Optimal Excess Risk Bounds for Empirical Risk Minimization on p-Norm Linear Regression

We study the performance of empirical risk minimization on the p-norm linear regression problem for p ∈ (1, ∞). 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 ∈ [2, ∞), 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 ∈ (1, 2) under mild assumptions that guarantee the existence of the Hessian of the risk at its minimizer.