  We investigate high-dimensional nonconvex penalized regression, where the
number of covariates may grow at an exponential rate. Although recent
asymptotic theory established that there exists a local minimum possessing the
oracle property under general conditions, it is still largely an open problem
how to identify the oracle estimator among potentially multiple local minima.
There are two main obstacles: (1) due to the presence of multiple minima, the
solution path is nonunique and is not guaranteed to contain the oracle
estimator; (2) even if a solution path is known to contain the oracle
estimator, the optimal tuning parameter depends on many unknown factors and is
hard to estimate. To address these two challenging issues, we first prove that
an easy-to-calculate calibrated CCCP algorithm produces a consistent solution
path which contains the oracle estimator with probability approaching one.
Furthermore, we propose a high-dimensional BIC criterion and show that it can
be applied to the solution path to select the optimal tuning parameter which
asymptotically identifies the oracle estimator. The theory for a general class
of nonconvex penalties in the ultra-high dimensional setup is established when
the random errors follow the sub-Gaussian distribution. Monte Carlo studies
confirm that the calibrated CCCP algorithm combined with the proposed
high-dimensional BIC has desirable performance in identifying the underlying
sparsity pattern for high-dimensional data analysis.
