Consider the {$\ell_{\alpha}$} regularized linear regression, also termed Bridge regression. For $\alpha\in (0,1)$, Bridge regression enjoys several statistical properties of interest such as sparsity and near-unbiasedness of the estimates (Fan and Li, 2001). However, the main difficulty lies in the non-convex nature of the penalty for these values of $\alpha$, which makes an optimization procedure challenging and usually it is only possible to find a local optimum. To address this issue, Polson et al. (2013) took a sampling based fully Bayesian approach to this problem, using the correspondence between the Bridge penalty and a power exponential prior on the regression coefficients. However, their sampling procedure relies on Markov chain Monte Carlo (MCMC) techniques, which are inherently sequential and not scalable to large problem dimensions. Cross validation approaches are similarly computation-intensive. To this end, our contribution is a novel \emph{non-iterative} method to fit a Bridge regression model. The main contribution lies in an explicit formula for Stein's unbiased risk estimate for the out of sample prediction risk of Bridge regression, which can then be optimized to select the desired tuning parameters, allowing us to completely bypass MCMC as well as computation-intensive cross validation approaches. Our procedure yields results in a fraction of computational times compared to iterative schemes, without any appreciable loss in statistical performance. An R implementation is publicly available online at: https://github.com/loriaJ/Sure-tuned_BridgeRegression .

翻译：考虑{$\ell_{\alpha}$}正则化线性回归，亦称为桥回归。当$\alpha\in (0,1)$时，桥回归具有若干理想的统计性质，如估计的稀疏性和近似无偏性（Fan and Li, 2001）。然而，主要困难在于该$\alpha$值对应的惩罚函数非凸性，这使得优化过程极具挑战性，通常只能找到局部最优解。为应对此问题，Polson等人（2013）采用基于抽样的完全贝叶斯方法，利用桥惩罚与回归系数幂指数先验之间的对应关系。但他们的抽样过程依赖于马尔可夫链蒙特卡洛（MCMC）技术，本质上是顺序执行的且无法扩展至大规模问题。交叉验证方法同样计算密集。为此，我们提出了一种新颖的\emph{非迭代}桥回归模型拟合方法。主要贡献在于推导出了桥回归样本外预测风险的斯坦无偏风险估计显式公式，通过优化该公式可选择所需的调优参数，从而完全绕过MCMC和计算密集型的交叉验证方法。与迭代方案相比，我们的方法在计算时间上实现了量级缩减，同时统计性能无明显损失。R语言实现已公开于：https://github.com/loriaJ/Sure-tuned_BridgeRegression。