Loss-Aware Selection of Dependence Information in Distributionally Robust Optimization

Distributionally robust optimization (DRO) is often formulated using partial distributional information, such as marginals or moments, rather than a full joint distribution. While this reduces statistical and computational burden, it leaves dependence directions that an adversary can exploit without changing the specified marginals. We study how to select additional dependence information that is relevant to the robust objective. In a finite-scenario DRO model, singleton and interaction views constrain overlapping marginal summaries of a common adversarial distribution. We characterize the reweighting directions left unidentified by a collection of views through the nullspace of the stacked view operator. For each candidate interaction, we derive two exact one-step criteria: a rank score measuring the number of unidentified directions removed and a loss-targeted score measuring the unidentified loss variation removed. We use the loss-targeted score in a greedy procedure that augments the ambiguity set with selected interaction views. For positive ambiguity radii, we derive an exact finite-dimensional dual that decomposes the robust objective into variation represented by the selected views and residual variation outside their span. Soft marginal views provide a finite-dimensional implementation for continuous and high-dimensional features. In controlled dependence-shift experiments, the loss-targeted rule recovers the relevant interactions; on the real-world CMU-MOSEI dataset, the selected interaction matches the oracle.