  Practitioners often use data from a randomized controlled trial to learn a
treatment assignment policy that can be deployed on a target population. A
recurring concern in doing so is that, even if the randomized trial was
well-executed (i.e., internal validity holds), the study participants may not
represent a random sample of the target population (i.e., external validity
fails)--and this may lead to policies that perform suboptimally on the target
population. We consider a model where observable attributes can impact sample
selection probabilities arbitrarily but the effect of unobservable attributes
is bounded by a constant, and we aim to learn policies with the best possible
performance guarantees that hold under any sampling bias of this type. In
particular, we derive the partial identification result for the worst-case
welfare in the presence of sampling bias and show that the optimal max-min,
max-min gain, and minimax regret policies depend on both the conditional
average treatment effect (CATE) and the conditional value-at-risk (CVaR) of
potential outcomes given covariates. To avoid finite-sample inefficiencies of
plug-in estimates, we further provide an end-to-end procedure for learning the
optimal max-min and max-min gain policies that does not require the separate
estimation of nuisance parameters.
