Estimators independent of the test-support bound

Construct estimators independent of the extrapolation-domain parameter \(\boldsymbol{\omega}\) that achieve the same out-of-distribution generalization bounds as the estimators depending on \(\boldsymbol{\omega}\), thereby explaining the domain-generalization behavior observed empirically.

Background

The theoretical polynomial, deep neural network, and deep neural operator estimators depend on ω\boldsymbol{\omega}, which specifies a hyperrectangle containing the support of the test distribution. This places the theory in a domain-adaptation setting because information about the test distribution is used when constructing the estimator.

By contrast, the numerical estimators do not use ω\boldsymbol{\omega} and nevertheless exhibit algebraic out-of-distribution convergence. The unresolved problem is therefore to obtain comparable guarantees for estimators that have no prior knowledge of the test-support parameter.

References

An important question is whether one can construct estimators independent of $\bm{\omega}$ that yield the same generalization bounds.. This would provide a theoretical explanation for the domain generalization behaviour observed in practice.

— Into the danger zone: stable extrapolation in high-dimensional function and operator learning  (2609.36709 - Adcock et al., 29 Sep 2026) in Section 7, subsection “From adaptation to generalization”; Remark “Unknown supports”

Currently, it is unknown whether the algebraic rates in the setting of Example \ref{ex:unbounded-side lengths} are also optimal.

— Into the danger zone: stable extrapolation in high-dimensional function and operator learning  (2609.36709 - Adcock et al., 29 Sep 2026) in Remark “Optimality of the rates,” Section 4.3; Section 7, subsection “Weaker conditions and sharper rates”