High-dimensional nonconvex LASSO-type M-estimators

A theory is developed to examine the convergence proper l1-norm penalized highdimensional M-estimators, with nonconvex risk and unrestricted domain. Under high-level root conditions, the estimators are shown to attain the rate of convergence s0 root log(nd)/n, where s0 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.