标题： 前向阶段回归和单调lasso 显示英文标题

标题： Forward stagewise regression and the monotone lasso

摘要： 我们研究了用于求解惩罚最小二乘回归问题的最小角回归和前向逐步算法。 在Efron、Hastie、Johnstone与Tibshirani（2004）中证明了，通过一个小修改后，最小角回归算法可以解决套索回归问题。 在这里，我们对增量前向逐步回归给出了一个类似的结论，表明它解决了强制单调性的套索问题的一个版本。 由此得出的一个推论如下：虽然套索在每单位增加的系数$\beta$的$L_1$-范数时能以最优方式减少残差平方和，但前向逐步方法则在每单位沿系数路径行进的$L_1$弧长上是最优的。 我们还研究了一个使套索的系数路径单调的条件，从而使得不同的算法一致。 最后，在涉及大量相关预测变量的模拟研究中，我们将套索和前向逐步算法进行了比较。 显示英文摘要

摘要： We consider the least angle regression and forward stagewise algorithms for solving penalized least squares regression problems. In Efron, Hastie, Johnstone & Tibshirani (2004) it is proved that the least angle regression algorithm, with a small modification, solves the lasso regression problem. Here we give an analogous result for incremental forward stagewise regression, showing that it solves a version of the lasso problem that enforces monotonicity. One consequence of this is as follows: while lasso makes optimal progress in terms of reducing the residual sum-of-squares per unit increase in $L_1$-norm of the coefficient $\beta$, forward stage-wise is optimal per unit $L_1$ arc-length traveled along the coefficient path. We also study a condition under which the coefficient paths of the lasso are monotone, and hence the different algorithms coincide. Finally, we compare the lasso and forward stagewise procedures in a simulation study involving a large number of correlated predictors.