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ANALYSIS OF GLOBAL AND LOCAL OPTIMA OF REGULARIZED QUANTILE REGRESSION IN HIGH DIMENSIONS: A SUBGRADIENT APPROACH

Published online by Cambridge University Press:  18 October 2022

Lan Wang Open the ORCID record for Lan Wang [Opens in a new window]  and
Xuming He Open the ORCID record for Xuming He [Opens in a new window]
Lan Wang*
Affiliation:
University of Miami
Xuming He
Affiliation:
University of Michigan
*
Address correspondence to Lan Wang, Department of Management Science, University of Miami, Coral Gables, FL 33146, USA; e-mail: [email protected].
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Abstract

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Regularized quantile regression (QR) is a useful technique for analyzing heterogeneous data under potentially heavy-tailed error contamination in high dimensions. This paper provides a new analysis of the estimation/prediction error bounds of the global solution of $L_1$-regularized QR (QR-LASSO) and the local solutions of nonconvex regularized QR (QR-NCP) when the number of covariates is greater than the sample size. Our results build upon and significantly generalize the earlier work in the literature. For certain heavy-tailed error distributions and a general class of design matrices, the least-squares-based LASSO cannot achieve the near-oracle rate derived under the normality assumption no matter the choice of the tuning parameter. In contrast, we establish that QR-LASSO achieves the near-oracle estimation error rate for a broad class of models under conditions weaker than those in the literature. For QR-NCP, we establish the novel results that all local optima within a feasible region have desirable estimation accuracy. Our analysis applies to not just the hard sparsity setting commonly used in the literature, but also to the soft sparsity setting which permits many small coefficients. Our approach relies on a unified characterization of the global/local solutions of regularized QR via subgradients using a generalized Karush–Kuhn–Tucker condition. The theory of the paper establishes a key property of the subdifferential of the quantile loss function in high dimensions, which is of independent interest for analyzing other high-dimensional nonsmooth problems.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 2 , April 2024 , pp. 233 - 277
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

Wang and He’s research is partly supported by NSF FRGMS-1952373. The authors are grateful to the Co-Editor and two anonymous referees, whose comments have helped to significantly improve the paper. They also thank Dr. Alexander Giessing for his helpful comments and Dr. Yunan Wu for her latex help on an earlier draft of the paper. Part of the results developed in this paper were made available as an earlier technical report (Wang, 2019).

References

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