Pathwise optimization for bridge-type estimators and its applications

Abstract Sparse parametric models are of great interest in statistical learning and are often analyzed by means of regularized estimators. Pathwise methods allow us to efficiently compute the full solution path for penalized estimators, for any possible value of the penalization parameter $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> . 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 $$\ell ^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> norms with $$q\in (0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> involving adaptive coefficients. For some loss functions this regularization achieves asymptotically the oracle properties (such as selection consistency). Nevertheless, since the objective function involves non-convex and non-differentiable terms, the minimization problem is computationally challenging. The aim of this paper is to apply some general algorithms, arising from non-convex 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. A pathwise scheme not only enables rapid, grid-based validation of the tuning parameter but also helps avoid spurious local minima in the underlying nonconvex 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. The latter represents a recent research topic in the field of statistics for time-dependent data.