We study subsampling-based ridge ensembles in the proportional asymptotics regime, where the feature size grows proportionally with the sample size such that their ratio converges to a constant. By analyzing the squared prediction risk of ridge ensembles as a function of the explicit penalty $\lambda$ and the limiting subsample aspect ratio $\phi_s$ (the ratio of the feature size to the subsample size), we characterize contours in the $(\lambda, \phi_s)$-plane at any achievable risk. As a consequence, we prove that the risk of the optimal full ridgeless ensemble (fitted on all possible subsamples) matches that of the optimal ridge predictor. In addition, we prove strong uniform consistency of generalized cross-validation (GCV) over the subsample sizes for estimating the prediction risk of ridge ensembles. This allows for GCV-based tuning of full ridgeless ensembles without sample splitting and yields a predictor whose risk matches optimal ridge risk.

翻译：我们研究比例渐近机制下的基于子采样的岭集成，其中特征维度与样本量按比例增长，使两者之比收敛于常数。通过分析岭集成的平方预测风险（作为显式惩罚参数 λ 与极限子采样宽高比 φ_s，即特征维度与子采样样本量之比的函数），我们刻画了(λ, φ_s)平面上任意可达风险对应的等高线。由此，我们证明最优全无岭集成（在所有可能子样本上拟合）的风险与最优岭预测器的风险相匹配。此外，我们证明了广义交叉验证（GCV）在子采样样本量上对岭集成预测风险估计的强一致收敛性。这使得无需样本分割即可对全无岭集成进行基于GCV的调参，并产生风险与最优岭风险相匹配的预测器。