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.

翻译：我们研究了经验风险最小化在$p$范数线性回归问题（其中$p \in (1, \infty)$）上的性能。我们证明，在可实现情形下，无需任何矩假设，且仅差一个依赖于分布的常数，$O(d)$个样本足以精确恢复目标。否则，对于$p \in [2, \infty)$，并在目标和协变量的弱矩假设下，我们证明了经验风险最小化器的高概率超额风险界，其主项与渐近精确率（仅差一个依赖于$p$的常数）相匹配。我们将此结果扩展到$p \in (1, 2)$的情形，其所需的温和假设保证了风险在其最小化点处海森矩阵的存在性。