  We consider data-driven approaches that integrate a machine learning
prediction model within distributionally robust optimization (DRO) given
limited joint observations of uncertain parameters and covariates. Our
framework is flexible in the sense that it can accommodate a variety of
regression setups and DRO ambiguity sets. We investigate asymptotic and finite
sample properties of solutions obtained using Wasserstein, sample robust
optimization, and phi-divergence-based ambiguity sets within our DRO
formulations, and explore cross-validation approaches for sizing these
ambiguity sets. Through numerical experiments, we validate our theoretical
results, study the effectiveness of our approaches for sizing ambiguity sets,
and illustrate the benefits of our DRO formulations in the limited data regime
even when the prediction model is misspecified.
