Retire: Robust Expectile Regression in High Dimensions

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.