Existing statistical learning guarantees for general kernel regressors often yield loose bounds when used with finite-rank kernels. Yet, finite-rank kernels naturally appear in a number of machine learning problems, e.g. when fine-tuning a pre-trained deep neural network's last layer to adapt it to a novel task when performing transfer learning. We address this gap for finite-rank kernel ridge regression (KRR) by deriving sharp non-asymptotic upper and lower bounds for the KRR test error of any finite-rank KRR. Our bounds are tighter than previously derived bounds on finite-rank KRR and, unlike comparable results, they also remain valid for any regularization parameters.

Existing statistical learning guarantees for general kernel regressors often yield loose bounds when used with finite-rank kernels. Yet, finite-rank kernels naturally appear in a number of machine learning problems, e.g. when fine-tuning a pre-trained deep neural network's last layer to adapt it to a novel task when performing transfer learning. We address this gap for finite-rank kernel ridge regression (KRR) by deriving sharp non-asymptotic upper and lower bounds for the KRR test error of any finite-rank KRR. Our bounds are tighter than previously derived bounds on finite-rank KRR and, unlike comparable results, they also remain valid for any regularization parameters.

Generalization is a central theme in statistical learning theory. The recent renewed interest in kernel methods, especially in Kernel Ridge Regression (KRR), is largely due to the fact that deep neural network (DNN) training can be approximated using kernels under appropriate conditions Jacot et al. [2018], Arora et al. [2019], Bordelon et al. [2020], in which the test error is more tractable analytically and thus enjoys stronger theoretical guarantees. However, many prior results have been derived under conditions incompatible with practical settings. For instance Liang and Rakhlin [2020], Liu et al. [2021a], Mei et al. [2021], Misiakiewicz [2022] give asymptotic bounds on the KRR test error, which requires the input dimension d to tend to infinity. In reality, the input dimension of the data set and the target function is typically finite.