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.

翻译：带有拒绝机制的学习是一种重要框架，它可以通过平衡预测与拒绝来避免作出关键性的错误预测。以往基于成本拒绝的研究仅关注分类场景，无法处理回归场景中连续且无限的目标空间。本文研究一种名为"基于成本拒绝的回归"的新回归问题，即模型在给定拒绝成本后可以拒绝某些样本的预测。为解决该问题，我们首先构建该问题的期望风险函数，进而推导出贝叶斯最优解。结果表明：当使用均方误差作为评估指标时，最优模型应拒绝方差大于拒绝成本的样本预测。此外，我们提出使用将拒绝视为二分类问题的代理损失函数来训练模型，并给出了模型一致性的条件，这证明所提代理损失能够恢复贝叶斯最优解。大量实验验证了该方法有效性。