A non-asymptotic theory of Kernel Ridge Regression: deterministic
  equivalents, test error, and GCV estimator
Theodor Misiakiewicz Basil Saeed
We consider learning an unknown target function $f_*$ using kernel ridge
regression (KRR) given i.i.d. data $(u_i,y_i)$, $i\leq n$, where $u_i \in U$ is
a covariate vector and $y_i = f_* (u_i) +\varepsilon_i \in \mathbb{R}$. A
recent string of work has empirically shown that the test error of KRR can be
well approximated by a closed-form estimate derived from an `equivalent'
sequence model that only depends on the spectrum of the kernel operator.
However, a theoretical justification for this equivalence has so far relied
either on restrictive assumptions -- such as subgaussian independent
eigenfunctions -- , or asymptotic derivations for specific kernels in high
dimensions.
  In this paper, we prove that this equivalence holds for a general class of
problems satisfying some spectral and concentration properties on the kernel
eigendecomposition. Specifically, we establish in this setting a non-asymptotic
deterministic approximation for the test error of KRR -- with explicit
non-asymptotic bounds -- that only depends on the eigenvalues and the target
function alignment to the eigenvectors of the kernel. Our proofs rely on a
careful derivation of deterministic equivalents for random matrix functionals
in the dimension free regime pioneered by Cheng and Montanari (2022).
  We apply this setting to several classical examples and show an excellent
agreement between theoretical predictions and numerical simulations. These
results rely on having access to the eigendecomposition of the kernel operator.
Alternatively, we prove that, under this same setting, the generalized
cross-validation (GCV) estimator concentrates on the test error uniformly over
a range of ridge regularization parameter that includes zero (the interpolating
solution). As a consequence, the GCV estimator can be used to estimate from
data the test error and optimal regularization parameter for KRR.