Tighter joint Wasserstein ambiguity sets exploiting dependence

Develop joint ambiguity sets for the lifted system–environment distribution that are tighter than the additive-radius construction \(\rho_y=\rho_x+\rho_z\) by exploiting additional structure or statistical dependence between the system state process and the environment process, while preserving the distributionally robust safety guarantees.

Background

The paper constructs a joint ambiguity radius for the lifted state by summing the system-side and environment-side Wasserstein radii, justified by worst-case coupling and triangle inequality arguments under independence. This leads to a conservative ambiguity set.

The authors point out that leveraging dependence or additional structure between the system and environment could yield tighter (less conservative) ambiguity sets, improving performance without sacrificing robustness. Formal methods to build and certify such refined sets remain to be developed.

References

The additive structure of $\rho_y$ arises from worst-case coupling arguments and the triangle inequality for the Wasserstein distance, and ensures robustness under independent uncertainties. Tighter ambiguity sets may be obtained by exploiting additional structure or dependence between $X_t$ and $Z_t$, which we leave for future work.

— $\mathcal{L}_1$-Certified Distributionally Robust Planning for Safety-Constrained Adaptive Control  (2603.28758 - Hakobyan et al., 30 Mar 2026) in Remark after Theorem “Closed-Loop Safety,” Section 3.3 “Distributional Certificates and Closed-Loop Safety”