Nonconvex Regularized Robust Regression with Oracle Properties in Polynomial Time

This paper investigates tradeoffs among optimization errors, statistical rates of convergence and the effect of heavy-tailed random errors for high-dimensional adaptive Huber regression with nonconvex regularization. When the additive errors in linear models have only bounded second moment, our results suggest that adaptive Huber regression with nonconvex regularization yields statistically optimal estimators that satisfy oracle properties as if the true underlying support set were known beforehand. Computationally, we need as many as O(log s + log log d) convex relaxations to reach such oracle estimators, where s and d denote the sparsity and ambient dimension, respectively. Numerical studies lend strong support to our methodology and theory.