I am changing the score to 7. The paper gives a new algorithm for learning the structure Restricted Boltzmann Machines (formalized using Markov blankets), which is claimed to work for larger parameter regimes than the previous work. This is done by considering the problem of predicting the spin of a node given the spins of all other nodes. This dependence is shown to be given by a one-hidden layer neural net (with somewhat non-standard activations). An algorithm for learning this network is given based on polynomial approximation of the neural net and using regression on degree-D monomial feature map (with \ell_1 constraint). The algorithm works under L_\inf constraint on the input vector which is different from the past work. Given the above algorithm for learning the dependence of one node on the rest, under suitable non-degeneracy conditions, an algorithm is given for learning the structure (Markov blanket) of the RBM. Nearly matching lower bounds are provided (under hardness assumptions or in the SQ model). The reduction to neural networks is also used for learning supervised RBMs, which can be thought of as a neural network under distributional assumptions on the data (in terms of "sparsity and nonnegative correlations among the input features 307 conditional on the output label"). This distributional assumptions seems to be new.

The initial scores in the four reviews were all in favour of accepting, although not strongly. The paper studies a relevant problem, presenting a new algorithm with performance guarantees and almost matching lower bounds. However some questions were raised regarding, for example, connections to other work and practical algorithms, and also more technical issues. The authors provided a detailed reply. After discussion among the reviewers, their concerns were partially answered, leading to somewhat stronger support for accepting.

Given the space constraint of the rebuttal, I will trust the authors to indeed incorporate the changes as promised, and given this I increased my score. However, at several places in this paper, it is too dense to follow. More detailed comments are as follows. First, this paper lacks a dedicated related work section. There is some brief discussion about how this work differs from existing literature, in the introduction, yet it is not enough.

This paper proposes a Bayesian formulation for SPN models. All reviewers felt this formulation had merit, and there was agreement with the authors on possible improvments.

Weaknesses: The paper emphasizes its focus on causal structure learning. In doing so it assumes "causal sufficiency", that is, it assumes that there are no latent confounders of the measured variables. Generally, there are many latent confounders of the measured variables in most domains. In the past 20 years, there has been substantial progress in developing graphical representations and algorithms for learning equivalence classes of causal networks from observational data. When causal sufficiency is assumed, the learning of DAG structure is generally called Bayesian network structure learning, not causal structural learning, as in the title of the paper. It would be helpful for the paper to more prominently highlight this assumption.

All of the reviewers agree that this paper is both theoretically and modeling-wise a solid contribution to NeurIPS. My only concerns are that some of the author rebuttal points have not made it into the paper -- all of them should be added I think, in particular the related work (extended), the causal sufficiency clarification, and the run times.

This theoretical paper considers the problem of computing optimal value function in entropy-regularized MDPs and two-player games. It shows that the smoothness property of the Bellman operator in the presence of entropy regularized policies (and possibly other forms of regularization), can be used to derive a sample complexity which is polynomial of order O((1/ε) {4 c}), with c being a problem independent constant and ε the precision of the value function estimate. The proof is built upon the proposed algorithm, SmoothCruiser, an algorithm motivated in the sparse sampling algorithm of Kearns et al that recursively estimates V through samples and subsequently aggregates the results. This sampling dynamic programming is done up to a depth when the required number of samples is no longer polynomial. The paper is very well written and provides a solid result.

The reviewers were in consensus that this is an interesting and well written paper with a significant theoretical contribution. While empirical results should not be strictly required for a paper that is strong theoretically, they would nonetheless greatly improve the paper, and thus the authors are strongly encouraged to include them in the final version, even if they are relegated to supplementary material.

The marginalisation of variables within some steps of an MCMC algorithm is delicate. The main proposal here appears well justified, but it would have been nice to see the argument made a little more explicitly. The type of marginalisation described here seems to be more or less what would be described as a (partially) collapsed Gibbs sampler in the sense of [David A Van Dyk and Taeyoung Park. "Partially collapsed Gibbs samplers: Theory and methods". It was less clear to me exactly how the "blocking" strategy detailed in Section 4.1 would be justified from a formal perspective, and I do think that this needs clarifying. I.e. the collection of variables to be sampled is divided into three parts -- x', x and theta and the decomposition of the kernel seems to involve sampling: x from a kernel invariant to its distribution conditional on both x' and theta (starting from the previous x) x' from a kernel invariant with respect to its distribution conditional only upon x (starting from the previous x') \theta from its full conditional distribution and it's not completely transparent how one knows that this is invariant with respect to the correct joint distribution.

This paper makes a solid contribution to improving inference in certain state space models that a used extensively in practice, particularly when implementing such models in a probabilistic programming language.