主成分回归在所有单调谱滤波器中占优（用于线性回归）
Principal Component Regression Dominates all Monotone Spectral Filters for Linear Regression
- University of California, Berkeley（加州大学伯克利分校）
- Google DeepMind（谷歌DeepMind）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明主成分回归（PCR）在所有单调谱滤波器中占优，包括梯度下降和岭回归，并建立新的上下界，扩展了Wu等人（2026）的结果。
AI中文摘要：
我们比较了线性回归中单调谱滤波器的逐实例有限样本风险，这是一类广泛的估计器，包括主成分回归（PCR）、梯度下降（GD）和岭回归。我们证明PCR在所有单调谱滤波器中占优：与任何此类滤波器相比，最优调参的PCR的风险在所有问题上不超过一个常数因子。此外，如果滤波器与阶跃函数分离（例如GD和岭），则这种占优是强占优：存在一些问题实例，其中PCR的风险在样本量依赖性上以多项式因子更小。我们的比较结果表明，PCR在单调滤波器中是最优的，因此是可采纳的，显著扩展了Wu等人（2026）关于GD强占优岭的结果。从技术角度来看，我们为一般谱滤波器建立了新的上下界，当专门应用于岭或GD时，这些界是逐实例尖锐的，恢复或改进了已知的最佳界。
英文摘要：
We compare the instance-wise, finite-sample risks of monotone spectral filters for linear regression, a broad class of estimators including principal component regression (PCR), gradient descent (GD), and ridge regression. We show that PCR dominates all monotone spectral filters: compared to any such filter, the risk of optimally tuned PCR is no bigger by a constant factor for all problems. Furthermore, the dominance is strong if the filter is separated from step functions (e.g., GD and ridge): there exist problem instances for which the risk of PCR is smaller by a polynomial factor in sample size dependence. Our comparison results show that PCR is optimal and thus admissible among monotone filters, significantly extending Wu et al. (2026)'s result that GD strongly dominates ridge. From a technical perspective, we establish new upper and lower bounds for general spectral filters, which are instance-wise sharp when specialized to ridge or GD, recovering or improving the best-known bounds.
