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A global two-stage algorithm for non-convex penalized high-dimensional linear regression problems

Author

Listed:
  • Peili Li

    (East China Normal University)

  • Min Liu

    (Wuhan University)

  • Zhou Yu

    (East China Normal University)

Abstract

By the asymptotic oracle property, non-convex penalties represented by minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD) have attracted much attentions in high-dimensional data analysis, and have been widely used in signal processing, image restoration, matrix estimation, etc. However, in view of their non-convex and non-smooth characteristics, they are computationally challenging. Almost all existing algorithms converge locally, and the proper selection of initial values is crucial. Therefore, in actual operation, they often combine a warm-starting technique to meet the rigid requirement that the initial value must be sufficiently close to the optimal solution of the corresponding problem. In this paper, based on the DC (difference of convex functions) property of MCP and SCAD penalties, we aim to design a global two-stage algorithm for the high-dimensional least squares linear regression problems. A key idea for making the proposed algorithm to be efficient is to use the primal dual active set with continuation (PDASC) method to solve the corresponding sub-problems. Theoretically, we not only prove the global convergence of the proposed algorithm, but also verify that the generated iterative sequence converges to a d-stationary point. In terms of computational performance, the abundant research of simulation and real data show that the algorithm in this paper is superior to the latest SSN method and the classic coordinate descent (CD) algorithm for solving non-convex penalized high-dimensional linear regression problems.

Suggested Citation

  • Peili Li & Min Liu & Zhou Yu, 2023. "A global two-stage algorithm for non-convex penalized high-dimensional linear regression problems," Computational Statistics, Springer, vol. 38(2), pages 871-898, June.
    • Handle: RePEc:spr:compst:v:38:y:2023:i:2:d:10.1007_s00180-022-01249-w
    DOI: 10.1007/s00180-022-01249-w
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    References listed on IDEAS

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    1. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    2. Shan Luo & Zehua Chen, 2014. "Sequential Lasso Cum EBIC for Feature Selection With Ultra-High Dimensional Feature Space," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1229-1240, September.
    3. Mazumder, Rahul & Friedman, Jerome H. & Hastie, Trevor, 2011. "SparseNet: Coordinate Descent With Nonconvex Penalties," Journal of the American Statistical Association, American Statistical Association, vol. 106(495), pages 1125-1138.
    4. Le Thi, H.A. & Pham Dinh, T. & Le, H.M. & Vo, X.T., 2015. "DC approximation approaches for sparse optimization," European Journal of Operational Research, Elsevier, vol. 244(1), pages 26-46.
    5. Jong-Shi Pang & Meisam Razaviyayn & Alberth Alvarado, 2017. "Computing B-Stationary Points of Nonsmooth DC Programs," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 95-118, January.
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