Kernel regression in high dimension: Refined analysis beyond double descent

In this paper, we provide a precise characterize of generalization properties of high dimensional kernel ridge regression across the under- and over-parameterized regimes, depending on whether the number of training data $n$ exceeds the feature dimension $d$. By establishing a novel bias-variance decomposition of the expected excess risk, we show that, while the bias is independent of $d$ and monotonically decreases with $n$, the variance depends on $n,d$ and can be unimodal or monotonically decreasing under different regularization schemes. Our refined analysis goes beyond the double descent theory by showing that, depending on the data eigen-profile and the level of regularization, the kernel regression risk curve can be a double-descent-like, bell-shaped, or monotonic function of $n$. Experiments on synthetic and real data are conducted to support our theoretical findings.