  Empirical risk minimization (ERM) and distributionally robust optimization
(DRO) are popular approaches for solving stochastic optimization problems that
appear in operations management and machine learning. Existing generalization
error bounds for these methods depend on either the complexity of the cost
function or dimension of the random perturbations. Consequently, the
performance of these methods can be poor for high-dimensional problems with
complex objective functions. We propose a simple approach in which the
distribution of random perturbations is approximated using a parametric family
of distributions. This mitigates both sources of complexity; however, it
introduces a model misspecification error. We show that this new source of
error can be controlled by suitable DRO formulations. Our proposed parametric
DRO approach has significantly improved generalization bounds over existing ERM
and DRO methods and parametric ERM for a wide variety of settings. Our method
is particularly effective under distribution shifts and works broadly in
contextual optimization. We also illustrate the superior performance of our
approach on both synthetic and real-data portfolio optimization and regression
tasks.
