  We study the performance of empirical risk minimization on the $p$-norm
linear regression problem for $p \in (1, \infty)$. We show that, in the
realizable case, under no moment assumptions, and up to a
distribution-dependent constant, $O(d)$ samples are enough to exactly recover
the target. Otherwise, for $p \in [2, \infty)$, and under weak moment
assumptions on the target and the covariates, we prove a high probability
excess risk bound on the empirical risk minimizer whose leading term matches,
up to a constant that depends only on $p$, the asymptotically exact rate. We
extend this result to the case $p \in (1, 2)$ under mild assumptions that
guarantee the existence of the Hessian of the risk at its minimizer.
