Iteratively Reweighted $\ell_1$-Penalized Robust Regression
This addresses robust statistical estimation for high-dimensional data with heavy-tailed noise, offering theoretical guarantees and computational efficiency, though it is incremental as it builds on existing robust regression methods.
The paper tackles robust regression with heavy-tailed errors in high-dimensional settings, showing that an iteratively reweighted ℓ₁-penalized Huber estimator achieves exponential deviation bounds, oracle convergence rates, and variable selection consistency with O(log s + log log d) iterations.
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.