Learning with rejection has been a prototypical model for studying the human-AI interaction on prediction tasks. Upon the arrival of a sample instance, the model first uses a rejector to decide whether to accept and use the AI predictor to make a prediction or reject and defer the sample to humans. Learning such a model changes the structure of the original loss function and often results in undesirable non-convexity and inconsistency issues. For the classification with rejection problem, several works develop consistent surrogate losses for the joint learning of the predictor and the rejector, while there have been fewer works for the regression counterpart. This paper studies the regression with rejection (RwR) problem and investigates a no-rejection learning strategy that uses all the data to learn the predictor. We first establish the consistency for such a strategy under the weak realizability condition. Then for the case without the weak realizability, we show that the excessive risk can also be upper bounded with the sum of two parts: prediction error and calibration error. Lastly, we demonstrate the advantage of such a proposed learning strategy with empirical evidence.

翻译：带拒绝学习一直是研究人机交互在预测任务中的典型模型。当样本实例到达时，模型首先使用拒绝器决定是否接受并使用AI预测器进行预测，或拒绝并将样本交由人类处理。学习此类模型改变了原始损失函数的结构，通常会导致非凸性和不一致性等不良问题。对于带拒绝分类问题，已有若干研究开发了用于联合学习预测器和拒绝器的一致性替代损失函数，而关于回归问题的研究则相对较少。本文研究了带拒绝回归（RwR）问题，并探究了一种使用全部数据学习预测器的无拒绝学习策略。我们首先证明了在弱可实现性条件下该策略具有一致性。随后，针对不存在弱可实现性的情况，我们证明了过度风险可以被上界约束为两部分之和：预测误差和校准误差。最后，我们通过实证证据展示了所提出学习策略的优势。