Abstract
The group folded concave penalization problems have been shown to process the satisfactory oracle property theoretically. However, it remains unknown whether the optimization algorithm for solving the resulting nonconvex problem can find such oracle solution among multiple local solutions. In this paper, we extend the well-known local linear approximation (LLA) algorithm to solve the group folded concave penalization problem for the linear models. We prove that, with the group LASSO estimator as the initial value, the two-step LLA solution converges to the oracle estimator with overwhelming probability, and thus closing the theoretical gap. The results are high-dimensional which allow the group number to grow exponentially, the true relevant groups and the true maximum group size to grow polynomially. Numerical studies are also conducted to show the merits of the LLA procedure.
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