In this work, we address this gap by introducing a novel form of conditional cross-fitting adapted to the dependence arising in survey sampling. Leveraging this approach, we develop a unified theory for the asymptotic behavior of model-assisted estimators based on general prediction methods. In particular, we establish the asymptotic normality of the proposed estimators and the consistency of a corresponding cross-fitting variance estimator, thereby enabling the construction of asymptotically valid confidence intervals. The proposed approach is shown to be asymptotically optimal in that it reaches the Godambe-Joshi lower bound. Simulation studies further demonstrate the favorable finite-sample performance of the proposed methodology.

Under holomorphic regularity assumptions on the target, we identify classes of functions and operators for which the OOD generalization error converges at algebraic rates. This phenomenon stems from the increasing smoothness of higher-index coordinates, leading to what we term a `blessing of high dimensionality’ We derive explicit rates for arbitrary test measures supported on suitable domains and quantify how the admissible domain depends on the underlying regularity. Our extrapolation guarantees are independent of the test distribution, depending only on its support. We establish matching rates for polynomial approximation, deep neural networks, and deep neural operators. In addition, we present extensive numerical experiments across a range of functions and operators that corroborate the predicted rates.