标题： 路径优化方法用于桥型估计量及其应用 显示英文标题

标题： Pathwise optimization for bridge-type estimators and its applications

摘要： 稀疏参数模型在统计学习中具有重要意义，并且通常通过正则化估计量进行分析。 路径方法允许高效计算惩罚估计量的完整解路径，对于惩罚参数$\lambda$的任何可能值。 在本文中，我们研究桥型问题的路径优化；即，我们感兴趣的是最小化一个损失函数，如负对数似然或残差平方和，加上涉及自适应系数的$\ell^q$范数之和，其中$q\in(0,1]$涉及自适应系数。 对于某些损失函数，这种正则化可以达到渐近的 oracle 属性（如选择一致性）。 然而，由于目标函数包含非凸和不可微项，最小化问题在计算上具有挑战性。 本文的目的是应用一些来自非凸优化理论的通用算法，以高效计算具有多重惩罚的自适应桥估计量的路径解。 特别是，我们考虑两种不同的方法：加速邻近梯度下降和块状交替优化。 讨论了这些算法的收敛性和路径一致性。 为了评估我们的方法，我们将这些算法应用于在离散时间观测到的扩散过程的惩罚估计。 后者是时间依赖数据统计领域的一个最新研究课题。 显示英文摘要

摘要： 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.