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Global optimality of nonconvex penalized estimators

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  • Yongdai Kim
  • Sunghoon Kwon

Abstract

Nonconvex penalties such as the smoothly clipped absolute deviation or minimax concave penalties have desirable properties such as the oracle property, even when the dimension of the predictive variables is large. However, checking whether a given local minimizer has such properties is not easy since there can be many local minimizers. In this paper, we give sufficient conditions under which a local minimizer is unique, and show that the oracle estimator becomes the unique local minimizer with probability tending to one. Copyright 2012, Oxford University Press.

Suggested Citation

  • Yongdai Kim & Sunghoon Kwon, 2012. "Global optimality of nonconvex penalized estimators," Biometrika, Biometrika Trust, vol. 99(2), pages 315-325.
    • Handle: RePEc:oup:biomet:v:99:y:2012:i:2:p:315-325
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    File URL: http://hdl.handle.net/10.1093/biomet/asr084
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    Cited by:

    1. Lee, Sangin & Kim, Yongdai & Kwon, Sunghoon, 2012. "Quadratic approximation for nonconvex penalized estimations with a diverging number of parameters," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1710-1717.
    2. Lian, Heng & Kim, Yongdai, 2016. "Nonconvex penalized reduced rank regression and its oracle properties in high dimensions," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 383-393.
    3. Xiang Zhang & Yichao Wu & Lan Wang & Runze Li, 2016. "Variable selection for support vector machines in moderately high dimensions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(1), pages 53-76, January.
    4. Kwon, Sunghoon & Lee, Sangin & Kim, Yongdai, 2015. "Moderately clipped LASSO," Computational Statistics & Data Analysis, Elsevier, vol. 92(C), pages 53-67.
    5. Jeon, Jong-June & Kwon, Sunghoon & Choi, Hosik, 2017. "Homogeneity detection for the high-dimensional generalized linear model," Computational Statistics & Data Analysis, Elsevier, vol. 114(C), pages 61-74.
    6. Yongdai Kim & Jong-June Jeon & Sangmi Han, 2016. "A Necessary Condition for the Strong Oracle Property," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(2), pages 610-624, June.
    7. Sunghoon Kwon & Jeongyoun Ahn & Woncheol Jang & Sangin Lee & Yongdai Kim, 2017. "A doubly sparse approach for group variable selection," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(5), pages 997-1025, October.

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