  Despite recent advances in transfer learning with multiple source data sets,
there still lacks developments for mixture target populations that could be
approximated through a composite of the sources due to certain key factors like
ethnicity in practice. To address this open problem under distributional shifts
of covariates and outcome models as well as the absence of accurate labels on
target, we propose a novel approach for distributionally robust transfer
learning targeting mixture population. It learns a set of covariate-specific
weights to infer the target outcome model with multiple sources, relying on a
joint source mixture assumption for the target population. Then our method
incorporates a group adversarial learning step to enhance the robustness
against moderate violation of the joint mixture assumption. In addition, our
framework allows the use of side information like small labeled sample as a
guidance to avoid over-conservative results. Statistical convergence and
predictive accuracy of our method are quantified through asymptotic studies.
Simulation and real-world studies demonstrate the out-performance of our method
over existing multi-source and transfer learning approaches.
