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 plug-in estimators under surprisingly weak conditions. These first-order corrections improve on plug-in estimators in a black-box fashion, and consequently are often used in conjunction with powerful off-the-shelf estimation methods. On the other hand, these first-order methods are provably suboptimal in a minimax sense for functional estimation when the nuisance functions live in Hölder-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 Hölder-type spaces, but in sharp contrast to first-order estimators, both the estimators which are minimax-optimal and their analyses are crucially tied to properties of the underlying function space. Along a similar vein, some work has considered n-consistent estimation of causal effects under weaker conditions than those required by first-order methods, once again relying on higher-order debiasing. More recent work in this area has focused on attempting to weaken the dependence of these higher-order estimators on the underlying nuisance function spaces, to make the resulting estimators and theory more robust. A central focus has been to try to make higher-order methods compatible with black-box nuisance estimators. 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. Particularly, we show that for several canonical integral functionals of interest it is impossible to improve on first-order estimators without making further, strong structural assumptions. 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 a better understanding of the structure of the underlying nuisance functions then carefully constructed higher-order estimators can outperform first-order estimators.
Balakrishnan et al. (Fri,) studied this question.