Iteratively reweighted ℓ1-penalized robust regression
Xiaoou Pan, Qiang Sun, Wen‐Xin Zhou
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 moments, we show that iteratively reweighted ℓ1-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(logs+loglogd) 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.