  We study the risk of minimum-norm interpolants of data in Reproducing Kernel
Hilbert Spaces. Our upper bounds on the risk are of a multiple-descent shape
for the various scalings of $d = n^{\alpha}$, $\alpha\in(0,1)$, for the input
dimension $d$ and sample size $n$. Empirical evidence supports our finding that
minimum-norm interpolants in RKHS can exhibit this unusual non-monotonicity in
sample size; furthermore, locations of the peaks in our experiments match our
theoretical predictions. Since gradient flow on appropriately initialized wide
neural networks converges to a minimum-norm interpolant with respect to a
certain kernel, our analysis also yields novel estimation and generalization
guarantees for these over-parametrized models.
  At the heart of our analysis is a study of spectral properties of the random
kernel matrix restricted to a filtration of eigen-spaces of the population
covariance operator, and may be of independent interest.
