Communication-constrained distributed quantile regression with optimal statistical guarantees
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Published version
Author(s)
Tan, Kean Ming
Battey, Heather
Zhou, Wen-Xin
Type
Journal Article
Abstract
We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression (QR) loss function, which invalidates the use of existing methodology. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric, thereby facilitating interpretation. In the low-dimensional regime, we establish a finite-sample theoretical framework for the sequentially defined distributed QR estimators. This reveals a trade-off between the communication cost and statistical error. We further discuss and compare several alternative confidence set constructions, based on inversion of Wald and score-type tests and resampling techniques, detailing an improvement that is effective for more extreme quantile coefficients. In high dimensions, a sparse framework is adopted, where the proposed doubly-smoothed objective function is complemented with an ℓ1-penalty. We show that the corresponding distributed penalized QR estimator achieves the global convergence rate after a near-constant number of communication rounds. A thorough simulation study further elucidates our findings.
Date Issued
2022-11-01
Date Acceptance
2022-08-01
Citation
Journal of Machine Learning Research, 2022, 23 (272), pp.1-61
ISSN
1532-4435
Publisher
Microtome Publishing
Start Page
1
End Page
61
Journal / Book Title
Journal of Machine Learning Research
Volume
23
Issue
272
Copyright Statement
©2022 Kean Ming Tan, Heather Battey, and Wen-Xin Zhou. This paper is distributed under a CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/. Attribution requirements are provided
at http://jmlr.org/papers/v23/21-1269.html
at http://jmlr.org/papers/v23/21-1269.html
License URL
Publication Status
Published