  Recent advances in machine learning have been achieved by using
overparametrized models trained until near interpolation of the training data.
It was shown, e.g., through the double descent phenomenon, that the number of
parameters is a poor proxy for the model complexity and generalization
capabilities. This leaves open the question of understanding the impact of
parametrization on the performance of these models. How does model complexity
and generalization depend on the number of parameters $p$? How should we choose
$p$ relative to the sample size $n$ to achieve optimal test error?
  In this paper, we investigate the example of random feature ridge regression
(RFRR). This model can be seen either as a finite-rank approximation to kernel
ridge regression (KRR), or as a simplified model for neural networks trained in
the so-called lazy regime. We consider covariates uniformly distributed on the
$d$-dimensional sphere and compute sharp asymptotics for the RFRR test error in
the high-dimensional polynomial scaling, where $p,n,d \to \infty$ while $p/
d^{\kappa_1}$ and $n / d^{\kappa_2}$ stay constant, for all $\kappa_1 ,
\kappa_2 \in \mathbb{R}_{>0}$. These asymptotics precisely characterize the
impact of the number of random features and regularization parameter on the
test performance. In particular, RFRR exhibits an intuitive trade-off between
approximation and generalization power. For $n = o(p)$, the sample size $n$ is
the bottleneck and RFRR achieves the same performance as KRR (which is
equivalent to taking $p = \infty$). On the other hand, if $p = o(n)$, the
number of random features $p$ is the limiting factor and RFRR test error
matches the approximation error of the random feature model class (akin to
taking $n = \infty$). Finally, a double descent appears at $n= p$, a phenomenon
that was previously only characterized in the linear scaling $\kappa_1 =
\kappa_2 = 1$.
