标题： 排序凹惩罚回归 显示英文标题

标题： Sorted Concave Penalized Regression

摘要： Lasso 存在偏差。 凹惩罚最小二乘估计（PLSE）利用信号强度来减少这种偏差，从而在预测、系数估计和变量选择方面产生更尖锐的误差界。 对于预测和估计，可以通过选择比选择一致性要求的更低的惩罚水平来减少 Lasso 的偏差，但这种较低的惩罚水平依赖于真实系数向量的稀疏性。 为了适应这种较低的惩罚水平，提出了排序 L1 惩罚估计（Slope）。 然而，凹 PLSE 和 Slope 的优势并不相互涵盖。 我们提出排序凹惩罚估计以结合凹惩罚和排序惩罚的优势。 我们证明了排序凹惩罚能够自适应地选择较低的惩罚水平，同时得益于信号强度，尤其是在大量信号强于自适应选择的惩罚水平时。 开发了一种局部凸逼近方法，该方法将局部线性和二次逼近扩展到排序凹惩罚，以促进排序凹 PLSE 的计算，并证明其具有所需的预测和估计误差界。 我们在一般优化框架下对惩罚函数进行了统一处理，包括上述排序惩罚和基于贝叶斯考虑的混合惩罚的惩罚水平和凹度。 我们的预测和估计误差分析需要设计满足受限特征值条件，且在所需最小信号强度条件下提供选择一致性。 因此，我们的结果通过移除稀疏 Riesz 条件中的稀疏上特征值部分，也使现有的凹 PLSE 结果更加精确。 显示英文摘要

摘要： The Lasso is biased. Concave penalized least squares estimation (PLSE) takes advantage of signal strength to reduce this bias, leading to sharper error bounds in prediction, coefficient estimation and variable selection. For prediction and estimation, the bias of the Lasso can be also reduced by taking a smaller penalty level than what selection consistency requires, but such smaller penalty level depends on the sparsity of the true coefficient vector. The sorted L1 penalized estimation (Slope) was proposed for adaptation to such smaller penalty levels. However, the advantages of concave PLSE and Slope do not subsume each other. We propose sorted concave penalized estimation to combine the advantages of concave and sorted penalizations. We prove that sorted concave penalties adaptively choose the smaller penalty level and at the same time benefits from signal strength, especially when a significant proportion of signals are stronger than the corresponding adaptively selected penalty levels. A local convex approximation, which extends the local linear and quadratic approximations to sorted concave penalties, is developed to facilitate the computation of sorted concave PLSE and proven to possess desired prediction and estimation error bounds. We carry out a unified treatment of penalty functions in a general optimization setting, including the penalty levels and concavity of the above mentioned sorted penalties and mixed penalties motivated by Bayesian considerations. Our analysis of prediction and estimation errors requires the restricted eigenvalue condition on the design, not beyond, and provides selection consistency under a required minimum signal strength condition in addition. Thus, our results also sharpens existing results on concave PLSE by removing the upper sparse eigenvalue component of the sparse Riesz condition.