Originality: I am not convinced that the contributions of this paper are more significant than that of [1], which have been cited in this paper already. Specifically, in comparison with [1] in Line 82, the authors state that these conclusions apply to a smaller set in weight space. I would appreciate it if the authors could quantify the difference here and have a discussion section to show the comparison with some form of mathematical comparison. Further, there have been quite a few papers that show convergence of GD on neural networks using something like strong convexity. Clarity The paper is written quite clearly and it is easy enough to follow the paper.

This paper studies the geometry of deep networks with piecewise affine and convex nonlinearities using the max affine spline operator (MASO) framework (BB18a, BB18b). The authors establish a connection between input space partitioning of a neural network and power diagrams. The paper is well-written, and presents interesting theoretical results. It would be great if the authors can provide more details on the computational aspects. Please see below for detailed comments/questions.

This is a very interesting paper with really impressive results! One thing I would have liked to see more is a "deconstruction" of the different elements in the proposed approach, to see which ingredients matter more. Of course, part of the difficulty in such an exercise is the current lack of a quantitative evaluation procedure for GANs (or other likelihood-free generative models). Something that concerns me is the complexity of the architecture and training procedure used in these experiments. As a condition for accepting this paper, I would like the authors to confirm that they will post their code for running these experiments.

GENERAL 1. accuracy vs. sample size: in statistics, "efficiency" is key (accuracy as a function of sample size). SPECIFICS 1. notation: i found it obtuse, perhaps there is no better way. "append" means put on the end. A) are these numbers good? B) assuming my comment #4 above is correct, then the fact that PfLDS does nearly identical to fLDS and the same is true for GCfLDS, suggests that f_\psi(.)

The authors propose a variational Bayes method for model selection in sigmoid belief networks. The method can eliminate nodes in the hidden layers of multilayer networks. The derivation of the criterion appears technically solid and a fair amount of experimental support for the good performance of the method is provided. I have to say I'm no expert in this area, and I hope other reviewers can comment on the level of novelty. You forgot to define b. - p. 5, ll.

This paper proposes ensemble sampling which essentially approximates Thompson Sampling for complex models in a tractable way. This will be useful in a variety of applications. There does not seem to be an obvious mistake in the proofs. However, there are a number of limitations of the proposed method. See below for detailed comments: 1.