🤖 AI Summary
This paper addresses the problem of estimating functionals of an unknown target function under a structure-agnostic setting—where no specific structural assumptions (e.g., Hölder smoothness) are imposed on the nuisance function, and only a generic convergence rate for nuisance estimation is assumed. Methodologically, it introduces the first formal framework for structure-agnostic estimation, operating under three simultaneous constraints: weak regularity conditions, compatibility with general-purpose nuisance estimators, and sample splitting. Theoretically, it establishes, for the first time, the essential optimality of first-order debiased estimators in this setting. Through minimax lower bound analysis, higher-order perturbation theory, and a unified debiasing framework, the paper precisely characterizes the optimal convergence rate and quantifies the fundamental trade-off between incorporating structural priors and improving estimation efficiency. These results provide foundational theoretical support for nonparametric and semiparametric inference.
📝 Abstract
Many recent developments in causal inference, and functional estimation problems more generally, have been motivated by the fact that classical one-step (first-order) debiasing methods, or their more recent sample-split double machine-learning avatars, can outperform plugin estimators under surprisingly weak conditions. These first-order corrections improve on plugin estimators in a black-box fashion, and consequently are often used in conjunction with powerful off-the-shelf estimation methods. These first-order methods are however provably suboptimal in a minimax sense for functional estimation when the nuisance functions live in Holder-type function spaces. This suboptimality of first-order debiasing has motivated the development of"higher-order"debiasing methods. The resulting estimators are, in some cases, provably optimal over Holder-type spaces, but both the estimators which are minimax-optimal and their analyses are crucially tied to properties of the underlying function space. In this paper we investigate the fundamental limits of structure-agnostic functional estimation, where relatively weak conditions are placed on the underlying nuisance functions. We show that there is a strong sense in which existing first-order methods are optimal. We achieve this goal by providing a formalization of the problem of functional estimation with black-box nuisance function estimates, and deriving minimax lower bounds for this problem. Our results highlight some clear tradeoffs in functional estimation -- if we wish to remain agnostic to the underlying nuisance function spaces, impose only high-level rate conditions, and maintain compatibility with black-box nuisance estimators then first-order methods are optimal. When we have an understanding of the structure of the underlying nuisance functions then carefully constructed higher-order estimators can outperform first-order estimators.
