Generalized cross-validation (GCV) is a widely-used method for estimating the squared out-of-sample prediction risk that employs a scalar degrees of freedom adjustment (in a multiplicative sense) to the squared training error. In this paper, we examine the consistency of GCV for estimating the prediction risk of arbitrary ensembles of penalized least squares estimators. We show that GCV is inconsistent for any finite ensemble of size greater than one. Towards repairing this shortcoming, we identify a correction that involves an additional scalar correction (in an additive sense) based on degrees of freedom adjusted training errors from each ensemble component. The proposed estimator (termed CGCV) maintains the computational advantages of GCV and requires neither sample splitting, model refitting, or out-of-bag risk estimation. The estimator stems from a finer inspection of ensemble risk decomposition and two intermediate risk estimators for the components in this decomposition. We provide a non-asymptotic analysis of the CGCV and the two intermediate risk estimators for ensembles of convex penalized estimators under Gaussian features and a linear response model. In the special case of ridge regression, we extend the analysis to general feature and response distributions using random matrix theory, which establishes model-free uniform consistency of CGCV.

翻译：广义交叉验证（Generalized cross-validation, GCV）是一种广泛使用的估计平方样本外预测风险的方法，它通过对平方训练误差进行标量自由度调整（乘法意义）来实现。本文研究了GCV在估计任意有限集成惩罚最小二乘估计器预测风险中的一致性。我们证明，对于任何规模大于1的有限集成，GCV是不一致的。针对这一缺陷，我们提出一种校正方法，该校正方法基于每个集成组分的自由度调整训练误差，引入额外的标量校正（加法意义）。所提出的估计器（称为CGCV）保留了GCV的计算优势，无需样本分割、模型重新拟合或袋外风险估计。该估计器源于对集成风险分解的精细检验以及针对该分解中两个中间风险估计量。对于凸惩罚估计器的集成，在高斯特征和线性响应模型下，我们提供了CGCV及两个中间风险估计器的非渐近分析。在岭回归的特例中，我们利用随机矩阵理论将分析扩展到一般特征和响应分布，从而建立了CGCV的无模型一致收敛性。