Recent studies have demonstrated theoretical attractiveness of a class of concave penalties in variable selection, including the smoothly clipped absolute deviation and minimax concave penalties. The computation of the concave penalized solutions in high-dimensional models, however, is a difficult task. We propose a majorization minimization by coordinate descent (MMCD) algorithm for computing the concave penalized solutions in generalized linear models. In contrast to the existing algorithms that use local quadratic or local linear approximation to the penalty function, the MMCD seeks to majorize the negative log-likelihood by a quadratic loss, but does not use any approximation to the penalty. This strategy makes it possible to avoid the computation of a scaling factor in each update of the solutions, which improves the efficiency of coordinate descent. Under certain regularity conditions, we establish theoretical convergence property of the MMCD. We implement this algorithm for a penalized logistic regression model using the SCAD and MCP penalties. Simulation studies and a data example demonstrate that the MMCD works sufficiently fast for the penalized logistic regression in high-dimensional settings where the number of covariates is much larger than the sample size.

中文翻译：

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凹面惩罚广义线性模型的坐标下降的专业化最小化。

最近的研究已经证明了变量选择中一类凹面惩罚的理论吸引力，包括平滑剪裁的绝对偏差和极小极大凹面惩罚。然而，高维模型中凹惩罚解的计算是一项艰巨的任务。我们提出了一种通过坐标下降（MMCD）算法来计算广义线性模型中的凹惩罚解的主要化最小化算法。与使用局部二次或局部线性逼近惩罚函数的现有算法相比，MMCD 试图通过二次损失来优化负对数似然，但不使用任何逼近惩罚。这种策略可以避免在每次更新解决方案时计算缩放因子，从而提高坐标下降的效率。在一定的规律性条件下，我们建立了MMCD的理论收敛性。我们使用 SCAD 和 MCP 惩罚为惩罚逻辑回归模型实现了该算法。模拟研究和数据示例表明，对于协变量数量远大于样本大小的高维设置中的惩罚逻辑回归，MMCD 的工作速度足够快。