Correctness: The main technical content seems to be correct. I have the following questions though: When using the linear assumption for the reward and the dynamics, the feature selection/setting is crutial. To relax the linear assumption, it is also mentioned, features can be pre-trained. What would be the recommended way to pre-learn it? For possible violation of the assumptions, how it would affect the results in practice?

This is a borderline paper. The paper is technically sound and addressing OPE in average-reward setting is an important problem. Despite that the work is an extension of Duan and Wang (for discounted setting) to the average-reward setting, the algorithm is somewhat different, as Duan and Wang uses FQE whereas the current paper performs stationary-distribution estimation. That said, there are a few weaknesses that the paper should try to address or at least discuss: 1. The entropy maximization is a novel algorithmic element which does not appear in previous approaches in the discounted setting.

Weaknesses: * Experiments: While the proposed method outperformed existing approaches in the results presented, the experiments seem rather limited in scope. For example, the VAE experiments included in the main text were done using linear encoder/decoder, which is very rarely used in practice. This is particularly concerning because for the nonlinear experiment included in the appendix, GS outperformed IGR on FMNIST (see Table 1). Although this was the only dataset on which IGR didn't outperform GS, it does raise the question of how IGR would scales to more difficult tasks (e.g. In this light, it would really strengthen this paper if the authors could demonstrate that IGR outperforms GS on more challenging tasks as well as compared to other methods such as VIMCO [1] and VQ-VAE [2].

This paper presents a simple alternative to the Gumbel-Softmax based on Gaussians and invertible transformations to the hypersimplex. As one reviewer noted, "the proposed approach is simple, has nice properties, and extensible". Many reviewers criticized the lack of experiments on non-linear models in the main text. Some reviewers felt that the clarity of the draft could be improved, in particular the motivation. This was a borderline paper, however I would like to recommend acceptance.

This paper addresses the important task of controlling the level sets that comprise the decision boundaries of a neural network. I think the proposed method is quite reasonable, well-described, and convincingly demonstrated to be quite useful across a number of tasks. Re level set sampling: - Doesn't the ReLU activation imply that D_x F(p; theta) is often 0 at many points p? How do you get around this issue when attempting to optimize p toward S(theta)? It seems this optimization might often get stuck in regions where D_x F(p;theta) 0 yet p lies far away from S(theta). Furthermore, the particular choice used by the authors should be a bit better motivated in the text as it's not clear to me. - It seems one can sample from level 0 set by instead just optimizing: min_x F(x;theta) via gradient descent in x.

This paper considers a communication efficient distributed optimization algorithm based on ADMM for stochastic optimization. The main idea is to perform multiple steps (can be timevarying) of stochastic gradient updates before the agents communicate, and therefore improving the communication efficiency. The proposed algorithm is shown to converge (in objective value & constraint violation) under a general non-smooth, non-strongly convex settings with O(1/eps) communication rounds and O(1/eps 2) unbiased gradient oracle calls. Other setting such as smooth strongly convex and non-smooth strongly convex are also analyzed and presented. Using multiple steps is however not novel.

After response----I keep my evaluation on the technical innovation and suboptimality of this paper. The basic Spider and SpiderBoost algorithms are both for first-order stationary point, they are almost the same, and both give n {1/2} rate. The simple way to modify both algorithms to escape saddle point is to add Negative Curvature Search (NCS) subroutine (which can be done in a very modular way, and is already shown in the Spider paper). I'd say it's almost trivial to also show SpiderBoost NCS to find second-order stationary point with n {1/2} rate. Comparing this paper with SpiderBoost NCS, there's no improvement from n {2/3} to n {1/2} (since Spiderboost is already n {1/2}), no simplification of Spider (as Spiderboost already did so). The only difference is replacing NCS by perturbations, which again requires some work, but most techniques are already there.

After extensive back and forth discussion by the reviewers, ultimately we felt that despite trivial solutions that are possible by mixing spider with negative curvature search, the approach of the present paper has its usefulness (as noted in the reviews too), and that the paper can be accepted. The authors are, however, strongly encouraged to look into the reviews carefully, and implement the points mentioned in the rebuttal -- because it would be a pity if after all the back-n-forth that this paper's review cycle has witnessed, if the authors did not update the paper to clarify its contribution, its value, and its contrasts with less implementable methods.

We prove the Fast Equilibrium Conjecture proposed by Li et al., (2020), i.e., stochastic gradient descent (SGD) on a scale-invariant loss (e.g., using networks with various normalization schemes) with learning rate \eta and weight decay factor \lambda mixes in function space in \mathcal{\tilde{O}}(\frac{1}{\lambda\eta}) steps, under two standard assumptions: (1) the noise covariance matrix is non-degenerate and (2) the minimizers of the loss form a connected, compact and analytic manifold. The analysis uses the framework of Li et al., (2021) and shows that for every T 0, the iterates of SGD with learning rate \eta and weight decay factor \lambda on the scale-invariant loss converge in distribution in \Theta\left(\eta {-1}\lambda {-1}(T \ln(\lambda/\eta))\right) iterations as \eta\lambda\to 0 while satisfying \eta \le O(\lambda)\le O(1) . Moreover, the evolution of the limiting distribution can be described by a stochastic differential equation that mixes to the same equilibrium distribution for every initialization around the manifold of minimizers as T\to\infty .

Originality: This paper develops asymptotics theory for high-dimensional regularized logistic regression (LR). The main result of the paper (Theorem 1) is proved for any locally-Lipschitz function \Psi which then in special cases provides asymptotics for common descriptive statistics like correlation, variance, mean-squared error. Special case results for L1 and L2 regularized LR are also derived and quantities highlighted in 1 above are derived. The paper also demonstrates that the numerical simulation results align with the theoretical relations. Quality: The paper contains high quality results and proofs, the notation and setup is well defined in section 2 before the main results.