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

ABSTRACT: 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: (i) Any local solution (stationary point) is a sparse estimator, under some conditions on the parameters of the folded concave penalties. (ii) Perhaps more importantly, any local solution satisfying a significant subspace second-order necessary condition (S3ONC), 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 S3ONC 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. (iii) We apply (ii) to the special case of FCPSLR with minimax concave penalty (MCP) and show that under the restricted eigenvalue condition, any S3ONC 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 S3ONC admits FPTAS.

SUBMITTER: Liu H

PROVIDER: S-EPMC5720392 | biostudies-literature | 2017 Nov

REPOSITORIES: biostudies-literature

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

Liu Hongcheng H Yao Tao T Li Runze R Ye Yinyu Y

Mathematical programming 20170210 1-2

This paper concerns the <i>folded concave penalized sparse linear regression</i> (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 schem ...[more]

PMID: 29225375

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Project description:BackgroundEnvironmental Enteropathy (EE) is a subclinical condition caused by constant fecal-oral contamination and resulting in blunting of intestinal villi and intestinal inflammation. Of primary interest in the clinical research is to evaluate the association between non-invasive EE biomarkers and malnutrition in a cohort of Bangladeshi children. The challenges are that the number of biomarkers/covariates is relatively large, and some of them are highly correlated.MethodsMany variable selection methods are available in the literature, but which are most appropriate for EE biomarker selection remains unclear. In this study, different variable selection approaches were applied and the performance of these methods was assessed numerically through simulation studies, assuming the correlations among covariates were similar to those in the Bangladesh cohort. The suggested methods from simulations were applied to the Bangladesh cohort to select the most relevant biomarkers for the growth response, and bootstrapping methods were used to evaluate the consistency of selection results.ResultsThrough simulation studies, SCAD (Smoothly Clipped Absolute Deviation), Adaptive LASSO (Least Absolute Shrinkage and Selection Operator) and MCP (Minimax Concave Penalty) are the suggested variable selection methods, compared to traditional stepwise regression method. In the Bangladesh data, predictors such as mother weight, height-for-age z-score (HAZ) at week 18, and inflammation markers (Myeloperoxidase (MPO) at week 12 and soluable CD14 at week 18) are informative biomarkers associated with children's growth.ConclusionsPenalized linear regression methods are plausible alternatives to traditional variable selection methods, and the suggested methods are applicable to other biomedical studies. The selected early-stage biomarkers offer a potential explanation for the burden of malnutrition problems in low-income countries, allow early identification of infants at risk, and suggest pathways for intervention.Trial registrationThis study was retrospectively registered with ClinicalTrials.gov, number NCT01375647, on June 3, 2011.

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

Publications

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

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