Generalization error rates in kernel regression: the crossover from the noiseless to noisy regime* *This article is an updated version of: Cui H, Loureiro B, Krzakala F and Zdeborová L 2021 Generalization error rates in kernel regression: the crossover from the noiseless to noisy regime Advances in Neural Information Processing Systems vol 34 ed M Ranzato, A Beygelzimer, Y Dauphin,

Abstract

In this manuscript we consider kernel ridge regression (KRR) under the Gaussian design. Exponents for the decay of the excess generalization error of KRR have been reported in various works under the assumption of power-law decay of eigenvalues of the features co-variance. These decays were, however, provided for sizeably different setups, namely in the noiseless case with constant regularization and in the noisy optimally regularized case. Intermediary settings have been left substantially uncharted. In this work, we unify and extend this line of work, providing characterization of all regimes and excess error decay rates that can be observed in terms of the interplay of noise and regularization. In particular, we show the existence of a transition in the noisy setting between the noiseless exponents to its noisy values as the sample complexity is increased. Finally, we illustrate how this crossover can also be observed on real data sets.