  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 Zhou et al. (2020)), 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.
