High-dimensional composite quantile regression: Optimal statistical guarantees and fast algorithms

IF 1 4区 数学 Q3 STATISTICS & PROBABILITY
Haeseong Moon, Wen-Xin Zhou
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引用次数: 1

Abstract

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 ℓ1-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 ℓ1-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.
高维复合分位数回归:最优统计保证和快速算法
综合分位数回归(CQR)是由邹和袁[Ann]提出的。统计学家。36(2008)1108-1126]作为具有重尾误差的线性模型的鲁棒回归方法,同时实现了高效率。最近,Gu和Zou [IEEE Trans]研究了高维稀疏模型的惩罚对应物。Inf. Theory 66(2020) 7132-7154],以及基于乘数交替直接法(ADMM)的专门优化算法。与各种一阶惩罚最小二乘算法相比,基于admm的算法不太适合大规模问题。为了克服这种计算困难,在本文中,我们对CQR采用了卷积平滑技术,并辅以迭代重加权的1-正则化。光滑复合损失函数是凸的、两次连续可微的、高概率的局部强凸。我们提出了一种基于梯度的惩罚平滑CQR算法,该算法通过最大-最小原则的变体获得了比ADMM更高的计算效率。理论上,我们证明了迭代重加权的1-惩罚光滑CQR估计器在没有任何矩约束的情况下在重尾误差下实现了近极小极大最优收敛速率,并且在较弱的最小信号强度条件下实现了比Gu和Zou(2020)所需的近oracle收敛速率。数值研究表明,与同时实现鲁棒性和高效率的两种最先进的方法相比,该方法在不影响统计性能的情况下具有显著的计算优势。
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来源期刊
Electronic Journal of Statistics
Electronic Journal of Statistics STATISTICS & PROBABILITY-
CiteScore
1.80
自引率
9.10%
发文量
100
审稿时长
3 months
期刊介绍: The Electronic Journal of Statistics (EJS) publishes research articles and short notes on theoretical, computational and applied statistics. The journal is open access. Articles are refereed and are held to the same standard as articles in other IMS journals. Articles become publicly available shortly after they are accepted.
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