Pathwise optimization for bridge-type estimators and its applications

Sparse parametric models are of great interest in statistical learning and are often analyzed by means of regularized estimators. Pathwise methods allow to efficiently compute the full solution path for penalized estimators, for any possible value of the penalization parameter $\lambda$. In this paper we deal with the pathwise optimization for bridge-type problems; i.e. we are interested in the minimization of a loss function, such as negative log-likelihood or residual sum of squares, plus the sum of $\ell^q$ norms with $q\in(0,1]$ involving adpative coefficients. For some loss functions this regularization achieves asymptotically the oracle properties (such as the selection consistency). Nevertheless, since the objective function involves nonconvex and nondifferentiable terms, the minimization problem is computationally challenging. The aim of this paper is to apply some general algorithms, arising from nonconvex optimization theory, to compute efficiently the path solutions for the adaptive bridge estimator with multiple penalties. In particular, we take into account two different approaches: accelerated proximal gradient descent and blockwise alternating optimization. The convergence and the path consistency of these algorithms are discussed. In order to assess our methods, we apply these algorithms to the penalized estimation of diffusion processes observed at discrete times. This latter represents a recent research topic in the field of statistics for time-dependent data.