High-dimensional nonconvex LASSO-type M-estimators

A theory is developed to examine the convergence properties of ℓ1-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 s0log(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.