  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.
