We propose a fast Newton algorithm for \(\ell _0\) regularized high-dimensional generalized linear models based on support detection and root finding. We refer to the proposed method as GSDAR. GSDAR is developed based on the KKT conditions for \(\ell _0\)-penalized maximum likelihood estimators and generates a sequence of solutions of the KKT system iteratively. We show that GSDAR can be equivalently formulated as a generalized Newton algorithm. Under a restricted invertibility condition on the likelihood function and a sparsity condition on the regression coefficient, we establish an explicit upper bound on the estimation errors of the solution sequence generated by GSDAR in supremum norm and show that it achieves the optimal order in finite iterations with high probability. Moreover, we show that the oracle estimator can be recovered with high probability if the target signal is above the detectable level. These results directly concern the solution sequence generated from the GSDAR algorithm, instead of a theoretically defined global solution. We conduct simulations and real data analysis to illustrate the effectiveness of the proposed method.

我们基于支持检测和根查找为\（\ ell _0 \）正则化高维广义线性模型提出了一种快速牛顿算法。我们将所提出的方法称为GSDAR。GSDAR是根据KKT条件为\（\ ell _0 \）开发的-对最大似然估计值进行惩罚，并迭代生成KKT系统的一系列解决方案。我们表明，GSDAR可以等效地表示为广义牛顿算法。在似然函数的受限可逆性条件和回归系数的稀疏性条件下，我们建立了GSDAR在极值范数下产生的解序列的估计误差的显式上限，并证明了它在有限迭代下达到了最佳阶。高概率。此外，我们表明，如果目标信号高于可检测的水平，则可以高概率恢复oracle估计量。这些结果直接涉及从GSDAR算法生成的解序列，而不是理论上定义的全局解。