Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: 
 This work is focus theoretically on the distributed learning of nonparametric quantile regression models in RKHS. 
 
There have been related works about the distributed learning for kernel ridge regression and also the non-distributed learning for quantile regression in RKHS.  The main idea of this paper is to approximate the quantile loss by a weighted least squares loss, then the quantile estimator can be estimated by the weighted least squares estimator.   Based on such approximation,  the authors prove the learning rates for the nonparametric quantile regression learning.
strengths_and_weaknesses: The ideas and logic of this paper are very clear and easy to follow. However, the format (especially for all the detailed proofs) seems not formal.The manuscript will benefit from reorganizing the proof details.
questions: 
Q1: The quantile level $\tau\in (0,1)$, however in the illustration section,  there are only results for $\tau = 0.5$ and $\tau =0.8$. Could you please provide more results for the entire range of $\tau$ , say setting $\tau$ 0.1 0.2,...0.8,0.9?

Q2:  What is the performance of this proposed method on real dataset? Quantile regression is a very classical problem and there should be many practical datasets, which have been used in the existing works, to illustrate the distributed quantile regression. 

limitations: yes
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper establishes the risk bounds for the divide-and-conquer version of the kernel ridge quantile regression estimator when the reproducing kernel has polynomially decaying eigenvalues. To prove the risk bounds, the paper introduces a novel second-order empirical process for quantile risk that can be used to establish a link between the quantile regression estimator and a regularized least-squares retrogression estimator.
strengths_and_weaknesses: Strengths: the proposed approach is novel by introducing a new theoretical tool that may be useful for other related problems.
Weaknesses: the paper is purely theoretical and lacks practical considerations.
questions: The paper is purely theoretical and introduces a theoretical new device (second-order empirical process) that can be potentially useful for other related problems. However, from a methodological point of view, the paper does not produce any change to the current practice of divide-and-conquer estimation for quantile regression in RHKS. 

From a practical point of view, the tuning parameter $\lambda$ is of critical importance and the new theoretical device can potentially be used to derive a data-driven criterion for the selection of $\lambda$, following similar approaches in [1], [2] and [3]. At least some discussions in this direction may strengthen the paper from a practical point of view.

Minor point: Please clarify the meaning of the notation $\mathcal L^r $ in equation (3).

[1] Xu, G., Shang, Z. and Cheng, G., 2018, July. Optimal tuning for divide-and-conquer kernel ridge regression with massive data. In International Conference on Machine Learning (pp. 5483-5491). PMLR.

[2] Xu, G., Shang, Z., and Cheng, G. (2019). Distributed generalized cross-validation for divide-and-conquer kernel ridge regression and its asymptotic optimality. Journal of computational and graphical statistics, 28(4), 891-908.

[3] Yuan, M., 2006. GACV for quantile smoothing splines. Computational statistics & data analysis, 50(3), pp.813-829.
limitations: none
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: The authors study distributed learning of nonparametric conditional quantiles with Tikhonov regularization in a reproducing kernel Hilbert space (RKHS). The authors establish the risk bounds using a second-order empirical process for quantile risk under the simple divide-and-conquer approach that partitions the data set into multiple parts and then takes an arithmetic average of the individual outputs. 
strengths_and_weaknesses: The manuscript includes enough information needed to support the claims it makes. The proposed method is novel in the sense that the authors investigate the theoretical underpinning of the divide-and-conquer strategy for distributed learning of nonparametric quantile regression models.  The method focuses on only an individual value of \tau without modeling different quantiles jointly.
questions: 1. My main concern is in the estimation efficiency when \tau is large/small. When \tau is large/small, the observations located on the tail of the distribution may have a large influence on the estimation results. However, when dividing the sample into multiple groups, such observations may be very few or even totally missing in the subgroup. In other words, the distribution of the subgroup is no longer the same/similar to the full, and the local estimation may be unreliable. How to deal with such cases in the divide-and-conquer approach for quantile regression?

2. Is there any discussion on the number of nodes m in the algorithm?

3. The experiments only consider the one-dimensional case. How about when the dimension of x increases?

limitations: The work shows the results for \tau=.5 and \tau=.8 for the one-dimensional case. It is better to discuss more cases for more choices of \tau and dimension.
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: The work focuses on analysis of distributed learning of nonparametric conditional quantiles with Tikhonov regularization in RKHS. A difficulty in this setting is decomposing the mean squared error into bias and variance terms; a necessity for establishing rates in the divide-and-conquer setting of distributed learning. The authors suggest combating this via approximating the quantile loss by a weighted square loss through a second-order empirical process where an explicit solution to the bias and variance terms is available. Furthermore, the authors suggest approximating the loss leads to approximating the estimator. Through this regime, learning rates for the central and distributed estimators are analyzed. The paper concludes with a simple numerical illustration demonstrating the efficacy of the divide-and-conquer method.
strengths_and_weaknesses: The subject of the paper is not my expertise, so I will leave validation of the mathematics to the other reviewers. Instead, my main concern of this work is that the presentation of the results is rough and seemingly rushed. The paper is mathematically dense and thus rather difficult to follow. The bulk of the main paper is composed of theorems/lemmas followed by their proofs with little explanations in between results for clarifications. The problem tackled in this work appears to have some significance as a thorough study of the risk bounds for quantile risk in the RKHS with the distributed learning setting is to my understanding not well studied. Lastly, the numerical results illustrate the efficacy of distributed learning in this setting, but it appears to be tangential to the analysis of the learning rate which was the focus of the bulk of the work (although the authors have brought up this concern).

Minor: syntactic - in the inequality and equality below line 180 and 183 respectively, replace $(1/2)$ with $\frac{1}{2}$. Same issue with $(1/N)$ in the inequality below line 111, 112, 118.
questions: I have no questions regarding the technical details of this work as it is not my area of expertise.
limitations: The work is theoretical and has no immediate potential negative societal impact. Limitations such as requiring $\alpha$ to be sufficiently large in their analysis were mentioned by the authors.
review_title: null

Meta-review: metareview: The paper investigates distributed quantile regression for RKHS-based estimators. Unlike in the usually considered case of distributed least squares learning, the quantile setup is more challenging from a technical perspective. Clearly, the strength of this paper is to tackle this challenge. 
The weakness of the paper is that it is a technically dense paper that might not be suited for a broader audience, as one of the reviews also suggests. For researchers more familiar with the overall material, however, the paper is nicely written, and for this reason I believe the strength outweighs the weakness. But in any case, a more gentle description of the assumptions plus examples would improve the paper.
Finally, from my personal reading I was surprised that the Bernoulli 2011 paper by Steinwart and Christmann on RKHS-based quantile regression was not mentioned. In particular a comparison of the results seems to be necessary. 

In summary, a technically sound paper that should be accepted provided the competition is not too fierce.   
confidence: Less certain
Decision: Accept