草图岭回归的自举误差估计与草图规模选择
Bootstrap Error Estimation and Sketch-Size Selection for Sketched Ridge Regression
AI总结：
本文将配对行自举法扩展至草图岭回归，推导了相关渐近性质，开发了多种自举改进方法，实验验证了其在误差估计与草图规模选择上的有效性。
AI中文摘要：
随机草图技术可降低大规模岭回归问题的计算成本，但系数误差依赖于实际生成的草图。本文将配对行自举法从随机最小二乘扩展至岭回归，仅通过压缩数据即可实现系数误差估计。在固定系数维度、数据与草图规模递增的条件下，本文推导了草图估计量及条件自举分布的渐近线性表示与高斯极限，证实了自举误差分布的一致一致性；当估计量与误差边界由同一张草图计算时，可实现渐近精确覆盖。对于具有独立、零均值、单位方差元素的草图，本文给出显式协方差公式，分离了残差、正则化及草图元素四阶矩的影响，且证明拉德马赫（Rademacher）元素在勒贝格序（Loewner order）下可最小化主导协方差矩阵。此外，本文还开发了快速线性化自举法、针对有限个自举重复的顺序统计量校正，以及从固定草图规模集合中选择的邦费罗尼（Bonferroni）规则。在两个真实数据集和两个合成数据集上的实验验证了所提方法的有效性：当草图规模为系数数量的15倍、自举重复次数为199、名义覆盖率为95%时，带 refitting 的自举法覆盖率为92.0%至95.3%；经顺序统计量校正后，覆盖率范围为94.7%至97.7%。
英文摘要：
Randomized sketching reduces the computational cost of large ridge-regression problems, but the coefficient error depends on the realized sketch. We extend the paired-row bootstrap from randomized least squares to ridge regression, enabling coefficient-error estimation using only compressed data. Under a fixed coefficient dimension and increasing data and sketch sizes, we derive asymptotic linear representations and Gaussian limits for the sketched estimator and the conditional bootstrap distribution. These results establish uniform consistency of the bootstrap error distribution and asymptotically exact coverage when the estimator and error bound are computed from the same sketch. For sketches with independent, mean-zero, variance-one entries, an explicit covariance formula separates the effects of the residual, regularization, and fourth moment of the sketch entries, and shows that Rademacher entries minimize the leading covariance matrix in the Loewner order. We also develop a fast linearized bootstrap, an order-statistic correction for finitely many bootstrap replicates, and a Bonferroni rule for selecting from a fixed set of sketch sizes. Experiments on two real and two synthetic data sets support the proposed methods. With a sketch size 15 times the number of coefficients, 199 bootstrap replicates, and nominal coverage of 95\%, the bootstrap with refitting attains coverage between 92.0\% and 95.3\%; after the order-statistic correction, coverage ranges from 94.7\% to 97.7\%.
