# Iteratively Reweighted $\ell_1$-Penalized Robust Regression

## TL;DR

This paper develops an iteratively reweighted $ ext{l}_1$-penalized robust regression method that achieves strong statistical guarantees and efficiency in high-dimensional settings with heavy-tailed errors.

## Contribution

It introduces a novel adaptive Huber regression estimator with theoretical guarantees under weak conditions and analyzes its computational complexity.

## Key findings

- Estimator satisfies exponential deviation bounds.
- Achieves oracle convergence rate and variable selection consistency.
- Requires $O( ext{log } s + ext{log log } d)$ iterations.

## Abstract

This paper investigates tradeoffs among optimization errors, statistical rates of convergence and the effect of heavy-tailed errors for high-dimensional robust regression with nonconvex regularization. When the additive errors in linear models have only bounded second moment, we show that iteratively reweighted $\ell_1$-penalized adaptive Huber regression estimator satisfies exponential deviation bounds and oracle properties, including the oracle convergence rate and variable selection consistency, under a weak beta-min condition. Computationally, we need as many as $O(\log s + \log\log d)$ iterations to reach such an oracle estimator, where $s$ and $d$ denote the sparsity and ambient dimension, respectively. Extension to a general class of robust loss functions is also considered. Numerical studies lend strong support to our methodology and theory.