Distributionally Robust Optimization via Targeted Integral Probability Metrics for General Data Processes

Abstract: Distributionally robust optimization (DRO) provides a principled framework for decision-making under distributional uncertainty. Classical data-driven DRO frameworks typically construct ambiguity sets from distributional information, such as moment constraints, divergence neighborhoods, or Wasserstein balls, specified before the downstream loss is considered. We propose a task-aware DRO framework based on targeted integral probability metrics. The ambiguity set is defined directly through the loss functions induced by feasible decisions, thereby controlling the loss discrepancy between an adversarial distribution and a data-driven reference distribution. This construction leads to an expected hinge-constrained formulation that is equivalent to an infinitely constrained loss-discrepancy formulation. It also yields finite-sample guarantees that bypass the ambient curse of dimensionality: whenever an appropriate scalar pointwise concentration inequality is available for the induced loss estimator, the ambiguity radius can be calibrated at the canonical $\widetilde{\mathcal O}(N^{-1/2})$ rate after uniformization over the decision class. As a result, the framework applies broadly to settings including heavier-tailed sub-Weibull losses, Markovian data, outlier-corrupted data, and incomplete data. We derive exact infinite-dimensional dual reformulations, establish out-of-sample and excess-risk guarantees, and develop a conservative Monte Carlo approximation scheme with convergence and suboptimality guarantees. For piecewise affine losses, the sampled problems admit tractable conic reformulations. Numerical experiments in inventory management under heavy-tailed demand and regression with outlier corruption demonstrate strong out-of-sample performance relative to existing approaches.