Failures and Successes of Cross-Validation for Early-Stopped Gradient Descent in High-Dimensional Least Squares

We analyze the statistical properties of generalized cross-validation (GCV) and leave-one-out cross-validation (LOOCV) applied to early-stopped gradient descent (GD) iterates in high-dimensional least squares regression. Surprisingly, our results show that GCV can be inconsistent for estimating the squared prediction risk, even under a well-specified linear model with isotropic design. In contrast, we prove that LOOCV converges uniformly along the GD trajectory to the prediction risk. Our theory holds under mild assumptions on the data distribution and does not require the underlying regression function to be linear. Furthermore, by suitably extending LOOCV, we construct consistent estimators for the entire prediction error distribution along the GD trajectory and for a wide class of its functionals. This in particular enables the construction of pathwise prediction intervals for the unknown response with asymptotically correct nominal coverage conditional on the training data.