## The Annals of Statistics

### Global solutions to folded concave penalized nonconvex learning

#### Abstract

This paper is concerned with solving nonconvex learning problems with folded concave penalty. Despite that their global solutions entail desirable statistical properties, they lack optimization techniques that guarantee global optimality in a general setting. In this paper, we show that a class of nonconvex learning problems are equivalent to general quadratic programs. This equivalence facilitates us in developing mixed integer linear programming reformulations, which admit finite algorithms that find a provably global optimal solution. We refer to this reformulation-based technique as the mixed integer programming-based global optimization (MIPGO). To our knowledge, this is the first global optimization scheme with a theoretical guarantee for folded concave penalized nonconvex learning with the SCAD penalty [J. Amer. Statist. Assoc. 96 (2001) 1348–1360] and the MCP penalty [Ann. Statist. 38 (2001) 894–942]. Numerical results indicate a significant outperformance of MIPGO over the state-of-the-art solution scheme, local linear approximation and other alternative solution techniques in literature in terms of solution quality.

#### Article information

Source
Ann. Statist., Volume 44, Number 2 (2016), 629-659.

Dates
Revised: June 2015
First available in Project Euclid: 17 March 2016

https://projecteuclid.org/euclid.aos/1458245730

Digital Object Identifier
doi:10.1214/15-AOS1380

Mathematical Reviews number (MathSciNet)
MR3476612

Zentralblatt MATH identifier
1337.62163

Subjects
Primary: 62J05: Linear regression
Secondary: 62J07: Ridge regression; shrinkage estimators

#### Citation

Liu, Hongcheng; Yao, Tao; Li, Runze. Global solutions to folded concave penalized nonconvex learning. Ann. Statist. 44 (2016), no. 2, 629--659. doi:10.1214/15-AOS1380. https://projecteuclid.org/euclid.aos/1458245730

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#### Supplemental materials

• Supplement to “Global solutions to folded concave penalized nonconvex learning”. This supplemental material includes the proofs of Proposition 2.1, 2.3 and Lemma 4.1, and some additional numerical results.