  Selective labels are a common feature of consequential decision-making
applications, referring to the lack of observed outcomes under one of the
possible decisions. This paper reports work in progress on learning decision
policies in the face of selective labels. The setting considered is both a
simplified homogeneous one, disregarding individuals' features to facilitate
determination of optimal policies, and an online one, to balance costs incurred
in learning with future utility. For maximizing discounted total reward, the
optimal policy is shown to be a threshold policy, and the problem is one of
optimal stopping. In contrast, for undiscounted infinite-horizon average
reward, optimal policies have positive acceptance probability in all states.
Future work stemming from these results is discussed.
