SURE调优的Bridge回归
SURE-tuned Bridge Regression
AI总结：
针对α∈(0,1)时Bridge回归非凸惩罚优化难、MCMC与交叉验证计算量大的问题，提出基于Stein无偏风险估计显式公式的非迭代调优方法，大幅提升计算效率且统计性能无明显损失，并公开R实现。
摘要：
考虑ℓ_α正则化线性回归，也称为Bridge回归。当α∈(0,1)时，Bridge回归具备若干受关注的统计特性，例如估计的稀疏性和近似无偏性（Fan和Li，2001）。然而，主要难点在于这些α取值下惩罚项的非凸性，这使得优化过程颇具挑战，通常仅能找到局部最优解。为解决这一问题，Polson等人（2013）采用基于采样的完全贝叶斯方法处理该问题，利用了Bridge惩罚与回归系数上的幂指数先验之间的对应关系。但他们的采样过程依赖马尔可夫链蒙特卡洛（MCMC）技术，该技术本质上是串行的，无法扩展到大规模问题维度。交叉验证方法同样计算密集。为此，我们的贡献是提出一种新颖的非迭代方法来拟合Bridge回归模型。核心贡献在于推导了Bridge回归样本外预测风险的Stein无偏风险估计的显式公式，随后可通过优化该公式来选择所需的调优参数，使我们能够完全绕开MCMC以及计算密集的交叉验证方法。与迭代方案相比，我们的过程仅需极少的计算时间即可得到结果，且统计性能无明显损失。对应的R语言实现已在https://github.com/loriaJ/Sure-tuned_BridgeRegression 公开提供。
英文摘要：
Consider the {$\ell_α$} regularized linear regression, also termed Bridge regression. For $α\in (0,1)$, 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 $α$, 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 .
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
