  Selective regression allows abstention from prediction if the confidence to
make an accurate prediction is not sufficient. In general, by allowing a reject
option, one expects the performance of a regression model to increase at the
cost of reducing coverage (i.e., by predicting on fewer samples). However, as
we show, in some cases, the performance of a minority subgroup can decrease
while we reduce the coverage, and thus selective regression can magnify
disparities between different sensitive subgroups. Motivated by these
disparities, we propose new fairness criteria for selective regression
requiring the performance of every subgroup to improve with a decrease in
coverage. We prove that if a feature representation satisfies the sufficiency
criterion or is calibrated for mean and variance, than the proposed fairness
criteria is met. Further, we introduce two approaches to mitigate the
performance disparity across subgroups: (a) by regularizing an upper bound of
conditional mutual information under a Gaussian assumption and (b) by
regularizing a contrastive loss for conditional mean and conditional variance
prediction. The effectiveness of these approaches is demonstrated on synthetic
and real-world datasets.
