# Retire: Robust Expectile Regression in High Dimensions

arXiv: https://arxiv.org/abs/2212.05562 · HTML: https://mlsanity.com/paper/2212.05562/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 70 references that are in the map.
- First posted on arXiv 11 Dec 2022.

## Builds on (most cited first, 7 of 7)

- [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013)
- [Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima](https://mlsanity.com/paper/1305.2436/) (2013)
- [Adaptive Huber Regression](https://mlsanity.com/paper/1706.06991/) (2017)
- [False Discoveries Occur Early on the Lasso Path](https://mlsanity.com/paper/1511.01957/) (2015)
- [Statistical consistency and asymptotic normality for high-dimensional robust M-estimators](https://mlsanity.com/paper/1501.00312/) (2015)
- [A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models](https://mlsanity.com/paper/1412.8765/) (2014)
- [Semismooth Newton Coordinate Descent Algorithm for Elastic-Net Penalized Huber Loss Regression and Quantile Regression](https://mlsanity.com/paper/1509.02957/) (2015)

## Abstract (arXiv)

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 ℓ1-penalization which reduces the estimation bias from ℓ1-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 sll nll d with s denoting the number of significant predictors. In the high-dimensional setting, we carefully characterize the solution path of the iteratively reweighted ℓ1-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 ℓ1-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.