广义岭回归再拟合用于Lasso及预测改进界
Generalized Ridge Refitting for the Lasso and Prediction Improvement Bounds
- Department of Pure and Applied Mathematics, Waseda University（早稻田大学纯应用数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一类基于Lasso的二次校正估计器，通过闭式分解和随机项控制，给出有限样本期望界，统一解释预测改进条件。
AI中文摘要：
我们研究一类基于Lasso的估计器，该类估计器通过对Lasso等相关集施加二次校正而获得。惩罚矩阵决定了校正的幅度和几何形状，并且包含各向同性Lasso-Ridge校正、最小二乘再拟合、Lasso与最小二乘之间的Gram比例插值以及坐标特定惩罚等情况。我们首先推导出闭式表示，并分离出由此产生的预测改进的正增益分量。然后，通过将随机符号等相关模型局部化在确定性参考支撑周围，我们在期望意义上控制剩余的随机线性项。这产生了一个有限样本期望界，该界明确考虑了Lasso模型选择引起的随机性。所得分解为理解基于Lasso的二次校正何时能改进预测提供了一个统一框架。
英文摘要：
We study a class of Lasso based estimators obtained by applying a quadratic correction on the Lasso equicorrelation set. The penalty matrix determines both the magnitude and geometry of the correction and contains, among other cases, the isotropic Lasso--Ridge correction, least squares refitting, Gram proportional interpolation between the Lasso and least squares, and coordinate specific penalties. We first derive a closed form representation and isolate the positive gain component of the resulting prediction improvement. We then control the remaining stochastic linear term in expectation by localizing the random signed equicorrelation model around a deterministic reference support. This yields a finite sample expectation bound that explicitly accounts for the randomness induced by Lasso model selection. The resulting decomposition provides a unified framework for understanding when Lasso based quadratic corrections can improve prediction.
