__ Summary and Contributions__: - The authors adapt a recent idea by Vaonik and Izmailov to the covariate shift setting, and provide theory on estimation of the V matrix (one component of the method) on the target task (classification here, specifically SVM, boosting).
- Their method considers the estimation of the CDF of the target distribution, which the authors empirically show is more stable in estimation of the conditional probabilities, without much parameter tuning compared to previous work.

__ Strengths__: - I am not an expert on the Vaonik and Izmailov and covariate shift literature, but as far as I know the methodology considered is new.
- The main contribution comes from replacing the V-matrix (indep of labels) by equation (11), computed using the target X for the purpose of covariate shift. The authors provide some theory that their estimator is MVUE and also provide some convergence rate.
- Empirically they show that their methodology has more robust prediction and is comparable/better on a variety of datasets (vs current approaches), though there are some comments below.

__ Weaknesses__: - I would like to see a bit more experiments, especially regarding its performance, as the performance of the proposed method is not beating other approaches by a wide margin (from looking at pure std, it seems to be overlapping mostly, although the mean is lower, and seems to be more stable on the synthetic example).
- I would like to see the result of the existing covariate shift methods with some tuning on the hyperparameters, as the authors proposed the cross val is wasteful of data. Here, they were fixed during experiment, perhaps it will be interesting to see with some tuning, how does the existing baselines compare, especially when authors claim their method is more stable and robust practically, compared to other methods. Even an oracle is ok.
- The method proposed only works on SVM and boosting (for classification), though the authors says this can be extended.

__ Correctness__: On the empirical side, perhaps the author can check whether the proposed method beat other methods given one set of problem consistently, using say a signed rank permutation test. This is because I imagine the problem can vary a lot in difficulty, and hence the std are likely to overlap.

__ Clarity__: Overall I follow the paper, and the diagrams were useful.

__ Relation to Prior Work__: Yes, the authors discuss this clearly in section 1, and the authors also clearly states their contribution vs that of Vaonik and Izmailov.

__ Reproducibility__: Yes

__ Additional Feedback__: I have a confusion about what is R in line 171, I cannot seem to find its definition in Theorem 3.3.
Minor: Typo in Table 1 caption
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After rebuttal:
Overall I think the methodology is nice, however I am still not convinced about the experimental results (and the scenarios that it is useful on real life applications).

__ Summary and Contributions__: For a setting of a covariate shift (different feature distributions but the same conditional distributions for source and target), authors look into approaches that train on S data while reweighing the points to account for the shift. Authors propose to estimate weights without having to estimate the densities (instead, using cdfs). The proposed method is more stable and not hyperparams (for densities) sensitive

__ Strengths__: - Derivations of V matrix and applications to gbdt (but it is in supplemental)
- Well written

__ Weaknesses__: - Limited impact. Authors demonstrate experiments with SVMs, compare with old methods for covariance shift (2007-2011)
- Small datasets experiments, many results are not significant
- Not clear how/if the method will scale to millions of instances, and larger feature dimensions (authors state that it won't scale to larger dimensions)

__ Correctness__: Yes

__ Clarity__: Yes, enough background on all the work they build on

__ Relation to Prior Work__: Yes but authors concentrate only on reweighing scheme to combat covariate shift. Any other modern methods for domain adaptation are not mentioned/not compared against

__ Reproducibility__: Yes

__ Additional Feedback__: Update: I read authors' responce, I am shifting my score to marginally above the reshold.
The idea is interesting and the vision is important. I do appreciate hyperparameter free solutions, as everyone does I suppose. I do still feel that this paper is not a clear accept. Applying it to non svm settings would strengthen its appeal immensely, and comparing with more modern covariate shift methods will go a long way. Also I don't find the claim that it gets the best performance in half of the experiments compelling enough.
=============
As I said before, the paper is ok but I am struggling to understand how useful it is/ what the impact is
1) Why do use SVMs, they are pretty old, do not work on large data and thousands of features. Does your method extends to any other models) like NN (I assume it should because it is weights for source that you are learning). Why not to test on NNs for example? Or gbdts?
2) All the methods you compare to are really old 2007-20011), why not to compare with new things like https://arxiv.org/abs/1505.07818, https://arxiv.org/pdf/1705.10667.pdf etc (nn based and works even with unlabelled target), or even fine tuning?
Minor:
Experiments:- Table 1 some rows don't have the best method bolden. Also you seem to bolden even if the results are not statistically significant (twonorm for replicating experiment etc)
Line 89 missing space before "In section"Section 2
introduce N number of training data points
Algorithm 1 should really go after section 3.2

__ Summary and Contributions__: This paper borrows the idea from Vapnik and Izmailov (2019) and proposes to construct a V matrix for solving the covariate shift (CS) problem. The main idea is to replace the uniform measure by the target set measure in Eq.(8) for evaluation. Consistency guarantees are provided, and experiments show that the proposed method can perform better than existing CS techniques.
=========== Update ===========
I thank the authors for their feedback. Overall, I understand and appreciate the idea of CS without parameter-tuning. However, as other reviewers pointed out, the performance gain is marginal in the experiments, and the applicability of the proposed method is limited in practice.

__ Strengths__: - The idea is simple and easy to understand.
- Theorems of consistency.

__ Weaknesses__: - The writing can be improved.
- Shortcomings are not discussed sufficiently.

__ Correctness__: Other than the technical inaccuracies of several statements (see below), most contents are correct.

__ Clarity__: - Eq.(1) does not involve the label variable Y (as opposite to L34). In fact, the readers may not be as familiar with the topic so it would be better to define the notations clearly and accurately, especially given that this is the first/introductory equation.
- When Fig.1c is first introduced in L57, there is no description of what the problem is.
- L39: "A key shortcoming of these methods is that they require estimating the probability density function of both the training and target data using density estimation techniques". This statement is invalid. For example, the [19] mentioned before this is not estimating individual densities.
- L42, L184, L214: it seems that this paper confuses KDE with linearly parameterized ratio models. Kernel density estimation is to estimate "density". Most existing methods for covariate shift estimate density "ratio". Even though the parametric form can be similar to KDE, it is NOT KDE.
- L78: "reduced L2 − prediction" is badly formatted.

__ Relation to Prior Work__: Well discussed.

__ Reproducibility__: Yes

__ Additional Feedback__: What if V(i,j)=0 for all i,j? Looking at (11), this can happen when every target point t<x for all x, especially for high-dimensional data.
About the experiments
- The loss has gamma in Eq.(4) as a parameter. Is this parameter sensitive? Since the main point of this work is to eliminate tuning hyperparameter, it would be necessary to see that the proposed method is indeed robust to this parameter to some extent.
- The experiment in Sec.3.3: if f is the V-SVM in Sec.2.4, how is the kernel width chosen?
- Table 1. Not every row has a bolded method. For the twonorm dataset, the proposed method is noticeably worse than other methods. It would be helpful to discuss why this case failed and when the proposed method will fail in general.

__ Summary and Contributions__: The paper introduces a new approach to address covariate shift using the empirical cumulative distribution function of the target data. It build heavily on recent work by Vapnik and Izmailov, is conceptually simple and shown to be effective on simulated and real-world tasks.

__ Strengths__: - Novel, conceptually simple and practically useful approach.
- Thorough comparison to existing methods.
- Transparent experiments. I much appreciate that the others admit that hyper-parameter tuning for competing methods may be sub-optimal.

__ Weaknesses__: - The paper could trade some technical details against giving intuitive explanations

__ Correctness__: Yes

__ Clarity__: Yes

__ Relation to Prior Work__: The paper builds heavily on the work by Vapnik and Izmailov which is clear throughout. It could be made clearer what the original contributions of this work are.

__ Reproducibility__: Yes

__ Additional Feedback__: I think the title is very non-informative. It does not say what the method does nor how.
Abstract
"Varying domains and biased datasets can lead to differences between the training and the target distributions, a problem termed covariate shift"
- There is a difference between covariance shift and more general distribution shift as the authors point out in to introduction.
- Not sure what "varying domains can lead to differences in the training and target distribution" should mean(?)
Introduction
- Eq. (1) assumes perfect generalization.
- Text annotations of Fig 1 are too small but there is a lot of whitespace below the figure. Maybe consider a 4x4 figure arrangement with 1/4 being the caption.
[Update] Thanks to the authors for their response. My rating remains.