Paper:
INVASE: INSTANCE-WISE VARIABLE SELECTION US- ING NEURAL NETWORKS
The advent of big data brings with it data with more and more dimensions and thus a growing need to be able to efficiently select which features to use for a variety of problems. While global feature selection has been a well-studied problem for quite some time, only recently has the paradigm of instance-wise feature selection been developed. In this paper, we propose a new instance-wise feature selection method, which we term INVASE. INVASE consists of 3 neural networks, a selector network, a predictor network and a baseline network which are used to train the selector network using the actor-critic methodology. Using this methodology, INVASE is capable of flexibly discovering feature subsets of a different size for each instance, which is a key limitation of existing state-of-the-art methods. We demonstrate through a mixture of synthetic and real data experiments that IN-VASE significantly outperforms state-of-the-art benchmarks.

Review:
This paper proposes a new instance-wise feature selection method, INVASE. It is closely related to the prior work L2X (Learning to Explain). There are three differences compared to L2X. The most important difference is about how to backpropagate through subset sampling to select features.  L2X use the Gumbel-softmax trick and this paper uses actor-critic models.
The paper is written well. It is easy to follow the paper. The contribution of this paper is that it provides a new way,  compared to L2X, to backpropagate through subset sampling in order to select features. The authors compare INVASE with L2X and several other approaches on synthetic data and show outperforming results. In the real-world experiments, the authors do not compare INVASE with other approaches. 
Regarding experiments, instance-wise feature selection is often applied on computer vision or natural language process applications, where global feature selection is not enough. This paper lacks experiments on CV or NLP applications. For the MAGGIC dataset, I expect to see subgroup patterns. The patterns that authors show in Figure 2 are very different for all randomly selected 20 patients. The authors do not explain why it is preferred to see very different feature patterns for all patients instead of subgroup patterns.
I have questions about other two differences from L2X, pointed by the authors. First, the selector function outputs a probability for selecting each feature \hat{S}^\theta(x). In the paper of L2X, it also produces a weight vector w_\theta(x) as described in section 3.4. I think the \hat{S}^\theta(x) has similar meaning as w_\theta(x) in L2X. In the synthetic data experiment, the authors fix the number of selected features for L2X so that it forces to overselect or underselect features in the example of Syn4. Did the author try to relax this constraint for L2X and use w_\theta(x) in L2X to select features as using \hat{S}^\theta(x) in INVASE? 
Second, I agree with the authors that L2X is inspired by maximizing mutual information between Y and X_S and INVASE is inspired by minimizing KL divergence between Y|X and Y|X_S. Both intuitions lead to similar objective functions that INVASE has an extra term \log p(y|x) and \lambda ||S(x)||. INVASE is able to add a l_0 penalty on S(x) since it uses the actor-critic models. For the \log p(y|x) term, as the author mentioned, it helps to reduce the variance in actor-critic models. This \log p(y|x) term is a constant in the optimization of S(x). In Algorithm 1, 12, the updates of \gamma does not depend on other parameters related to the predictor network and selector network. Could the authors first train a baseline network and use it as a fixed function in Algorithm 1? I don't understand the meaning of updates for \gamma iteratively with other parameters since it does not depend on the learning of other parameters. Does this constant term \log p(y|x) have other benefits besides reducing variance in actor-critic models?
I have another minor question about scaling. How does the scaling of X affect the feature importance learned by INVASE?
Note: I have another concern about the experiments. Previous instance-wise variable selection methods are often tested on CV or NLP applications, could the authors present those experiments as previous works?