As such, there's no reason to believe priori that the guided samplers are sampling from the intended titled conditionals. They 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. To 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.  The 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.

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. minor: - explain second equality in (1) to the reader (that the coef. gets normalized out + dropped due to grads) - broken ref right before (2)

Questions

Nothing notable for now, but I will add additional comments when some questions come up during the discussion period.

Rating

9

Confidence

4

Soundness

4

Presentation

4

Contribution

4

Limitations

Nothing notable.

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. They 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.

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: - 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). - 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. - 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. - 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? [1] Ma et. al. (2024) https://arxiv.org/pdf/2401.08740

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.

Rating

5

Confidence

3

Soundness

3

Presentation

2

Contribution

3

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.

Response to rebuttal

Thanks kindly to the authors for the substantive response. It took a day for the general reply to load but I see it now. The clarification regarding the mechanism of noising the distribution, as well as its disparity from the tilted distribution is now nice and clear to me. I hope that the authors agree that being explicit like this in the paper will make it much more legible and insightful. I am happy to raise my score given the proposed reworkings -- I really think it will help the paper! One little caveat -- the MNIST experiments are still pretty heuristic, but I understand the challenge in making them less so. It is something worth a bit more thought, perhaps down the road, because the theoretical analysis in this work is insightful. Thanks.