标题： 最小绝对梯度选择器：通过伪硬阈值的统计回归 显示英文标题

标题： Least Absolute Gradient Selector: Statistical Regression via Pseudo-Hard Thresholding

摘要： 变量选择在线性模型中在现代统计学中起着关键作用。硬阈值方法，如$l_0$正则化，在理论上是理想的但计算上不可行。在本文中，我们提出了一种新方法，称为\textbf{滞后}，即“最小绝对梯度选择器”，通过模仿$l_0$正则化的离散选择过程来解决这个具有挑战性且有趣的问题。 为了在噪声影响下估计$\beta$，我们仍然考虑以下凸程序 \[\hat{\beta} = \textrm{arg min}\frac{1}{n}\|X^{T}(y - X\beta)\|_1 + \lambda_n\sum_{i = 1}^pw_i(y;X;n)|\beta_i|\] $\lambda_n > 0$控制稀疏性，$w_i > 0$依赖于$y, X$和 $n$是不同$\beta_i$的权重；$n$是样本量。 令人惊讶的是，我们在论文中将通过几何和分析方法展示，LAGS具有两个吸引人的特性：(1) LAGS在选择离散变量和硬阈值属性方面表现出与$l_0$正则化相似的行为，这是通过精心选择的$w_i$实现的，我们称这一特性为\emph{“伪硬阈值”}；(2) 渐近地，LAGS是一致的，并能够发现真实模型；非渐近地，LAGS能够识别模型中的稀疏性，系数的预测误差在噪声水平上被限制，最多是一个对数因子---$\log p$，其中$p$是预测变量的数量。 在计算方面，由于凸性，LAGS可以通过凸程序方法高效求解，或者在将其重新表述为线性规划后通过单纯形算法求解。 数值模拟显示，在均方误差和模型简洁性方面，LAGS优于软阈值方法。 显示英文摘要

摘要： Variable selection in linear models plays a pivotal role in modern statistics. Hard-thresholding methods such as $l_0$ regularization are theoretically ideal but computationally infeasible. In this paper, we propose a new approach, called the \textbf{LAGS}, short for "least absulute gradient selector", to this challenging yet interesting problem by mimicking the discrete selection process of $l_0$ regularization. To estimate $\beta$ under the influence of noise, we consider, nevertheless, the following convex program \[\hat{\beta} = \textrm{arg min}\frac{1}{n}\|X^{T}(y - X\beta)\|_1 + \lambda_n\sum_{i = 1}^pw_i(y;X;n)|\beta_i|\] $\lambda_n > 0$ controls the sparsity and $w_i > 0$ dependent on $y, X$ and $n$ is the weights on different $\beta_i$; $n$ is the sample size. Surprisingly, we shall show in the paper, both geometrically and analytically, that LAGS enjoys two attractive properties: (1) LAGS demonstrates discrete selection behavior and hard thresholding property as $l_0$ regularization by strategically chosen $w_i$, we call this property \emph{"pseudo-hard thresholding"}; (2) Asymptotically, LAGS is consistent and capable of discovering the true model; nonasymptotically, LAGS is capable of identifying the sparsity in the model and the prediction error of the coefficients is bounded at the noise level up to a logarithmic factor---$\log p$, where $p$ is the number of predictors. Computationally, LAGS can be solved efficiently by convex program routines for its convexity or by simplex algorithm after recasting it into a linear program. The numeric simulation shows that LAGS is superior compared to soft-thresholding methods in terms of mean squared error and parsimony of the model.