Mathematical Programming

, Volume 166, Issue 1–2, pp 207–240 | Cite as

Folded concave penalized sparse linear regression: sparsity, statistical performance, and algorithmic theory for local solutions

  • Hongcheng Liu
  • Tao Yao
  • Runze LiEmail author
  • Yinyu Ye
Full Length Paper Series A


This paper concerns the folded concave penalized sparse linear regression (FCPSLR), a class of popular sparse recovery methods. Although FCPSLR yields desirable recovery performance when solved globally, computing a global solution is NP-complete. Despite some existing statistical performance analyses on local minimizers or on specific FCPSLR-based learning algorithms, it still remains open questions whether local solutions that are known to admit fully polynomial-time approximation schemes (FPTAS) may already be sufficient to ensure the statistical performance, and whether that statistical performance can be non-contingent on the specific designs of computing procedures. To address the questions, this paper presents the following threefold results: (1) Any local solution (stationary point) is a sparse estimator, under some conditions on the parameters of the folded concave penalties. (2) Perhaps more importantly, any local solution satisfying a significant subspace second-order necessary condition (S\(^3\)ONC), which is weaker than the second-order KKT condition, yields a bounded error in approximating the true parameter with high probability. In addition, if the minimal signal strength is sufficient, the S\(^3\)ONC solution likely recovers the oracle solution. This result also explicates that the goal of improving the statistical performance is consistent with the optimization criteria of minimizing the suboptimality gap in solving the non-convex programming formulation of FCPSLR. (3) We apply (2) to the special case of FCPSLR with minimax concave penalty and show that under the restricted eigenvalue condition, any S\(^3\)ONC solution with a better objective value than the Lasso solution entails the strong oracle property. In addition, such a solution generates a model error (ME) comparable to the optimal but exponential-time sparse estimator given a sufficient sample size, while the worst-case ME is comparable to the Lasso in general. Furthermore, to guarantee the S\(^3\)ONC admits FPTAS.


Sparse recovery Non-convex programming NP-completeness Folded concave penalty Lasso 

Mathematics Subject Classification

90C26 90C90 62J05 62J07 68Q25 



The authors thank the AE and referees for their valuable comments, which significantly improve the paper. This work was supported by Penn State Grace Woodward Collaborative Research Grant, NSF grants CMMI 1300638 and DMS 1512422, NIH grants P50 DA036107 and P50 DA039838, Marcus PSU-Technion Partnership grant, Air Force Office of Scientific Research grant FA9550-12-1-0396, and Mid-Atlantic University Transportation Centers grant. This work was also partially supported by NNSFC grants 11690014 and 11690015. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NSF, the NIDA, the NIH, the AFOSR, the MAUTC or the NNSFC.

Supplementary material

10107_2017_1114_MOESM1_ESM.pdf (136 kb)
Supplementary material 1 (pdf 136 KB)


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Harold and Inge Marcus Department of Industrial and Manufacturing EngineeringThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of Statistics and the Methodology CenterThe Pennsylvania State UniversityUniversity ParkUSA
  3. 3.Department of Management Science and EngineeringStanford UniversityStanfordUSA

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