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。