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Group Regularized Estimation Under Structural Hierarchy

Author

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  • Yiyuan She
  • Zhifeng Wang
  • He Jiang

Abstract

Variable selection for models including interactions between explanatory variables often needs to obey certain hierarchical constraints. Weak or strong structural hierarchy requires that the existence of an interaction term implies at least one or both associated main effects to be present in the model. Lately, this problem has attracted a lot of attention, but existing computational algorithms converge slow even with a moderate number of predictors. Moreover, in contrast to the rich literature on ordinary variable selection, there is a lack of statistical theory to show reasonably low error rates of hierarchical variable selection. This work investigates a new class of estimators that make use of multiple group penalties to capture structural parsimony. We show that the proposed estimators enjoy sharp rate oracle inequalities, and give the minimax lower bounds in strong and weak hierarchical variable selection. A general-purpose algorithm is developed with guaranteed convergence and global optimality. Simulations and real data experiments demonstrate the efficiency and efficacy of the proposed approach. Supplementary materials for this article are available online.

Suggested Citation

  • Yiyuan She & Zhifeng Wang & He Jiang, 2018. "Group Regularized Estimation Under Structural Hierarchy," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(521), pages 445-454, January.
    • Handle: RePEc:taf:jnlasa:v:113:y:2018:i:521:p:445-454
    DOI: 10.1080/01621459.2016.1260470
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    Citations

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    Cited by:

    1. Feng Li & Yajie Li & Sanying Feng, 2021. "Estimation for Varying Coefficient Models with Hierarchical Structure," Mathematics, MDPI, vol. 9(2), pages 1-18, January.
    2. Yao Dong & He Jiang, 2018. "A Two-Stage Regularization Method for Variable Selection and Forecasting in High-Order Interaction Model," Complexity, Hindawi, vol. 2018, pages 1-12, November.
    3. He Jiang, 2022. "A novel robust structural quadratic forecasting model and applications," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 41(6), pages 1156-1180, September.
    4. Dewei Zhang & Yin Liu & Sam Davanloo Tajbakhsh, 2022. "A First-Order Optimization Algorithm for Statistical Learning with Hierarchical Sparsity Structure," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1126-1140, March.
    5. Wang, Cheng & Chen, Haozhe & Jiang, Binyan, 2024. "HiQR: An efficient algorithm for high-dimensional quadratic regression with penalties," Computational Statistics & Data Analysis, Elsevier, vol. 192(C).
    6. Sanying Feng & Menghan Zhang & Tiejun Tong, 2022. "Variable selection for functional linear models with strong heredity constraint," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(2), pages 321-339, April.

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