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We provide theoretical analysis of the statistical and computationalproperties of penalized $M$-estimators that can be formulated as the solutionto a possibly nonconvex optimization problem. Many important estimators fall inthis category, including least squares regression with nonconvexregularization, generalized linear models with nonconvex regularization andsparse elliptical random design regression. For these problems, it isintractable to calculate the global solution due to the nonconvex formulation.In this paper, we propose an approximate regularization path-following methodfor solving a variety of learning problems with nonconvex objective functions.Under a unified analytic framework, we simultaneously provide explicitstatistical and computational rates of convergence for any local solutionattained by the algorithm. Computationally, our algorithm attains a globalgeometric rate of convergence for calculating the full regularization path,which is optimal among all first-order algorithms. Unlike most existing methodsthat only attain geometric rates of convergence for one single regularizationparameter, our algorithm calculates the full regularization path with the sameiteration complexity. In particular, we provide a refined iteration complexitybound to sharply characterize the performance of each stage along theregularization path. Statistically, we provide sharp sample complexity analysisfor all the approximate local solutions along the regularization path. Inparticular, our analysis improves upon existing results by providing a morerefined sample complexity bound as well as an exact support recovery result forthe final estimator. These results show that the final estimator attains anoracle statistical property due to the usage of nonconvex penalty.

Optimal computational and statistical rates of convergence for sparse nonconvex learning problems