  Learning to defer with multiple experts is a framework where the learner can
choose to defer the prediction to several experts. While this problem has
received significant attention in classification contexts, it presents unique
challenges in regression due to the infinite and continuous nature of the label
space. In this work, we introduce a novel framework of regression with
deferral, which involves deferring the prediction to multiple experts. We
present a comprehensive analysis for both the single-stage scenario, where
there is simultaneous learning of predictor and deferral functions, and the
two-stage scenario, which involves a pre-trained predictor with a learned
deferral function. We introduce new surrogate loss functions for both scenarios
and prove that they are supported by $H$-consistency bounds. These bounds
provide consistency guarantees that are stronger than Bayes consistency, as
they are non-asymptotic and hypothesis set-specific. Our framework is
versatile, applying to multiple experts, accommodating any bounded regression
losses, addressing both instance-dependent and label-dependent costs, and
supporting both single-stage and two-stage methods. A by-product is that our
single-stage formulation includes the recent regression with abstention
framework (Cheng et al., 2023) as a special case, where only a single expert,
the squared loss and a label-independent cost are considered. Minimizing our
proposed loss functions directly leads to novel algorithms for regression with
deferral. We report the results of extensive experiments showing the
effectiveness of our proposed algorithms.
