{"title":"Nonconvex Dantzig selector and its parallel computing algorithm","authors":"Jiawei Wen, Songshan Yang, Delin Zhao","doi":"10.1007/s11222-024-10492-8","DOIUrl":null,"url":null,"abstract":"<p>The Dantzig selector is a popular <span>\\(\\ell _1\\)</span>-type variable selection method widely used across various research fields. However, <span>\\(\\ell _1\\)</span>-type methods may not perform well for variable selection without complex irrepresentable conditions. In this article, we introduce a nonconvex Dantzig selector for ultrahigh-dimensional linear models. We begin by demonstrating that the oracle estimator serves as a local optimum for the nonconvex Dantzig selector. In addition, we propose a one-step local linear approximation estimator, called the Dantzig-LLA estimator, for the nonconvex Dantzig selector, and establish its strong oracle property. The proposed regularization method avoids the restrictive conditions imposed by <span>\\(\\ell _1\\)</span> regularization methods to guarantee the model selection consistency. Furthermore, we propose an efficient and parallelizable computing algorithm based on feature-splitting to address the computational challenges associated with the nonconvex Dantzig selector in high-dimensional settings. A comprehensive numerical study is conducted to evaluate the performance of the nonconvex Dantzig selector and the computing efficiency of the feature-splitting algorithm. The results demonstrate that the Dantzig selector with nonconvex penalty outperforms the <span>\\(\\ell _1\\)</span> penalty-based selector, and the feature-splitting algorithm performs well in high-dimensional settings where linear programming solver may fail. Finally, we generalize the concept of nonconvex Dantzig selector to deal with more general loss functions.</p>","PeriodicalId":22058,"journal":{"name":"Statistics and Computing","volume":"1 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Statistics and Computing","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11222-024-10492-8","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
The Dantzig selector is a popular \(\ell _1\)-type variable selection method widely used across various research fields. However, \(\ell _1\)-type methods may not perform well for variable selection without complex irrepresentable conditions. In this article, we introduce a nonconvex Dantzig selector for ultrahigh-dimensional linear models. We begin by demonstrating that the oracle estimator serves as a local optimum for the nonconvex Dantzig selector. In addition, we propose a one-step local linear approximation estimator, called the Dantzig-LLA estimator, for the nonconvex Dantzig selector, and establish its strong oracle property. The proposed regularization method avoids the restrictive conditions imposed by \(\ell _1\) regularization methods to guarantee the model selection consistency. Furthermore, we propose an efficient and parallelizable computing algorithm based on feature-splitting to address the computational challenges associated with the nonconvex Dantzig selector in high-dimensional settings. A comprehensive numerical study is conducted to evaluate the performance of the nonconvex Dantzig selector and the computing efficiency of the feature-splitting algorithm. The results demonstrate that the Dantzig selector with nonconvex penalty outperforms the \(\ell _1\) penalty-based selector, and the feature-splitting algorithm performs well in high-dimensional settings where linear programming solver may fail. Finally, we generalize the concept of nonconvex Dantzig selector to deal with more general loss functions.
期刊介绍:
Statistics and Computing is a bi-monthly refereed journal which publishes papers covering the range of the interface between the statistical and computing sciences.
In particular, it addresses the use of statistical concepts in computing science, for example in machine learning, computer vision and data analytics, as well as the use of computers in data modelling, prediction and analysis. Specific topics which are covered include: techniques for evaluating analytically intractable problems such as bootstrap resampling, Markov chain Monte Carlo, sequential Monte Carlo, approximate Bayesian computation, search and optimization methods, stochastic simulation and Monte Carlo, graphics, computer environments, statistical approaches to software errors, information retrieval, machine learning, statistics of databases and database technology, huge data sets and big data analytics, computer algebra, graphical models, image processing, tomography, inverse problems and uncertainty quantification.
In addition, the journal contains original research reports, authoritative review papers, discussed papers, and occasional special issues on particular topics or carrying proceedings of relevant conferences. Statistics and Computing also publishes book review and software review sections.