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This paper proposes a theory for $\ell_1$-norm penalized high-dimensional$M$-estimators, with nonconvex risk and unrestricted domain. Under high-levelconditions, 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 ofthe parameter of interest. Sufficient conditions for our main assumptions arethen developed and finally used in several examples including robust linearregression, binary classification and nonlinear least squares.

High-dimensional nonconvex lasso-type $M$-estimators