Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 3: reject, not good enough
review_content: summary_of_the_paper: This paper studied the failure of the generalization of the quantum kernel method by focusing on the major cosine kernel function. The authors proposed introducing the bandwidth to control the spectrum of the cosine kernel. The authors derived the formula of the spectrum for that kernel. Then the authors showed that introducing the bandwidth improves the generalization by combining the generalization error bound of the existing work and deriving the spectrum formula. 
strength_and_weaknesses: # Strength
- The authors rigorously evaluated the eigenvalues of cosine kernel under the large qubit settings and showed how the bandwidth affects the spectrum. Then the authors also derived the scaling laws of the spectrum. As far as I know, the derived scaling laws are new and meaningful.
- Combined with the spectrum, the authors derived the generalization error of the kernel regression task and showed that introducing the bandwidth improves the generalization. 

# Weakness
- The idea of using bandwidth is not new.
- The generalization analysis is not novel; it just substitutes the derived spectrum formula into the Theorem in Canatar 2021.
- The analysis and numerical experiments are not consistent. The generalization analysis focuses on the regression task, but numerical experiments are the classification task. Also, the authors used different kernel functions from the theoretical analysis. Thus, it is hard to judge how significant the theoretical analysis is in a real-world application.
clarity,_quality,_novelty_and_reproducibility: I think the layout of tables and figures should be improved. For example, the caption of Table 2 and Figure 3 are hard to distinguish from the main text.
summary_of_the_review: As I wrote in the weakness, the contribution of this paper is not significant and novel. Using the bandwidth is not new and the derived generalization error bound is the simple application of existing work. The derived scaling law of the spectrum seems new and interesting, but the numerical experiments are conducted with different kernel functions from the theory; thus hard to see the usefulness of the theoretical analysis.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: Prior works have shown that quantum kernel methods may not generalize well on machine learning tasks because the spectrum of the kernel operator is too flat. This work shows that by tuning an appropriate bandwidth parameter in the kernel, the spectrum can be made less flat, hence enabling generalization.
strength_and_weaknesses: The paper is very well-written with clear, intuitive explanations, and gives compelling reasoning for the use of bandwidth in quantum kernels.

The main weakness of the work is that the theoretical results are very specific, pertaining to a specific kernel, and the analysis seems hard to generalize to more realistic kernels; also, the result is not a fully rigorous analysis of the generalization error. Nevertheless I believe that addressing these issues is best left as future work, and the current paper already makes a solid contribution by providing elucidating intuition.

clarity,_quality,_novelty_and_reproducibility: The paper is very clear and of high quality. The novelty of the paper is not that high, since the idea of using bandwidth has appeared in prior literature, but the spectral analysis in this paper is to the best of my knowledge new. I have hardly found any typos which is impressive, but I point out some suggestions for clarity:
- The ICLR submission title has a typo “bandwith”
- On pg. 2 near the top, “show that the generalization is impossible” should be changed to “show that generalization is impossible”
- On pg. 3, what is the meaning of the equation $\bar{\mathbf y}_\mu = \bar y^\mu$? Namely, why are there bars?
- On pg. 3, the explanation of the Bloch sphere is hard to understand without explicitly writing out the parameterization of the density matrix in terms of n.
- Pg. 3, what is the meaning of $\psi(\mathbf x) \to \rho(\mathbf x)$?
summary_of_the_review: Although the analysis in this work should be viewed as preliminary and awaits more detailed study, nevertheless I found the submission to be illuminating and well-written, and hence I recommend acceptance.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper studies quantum kernel learning. Their observation seems that the dimension of the feature maps associated with quantum kernels is exponential but can be processed by quantum boxes. The problem with these kernel is that their leading eigenvalues are too small so that these models do not generalize well. The authors then observe that if one change the ``bandwidth’’ of these kernel functions, then distribution of the eigenvalues becomes more benign. I am not sure whether I interpret the result correctly, but it appears that the authors provide analysis for one specific kernel (with closed forms) and argued that tuning bandwidth could be beneficial for other kernels. 

The paper is quite well written and provided careful exposition to both kernel learning and quantum computing (I was able to follow more than 50% of the paper). The results also appear to be quite significant, especially in a new area. I am leaning towards acceptance.


Is the number of cubits needed the same as the dimension of the data? That sounds quite painful? 
For the problem analyzed (eq. 5), because the kernel can be explicitly written, what’s the reason that this kernel cannot be solved by classical computers? 
In the experiment, it was stated that pre-computed kernels are used. But if kernel bandwidth needs to be tuned, wouldn’t that just mean pre-computed values could be mostly irrelevant? 

strength_and_weaknesses: 
Strength: the paper gave a quite clear exposition to quantum kernel learning and the paper’s result appears to be quite significant. 


Weakness: I am still not sure about whether I understand the big picture (perhaps just me not knowing enough about quantum). See the questions above. 

clarity,_quality,_novelty_and_reproducibility: Mostly clear; 
but quantum is still too foreign to me. 
summary_of_the_review: (+) it gives a fairly reasonable overview of quantum kernel learning. 
(+) it appears to be a significant result for quantum kernel learning. 

(-) Probably most people in iclr wont understand quantum part. 
(-) I cannot calibrate the technical depth of the result.
review_title: null

final_score: rating: 3: reject, not good enough
review_content: summary_of_the_paper: The authors propose a way to make quantum kernels generalizable through a bandwidth parameter that can restrict features in the Bloch sphere. They also provide some evidence on real datasts that these kernels can generalize unlike the ones without the bandwidth parameter. 
strength_and_weaknesses: + it's a simple but possibly powerful way to increase the ability of quantum kernels to generalize
- the idea of bandwith for quantum kernels has appeared before (Shaydulin & Wild, 2021).
- the experimental evidence is not strong enough to be able to draw any significant conclusions
clarity,_quality,_novelty_and_reproducibility: The results are clear, quite straightforward, and they follow from previous work. The results are reproducible.
summary_of_the_review: A simple and interesting idea but given that the theoretical part is not really novel and the experimental evidence is preliminary the work is not strong enough for ICLR.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: The authors establish a new bound on the spectrum associated with a quantum kernel, in terms of a bandwidth parameter recommended for use in earlier work.  This leads to a new generalization upper bound, which is consistent with the previous observation that the bandwidth parameter could be used to improve generalization in practice.  

The consensus view that emerged was that this paper did not have enough fundamentally novel content for publication in ICLR.

justification_for_why_not_higher_score: Please see the above.

justification_for_why_not_lower_score: N/A
Decision: Reject