Generalization error of random features and kernel methods: hypercontractivity and kernel matrix concentration

Consider the classical supervised learning problem: we are given data (yi,boldsymbol xi), i≤ n, with yi a response and boldsymbol xi∈ mathcal X a covariates vector, and try to learn a model f:mathcal X→mathbb R to predict future responses. Random features methods map the covariates vector boldsymbol xi to a point boldsymbol φ(boldsymbol xi) in a higher dimensional space mathbb RN, via a random featurization map boldsymbol φ. We study the use of random features methods in conjunction with ridge regression in the feature space mathbb RN. This can be viewed as a finite-dimensional approximation of kernel ridge regression (KRR), or as a stylized model for neural networks in the so called lazy training regime. We define a class of problems satisfying certain spectral conditions on the underlying kernels, and a hypercontractivity assumption on the associated eigenfunctions. These conditions are verified by classical high-dimensional examples. Under these conditions, we prove a sharp characterization of the error of random features ridge regression. In particular, we address two fundamental questions: (1)~What is the generalization error of KRR? (2)~How big N should be for the random features approximation to achieve the same error as KRR? In this setting, we prove that KRR is well approximated by a projection onto the top ℓ eigenfunctions of the kernel, where ℓ depends on the sample size n. We show that the test error of random features ridge regression is dominated by its approximation error and is larger than the error of KRR as long as N≤ n1-δ for some δ>0. We characterize this gap. For N≥ n1+δ, random features achieve the same error as the corresponding KRR, and further increasing N does not lead to a significant change in test error.