  Random Feature (RF) models are used as efficient parametric approximations of
kernel methods. We investigate, by means of random matrix theory, the
connection between Gaussian RF models and Kernel Ridge Regression (KRR). For a
Gaussian RF model with $P$ features, $N$ data points, and a ridge $\lambda$, we
show that the average (i.e. expected) RF predictor is close to a KRR predictor
with an effective ridge $\tilde{\lambda}$. We show that $\tilde{\lambda} >
\lambda$ and $\tilde{\lambda} \searrow \lambda$ monotonically as $P$ grows,
thus revealing the implicit regularization effect of finite RF sampling. We
then compare the risk (i.e. test error) of the $\tilde{\lambda}$-KRR predictor
with the average risk of the $\lambda$-RF predictor and obtain a precise and
explicit bound on their difference. Finally, we empirically find an extremely
good agreement between the test errors of the average $\lambda$-RF predictor
and $\tilde{\lambda}$-KRR predictor.
