  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.
