Learning with rejection is an important framework that can refrain from
making predictions to avoid critical mispredictions by balancing between
prediction and rejection. Previous studies on cost-based rejection only focused
on the classification setting, which cannot handle the continuous and infinite
target space in the regression setting. In this paper, we investigate a novel
regression problem called regression with cost-based rejection, where the model
can reject to make predictions on some examples given certain rejection costs.
To solve this problem, we first formulate the expected risk for this problem
and then derive the Bayes optimal solution, which shows that the optimal model
should reject to make predictions on the examples whose variance is larger than
the rejection cost when the mean squared error is used as the evaluation
metric. Furthermore, we propose to train the model by a surrogate loss function
that considers rejection as binary classification and we provide conditions for
the model consistency, which implies that the Bayes optimal solution can be
recovered by our proposed surrogate loss. Extensive experiments demonstrate the
effectiveness of our proposed method.

针对具有连续和无限目标空间的回归问题，提出了一种新颖的基于成本拒绝的回归模型，其可以通过考虑拒绝成本来拒绝对某些示例进行预测。该研究首先建立了该问题的期望风险模型，然后导出了贝叶斯最优解，表明当使用均方误差作为评估指标时，最优模型应拒绝对方差大于拒绝成本的示例进行预测。此外，提出使用考虑拒绝作为二分类的替代损失函数进行模型训练，并提供了模型一致性的条件，表明我们提出的替代损失函数可以恢复贝叶斯最优解。大量实验证明了我们提出方法的有效性。