标题： 坐标下降算法用于套索惩罚回归 显示英文标题

标题： Coordinate descent algorithms for lasso penalized regression

摘要： 施加套索惩罚会将参数估计值收缩到零并执行连续模型选择。 套索惩罚回归能够处理预测变量数量远超过案例数量的线性回归问题。 本文测试了两种极其快速的算法，用于估计带有套索惩罚的回归系数。 已知的$\ell_2$算法基于循环坐标下降。 我们新的$\ell_1$算法基于贪婪坐标下降和埃奇沃思针对普通$\ell_1$回归的算法。 每个算法都依赖一个可以通过交叉验证选择的调优常数。 在一些回归问题中，自然地将参数分组，并按组而不是单独地对参数进行惩罚。 如果组惩罚与该组参数的欧几里得范数成比例，则可以最大化范数并将参数估计简化为带有套索惩罚的$\ell_2$回归。 因此，现有的算法可以扩展到新的设置。 讨论中的每个算法都通过模拟数据、真实数据或两者进行了测试。 附录证明了$\ell_2$算法的一种贪婪形式收敛于目标函数的最小值。 显示英文摘要

摘要： Imposition of a lasso penalty shrinks parameter estimates toward zero and performs continuous model selection. Lasso penalized regression is capable of handling linear regression problems where the number of predictors far exceeds the number of cases. This paper tests two exceptionally fast algorithms for estimating regression coefficients with a lasso penalty. The previously known $\ell_2$ algorithm is based on cyclic coordinate descent. Our new $\ell_1$ algorithm is based on greedy coordinate descent and Edgeworth's algorithm for ordinary $\ell_1$ regression. Each algorithm relies on a tuning constant that can be chosen by cross-validation. In some regression problems it is natural to group parameters and penalize parameters group by group rather than separately. If the group penalty is proportional to the Euclidean norm of the parameters of the group, then it is possible to majorize the norm and reduce parameter estimation to $\ell_2$ regression with a lasso penalty. Thus, the existing algorithm can be extended to novel settings. Each of the algorithms discussed is tested via either simulated or real data or both. The Appendix proves that a greedy form of the $\ell_2$ algorithm converges to the minimum value of the objective function.