Pivotal estimation in high-dimensional regression via linear programming

We propose a new method of estimation in high-dimensional linear regression models. It allows for very weak distributional assumptions, including heteroscedasticity, and does not require knowledge of the variance of random errors. The method is based on linear programming only, so that its numerical implementation is faster than for previously known techniques using conic programs, and it allows one to deal with higher-dimensional models. We provide upper bounds for estimation and prediction errors of the proposed estimator, showing that it achieves the same rate as in the more restrictive situation of fixed design and i.i.d. Gaussian errors with known variance. Following Gautier and Tsybakov (High-dimensional instrumental variables regression and confidence sets. ArXiv e-prints 1105.2454, 2011), we obtain the results under weaker sensitivity assumptions than the restricted eigenvalue or assimilated conditions.