  We investigate the power of censoring techniques, first developed for
learning {\em fair representations}, to address domain generalization. We
examine {\em adversarial} censoring techniques for learning invariant
representations from multiple "studies" (or domains), where each study is drawn
according to a distribution on domains. The mapping is used at test time to
classify instances from a new domain. In many contexts, such as medical
forecasting, domain generalization from studies in populous areas (where data
are plentiful), to geographically remote populations (for which no training
data exist) provides fairness of a different flavor, not anticipated in
previous work on algorithmic fairness.
  We study an adversarial loss function for $k$ domains and precisely
characterize its limiting behavior as $k$ grows, formalizing and proving the
intuition, backed by experiments, that observing data from a larger number of
domains helps. The limiting results are accompanied by non-asymptotic
learning-theoretic bounds. Furthermore, we obtain sufficient conditions for
good worst-case prediction performance of our algorithm on previously unseen
domains. Finally, we decompose our mappings into two components and provide a
complete characterization of invariance in terms of this decomposition. To our
knowledge, our results provide the first formal guarantees of these kinds for
adversarial invariant domain generalization.
