  The (gradient-based) bilevel programming framework is widely used in
hyperparameter optimization and has achieved excellent performance empirically.
Previous theoretical work mainly focuses on its optimization properties, while
leaving the analysis on generalization largely open. This paper attempts to
address the issue by presenting an expectation bound w.r.t. the validation set
based on uniform stability. Our results can explain some mysterious behaviours
of the bilevel programming in practice, for instance, overfitting to the
validation set. We also present an expectation bound for the classical
cross-validation algorithm. Our results suggest that gradient-based algorithms
can be better than cross-validation under certain conditions in a theoretical
perspective. Furthermore, we prove that regularization terms in both the outer
and inner levels can relieve the overfitting problem in gradient-based
algorithms. In experiments on feature learning and data reweighting for noisy
labels, we corroborate our theoretical findings.
