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.

翻译：我们针对回归问题中的$H$-一致性界进行了详细研究。首先，我们提出了新的定理，对先前用于建立$H$-一致性界的工具进行了推广。这一推广对于分析回归问题特有的$H$-一致性界至关重要。随后，在对称分布和有界假设集的条件下，我们证明了平方损失代理损失函数的一系列新型$H$-一致性界。这包括Huber损失、所有$p \geq 1$的$\ell_p$损失、平方$\epsilon$-不敏感损失的正向结果，以及平方支持向量回归（SVR）中使用的$\epsilon$-不敏感损失的反向结果。我们进一步利用回归中$H$-一致性的分析结果，推导出对抗性回归的原则性代理损失函数（第5节）。这直接建立了对抗性回归的新算法，我们在第6节中报告了有利的实验结果。