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2023 High-dimensional composite quantile regression: Optimal statistical guarantees and fast algorithms
Haeseong Moon, Wen-Xin Zhou
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Electron. J. Statist. 17(2): 2067-2119 (2023). DOI: 10.1214/23-EJS2147


The composite quantile regression (CQR) was introduced by Zou and Yuan [Ann. Statist. 36 (2008) 1108–1126] as a robust regression method for linear models with heavy-tailed errors while achieving high efficiency. Its penalized counterpart for high-dimensional sparse models was recently studied in Gu and Zou [IEEE Trans. Inf. Theory 66 (2020) 7132–7154], along with a specialized optimization algorithm based on the alternating direct method of multipliers (ADMM). Compared to the various first-order algorithms for penalized least squares, ADMM-based algorithms are not well-adapted to large-scale problems. To overcome this computational hardness, in this paper we employ a convolution-smoothed technique to CQR, complemented with iteratively reweighted 1-regularization. The smoothed composite loss function is convex, twice continuously differentiable, and locally strong convex with high probability. We propose a gradient-based algorithm for penalized smoothed CQR via a variant of the majorize-minimization principal, which gains substantial computational efficiency over ADMM. Theoretically, we show that the iteratively reweighted 1-penalized smoothed CQR estimator achieves near-minimax optimal convergence rate under heavy-tailed errors without any moment constraint, and further achieves near-oracle convergence rate under a weaker minimum signal strength condition than needed in Gu and Zou (2020). Numerical studies demonstrate that the proposed method exhibits significant computational advantages without compromising statistical performance compared to two state-of-the-art methods that achieve robustness and high efficiency simultaneously.


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Haeseong Moon. Wen-Xin Zhou. "High-dimensional composite quantile regression: Optimal statistical guarantees and fast algorithms." Electron. J. Statist. 17 (2) 2067 - 2119, 2023.


Received: 1 April 2022; Published: 2023
First available in Project Euclid: 2 October 2023

MathSciNet: MR4649035
Digital Object Identifier: 10.1214/23-EJS2147

Primary: 62J07
Secondary: 62A01

Keywords: Asymptotic efficiency , composite quantile regression , convolution smoothing , High-dimensional data , oracle property , Sparsity

Vol.17 • No. 2 • 2023
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