Abstract

This dissertation develops novel methodologies for distributed quantile regression analysisfor big data by utilizing a distributed optimization algorithm called the alternating directionmethod of multipliers (ADMM). Specifically, we first write the penalized quantile regressioninto a specific form that can be solved by the ADMM and propose numerical algorithmsfor solving the ADMM subproblems. This results in the distributed QR-ADMMalgorithm. Then, to further reduce the computational time, we formulate the penalizedquantile regression into another equivalent ADMM form in which all the subproblems haveexact closed-form solutions and hence avoid iterative numerical methods. This results in thesingle-loop QPADM algorithm that further improve on the computational efficiency of theQR-ADMM. Both QR-ADMM and QPADM enjoy flexible parallelization by enabling datasplitting across both sample space and feature space, which make them especially appealingfor the case when both sample size n and feature dimension p are large.Besides the QR-ADMM and QPADM algorithms for penalized quantile regression, wealso develop a group variable selection method by approximating the Bayesian informationcriterion. Unlike existing penalization methods for feature selection, our proposed gMICalgorithm is free of parameter tuning and hence enjoys greater computational efficiency.Although the current version of gMIC focuses on the generalized linear model, it can benaturally extended to the quantile regression for feature selection.We provide theoretical analysis for our proposed methods. Specifically, we conduct numericalconvergence analysis for the QR-ADMM and QPADM algorithms, and provideasymptotical theories and oracle property of feature selection for the gMIC method. Allour methods are evaluated with simulation studies and real data analysis.

Committee Chair

Nan Lin

Committee Members

Yixin Chen, Jimin Ding, Jose Figueroa-Lopez, Todd Kuffner,

Comments

Permanent URL: https://doi.org/10.7936/K7F47NKS

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

Spring 5-15-2018

Language

English (en)

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Mathematics Commons

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