Hedging against Complexity: Distributionally Robust Optimization with Parametric Approximation

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 uncertain parameters; consequently, the performance of these methods is poor for high-dimensional problems with objective functions under high complexity. We propose a simple approach in which the distribution of uncertain parameters 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 / DRO methods and parametric ERM for a wide variety of settings. Our method is particularly effective under distribution shifts. We also illustrate the superior performance of our approach on both synthetic and real-data portfolio optimization and regression tasks.