NeurIPS 2020

### Review 1

Summary and Contributions: This paper has a theoretical analysis on the knowledge distillation of a wide nerual network. In particular, they give a transfer risk bound, and discussed data inefficiency and imperfect teacher.

Strengths: The authors have done an in-depth analysis on the knowledge distillation. Further it is nice to see the discussion on the data inefficiency issue and imperfect teacher.

Weaknesses: As discussed in Section 3.2, the authors mentioned that the bound in [18] could show looseness, but the proposed bound could be loose with small sample size. Hence, it is hard to justify the advantages of the proposed analysis over the existing work [18]. Above Eq. (2), the authors assumed that the student is over-parameterized and wide. But recall that the motivation of studying student network is to have some lightweight network that can achieve a similar performance as that of the teacher network. The assumption of this paper can thus be questionable in practice. In Section 4, the authors had data inefficiency analysis. However if the sample size is small, the training of teacher network could be influenced as well. The analysis here then will become even complex.

Correctness: Most of the claims are right. But the assumption on the over-parameterized and wide student network could be rigorous in practice.

Clarity: The paper has been cleary written.

Relation to Prior Work: The authors have included the discussion of related works and explained the difference.

Reproducibility: Yes

### Review 2

Summary and Contributions: Knowledge distillation is a successful method of model compression and knowledge transfer. However, current knowledge distillation lacks a convincing theoretical understanding. This paper has theoretically analyzed the knowledge distillation of a wide neural network. First, a transfer risk bound is provided for the linearized model of the network. Then, a metric of the task’s training difficulty, (data ineffificiency) is proposed. Based on this metric, this paper finds that for a perfect teacher, a high ratio of teacher’s soft labels can be beneficial. Finally, for the case of an imperfect teacher, this paper finds that hard labels can correct the wrong prediction of the teacher, which explains the practice of mixing hard and soft labels.

Strengths: This paper has explained knowledge distillation in the setting of wide network linearization. This paper has provided a transfer risk bound based on angle distribution in random feature space. Early stopping of teacher and distillation with a higher soft ratio are both benefificial in making effificient use of data. Even if hard labels are data ineffificient, this paper has demonstrated that they can correct an imperfect teacher’s mistakes, and therefore a little portion of hard labels are needed in practical distillation.

Weaknesses: This paper investigated knowledge distillation in the setting of wide network linearization instead of a practical nonlinear neural network. There is no experiment on ImageNet. It is very hard to check the effectiveness of the proposed method.

Correctness: Following the logic of the paper, the analysis looks correct.

Clarity: The paper is understandable but it is not very straightforward.

Relation to Prior Work: [18] considers distillation of linear models and gives a loose transfer risk bound. The proposed method in this paper has improved the bound and generalize [18] to the case of linearization of an actual neural network.

Reproducibility: Yes

Additional Feedback: There is no experiment on ImageNet. It is very hard to check the effectiveness of the proposed method.

### Review 3

Summary and Contributions: This paper gives a theoretical understanding of knowledge distillation, using a linear approximation of a general wide neural network for binary classification, as given by the neural tangent kernel. Assuming a perfect teacher, this work provides a tighter transfer risk bound than previous work, and for a linearized NN. Under the perfect teacher, they also provide a metric for the difficulty of the student to recover the weights of the teacher, under the number of samples in the training task. This metric is used to show that early stopping of the teacher improves data efficiency for the knowledge distillation task. Finally, allowing for an imperfect teacher, they show that the presence of hard (ground truth) labels can be helpful for correcting a wrong prediction (soft label) by the teacher, even as they worsen data efficiency. Hence, the recommendation is to use a few hard labels, as is standard practice in knowledge distillation. Overall, this work shows that the benefit of knowledge distillation comes from the ability of the teacher to provide the student network with a smoothened output function, but hard labels result in the loss of this smoothness, even as they promote better students.

Strengths: The submission presents useful results which help the theoretical understanding of knowledge distillation. Though there are no empirical results, the work seems to be of significance to the theoretical ML community. Moreover, it draws connections between KD and NTK, the latter being a recent, but impactful development. This paper is also one of the few in this conference that acknowledge the risk of negative impact of deep learning, if proper restrictions are not employed on its abuse.

Weaknesses: Since this paper is outside of my area of expertise, I can only comment on a couple of things I would have liked to see in this work, as follows: - Perhaps the inclusion of small scale synthetic experiments which empirically show the trade off of the performance benefit of hard labels vs. data efficiency would have made this work even stronger. - It is not clear what the practical implications of the data inefficiency metrics are; even though the conclusions from this metric seem to align well with findings from prior work.

Correctness: I am not very familiar with the theoretical background of neural tangent kernels, hence I'm not able to assess the correctness of bound presented in the paper.

Clarity: Yes.

Relation to Prior Work: Yes

Reproducibility: Yes

Additional Feedback: I would recommend a larger figure 1, even as the aspect ratio might need to change, since this figure is important to understanding this work. *Post author response* I thank the authors for providing a synthetic experiments, which makes this paper stronger. I stand by my original assessment of this paper.

### Review 4

Summary and Contributions: Summary: The paper presents a theoretical analysis on knowledge distillation, where the student network is overparameterized so that it behaves as an NTK. In this case the paper proves a tighter transfer risk bound than a previous model in [18]. Then, the paper presents a metric to evaluate data efficiency in knowledge distillation and shows that early stopping / high soft ratio leads to higher efficiency. Finally, the paper analyzes the use of hard labels when the teacher network is not perfect.

Strengths: Pros: - The paper is well written and clear. The mathematical setup and analysis are solid. - The use of NTK is innovative in the area of knowledge distillation. - The risk bound is not based on (and better than) the classical Rademacher complexity, which is also novel. - The definition for data inefficiency is clear and makes a lot of sense. The result that higher soft ratio / early stopping can help obtain higher data efficiency is natural and important in explaining why they are useful. - The paper also talks about the use of hard labels for imperfect teacher networks, making the topic complete.

Weaknesses: Cons: - The paper assumes the student network f is overparameterized. However, there is no definition for f. There is also no comment on why you choose an overparameterized f (other than making it easier to analyze). In practice, student networks are usually small, which is mentioned in your introduction. Then, why is it meaningful to look at an overparameterized student network? More importantly, the paper assumes convergence when f is overparameterized, which is in fact not guaranteed. In [7], it is shown that overparameterized networks trained with sgd/gd can reach small l2 loss (because optimization is near convex). However, this may not be true for a completely different distillation loss. Therefore, it is necessary to prove convergence in for the distillation loss to make your assumption solid. - What is b at line 140? It seems the bound is tighter than classical bounds only when $b>1$. However, I don't see a definition for b. - As a theory paper, there is no formal theorem in both section 4 and section 5. It is necessary to extract the core ideas in these two sections into concise and rigorous theorems, even if they are expressed by words. - Finally, it is unclear how the three points made in this paper relate with each other. I don't see an overall picture after reading this paper; it seems like the three sections (3,4,5) are disjoint pieces and don't form a well-connected story. It would be clear if the authors can write a separate paragraph in introduction about how these points are related and together contribute to a bigger idea. There may be some points where I misunderstood or made mistakes. I am willing to increase my score if the above questions are resolved and made clear.

Correctness: Yes.

Clarity: Yes.

Relation to Prior Work: Yes.

Reproducibility: Yes