Failures and Successes of Cross-Validation for Early-Stopped Gradient
  Descent
Pratik Patil Yuchen Wu Ryan J. Tibshirani
We analyze the statistical properties of generalized cross-validation (GCV)
and leave-one-out cross-validation (LOOCV) applied to early-stopped gradient
descent (GD) in high-dimensional least squares regression. We prove that GCV is
generically inconsistent as an estimator of the prediction risk of
early-stopped GD, even for a well-specified linear model with isotropic
features. In contrast, we show that LOOCV converges uniformly along the GD
trajectory to the prediction risk. Our theory requires only mild assumptions on
the data distribution and does not require the underlying regression function
to be linear. Furthermore, by leveraging the individual LOOCV errors, we
construct consistent estimators for the entire prediction error distribution
along the GD trajectory and consistent estimators for a wide class of error
functionals. This in particular enables the construction of pathwise prediction
intervals based on GD iterates that have asymptotically correct nominal
coverage conditional on the training data.