Iteratively reweighted ?1-penalized robust regression

This paper investigates tradeoffs among optimization errors, statistical rates of convergence and the effect heavy-tailed errors for high-dimensional robust regression with nonconvex regularization. When additive in linear models have only bounded second moments, we show that iteratively reweighted ?1-penalized adaptive Huber estimator satisfies exponential deviation bounds oracle properties, including rate variable selection consistency, under a weak beta-min condition. Computationally, need as many O(logs+loglogd) iterations to reach such an estimator, where s d denote sparsity ambient dimension, respectively. Extension general class loss functions is also considered. Numerical studies lend strong support our methodology theory.