  Despite their many appealing properties, kernel methods are heavily affected
by the curse of dimensionality. For instance, in the case of inner product
kernels in $\mathbb{R}^d$, the Reproducing Kernel Hilbert Space (RKHS) norm is
often very large for functions that depend strongly on a small subset of
directions (ridge functions). Correspondingly, such functions are difficult to
learn using kernel methods. This observation has motivated the study of
generalizations of kernel methods, whereby the RKHS norm -- which is equivalent
to a weighted $\ell_2$ norm -- is replaced by a weighted functional $\ell_p$
norm, which we refer to as $\mathcal{F}_p$ norm. Unfortunately, tractability of
these approaches is unclear. The kernel trick is not available and minimizing
these norms requires to solve an infinite-dimensional convex problem.
  We study random features approximations to these norms and show that, for
$p>1$, the number of random features required to approximate the original
learning problem is upper bounded by a polynomial in the sample size. Hence,
learning with $\mathcal{F}_p$ norms is tractable in these cases. We introduce a
proof technique based on uniform concentration in the dual, which can be of
broader interest in the study of overparametrized models. For $p= 1$, our
guarantees for the random features approximation break down. We prove instead
that learning with the $\mathcal{F}_1$ norm is $\mathsf{NP}$-hard under a
randomized reduction based on the problem of learning halfspaces with noise.
