Strengths: Comments about the paper: This paper presents convergence analysis of primal-dual natural policy gradient methods under the CMDP framework. Several recent works have shown convergence of policy gradients and optimality bounds (e.g Agarwal et al., Mei et al), but the paper extends similar analysis to (a) natural policy gradients (b) CMDP framework with constraints. Overall, it archives a sublinear rate of convergence in the CMDP framework, similar to other related works with convergence analysis. The analysis of the paper is done for the general MDP case with function approximation and restricted policy classes. It is a very well written paper that is easy to follow with significant theoretical derivation and proof details.

After reading the authors' rebuttal, the reviewers discussed their concerns about this paper. Ultimately, a consensus was not reached asreviewer #1 feels that the issues raised in her/his review were not properly addressed in the authors' feedback. The other reviewers also share some of the concerns raised by reviewer #1, but, given the rebuttals, they believe the authors can fix them in the final version and make the contribution of their paper clearer. I agree with them and so I suggest to accept the paper, but I recommend that the authors take into consideration the issues raised in the reviews and address them carefully in the final version of the paper.

Post-rebuttal: The authors have promised to incorporate an exposition of the sampler in the revised paper, I believe that will make the paper a more self-contained read. I maintain my rating of strong accept (8). I think this paper makes very nice contributions to the fundamental question of estimating the MLE distribution given a bunch of observations. I think the key contributions can be broken up into two key parts: - A bunch of simple but elegant structural results for the MLE distribution in terms of'tent distributions' -- distributions such that its log-density is piecewise linear, and is supported over subdivisions of the convex hull of the datapoints. This allows them to write a convex program for optimizing over tent distributions.

The submission provides a polynomial-time approximation algorithm for finding the maximum-likelihood log-concave density for a given set of data points in R d, for arbitrary d. The work is theoretical in nature, with proofs and no experiments. The problem is very interesting, since log-concave distributions include may of the commonly used parametric families (such as Gaussian), and the log-concave MLE has also other interesting properties. Previously the sample-complexity of learning a log-concave distribution has been studied, but a polynomial-time algorithm has been lacking. The present work provides such an algorithm.

Since everything is parametric, I'd expect explicit rates of convergence involvind all probalem complexity parameters (n, m, p, etc.) To make the rest of my points clear, let me recall the following notations are used in the paper: - n: the dimensionality of the covariate (i.e feature vector) X. Thus X is random vector in R n. BTW, in the context of ML or stats, I'd use another notation here, as n conventionally stands for "sample size".

This paper proposes a method for distributionally robust optimization under KL ambiguity sets for exponential families. Although KL ambiguity sets have their drawbacks, in particular not covering any changes in the inputs x, the present work produces a standard conic problem for a wide problem class via a novel analysis, provides good theoretical analysis, and yields good numerical results for a variety of small-scale classification problems. With the various clarifications that came up in the reviews, this paper makes a solid contribution to the DRO literature and will be quite welcome to the NeurIPS audience.

This paper develops a novel method to infer directional relationships between cortical areas of the brain based on simultaneously acquired EEG and fMRI data. Specifically, the fMRI activations are used to select ROIs related to the paradigm of interest. This information is used in a coupled state-space and forward propagation model to identify robust spatial sources and directional connectivity. The authors use a variational Bayesian framework to infer the latent posteriors and noise covariances. They demonstrate the power of joint EEG/fMRI analysis using two simulated experiments and a real-world dataset.

Restricted Boltzmann Machines (RBMs) are a common family of undirected graphical models with latent variables. An RBM is described by a bipartite graph, with all observed variables in one layer and all latent variables in the other. We consider the task of learning an RBM given samples generated according to it. The best algorithms for this task currently have time complexity \tilde{O}(n 2) for ferromagnetic RBMs (i.e., with attractive potentials) but \tilde{O}(n d) for general RBMs, where n is the number of observed variables and d is the maximum degree of a latent variable. Let the \textit{MRF neighborhood} of an observed variable be its neighborhood in the Markov Random Field of the marginal distribution of the observed variables.

Originality: First-order Gradient-based MCMC methods have to deal with determining an appropriate length scale for each variable. NUTS is one approach and this paper gives another approach whereby a parameter theta of a proposal distribution is adaptively improved to account for the covariance structure. At the same time theta is adapted to consider the entropy of the proposal distribution. This trade off for theta is rolled into a new speed measure which is the central point of this paper. The paper includes a lower bound of the speed measure that can be directly differentiated resulting in a practical algorithm. The paper also includes a heuristic that makes this adaptive MCMC algorithm applicable to MALA as well.

Proposes methods to update proposal distributions for MCMC using variational-type bounds and showed it is competitive with state of the art MCMC methods. Reviewers agreed this was an important contribution.