We derive high-dimensional Gaussian comparison results for the standard $V$-fold cross-validated risk estimates. Our results combine a recent stability-based argument for the low-dimensional central limit theorem of cross-validation with the high-dimensional Gaussian comparison framework for sums of independent random variables. These results give new insights into the joint sampling distribution of cross-validated risks in the context of model comparison and tuning parameter selection, where the number of candidate models and tuning parameters can be larger than the fitting sample size. As a consequence, our results provide theoretical support for a recent methodological development that constructs model confidence sets using cross-validation.

翻译：我们推导了标准$V$折交叉验证风险估计的高维高斯比较结果。该结果融合了近期基于稳定性的低维交叉验证中心极限定理论证与独立随机变量和的高维高斯比较框架。这些研究为模型比较与调参选择中交叉验证风险的联合抽样分布提供了新见解，尤其适用于候选模型与调参数量超过拟合样本量的情形。最终，我们的结果为近期提出的基于交叉验证构建模型置信集的方法学进展提供了理论支撑。