  This paper proposes a theory for $\ell_1$-norm penalized high-dimensional
$M$-estimators, with nonconvex risk and unrestricted domain. Under high-level
conditions, the estimators are shown to attain the rate of convergence
$s_0\sqrt{\log(nd)/n}$, where $s_0$ is the number of nonzero coefficients of
the parameter of interest. Sufficient conditions for our main assumptions are
then developed and finally used in several examples including robust linear
regression, binary classification and nonlinear least squares.
