We present a detailed study of $H$-consistency bounds for regression. We first present new theorems that generalize the tools previously given to establish $H$-consistency bounds. This generalization proves essential for analyzing $H$-consistency bounds specific to regression. Next, we prove a series of novel $H$-consistency bounds for surrogate loss functions of the squared loss, under the assumption of a symmetric distribution and a bounded hypothesis set. This includes positive results for the Huber loss, all $\ell_p$ losses, $p \geq 1$, the squared $\epsilon$-insensitive loss, as well as a negative result for the $\epsilon$-insensitive loss used in squared Support Vector Regression (SVR). We further leverage our analysis of $H$-consistency for regression and derive principled surrogate losses for adversarial regression (Section 5). This readily establishes novel algorithms for adversarial regression, for which we report favorable experimental results in Section 6.

我们对回归的H一致性界限进行了详细研究，提出了适用于回归的H一致性界限的新定理，并针对对称分布和有界假设集的平方损失函数的代理损失函数证明了一系列新的H一致性界限，包括Huber损失、所有的lp损失(p≥1)、平方ε-insensitive损失，以及在平方支持向量回归(SVR)中使用的ε-insensitive损失的负面结果。我们进一步利用我们对回归H一致性的分析，并推导出对抗性回归的合理代理损失函数，从而建立了新的对抗性回归算法，并在第6节报道了有利的实验结果。

$H$-一致性保证用于回归