Abstract: 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.\n \nAt 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.