We employ random matrix theory to establish consistency of generalized cross validation (GCV) for estimating prediction risks of sketched ridge regression ensembles, enabling efficient and consistent tuning of regularization and sketching parameters. Our results hold for a broad class of asymptotically free sketches under very mild data assumptions. For squared prediction risk, we provide a decomposition into an unsketched equivalent implicit ridge bias and a sketching-based variance, and prove that the risk can be globally optimized by only tuning sketch size in infinite ensembles. For general subquadratic prediction risk functionals, we extend GCV to construct consistent risk estimators, and thereby obtain distributional convergence of the GCV-corrected predictions in Wasserstein-2 metric. This in particular allows construction of prediction intervals with asymptotically correct coverage conditional on the training data. We also propose an "ensemble trick" whereby the risk for unsketched ridge regression can be efficiently estimated via GCV using small sketched ridge ensembles. We empirically validate our theoretical results using both synthetic and real large-scale datasets with practical sketches including CountSketch and subsampled randomized discrete cosine transforms.

翻译：我们利用随机矩阵理论建立了广义交叉验证（GCV）在估计草图岭回归集成预测风险时的一致性，从而实现对正则化参数和草图参数的高效且一致的调优。我们的结果适用于一类广泛的渐近自由草图，且数据假设条件非常宽松。针对平方预测风险，我们将其分解为无草图等效隐式岭偏置与基于草图的方差两部分，并证明在无限集成中仅通过调整草图大小即可全局优化该风险。对于一般的次二次预测风险泛函，我们将GCV扩展为一致的风险估计量，进而获得经GCV校正的预测在Wasserstein-2度量下的分布收敛性。这尤其允许在训练数据条件下构建渐近覆盖正确的预测区间。我们还提出了一种“集成技巧”，通过使用小型草图岭集成的GCV高效估计无草图岭回归的风险。我们使用包含CountSketch和子采样随机离散余弦变换等实际草图的合成与大规模真实数据集，对理论结果进行了实证验证。