__ Summary and Contributions__: This paper studies active learning with label and comparison queries, extending the works of (Kane et al, 2017, Xu et al, 2017). It gives a new set of fundamental upper and lower bounds on halfspace learning in passive/active, label/label+comparison queries, PAC/RPU(Reliably Probably Useful) models, summarized in Tables 1 and 2 in the paper.
- On the lower bound side, it shows that any label-only active learning algorithm must have polynomial query complexity in PAC learning setting, and exponential sample complexity in the RPU learning setting. The main techniques are using existing combinatorial geometry results.
- On the upper bound side, it shows that under mild distributional assumptions, label+comparison based active learning algorithms can have logarithmic query complexity in both PAC and RPU learning settings. The main technique is by developing a distribution-dependent variant of inference dimension (Kane et al, 2017) and a novel online-to-batch conversion (Thm 4.11).

__ Strengths__: Overall, I think it is a solid paper that greatly advances the understanding of RPU learning and learning with comparison queries. Specifically:
- The development of distribution-dependent inference dimension is an important contribution, which can helps spur more works in distribution-dependent analysis of machine learning (similar to Hanneke's disagreement coefficient / Alexander's capacity function for active and passive learning)
- The proof of Theorem 4.10 (distribution-dependent inference dimension for halfspaces) is not trivial, in my understanding. I was initially thinking if some arguments therein can be simplified by using (normalized) VC inequalities, but it does not lead to g(n) = 2^{-Omega(n^1+\alpha)} for some positive alpha, which is crucial for obtaining the results in this paper.

__ Weaknesses__: - Many of the lower bounds seem to be direct applications of existing results of combinatorial geometry (although the tightness of these results are also discussed, and they nicely extend the early negative results of Kivinen on RPU learning)

__ Correctness__: They are correct as far as I have checked, although there are a few subtle points that I need the authors' clarification.

__ Clarity__: While I can understand the statements in this paper, I think the presentation can be much improved. For example:
- As RPU learning implies PAC learning, is there really a need to present Algorithm 1 and Theorem 3.5? Aren't we already happy with Theorem 4.11?
- in Algorithm 1, Threshold(S) is only informally defined, and an elaboration is needed. I think basically, the algorithm can successfully approximately recover b if it can find two neighboring + and - examples?
- In the proof of Proposition 4.2, no high-level idea of the proof was given, which makes it a bit hard to follow (although I agree that it is correct)
- In the second half of Corollary 4.7, is the goal only to identify the labels of all examples S drawn?
Also, what is the active learning algorithm used here? I am also confused about the statement that "the boosting algorithm in KLMZ [5] is reliable even when when given the wrong inference dimension as input". Are you referring the algorithm in page 16 of that paper (arxiv version)?
- In Theorem 4.9, is H here H_{d, \eta}?
- In the proof of Theorem 4.10, what is h \times [-\gamma, \gamma]? Is it {x: h(x) \in [-\gamma, \gamma]}?
- At the end of the proof of Theorem 4.10, it is said that "\forall h there must exists x, s.t. Q(S_h' - {x}) infers x." Would it be still possible that there exist two hypotheses h_1, h_2 that have _very small minimal-ratio_, and they agree with the queries in Q(S_h' - {x}), but disagree on x?
- In Algorithm 2, if g is a constant function, can be just replace it with a constant?
To align with the terminology of KLMZ as much as possible, I suggest changing the name "average inference dimension" to e.g.. "inference dimension tail bound". I think people usually use "dimension" to denote values that take integers.

__ Relation to Prior Work__: Yes. It also discusses a subsequent paper that benefits from the techniques in this paper.

__ Reproducibility__: Yes

__ Additional Feedback__: I thank the authors for the reply. My opinion has not changed. But I have follow-up questions that I hope the authors can clarify in the final version:
1. In the second half of Corollary 4.7, in the active RPU learning algorithm, do we need to first compute the value of k that minimizes the right hand side of the equation, then apply KLMZ's algorithm with that value of k? It might be interesting to extend KLMZ's algorithm and analysis to achieve adaptivity to the data-dependent inference dimension.
2. In Theorem 4.10, it would be very helpful to point out that what the inference algorithm used here is (it is still not clear to me if the inference algorithm needs to know the minimal-ratio or not - specifically, in the linear program alluded in page 16 of KLMZ, in combination with Claim 4.10 therein, it seems like the knowledge of minimal-ratio is needed). The algorithm right below Theorem 4.11 should be revised accordingly to incorporate this.

__ Summary and Contributions__: This paper studies the power of comparisons in the problem of actively learning (non-homogenous) linear classifiers. There are three main results in the paper: 1) in the PAC learning model, neither active learning nor comparison queries alone provide a significant speed-up over passive learning; 2) in the PUR-learning model, the paper confirms that passive learning with label queries is intractable information-theoretically, and active learning alone provides little improvement; 3) in the PUR-learning model, the comparison oracle provides a significant improvement in both active and passive learning scenarios.
In the context of previous work, the techniques of this paper are heavily based on a combination of the inference dimension (Kane, Lovett, Moran, and Zhang [5]) and the (non-efficient version of) margin-based active learning (Balcan and Long [2]). The paper also extends the analysis to the s-concave distribution based on a concentration result in Balcan and Zhang [3].

__ Strengths__: 1. Learning of linear classifiers with both comparison and label oracles were rarely studied in the literature; to the best of my knowledge, existing work includes Kane, Lovett, Moran, and Zhang [5] and Xu, Zhang, Miller, Singh, and Dubrawski [7], but many fundamental questions remain open in the community. Using existing techniques, this paper answers many of these questions.
2. The techniques are novel for the NeurIPS standard. The theoretical analysis seem solid. Experiments are available as a theoretical paper.

__ Weaknesses__: The paper does not consider computational efficiency and noise tolerance (given that the paper is built upon [2], which is a computationally-inefficient algorithm), while the techniques of achieving computational efficiency and noise tolerance in the existing work of active learning are available.

__ Correctness__: The claims and method are correct to me.

__ Clarity__: The paper is well-written and easy to understand.

__ Relation to Prior Work__: The paper did a good job on discussing how this work differs from previous contributions.

__ Reproducibility__: Yes

__ Additional Feedback__: ==========after rebuttal==========
I have read the rebuttal, and I am happy to recommend acceptance of this paper.

__ Summary and Contributions__: I have reviewed this paper before, the following is an adaptation of my past review.
This paper considers the problem of learning non-homogenous linear classifiers using comparison queries over distributions that are weakly concentrated (such as s-concave). The main focus of the paper is on the power of comparison queries for distribution-dependent and Reliable and Probably Useful learning.
Comparison queries in addition to the labels reveal which of two instances are closer to the boundary of the classifier. Comparison queries were shown to be useful for improving the query complexity of a learning task. Kane et al.’17 considered distribution-independent setting and showed that under assumptions, such as large margin assumption, comparison queries gain an exponential improvement in query complexity over (label) active learning. The same paper also showed that in some cases, query complexity won’t have an asymptotic improvement over passive or active learning, based on a definition of a notion of “inference dimension”. An example of this is learning halfspaces in a distribution independent setting even in R^3.
This paper first shows that learning non-homogenous halfspaces over uniform distribution active or passive-comparison learning individually requires poly(1/eps) queries. Similarly, in the RPU setting passive or active learning individually require (1/eps)^O(d) and comparison passive learning requires (1/eps) and comparison pool based only uses (d polylog(1/eps)).

__ Strengths__:
Overall, I like the results of the paper and I think studying comparison and label active learning learning for distribution-dependent settings is very valuable.
I also like that the paper studies RPU setting, where the algorithm has know what it’s uncertain about. Here, as mentioned above you need the power of comparison queries to get to d polylog(1/eps) as opposed to 1/eps^d. This model is not as well studied in modern literature, but it’s an important learning model when it comes to robustness learning.

__ Weaknesses__: Don't see a particular weakness.

__ Correctness__: Seems correct.

__ Clarity__: Good

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This paper investigates several scenarios in active learning. Namely, both PAC learning and RPU learning are considered for the model; both pool-based and membership query-based settings are considered for the active learning paradigm; and, both labeling and comparisons are considered for the queries. Several new lower bounds and upper bounds are obtained for the query complexity in those scenarios. Remarkably, it is shown that for comparison queries, a query complexity polynomial in log(1/epsilon) is obtained for both the PAC setting and the RPU setting, under relatively weak assumptions for the distributions. Synthetic experiments are performed for corroborating such results.

__ Strengths__: This is a well-written, well-positioned, and well-motivated paper, with new nontrivial results in active learning. Although the proof for the upper-bound in the PAC setting (Theorem 3.3) is inspired from [2], the upper bound in the RPU setting (Theorems 3.7 & 3.8) relies on advanced techniques in high-dimensional geometry (notably the concept of average inference dimension). In essence, this is a very good theoretical contribution.

__ Weaknesses__: I found no real weaknesses.

__ Correctness__: I am not an expert in high-dimensional geometry, and techniques related to the interesting concept of inference dimension. Yet, as far as I could check, the proofs look correct.

__ Clarity__: As mentioned above, this paper is very well-written: all notations and definitions are clearly presented, and Section 4 is particularly useful for understanding the main concepts and tools used in the proofs.

__ Relation to Prior Work__: To the best of my knowledge, the paper is well-positioned with respect to related work, and clearly explains the main improvements obtained for different scenarios.

__ Reproducibility__: Yes

__ Additional Feedback__: As a minor comment, I would suggest to mention some perspectives of further research, such as active learning with comparison queries in agnostic settings.
But a detailed conclusion is already provided in the extended version of the paper.