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We study the risk of minimum-norm interpolants of data in Reproducing KernelHilbert Spaces. Our upper bounds on the risk are of a multiple-descent shapefor the various scalings of $d = n^{\alpha}$, $\alpha\in(0,1)$, for the inputdimension $d$ and sample size $n$. Empirical evidence supports our finding thatminimum-norm interpolants in RKHS can exhibit this unusual non-monotonicity insample size; furthermore, locations of the peaks in our experiments match ourtheoretical predictions. Since gradient flow on appropriately initialized wideneural networks converges to a minimum-norm interpolant with respect to acertain kernel, our analysis also yields novel estimation and generalizationguarantees for these over-parametrized models.  At the heart of our analysis is a study of spectral properties of the randomkernel matrix restricted to a filtration of eigen-spaces of the populationcovariance operator, and may be of independent interest.

On the Multiple Descent of Minimum-Norm Interpolants and Restricted Lower Isometry of Kernels