__ Summary and Contributions__: This paper considers the problem of learning a Restricted Boltzmann Machine when the number of latent variables connected to any Markov Random Field neighborhood is small. Specifically, if this number is bounded by s, then the authors give an algorithm to learn the RBM in time O(n^(2^s + 1)), where n is the number of visible nodes in the RBM. This represents an improvement over the current state of the art whenever s < log(d-1), where d is the maximum degree of the latent variables in the RBM.
The key ingredient of the argument is a structural result on RBMs, showing that whenever there is a set of nodes with large mutual information with a node, there must exist a subset of those nodes of size 2^s that has large mutual information with the node. Thus, when constructing the neighborhoods of the RBM, it is sufficient to search over subsets of size at most 2^s.
The paper also considers the problem of learning such RBMs with respect to prediction error. That is, conditioned on the rest of the nodes, give the probability that a particular node will be set to 1. The authors show that this can be done in much the same way as the other result, with the exception that they do not have any dependence on the minimum potential of the underlying MRF, which can cause a large blow up in the other setting.

__ Strengths__:
1. This paper clearly identifies settings in which they can improve upon the best known time for learning RBMs. They also do a nice job of providing examples where such setting can occur.
2. The paper also identifies an issue with their result (dependence on the minimum potential) and provides a different learning objective where such a dependence can be avoided.

__ Weaknesses__: As this paper builds upon a previously established approach to learning RBMs via MRFs, one may ask whether or not there is enough here that is novel for NeurIPS. I am inclined to believe that this work is novel enough; however, I think that the authors could help their case here by pointing out the difficulties that arise in proving the paper's main structural result that do not arise in other related structural results such as Hamilton et al. (2017).

__ Correctness__: I have not verified all the details of the proofs, but what I have looked at appears to be correct.

__ Clarity__: The paper is clear and well-written.

__ Relation to Prior Work__: This paper does a good job of describing related work, particularly those results which it builds on. As I pointed out in "weaknesses", this paper could be further improved by pointing out the new challenges that arise in proving their structural result that are not present in previous works.

__ Reproducibility__: Yes

__ Additional Feedback__: One question I have about Theorem 11: is it possible to give a similar result when we condition on a smaller subset of nodes (instead of everything except u)?
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After reading the other reviews and the author response, I still feel that my original score is appropriate. I do feel that the authors should include a brief discussion in their paper about the difficulties that arise in proving their structural result (the author response on this point would be a great starting point). This would help reinforce the novelty of the paper and hopefully inspire future work.

__ Summary and Contributions__: The authors describe theoretical rates of learning for RBMs with "few latent variables" (connected to any particular visible node). This is a relatively novel task, and their bounds follow by applying some existing work on structure recovery with the addition of a few key insights.

__ Strengths__: * Appears to be a novel task, potentially interesting to the community

__ Weaknesses__: * Precise connection to alternative approaches is unclear (see additional notes)
* Significance and usefulness not clear (see additional notes)

__ Correctness__: I think so

__ Clarity__: Mostly. It gives precise definitions, but I found aspects of it confusing (see additional notes).

__ Relation to Prior Work__: Mostly. A notable exception is whether other works could also be applied directly using the authors' "small Markov blanket" assumptions (see notes).

__ Reproducibility__: Yes

__ Additional Feedback__: I found the phrase "few latent variables" to be confusing, since this is not really what the authors mean. In particular, their bounds use "s", which is bounded by the number of latent variables connected to any variable in the Markov blanket of Xi in the marginal distribution over the visible X. This was not clear, in my view, until the formal definition (150-155).
Since this style of RBM is not well studied, the practical significance of learning rates is not clear. The type of model is intuitively appealing (many models use "local" latent variables or encourage sparseness), and perhaps the work would spur application of these styles of RBM, but it's difficult to say that this is improving the theory for a well-established problem of importance.
In my opinion, the work could also use more intuition about what properties are being leveraged to provide the improved bounds. In particular, one view is that in this setting, we can estimate the Markov blanket of X (which cannot be too big), and then use these cliques to determine the set of latent variables. From that perspective, almost any method that uses independence tests could be applied and achieve "similar" bounds. These would likely not be equivalent, since the authors' method explicitly takes into account the number of latent variables (used in e.g. Lemma 6 to define the mutual information as a weighted sum of 2^s terms), but it is unclear to me how these approaches would be related, or whether the authors' results in better statistical efficiency.
Additionally, Section 5 explictly shows how learning such models can be extremely difficult, due to cancellations of correlation in X from different latent variables. The authors' argument appears to be that such errors in the model do not affect the accuracy of conditional predictions, which they formalize using the same bounding techniques.
Line 203: connected to observed variables in I : should this also include u?

__ Summary and Contributions__:
The paper provides an algorithm that correctly recovers the neighborhood of each observed variable, and has better time complexity, for restricted Boltzmann machines with "few latent variables".

__ Strengths__:
The theoretical results seem sound.

__ Weaknesses__:
Relevance might be the main weakness. (Please see my additional feedback below.)

__ Correctness__:
The main claims seem correct.

__ Clarity__:
The paper is clear.

__ Relation to Prior Work__:
Relation to prior work is properly discussed.

__ Reproducibility__: Yes

__ Additional Feedback__:
While the authors properly address the issue that an algorithm better than O(n^d) might not be possible in general (Lines 59-66), a reader might still wonder why the regime s < log(d-1) is relevant for their O(n^{2^s+1}) result to be better than prior work. It might better to motivate this somewhat tangentially from some applied areas.
=== AFTER REBUTTAL
According to the author response, "a bound of d on the maximum degree in an RBM implies that s <= d^3". The authors then assume the regime s < log(d-1). I was expecting some sort of motivation in either theoretical prior work in this or other ML problems, or motivation tangentially from some applied areas. Still, some illustrative examples are given in Figure 1.
Given the above, I keep my initial evaluation of 6.