An Alternating Direction Method of Multipliers for MCP-penalized Regression with High-dimensional Data

Shi, Yue; Jiao, Yu; Cao, Yong Xiu; Liu, Yan Yan

doi:10.1007/s10114-018-7096-8

Cited by 5 publications

(1 citation statement)

References 24 publications

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“…To select an "optimal" value of the tuning parameter λ, a high-dimensional Bayesian information criterion (HBIC) [33], [34] can be utilized. In this paper, we also propose to use a novel voting criterion (VC) [32], [35] for choosing the optimal value of λ.…”

Section: Solution Pathmentioning

confidence: 99%

A Semismooth Newton Algorithm for High-Dimensional Nonconvex Sparse Learning

Shi

Huang

Jiao

et al. 2020

IEEE Trans. Neural Netw. Learning Syst.

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The smoothly clipped absolute deviation (SCAD) and the minimax concave penalty (MCP) penalized regression models are two important and widely used nonconvex sparse learning tools that can handle variable selection and parameter estimation simultaneously, and thus have potential applications in various fields such as mining biological data in high-throughput biomedical studies. Theoretically, these two models enjoy the oracle property even in the high-dimensional settings, where the number of predictors p may be much larger than the number of observations n. However, numerically, it is quite challenging to develop fast and stable algorithms due to their non-convexity and non-smoothness. In this paper we develop a fast algorithm for SCAD and MCP penalized learning problems. First, we show that the global minimizers of both models are roots of the nonsmooth equations. Then, a semi-smooth Newton (SSN) algorithm is employed to solve the equations. We prove that the SSN algorithm converges locally and superlinearly to the Karush-Kuhn-Tucker (KKT) points. Computational complexity analysis shows that the cost of the SSN algorithm per iteration is O(np). Combined with the warm-start technique, the SSN algorithm can be very efficient and accurate. Simulation studies and a real data example suggest that our SSN algorithm, with comparable solution accuracy with the coordinate descent (CD) and the difference of convex (DC) proximal Newton algorithms, is more computationally efficient.

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