Sharp Structure-Agnostic Lower Bounds for General Functional Estimation

The design of efficient nonparametric estimators has long been a central problem in statistics, machine learning, and decision making. Classical optimal procedures often rely on strong structural assumptions, which can be misspecified in practice and complicate deployment. This limitation has sparked growing interest in structure-agnostic approaches -- methods that debias black-box nuisance estimates without imposing structural priors. Understanding the fundamental limits of these methods is therefore crucial. This paper provides a systematic investigation of the optimal error rates achievable by structure-agnostic estimators. We first show that, for estimating the average treatment effect (ATE), a central parameter in causal inference, doubly robust learning attains optimal structure-agnostic error rates. We then extend our analysis to a general class of functionals that depend on unknown nuisance functions and establish the structure-agnostic optimality of debiased/double machine learning (DML). We distinguish two regimes -- one where double robustness is attainable and one where it is not -- leading to different optimal rates for first-order debiasing, and show that DML is optimal in both regimes. Finally, we instantiate our general lower bounds by deriving explicit optimal rates that recover existing results and extend to additional estimands of interest. Our results provide theoretical validation for widely used first-order debiasing methods and guidance for practitioners seeking optimal approaches in the absence of structural assumptions. This paper generalizes and subsumes the ATE lower bound established in \citet{jin2024structure} by the same authors.