Retire: Robust Expectile Regression in High Dimensions

• URL: http://arxiv.org/abs/2212.05562v2
• Date: Wed, 22 Mar 2023 15:06:30 GMT
• Title: Retire: Robust Expectile Regression in High Dimensions
• Authors: Rebeka Man, Kean Ming Tan, Zian Wang, and Wen-Xin Zhou
• Abstract summary: Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. We propose and study (penalized) robust expectile regression (retire) We show that the proposed procedure can be efficiently solved by a semismooth Newton coordinate descent algorithm.
• Score: 3.9391041278203978
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects. Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. The former is computationally challenging due to the non-smooth nature of the check loss, and the latter is sensitive to heavy-tailed error distributions. In this paper, we propose and study (penalized) robust expectile regression (retire), with a focus on iteratively reweighted $\ell_1$-penalization which reduces the estimation bias from $\ell_1$-penalization and leads to oracle properties. Theoretically, we establish the statistical properties of the retire estimator under two regimes: (i) low-dimensional regime in which $d \ll n$; (ii) high-dimensional regime in which $s\ll n\ll d$ with $s$ denoting the number of significant predictors. In the high-dimensional setting, we carefully characterize the solution path of the iteratively reweighted $\ell_1$-penalized retire estimation, adapted from the local linear approximation algorithm for folded-concave regularization. Under a mild minimum signal strength condition, we show that after as many as $\log(\log d)$ iterations the final iterate enjoys the oracle convergence rate. At each iteration, the weighted $\ell_1$-penalized convex program can be efficiently solved by a semismooth Newton coordinate descent algorithm. Numerical studies demonstrate the competitive performance of the proposed procedure compared with either non-robust or quantile regression based alternatives.

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