  We propose a new method of estimation in high-dimensional linear regression
model. It allows for very weak distributional assumptions including
heteroscedasticity, and does not require the 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 (2011), we obtain the results under weaker
sensitivity assumptions than the restricted eigenvalue or assimilated
conditions.
