Extension of the empirical convergence theorem to unbounded domains

Establish whether the empirical adversarial training convergence theorem extends from bounded connected domains Ω with Lipschitz boundary to the whole Euclidean space R^d, by localizing Wasserstein convergence to regions where the conditional densities are bounded away from zero.

Background

The empirical convergence theorem is proved under the assumption that the conditional densities are supported in a bounded connected domain Ω with Lipschitz boundary and are uniformly bounded below there. These assumptions enable the use of Wasserstein convergence rates for empirical distributions.

The authors indicate that an analogous result on Rd is plausible but unresolved. Such an extension would require additional localization arguments because the densities may approach zero outside the regions where the existing lower-bound assumptions hold.

References

It is likely that Theorem \ref{thm:mainEmpirical} holds in $Rd$ (analogous to the statement of Theorem \ref{thm:main}) with more work spent localizing the Wasserstein convergence to regions where the densities are bounded away from $0$. However, as this would distract from the essential difficulties, we do not pursue this here.

— On the tradeoff between robustness and geometric accuracy in adversarial training  (2610.22955 - Morris et al., 16 Oct 2026) in Section 2, Subsection “Convergence for the empirical adversarial training problem,” Remark “The domain Ω”