We establish precise structural and risk equivalences between subsampling and ridge regularization for ensemble ridge estimators. Specifically, we prove that linear and quadratic functionals of subsample ridge estimators, when fitted with different ridge regularization levels $\lambda$ and subsample aspect ratios $\psi$, are asymptotically equivalent along specific paths in the $(\lambda, \psi )$-plane (where $\psi$ is the ratio of the feature dimension to the subsample size). Our results only require bounded moment assumptions on feature and response distributions and allow for arbitrary joint distributions. Furthermore, we provide a datadependent method to determine the equivalent paths of $(\lambda, \psi )$. An indirect implication of our equivalences is that optimally-tuned ridge regression exhibits a monotonic prediction risk in the data aspect ratio. This resolves a recent open problem raised by Nakkiran et al. under general data distributions and a mild regularity condition that maintains regression hardness through linearized signal-to-noise ratios.

翻译：我们建立了集成岭估计器中子采样与岭正则化之间精确的结构和风险等价性。具体而言，我们证明当使用不同岭正则化水平 $\lambda$ 和子采样纵横比 $\psi$ 拟合时，子采样岭估计量的线性与二次泛函在 $(\lambda, \psi)$-平面（其中 $\psi$ 为特征维度与子样本量之比）的特定路径上渐近等价。该结论仅要求特征与响应分布满足有界矩假设，并允许任意联合分布。此外，我们提出了一种数据驱动方法以确定 $(\lambda, \psi)$ 的等价路径。这些等价性的一个间接推论是：最优调参的岭回归在数据纵横比上呈现单调预测风险。这一结果解决了 Nakkiran 等人近期提出的一个公开问题，适用于一般数据分布及通过线性化信噪比维持回归难度的温和正则性条件。