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Learning with rejection has been a prototypical model for studying thehuman-AI interaction on prediction tasks. Upon the arrival of a sampleinstance, the model first uses a rejector to decide whether to accept and usethe 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 andoften results in undesirable non-convexity and inconsistency issues. For theclassification with rejection problem, several works develop consistentsurrogate losses for the joint learning of the predictor and the rejector,while there have been fewer works for the regression counterpart. This paperstudies the regression with rejection (RwR) problem and investigates ano-rejection learning strategy that uses all the data to learn the predictor.We first establish the consistency for such a strategy under the weakrealizability condition. Then for the case without the weak realizability, weshow that the excessive risk can also be upper bounded with the sum of twoparts: prediction error and calibration error. Lastly, we demonstrate theadvantage of such a proposed learning strategy with empirical evidence.

When No-Rejection Learning is Consistent for Regression with Rejection