Exact Gap between Generalization Error and Uniform Convergence in Random Feature Models

Recent work showed that there could be a large gap between the classical uniform convergence bound and the actual test error of zero-training-error predictors (interpolators) such as deep neural networks. To better understand this gap, we study the uniform convergence in the nonlinear random feature model and perform a precise theoretical analysis on how uniform convergence depends on the sample size and the number of parameters. We derive and prove analytical expressions for three quantities in this model: 1) classical uniform convergence over norm balls, 2) uniform convergence over interpolators in the norm ball (recently proposed by~\citet{zhou2021uniform}), and 3) the risk of minimum norm interpolator. We show that, in the setting where the classical uniform convergence bound is vacuous (diverges to $\infty$), uniform convergence over the interpolators still gives a non-trivial bound of the test error of interpolating solutions. We also showcase a different setting where classical uniform convergence bound is non-vacuous, but uniform convergence over interpolators can give an improved sample complexity guarantee. Our result provides a first exact comparison between the test errors and uniform convergence bounds for interpolators beyond simple linear models.