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 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 about 1/8th to 1/10th of the computational time 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} 正常的线性回归, 也称为 Bridge 回归 。 对于 $\ alpha\ in ( 0, 1, 1 美元), Bridge 回归具有若干值得关注的统计属性, 如估计数的夸大性和近乎无偏差性( Fan 和 Li, 2001 ) 。 但是,主要困难在于这些值的罚款的非隐含性质 $\ alpha$, 这使得优化程序具有挑战性, 通常只能找到一个地方最佳的地方。为了解决这个问题, Polson 等人 等人 2013 采用了完全基于巴耶西亚的抽样方法, 使用Bridge 罚款与 回归系数 之前的指数指数指数指数指数指数指数之间的对应。 但是, 他们的取样程序依赖于Markov 链 Monte Carlo (MC ) 技术, 这些技术本质上是序列性的, 无法向大问题层面伸缩。 交叉验证方法同样是 。 为此, 我们的贡献是一种新的不触动性的方法, 适合一个桥梁回归模式。 主要贡献于 Stein' 公正的风险估计 校正 校正 校正 校正 校正 校正 的校正 的校正 的校正的校正的校正的校正的校正的校正的校正 。