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Variable selection with false discovery rate control in deep neural networks

Nature Machine Intelligence volume 3, pages 426–433 (2021)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

Deep neural networks are famous for their high prediction accuracy, but they are also known for their black-box nature and poor interpretability. We consider the problem of variable selection, that is, selecting the input variables that have significant predictive power on the output, in deep neural networks. Most existing variable selection methods for neural networks are only applicable to shallow networks or are computationally infeasible on large datasets; moreover, they lack a control on the quality of selected variables. Here we propose a backward elimination procedure called SurvNet, which is based on a new measure of variable importance that applies to a wide variety of networks. More importantly, SurvNet is able to estimate and control the false discovery rate of selected variables empirically. Further, SurvNet adaptively determines how many variables to eliminate at each step in order to maximize the selection efficiency. The validity and efficiency of SurvNet are shown on various simulated and real datasets, and its performance is compared with other methods. Especially, a systematic comparison with knockoff-based methods shows that although they have more rigorous false discovery rate control on data with strong variable correlation, SurvNet usually has higher power.

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Fig. 1: Flowchart of SurvNet.
Fig. 2: Variable selection process on dataset 1.
Fig. 3: Results of SurvNet on the two real datasets.

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Data availability

The simulated data (datasets 1–4) were generated using the code at https://github.com/zixuans/SurvNet/tree/master/Data. The MNIST data (dataset 5) is available at http://yann.lecun.com/exdb/mnist/. The single-cell RNA-Seq data (dataset 6) is available at the GEO repository https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE87544. The synthetic data used in the NeurIPS paper16 were simulated using the code on https://github.com/zixuans/SurvNet/tree/master/Comparisons%20with%20knockoffs/Scenario%203, and the real datasets were provided by request from its author, Y. Lu. The synthetic data used in the AISTATS paper17 were simulated using the code at https://github.com/zixuans/SurvNet/tree/master/Comparisons%20with%20knockoffs/Scenario%204, and the two real datasets are available at https://archive.ics.uci.edu/ml/datasets/Bank+Marketing and https://archive.ics.uci.edu/ml/datasets/Polish+companies+bankruptcy+data.

Code availability

The code developed for the study of SurvNet is publicly available at the Github repository https://github.com/zixuans/SurvNet. The code for GL and SGL13 is publicly available at https://bitbucket.org/ispamm/group-lasso-deep-networks/src/master/. The code used to construct second-order knockoffs19 and deep knockoffs26 is available at https://github.com/msesia/knockoff-filter and https://github.com/msesia/deepknockoffs, respectively. The code of the algorithm proposed in the AISTATS paper17 is publicly available at https://github.com/jroquerogimenez/ConditionallySalientFeatures.

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Acknowledgements

This work was supported by the National Institutes of Health (R01GM120733 to J.L.), the American Cancer Society (RSG-17-206-01-TBG to J.L.) and the National Science Foundation (1925645 to J.L.).

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Authors and Affiliations

  1. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA

    Zixuan Song & Jun Li

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  1. Zixuan Song
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  2. Jun Li
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Contributions

J.L. conceived and supervised the study. J.L. and Z.S. proposed the methods. Z.S. implemented the methods and constructed the data analysis. Z.S. drafted the manuscript and J.L. substantively revised it.

Corresponding author

Correspondence to Jun Li.

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The authors declare no competing interests.

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Peer review information Nature Machine Intelligence thanks the anonymous reviewers for their contribution to the peer review of this work.

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Supplementary Information

Supplementary methods, results, discussions, Figs. 1 and 2, and Tables 1–11.

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Song, Z., Li, J. Variable selection with false discovery rate control in deep neural networks. Nat Mach Intell 3, 426–433 (2021). https://doi.org/10.1038/s42256-021-00308-z

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