From: ICLR.cc/2023/Conference/Program_Chairs
title: Paper Decision
decision: Reject
metareview:_summary,_strengths_and_weaknesses: This paper presents a new approach named sparse learnable masks (SLM) for feature selection. It maximizes the mutual information between features and labels, and optimizes the training objective in an end-to-end manner. Empirical results on eight datasets are reported and discussed. 

Overall, the proposed method is well motivated. Some of the presented ideas, such as the quadratic relaxation of mutual information, are very interesting. Experimental results on multiple benchmarks demonstrate the effectiveness of the proposed method.

Reviewers raised concerns regarding novelty, experiments, paper writing, and technical details. The authors have provided detailed responses, which addressed some of the concerns from reviewers. However, during the discussion phase, reviewers pointed out some remaining issues might not be easily addressed with minor revisions. For instance, the paper is not easy to follow, and evaluating the proposed method at different levels of selected features. Hopefully these issues could be addressed in the next version of this work.
justification_for_why_not_higher_score: Reviewers raised some valid concerns on paper writing and experiments, and they have consistent negative ratings. Reviewers found that the responses could not fully address the concerns.
justification_for_why_not_lower_score: N/A
summary_of_AC-reviewer_meeting: N/A

From: ICLR.cc/2023/Conference/Paper3480/Authors
title: Interesting point -- the sparsemax input scaling (Lemma 3.2) can address this phenomenon.
comment: This is indeed another interesting aspect that our work, in particular scaling-before-sparsemax, addresses. Dead neurons can indeed be a practical issue of vanilla sparsemax: as the support of sparsemax is determined by the distances between the top-ranking elements, having non-zero gradients only for the elements in the support of sparsemax causes only those elements to drift. As the inputs to sparsemax can vary without bound, this drift becomes larger over time, i.e. the distance between the top elements, which are in sparsemax's support, becomes larger, making the sparsemax support smaller, and blocking new elements from entering the support. Note that simply not receiving gradients does not automatically cause a neuron to die, as the top elements can become smaller from their gradient updates, hence reordering the sparsemax input. This is more likely to happen if the input to sparsemax were bounded.

However, when we scale the input to sparsemax (Lemma 3.2), we control this drift, and can either lengthen or shorten the distances between the top elements, hence avoiding the dead neurons by allowing new elements into the sparsemax support. 

To verify these two cases with dwindling and non-dwindling support, i.e. without and with scaling, we prepared a colab: https://colab.research.google.com/drive/11Bo3xqllUsTRmjlrsoSTX0sG9eqASOzy?usp=sharing.
This shows that with scaling, sparsemax support does not decrease. In fact, the same code also shows that while sparsemax loses support in the case without scaling, new indices are added to the sparsemax support, avoiding the dead neuron phenomenon.

We included this discussion in the Appendix. It can be made rigorous by using theory from Brownian motion, and is an interesting new direction to explore in followup work.

From: ICLR.cc/2023/Conference/Paper3480/Reviewer_BDBF
title: Comment
comment: A practical downside of sparsemax is the problem of "dead neurons". 

That is, since the gradient of sparsemax is zero for elements outside the support, the $i$th input that was initially mapped to 0 will always receive 0 probability until the end of training. Did you observe any stability issues during training?

From: ICLR.cc/2023/Conference/Paper3480/Authors
title: Thank you for reviewing the revisions. We made some updates regarding your comments.
comment: Thank you for reviewing the revisions and raising the score. We appreciate working together to improve this paper.

**Q**: One example that was also raised by other reviewers is the evaluation of the method for different levels of selected features and stability over different runs (initialization).

A: Regarding the stability over different runs, we’d like to emphasize that the different runs (e.g. in Fig. 2.3 as you pointed out earlier) have different randomly generated _hyperparameter_ settings, hence a degree of variability is expected, like for all neural network models [1, 2]. In general, feature selection variability is very low – similar features are selected when given the same hyperparameters but different random seeds. We have added a clarification on this. 

Regarding different levels of selected features, Table 2 shows the results at different levels of selected features on the largest dataset, and Fig. 2.3 shows performance of individual runs as a function of different levels of selected features. We’d be happy to add more results if you have any specific suggestions. 

[1] Snoek et al, Practical Bayesian Optimization of Machine Learning Algorithms. NeurIPS 2012.

[2] Chen et al, Towards Learning Universal Hyperparameter Optimizers with Transformers, 2022.

From: ICLR.cc/2023/Conference/Paper3480/Authors
title: Thank you! We updated the paper with further explanations, clarifications, and references.
comment: We appreciate your insightful comments and multiple thoughtful reviews of this work. 

**Q1**: This is indeed a common misunderstanding in the literature of feature selection when making claims about interpretability. It is ok to claim transparency with feature selection; however, there is a large body of work analyzing how the selected features align with human intuition and social attribution, a crucial aspect of interpretability.

A1: Thank you for this clarification. Indeed, while it is not our focus, there is a rich body of literature analyzing feature importance. We have reviewed and added some relevant references (Lundberg & Lee, 2017 [1]; Shrikumar et al., 2017 [2]; Zintgraf et al., 2017 [3]; Chang et al., 2019 [4]) to the beginning of Sec. 4.2. And as you mentioned, SLM achieves transparency (besides superior task performance) with feature selection, i.e. the user knows exactly which features are being used.

It is interesting to consider how the perceptual quality of selected features align with human intuition and social attribution. Various experimental studies (e.g. [5]) compare different feature selection methods for such alignment. In general, it is an interesting challenge to optimize feature selection with an objective that represents human intuition directly.

[1] Scott M Lundberg and Su-In Lee. A unified approach to interpreting model predictions. In Advances in Neural Information Processing Systems, 2017.

[2] Avanti Shrikumar, Peyton Greenside, and Anshul Kundaje. Learning important features through propagating activation differences. ICML 2017.

[3] Luisa M. Zintgraf, Taco S. Cohen, Tameem Adel, and Max Welling. Visualizing deep neural network decisions: Prediction difference analysis. ICLR 2017.

[4] Chun-Hao Chang, Elliot Creager, Anna Goldenberg, and David Duvenaud. Explaining image classifiers by counterfactual generation. ICLR 2019.

[5] I. Lage et al. “Human Evaluation of Models Built for Interpretability”, AAAI 2019.

**Q2**: Regarding Equation 10, what is the impact of $r_{cs}$ in the final loss (Eq. 9)?

It is a good question about the consistency term $r_{cs}$, which ensures that if two samples have the same values in their selected features, their model predictions are the same as well. In theory, if we imagine giving $r_{cs}$ a weight coefficient $\alpha$ in Eq. 9, with the interpretation that in the limit where $\alpha\to \infty$, this consistency is strictly enforced; and in the limit where $\alpha\to 0$, not at all. In practice, given that they have the same orders on terms such as model predictions $R(X,Y)$ by design, $r_{cs}$ and the remaining term in Eq. 9 have the same order of magnitude. Furthermore, $r_{cs}$ indeed is the most compute-intensive part in Eq. 9, as $r_{cs}$ requires pairwise comparisons within the batch (for each pair $X_{i1}, X_{i2}$ it is computed over the feature indices $j$ where $X_{i1}^{(j)} \ne X_{i2}^{(j)}$). Experimentally, on the Ames dataset with a batch size of 128, the $r_{cs}$ computation takes up 1.51 $\pm$ 0.012  ms, out of 1.94 $\pm$ 0.017 ms for the entire MI regularizer computation described in Eq. 9. This reveals an interesting accuracy-compute tradeoff, where users may want to skip the $r_{cs}$ computation for an even faster, approximate MI regularizer computation. We have added this explanation to the Appendix section on computational complexity.

**Q3**: In Section A.5: "whereas sparsemax can pass gradients through all of sparsemax(v)." This is not entirely true because the gradient of sparsemax is zero for entries outside the support.

A3: We appreciate you for pointing out the sparsemax gradient mistake (especially so as it's in the Appendix), the gradient is indeed 0 outside the support of sparsemax, hence not providing an advantage over top-k with softmax. One gradient related advantage sparsemax(z) does have is that it allows a faster computation of the Jacobian-vector product -- which typically suffices for backpropagation -- in $O(S(z))$ time, $S(z)$ being the support of $z$ (Martins and Astudillo, 2016), compared with linear time for softmax. We have updated this in A.5.

From: ICLR.cc/2023/Conference/Paper3480/Reviewer_BDBF
title: Response
comment: I want to thank the authors for their careful and thorough responses.

> A3: For real-world scenarios, we often do not know the ground truth feature importance, so directly assessing the accuracy of the selected features would not be possible. Hence academic literature (including LassoNet and other cited papers) on feature selection focuses on evaluations based on the downstream task accuracy, similar to our paper.

This is indeed a common misunderstanding in the literature of feature selection when making claims about interpretability. It is ok to claim transparency with feature selection; however, there is a large body of work analyzing how the selected features align with human intuition and social attribution, a crucial aspect of interpretability.

Regarding Equation 10, what is the impact of $r_{cs}$ in the final loss (Eq. 9)?

Some notes:

- In Section A.5: "whereas sparsemax can pass gradients through all of sparsemax(v)." This is not entirely true because the gradient of sparsemax is zero for entries outside the support.

From: ICLR.cc/2023/Conference/Paper3480/Reviewer_DZWY
title: After rebutal
comment: First, I want to thank the authors for the hard work spent in addressing all comments raised by the reviewers.
After reading the revised manuscript, I think that the paper has been improved.
I, therefore, raise my score to 5. 
In general, I like the ideas presented in the paper, but I still feel that some aspects should be improved before publication. 
One example that was also raised by other reviewers is the evaluation of the method for different levels of selected features and stability over different runs (initialization).

From: ICLR.cc/2023/Conference/Paper3480/Authors
title: Continuation of the previous response
comment: 
**Q19**: I would expect a comparison between the proposed method and a baseline that incorporates the sparsemax mask into a cross-entropy loss guiding the selection of features.

**A19**: Having a learned sparsemax mask as the way of filtering out non-selected features is the core building block of SLM, (other vital components such as the MI regularizer and tempering build on top of it). Note that there is no ground truth for the selected features, so using the supervision on feature selection accuracy would not be practical. Instead, SLM uses the supervision from downstream task accuracy, and there is cross entropy loss defined over class likelihoods.

From: ICLR.cc/2023/Conference/Paper3480/Authors
title: Continuation of the previous response
comment: **Q9**: Why do you choose specifically 50 features for all datasets? This is very restrictive, and limits the analysis.

**A9**: We consistently select 50 selected features to adhere to conventions set by literature, e.g. LassoNet reports the performance with 50 selected features in its main results. We note that this represents a small fraction of the total features for most datasets, similar to real world application scenarios where the number of features might be limited to a certain amount due to inference latency or feature storing cost. Furthermore, we deliberately kept this consistent across datasets without tuning this number to any method, to avoid favoring any given method. 

**Q10**: Figure 2 -subfigure 3 is confusing. The variability between different models seems HUGE, this means that the model strongly depends on initialization?

**A10**: That subfigure is showing the results for different hyperparameter combinations. For some hyperparameters, the optimal number of features might be a smaller intermediate value (e.g. if the model capacity is small). For that figure, we do not reoptimize hyperparameters for every number of selected features. That’s why the variability is observed for different hyperparameter trials. Our goal is to emphasize the general trend to converge at an intermediate value for the number of selected features. We also note that the y-axis in Figure 2 subfigure 3 is zoomed in to focus on the range from 0.84 to 0.95, precisely to make the differences across trials span across the vertical range.

**Q11**: Are the results in table 1 the average over multiple runs? Or the best run out of 300 runs based on the validation accuracy? Statistics are required to understand the performance of all methods.

**A11**: The results in Table 1 are the performance on the test split based on the best validation performance across 300 runs. This adheres to what’s often done in practice when deploying ML models, in terms of taking the model with the best validation performance. In particular, to ensure sufficient coverage of the hyperparameter space, we are using a large number of trials.

**Q13**: What are the train/ test validation sizes?

**A13**: We use a 70-10-20 train/validation/test split; and when available, we use the exact same train/validation/test samples as LassoNet for fair comparison. We have clarified this in the paper.

**Q14**: At least one simulation evaluating the FS capabilities of the method is required.

**A14**: We have added additional experiments on synthetic data, in Section A.8, further demonstrating that SLM scales well with respect to both the number of features and the number of samples.

**Q15**: Page 9 “Feature importance interpretability” the model is global; how do you interpret the FS locally for each sample? Also, nor example is given.

**A15**: Feature selection provides global explanations, rather than local. For a given task, we can understand what features contain the most relevant information for prediction. For local explanations, inherently interpretable models like TabNet (which contain sample-dependent masking) or post-hoc methods like Shapley values can be used. We have added a feature interpretability example in Appendix A.6.

**Q16**: How do I interpret figure 2? Are all comparisons using FS at the same level? If so, how are features selected without the MI without tempering? I can’t judge these results without this explanation.

**A16**: Correct, all comparisons in Fig. 2 are for the same number of features selected (50) for fairness. We clarified this. The MI and tempering components are not required for feature selection, as SLM can learn the sparse feature selection mask without either component: the MI regularizer can be removed from the training objective, and tempering can be disabled by keeping the number of selected features constant throughout training.

**Q17**: both probabilistic sparsity and tempering were previously studied for FS, and the authors don’t mention those prior works [2] and [3].

**A17**: Thanks for bringing this to our attention. We have added those references. 

Tempering is not a novel idea, and as you mentioned, other works have proposed it as well. We have reframed our contributions to not propose it as a novel contribution.

Regarding sparsity, other works typically have different regularization approaches rather than sparse masking of SLM - i.e. rather than the idea of using sparsity, our novelty resides on the way we achieve sparsity compared to the literature. We have clarified this point.

**Q18**: The MI component is interesting, but it’s hard to judge its value since figure 2 is not well explained.

**A18**: Thank you for pointing this out, we have updated the explanation of Fig. 2 and the Ablation Studies section.

From: ICLR.cc/2023/Conference/Paper3480/Authors
title: Thanks for your feedback! We have added scalability experiments, interpretability examples, writing clarifications, all the presentation fixes, and more complete comparison with literature
comment: Thanks for your valuable comments that have helped us improve our paper. We especially appreciate the positive comments on the paper writing, evaluation and the proposed components, and the constructive comments on the framing, the experiments, and related works. We have tried our best to address each of your comments, and the corresponding edits in the paper are marked with a navy blue “R3”. Please let us know if there are any remaining questions or concerns.  

**Q1**: I don’t see any analysis of how scalable the method is. In fact, the authors only apply it to dimensions that are not so large (D<1000).

**A1**: We describe SLM’s scalability with respect to the number of features and samples in the “Computational Complexity” section, as well as the fact that SLM amortizes feature selection across batches, and is therefore scalable with respect to the number of samples. To further corroborate these points given your feedback, we have added a new Section, Appendix A.7, to highlight the computational runtime advantage of SLM. In the example presented, we show that SLM is over an order of magnitude faster than the strong baseline LassoNet. 

We have also added new results on Synthetic datasets with 3000 features in Appendix A.8. The superior performance of SLM is observed consistently, for a high number of features even when only a small subset of them are salient. 

**Q2**: It is worthwhile to add the dimensions and sizes of all datasets.

**A2**: Further dataset details, including dimensions and sizes of all datasets, are already included in Appendix A.1 (Mentioned in the “Datasets and Settings” section.).

**Q3**: Furthermore, some strong, highly related baselines are missing.

**A3**:We indeed use LassoNet as a representative strong baseline, as well as a reference point for experimentation, throughout the paper. E.g. see results in Table 1.  Thank you for the additional references, we have included them in the related works section. We further analyzed comparison to these baselines in Appendix A.9, as well as conducted additional experiments against STG, the most closely related, strongest baseline (further details on this choice are in A.9. In summary, FsNet focuses on the high-dimension low-sample size setting often found in biological data, and Concrete Autoencoder is unsupervised). Whereas STG learns each Bernoulli variable gate independently, one advantage SLM has is accounting for interdependence amongst selected features, through both the fact that the probabilities in the mask are interdependent, and through the MI regularizer (discussed in Discussion section).

**Q4**: Introduction:“vast majority of the feature selection methods cannot take advantage of end-to-end learning.” This sentence is wrong and misleading, there are countless embedded methods that DO take advantage of this type of training, including 1-3, and many, many more.

**A4**: We have modified this sentence to be more precise. We already use LassoNet as a main baseline, and follow many of its conventions (e.g. train-validation-test splits, number of features selected, etc), we added the additional references you suggest. 

**Q5**: On page 2, the authors say that local feature selection models can improve accuracy, in the next sentence, they say it doesn’t alleviate overfitting. This is somewhat of a contradiction.

**A5**: The claim on the potential to improve the accuracy refers to the situation when train and test sets align perfectly in distribution and the datasets would be sufficiently large to train them. In this scenario, the higher modeling capacity of local feature selection can yield superior accuracy. However, for most real world datasets, this is rarely the scenario - many datasets may have distribution shifts, and thus high capacity models suffer from generalization on test dataset. In these settings, global feature selection could bring extra benefits as they would restrict the model capacity with a  judicious inductive bias. We have updated the writing to clarify this.

**Q6**: In the algorithm please mention where $\ell$ and E losses are explained in the paper.

**A6**: Currently the mutual information loss $E$ is derived in Eq. 9 as indicated in the algorithm. Following your feedback, we added an explanation for the $\ell$ loss right by the algorithm, in addition to its current explanation in the “Datasets and Settings” section.

**Q7**: Section 3.1 is called Sparse attention, but it has nothing to do with attention. Also this section is not very clear.

**A7**: We have updated the section title to “Mask Sparsity”, and improved the writing of this section. 
The original mention of attention refers to the analog with local attention described in the related works section (e.g. in Arik & Pfister 2019), where element-wise multiplication is used for one input vector to attend to the other.

**Q8**: Commas missing in Eq.7. and after theorem 3.3

**A8**: Thanks, we have added them.

From: ICLR.cc/2023/Conference/Paper3480/Authors
title: Thank you! We have clarified the novelty and contributions, improved writing and added more experimental results
comment: Thanks for your constructive comments. We especially appreciate the positive comments on the results and the quadratic relaxation effectiveness. We have tried to address each of your comments, and the corresponding edits in the paper are marked with a purple “R2”. Please let us know if there are any remaining questions or concerns.  

**Q1**:The novelty seems a little limited.

**A1**:Our paper proposes a new, simple method for effective feature selection, with principled techniques such as mask scaling for precise sparsity control and mutual information maximization. There is no prior work containing these methods for feature selection - if we are missing any past relevant works, please do let us know. Thanks to these innovations, we demonstrate SOTA results on eight real-world datasets along with ablation studies on the key components. We hope these help clarify the novelty. 

**Q2**:What is the primary advantage of the quadratic relaxation? Did the authors consider different estimators of MI, which have been extensively studied in representation learning? 

**A2**:The quadratic relaxation was derived for our mask-based feature selector, based on first principles, in that 1) naively computing the mutual information between features and labels involves computing an exponential number of probabilities, the optimization of which is intractable (Fleuret, 2004), the quadratic relaxation computes a tractable and provably close approximation; and 2) the quadratic relaxation allows for mutual information maximization _conditioned_ on the learned feature mask. These points are discussed in Section 3.4 - we have further clarified some explanations. Indeed mutual information approximators have been extensively studied in representation learning, we do consider them, and compare and contrast in the “Relation to other MI estimations in deep learning” section under Discussion. 
 
**Q3**:the formal definitions of mathematical notations could be summarized in an individual paragraph or section

**A3**:Thanks for this suggestion. We have added a central notations section (see Sec. 3) to improve the readability of the paper. We have also improved notation throughout the paper, such as updating the mask notation as you suggested. 

**Q4**:The writing and poor motivation extremely limits the readability of this paper.

**A4**:Regarding the writing, we have put significant effort into improving the paper writing:
We have improved the Introduction section, more clearly positioning the novelty of our work and contributions. 
We have introduced a new section to summarize the math notations in Sec. 3.
We have systematized the math notation addressing the comments.
We have improved the motivation for the methods.
We have added extra relevant references and highlighted the difference of our work. 
We have added missing details for the experiments, and improved the explanations of the results.

Regarding the motivation, we highlight that SLM is a novel technique for end-to-end global feature selection, which is a well-motivated problem with many prior studies, as outlined in the Related Works section. SLM incorporates methodological innovations on integration of sparse masking and mutual information regularization, and outperforms other feature selection techniques on a wide range of tasks. 
If you have further feedback on paper writing, please let us know and we would be happy to incorporate.

**Q5**:The experiments are not sufficient.

**A5**:We currently have extensive experiments on feature selection performance on 8 real-world datasets against 9 competitive baselines; ablation studies on the MI regularizer and tempering, as well as two studies on feature selection performance as the number of features varies. In addition, we have added additional experiments: 
Feature interpretability experiments in Appendix A.6, where we study the selected features on the Ames housing dataset.
Computational time experiments in Appendix A.7, where we compare SLM’s runtime against a strong end-to-end baseline. 
Synthetic data experiments in Appendix A.8, to highlight the strong performance of SLM even with a very high number of features.
Extra comparisons with other end-to-end baselines in Appendix A.9, to better highlight the strengths of SLM.

We hope the improved clarity and our responses addressed your concerns for “Correctness” and “Technical/Empirical Novelty And Significance”. Please let us know if you have any further feedback.

From: ICLR.cc/2023/Conference/Paper3480/Authors
title: ...above comment continued.
comment: **Q9**: What is b in its time complexity, batch size? What is n? What is N (in sec. 3.2)?

A9: We have clarified the notation in the computational complexity section (which is consistent with the overall notation). $n$ is indeed the sample size, $b$ the batch size, and $N$ the total number of train steps.

**Q10**: How exactly were the different methods of feature selection tested?

A10: For each feature selection method, we employ the corresponding method to select features, and we pass only those selected features as inputs to the downstream predictor (which is fixed to be an MLP). While this was alluded to elsewhere, we have added this description to Sec. 4.1 to improve clarity.

**Q11**: The caption of Figure 2 is too close to the main text.

A11: We have updated Fig. 2 caption placement.

**Q12**: This arrives from the property of temperature-scaling of Fenchel-Young Losses, which can be used to cast sparsemax as a regularized prediction function.

A12: Thanks for pointing out this interesting connection with Fenchel-Young losses – this is indeed a nice generalization. We have added this connection and the suggested references in Sec. 3.2. 

**Q13**: Although the sparsity can be exploited to interpret the model globally, the paper does not provide examples or evaluations toward this goal.

A13: We have added feature interpretability experiments in Sec. A.6 on the Ames housing dataset, and provide an analysis on the plausibility of selected features.

From: ICLR.cc/2023/Conference/Paper3480/Authors
title: Thank you. We streamlined the writing, improved notation, added experiments & references.
comment: Thank you for your valuable comments, which have helped us improve the paper. We especially appreciate the positive comments on the methodology and results, and the constructive comments regarding notation, connections to other works, and additional experiment suggestions. We have tried to address each of your comments, and hope they resolve your concerns. The corresponding edits in the paper are marked with a blue “R1”. Please let us know if there are any remaining questions or concerns. 

**Q1**: The math notation makes the paper hard to follow.

A1: Thanks for noting this point. We have added a central notations section (see Sec. 3) to improve the readability of the paper. We have also improved notation throughout the paper, such as updating the mask notation as you suggested. 

**Q2**: the related work on masking covers a small number of recent works.

A2: Thanks for bringing these works to our attention. We have added all of these to our paper. We agree that those works are not proposed for global feature selection, and the rationale extraction task they are focusing on corresponds to local explainability via local feature selection, similar to the TabNet paper that we cite. As you point out, those papers also propose mechanisms to control sparsity, for example by optimizing the KL divergence between the mask distribution and a prior distribution. Correia et al relates to our work the most with regard to controlling sparsity, as it also provides a modification to the sparsemax operator, but it is limited to reducing the number of selected features.

**Q3**:It is hard to assess if the subset of selected features is meaningful since the ground truth feature importance is unknown for the experimented datasets.

A3: For real-world scenarios, we often do not know the ground truth feature importance, so directly assessing the accuracy of the selected features would not be possible. Hence academic literature (including LassoNet and other cited papers) on feature selection focuses on evaluations based on the downstream task accuracy, similar to our paper. 

As qualitative analysis on the meaningfulness of the selected features, we have added a new interpretability section to Appendix A.6. On the Ames housing dataset, we show that the chosen features and their importance ranking (via mask scores) conform to common sense about the task. On the other hand, we note that the mask scores are just first order approximations to feature importance, as the downstream models are nonlinear and combinations of features can be more important than sum of individual importances (i.e. the importance of each individual feature is not well-defined due to feature interactions.). However, sparse selection of features, and elimination of unused features already yield useful insights about the task. We have emphasized this point in the paper.

**Q4**: there are no concrete examples of the interpretability of the selected features. Are the selected features plausible?

A4: Thanks for the suggestion. We have added a new section in Appendix A.6, on the interpretability of selected features for the Ames housing dataset. As we show, SLM selects the salient features that are expected to be related to the downstream task. 

**Q5**: the paper can be improved by reporting the running costs.

A5: This is a good idea. We have added running time cost experiments in Appendix A.7, where we repeatedly measure the runtime of one epoch on the mice dataset, and compare against LassoNet, a strong end-to-end learning baseline. Our results show that SLM’s low computational complexity in theory also translates to fast execution in practice - SLM is an order of magnitude faster than LassoNet in this experiment, this is consistent with our observations throughout experiments. 

**Q6**: Overall I found the paper hard to follow.

A6: We appreciate this point. We have put significant effort into improving the readability of our paper. We have improved the motivations, systematized and improved the math notations (e.g. lowercase bold letters refer to vectors only), and improved the writing in every section.

**Q7**: The introduction is too overwhelming and disconnected from the main part of the paper.

A7: We have improved the flow of the Introduction section and minimized the irrelevant discussions. We have also added more connections with the rest of the paper. 

**Q8**: while the math notation is confusing.

A8: Thank you for pointing this out about the notation. We have improved the notation such as making bold lowercase letters denote vectors only, and defined a custom $\odot$ element-wise multiplication between the input data and the learned mask $X\odot m_{sp}$, instead of using the overloaded dot product symbol.

From: ICLR.cc/2023/Conference/Paper3480/Reviewer_BDBF
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: In this paper, the authors propose a method to dynamically select a subset of features in a neural network. In contrast to standard approaches, this set of features is _global_, and therefore the same subset of features will be ignored for all samples in the dataset.
The proposed method consists of three ingredients:
1. Scaling sparsemax to the desired sparsity level.
2. Linearly increase the desired level of sparsity during training.
3. Optimize the task loss with a regularization term that captures the mutual information between the selected features and the label.

Results on 8 datasets showcase the method's effectiveness, and ablation studies are performed to understand the impact of the sparsity level and the regularization term.
strength_and_weaknesses: Things that I liked in this paper:
- The motivation is clear.
- The idea of scaling inputs by sparsemax globally is exciting.
- Lemma 3.2 is incredibly useful.
- The results are impressive, and the ablation studies are very informative.


However, there are many things to be improved:

- The math notation makes the paper hard to follow. 
- Although I liked the connection to the classical literature on feature selection, the related work on masking covers a small number of recent works. See some important missing references below.
- It is hard to assess if the subset of selected features is meaningful since the ground truth feature importance is unknown for the experimented datasets.
- Linked to the above point, there are no concrete examples of the interpretability of the selected features. Are the selected features plausible? 
- Although the authors claim the regularizer can be computed efficiently, the paper can be improved by reporting the running costs.

On sparse masks with a controlled sparsity level:
- Lei, Tao, Regina Barzilay, and Tommi Jaakkola. "Rationalizing neural predictions." arXiv preprint arXiv:1606.04155 (2016). 
- Bastings, Jasmijn, Wilker Aziz, and Ivan Titov. "Interpretable neural predictions with differentiable binary variables." arXiv preprint arXiv:1905.08160 (2019).
- Guerreiro, Nuno Miguel, and André FT Martins. "Spectra: Sparse structured text rationalization." arXiv preprint arXiv:2109.04552 (2021).
- Paranjape, Bhargavi, Mandar Joshi, John Thickstun, Hannaneh Hajishirzi, and Luke Zettlemoyer. "An information bottleneck approach for controlling conciseness in rationale extraction." arXiv preprint arXiv:2005.00652 (2020)
- Correia, Gonçalo, Vlad Niculae, Wilker Aziz, and André Martins. "Efficient marginalization of discrete and structured latent variables via sparsity." Advances in Neural Information Processing Systems 33 (2020): 11789-11802.
clarity,_quality,_novelty_and_reproducibility: Overall I found the paper hard to follow. The introduction is too overwhelming and disconnected from the main part of the paper, while the math notation is confusing. For example: bold symbols are used to denote scalars (e.g., $\mathbf{m}$), and $\cdot$ between vectors is used to denote element-wise multiplication (e.g., $\mathcal{M}_{sp} \cdot \mathbf{x}$). Moreover, it is unclear how equation 10 is computed efficiently. What is $b$ in its time complexity, batch size? What is $n$? What is $N$ (in sec. 3.2)? 

Clarification question:
How exactly were the different methods of feature selection tested? Taking XGBoost as an example: were the "selected" features first discovered with XGBoost, and then later only these features were passed as input to an MLP?

Other things:
- The caption of Figure 2 is too close to the main text.
- Multiplying the argument of sparsemax by $m$ is equivalent to use a temperature parameter $\tau = 1/m$. This arrives from the property of temperature-scaling of Fenchel-Young Losses, which can be used to cast sparsemax as a regularized prediction function. See more in [1] and [2]. 

[1] Peters, Ben, Vlad Niculae, and André FT Martins. "Sparse sequence-to-sequence models." arXiv preprint arXiv:1905.05702 (2019).

[2] Blondel, Mathieu, André FT Martins, and Vlad Niculae. "Learning with Fenchel-Young losses." J. Mach. Learn. Res. 21, no. 35 (2020): 1-69.
summary_of_the_review: Overall this paper provides a new method to induce a sparse global model. Although the sparsity can be exploited to interpret the model globally, the paper does not provide examples or evaluations toward this goal. However, the predictive performance is impressive across a wide range of datasets, and the ablation studies help illuminate the proposed approach's main contributions. 
correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
empirical_novelty_and_significance: 3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.
flag_for_ethics_review: NO.
recommendation: 5: marginally below the acceptance threshold

From: ICLR.cc/2023/Conference/Paper3480/Reviewer_1pMd
confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
summary_of_the_paper: This paper proposes a method with the automatic mask scaling which can gradually temper this sparsity. The proposed SLM maximize a quadratic relaxation (with the help of Taylor approximation) of the mutual information between the selected features and the labels.
strength_and_weaknesses: ### Strength
- The quadratic relaxation of MI is interesting to some extent. 
- The experiments seems effective (even though the experiments may be insufficient). 

### Weakness
- The novelty seems a little limited. Maybe I misunderstood something important due to the poor description of the motivation. Sections 3.2-3.4 are not attractive enough. The quadratic relaxation of MI may be interesting to some extent. Did the authors consider different estimators of MI, which have been extensively studied in representation learning? What is the primary advantage of the quadratic relaxation?
- The writing of this paper could be further improved. For example, the formal definitions of mathematical notations could be summarized in an individual paragraph or section. The writing and poor motivation extremely limits the readability of this paper. 
- The experiments are not sufficient. Although the space is limited, the extra experiments with different selected features could be reported in appendix. 
clarity,_quality,_novelty_and_reproducibility: The writing quality of this paper could be improved. For example, a summary of mathematical notations will help a lot to improve the readability. 

The novelty is a little limited.  
summary_of_the_review: The main concerns come from the limited novelty. However, I would like to update score after discussion, if my concerns can be addressed.  
correctness: 2: Several of the paper’s claims are incorrect or not well-supported.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 3: reject, not good enough

From: ICLR.cc/2023/Conference/Paper3480/Reviewer_DZWY
confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
summary_of_the_paper: The authors address a challenging and important problem of feature selection. The new approach is an embedded-type method that relies on sparse max to impose sparsity on the input features of a model. They use 8 real datasets and compare to several baselines to demonstrate the effectiveness of the method. 
strength_and_weaknesses: The problem is well-studied but still important for the community. The English level is satisfactory, and overall I enjoyed reading the paper. The problem statement and solutions are clear. The evaluation seems clear, and they further provide an ablation study to demonstrate that the proposed components are beneficial. Related work is mostly well-written. Another strength is the ability of the method to select exactly k features. 

The major weakness of the paper is its framing; the authors name their method Scalable FS… but I don’t see any analysis of how scalable the method is.
In fact, the authors only apply it to dimensions that are not so large (D<1000). It is worthwhile to add the dimensions and sizes of all datasets. Furthermore, some strong, highly related baselines are missing, for example: 

[1] Lemhadri et al. Lassonet: Neural networks with feature sparsity

[2] Yamada et al. Feature Selection using stochastic gates, 2020

[3] Balin et al .Concrete autoencoders: Differentiable feature selection and reconstruction

[4] Singh et al. Fsnet: Feature selection network on high-dimensional biological data

In the following, I provide detailed comments to help the authors improve the paper:
Introduction: “vast majority of the feature selection methods cannot take advantage of end-to-end learning.” This sentence is wrong and misleading, there are countless embedded methods that DO take advantage of this type of training, including 1-3, and many, many more.

Missing works in related work 1-3.

On page 2, the authors say that local feature selection models can improve accuracy, in the next sentence, they say they it doesn’t alleviate overfitting. This is somewhat of a contradiction.

In the algorithm please mention where \ell and E losses are explained in the paper.

Section 3.1 is called Sparse attention, but it has nothing to do with attention.
Also this section is not very clear.

Commas missing in Eq.7. and after theorem 3.3

Why do you choose specifically 50 features for all datasets? This is very restrictive, and limits the analysis.

Figure 2 -subfigure 3 is confusing. The variability between different models seems HUGE, this means that the model strongly depends on initialization? You need to elaborate on this and demonstrate.

Are the results in table 1 the average over multiple runs? Or just the best run out of 300 runs based on the validation accuracy?

Statistics are required to understand the performance of all methods.

What are the train/ test validation sizes?

At least one simulation evaluating the FS capabilities of the method is required. For example, using synthetic data or datasets from the Nuerips 2003 FS challenge. Thus providing information to the reader about the model's ability to recover the informative features.

 Page 9 “Feature importance intepretability” the model is global; how do you interpret the FS locally for each sample? Also, nor example is given.

How do I interpret figure 2? Are all comparisons using FS at the same level?
If so, how are features selected without the MI without tempering? I can’t judge these results without this explanation. 

clarity,_quality,_novelty_and_reproducibility: The introduction and problem statement is clear, the paper does provide some novelty tempering procedures. However, both probabilistic sparsity and tempering were previously studied for FS, and the authors don’t mention those prior works [2] and [3]. The evaluation is very limited since the authors don’t show averages and standard deviations over many experiments, also such results are hard to reproduce.
summary_of_the_review: Overall the authors present a new method for an important problem. They use sparse max with tempering and demonstrate the approach on 8 datasets. However, the experimental results are quite limited as they don’t provide statistics and don’t compare to highly related works (mentioned above). The MI component is interesting, but it’s hard to judge its value since figure 2 is not well explained. I would expect a comparison between the proposed method and a baseline that incorporates the sparsemax mask into a cross-entropy loss guiding the selection of features. Based on these reasons, I can’t recommend accepting the paper.

correctness: 3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.
technical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
empirical_novelty_and_significance: 2: The contributions are only marginally significant or novel.
flag_for_ethics_review: NO.
recommendation: 5: marginally below the acceptance threshold