论文标题
估计$ \ ell_ {1} $的罚款水平 - 通过两种高斯近似方法最小化
Estimating the Penalty Level of $\ell_{1}$-minimization via Two Gaussian Approximation Methods
论文作者
论文摘要
在本文中，我们旨在给出$ \ ell_ {1} $ - 正则化问题的罚款水平的理论近似。与传统方法（例如交叉验证）相比，这可以节省很多时间。为了实现这一目标，我们开发了两种高斯近似方法，这些方法分别基于中等偏差定理和Stein的方法。他们俩都提供了有效的近似值，并且在模拟中具有良好的性能。我们将两种高斯近似方法应用于三种类型的超高维度$ \ ell_ {1} $惩罚回归：lasso，square-root lasso和加权$ \ ell_ {1} $惩罚poisson回归。数值结果表明，我们估计惩罚水平的两种方法达到了高计算效率。此外，我们的预测错误优于基于10倍交叉验证的表现。
In this paper, we aim to give a theoretical approximation for the penalty level of $\ell_{1}$-regularization problems. This can save much time in practice compared with the traditional methods, such as cross-validation. To achieve this goal, we develop two Gaussian approximation methods, which are based on a moderate deviation theorem and Stein's method respectively. Both of them give efficient approximations and have good performances in simulations. We apply the two Gaussian approximation methods into three types of ultra-high dimensional $\ell_{1}$ penalized regressions: lasso, square-root lasso, and weighted $\ell_{1}$ penalized Poisson regression. The numerical results indicate that our two ways to estimate the penalty levels achieve high computational efficiency. Besides, our prediction errors outperform that based on the 10-fold cross-validation.
