  We study the problem of learning with selectively labeled data, which arises
when outcomes are only partially labeled due to historical decision-making. The
labeled data distribution may substantially differ from the full population,
especially when the historical decisions and the target outcome can be
simultaneously affected by some unobserved factors. Consequently, learning with
only the labeled data may lead to severely biased results when deployed to the
full population. Our paper tackles this challenge by exploiting the fact that
in many applications the historical decisions were made by a set of
heterogeneous decision-makers. In particular, we analyze this setup in a
principled instrumental variable (IV) framework. We establish conditions for
the full-population risk of any given prediction rule to be point-identified
from the observed data and provide sharp risk bounds when the point
identification fails. We further propose a weighted learning approach that
learns prediction rules robust to the label selection bias in both
identification settings. Finally, we apply our proposed approach to a
semi-synthetic financial dataset and demonstrate its superior performance in
the presence of selection bias.
