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

Black-Box Model Confidence Sets Using Cross-Validation with High-Dimensional Gaussian Comparison