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.

翻译：稀疏参数模型在统计学习中具有重要意义，通常通过正则化估计器进行分析。路径方法能够高效计算惩罚估计器的完整解路径，适用于惩罚参数 $\lambda$ 的所有可能取值。本文研究桥型问题的路径优化方法，即关注损失函数（如负对数似然或残差平方和）与包含自适应系数的 $\ell^q$ 范数之和（其中 $q\in(0,1]$）的最小化问题。对于某些损失函数，该正则化方法能够渐近达到预言机性质（如选择一致性）。然而，由于目标函数包含非凸且不可微项，该最小化问题在计算上具有挑战性。本文旨在应用非凸优化理论中的通用算法，高效计算具有多重惩罚的自适应桥估计器的路径解。特别地，我们考虑两种不同方法：加速近端梯度下降法和分块交替优化法。文中讨论了这些算法的收敛性与路径一致性。为评估所提方法，我们将这些算法应用于离散时间观测扩散过程的惩罚估计问题，该问题是时间依赖数据统计领域的最新研究方向。