Penalization is a popular tool for multi- and high-dimensional data. Most of the existing computational algorithms have been developed for convex loss functions. Nonconvex loss functions can sometimes generate more robust results and have important applications. Motivated by the BLasso algorithm, this study develops the Forward and Backward Stagewise (Fabs) algorithm for nonconvex loss functions with the adaptive Lasso (aLasso) penalty. It is shown that each point along the Fabs paths is a δ-approximate solution to the aLasso problem and the Fabs paths converge to the stationary points of the aLasso problem when δ goes to zero, given that the loss function has second-order derivatives bounded from above. This study exemplifies the Fabs with an application to the penalized smooth partial rank (SPR) estimation, for which there is still a lack of effective algorithm. Extensive numerical studies are conducted to demonstrate the benefit of penalized SPR estimation using Fabs, especially under high-dimensional settings. Application to the smoothed 0-1 loss in binary classification is introduced to demonstrate its capability to work with other differentiable nonconvex loss function.

中文翻译：

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具有自适应套索的非凸损失函数的前向和后向阶段算法

惩罚是多维和高维数据的流行工具。大多数现有的计算算法都是针对凸损失函数开发的。非凸损失函数有时可以产生更稳健的结果并具有重要的应用。本研究受 BLasso 算法的启发，开发了具有自适应套索 (aLasso) 惩罚的非凸损失函数的前向和后向阶段 (Fabs) 算法。结果表明，Fabs 路径上的每个点都是 aLasso 问题的 δ 近似解，并且当 δ 变为零时，Fabs 路径收敛到 aLasso 问题的平稳点，假设损失函数具有二阶导数有界从上面。本研究举例说明了 Fabs 应用于惩罚平滑部分秩 (SPR) 估计，目前还缺乏有效的算法。进行了广泛的数值研究，以证明使用 Fab 进行惩罚 SPR 估计的好处，尤其是在高维设置下。引入了在二元分类中平滑 0-1 损失的应用，以证明其与其他可微非凸损失函数一起工作的能力。