  The generalization performance of kernel ridge regression (KRR) exhibits a
multi-phased pattern that crucially depends on the scaling relationship between
the sample size $n$ and the underlying dimension $d$. This phenomenon is due to
the fact that KRR sequentially learns functions of increasing complexity as the
sample size increases; when $d^{k-1}\ll n\ll d^{k}$, only polynomials with
degree less than $k$ are learned. In this paper, we present sharp asymptotic
characterization of the performance of KRR at the critical transition regions
with $n \asymp d^k$, for $k\in\mathbb{Z}^{+}$. Our asymptotic characterization
provides a precise picture of the whole learning process and clarifies the
impact of various parameters (including the choice of the kernel function) on
the generalization performance. In particular, we show that the learning curves
of KRR can have a delicate "double descent" behavior due to specific
bias-variance trade-offs at different polynomial scaling regimes.
