Structurally Compatible Generative Ambiguity Sets via Exogenous-Space Factorization

Distributionally robust optimization (DRO) critically depends on the ambiguity set used to characterize plausible distribution shifts. Recent generative approaches parameterize adversarial distributions with trainable generative models, enabling flexible shifts beyond the empirical support, but they do not explicitly preserve known causal structure. In contrast, structural causal optimal transport restricts admissible distributions according to a structural causal model (SCM), but is not naturally formulated as a trainable generative adversary. We bridge these perspectives with exoDRO, which constructs generative ambiguity sets in the exogenous space of a bijective SCM. Rather than perturbing the endogenous distribution directly, we independently parameterize the exogenous marginals and push their product distribution through the fixed structural map. This construction guarantees structural compatibility for every adversarial parameter value by design. We further show that the inclusive KL divergence between the nominal and adversarial endogenous distributions decomposes exactly into the sum of the corresponding exogenous marginal KL divergences. This identity holds for any bimeasurable bijective structural map and does not depend on causal depth, Jacobian volume preservation, or additive-noise structure. Combined with likelihood-based generative consistency bounds, this identity yields a global reconstruction-based certificate and finite-sample control using a single dataset shared across all exogenous coordinates. To optimize the resulting coupled adversarial problem, we develop a cyclic block-coordinate scheme whose fixed-block subproblems reduce to single-model generative-DRO inner problems. Experiments verify the predicted structural properties and show that, on the predictive benchmark, exoDRO achieves the lowest prediction error across the evaluated nominal and held-out OOD settings compared with classical, generative, and structure-aware DRO baselines.