Communication-Efficient Distributed Quantile Regression with Optimal Statistical Guarantees

• URL: http://arxiv.org/abs/2110.13113v1
• Date: Mon, 25 Oct 2021 17:09:59 GMT
• Title: Communication-Efficient Distributed Quantile Regression with Optimal Statistical Guarantees
• Authors: Heather Battey, Kean Ming Tan, and Wen-Xin Zhou
• Abstract summary: We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. 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.
• Score: 2.064612766965483
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 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 discuss and compare several alternative confidence set constructions, based on inversion of Wald and score-type tests and resam-pling 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 $\ell_1$-penalty. A thorough simulation study further elucidates our findings. Finally, we provide estimation theory and numerical studies for sparse quantile regression in the high-dimensional setting.

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