Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference

A moment function is called doubly robust if it is comprised of two nuisance functions and has the desired property that the estimator based on it is a consistent estimator of the target parameter even if one of the nuisance functions is misspecified. A common approach for obtaining such a moment function is based on using the influence function (IF) of the parameter of interest. Robins et al. (2008) introduced a large class of doubly robust IFs. However, that class does not include the IF of functionals for which the nuisance functions are solutions to integral equations. Such functionals are particularly important in the field of causal inference, specifically in the recently proposed proximal inference framework of (Miao et al., 2018; Tchetgen Tchetgen et al., 2020), which allows for estimating the average causal effect when unobserved confounders are present in the system. Motivated by the proximal inference framework, in this paper, we first extend the class of Robins et al. to include doubly robust IFs in which the nuisance functions are solutions to integral equations. Then we demonstrate that the double robustness property of these IFs can be leveraged to construct estimating equations for the nuisance functions, which enables us to solve the integral equations without resorting to parametric models. The main idea is to choose each nuisance function such that it minimizes the dependency of the expected value of the moment function to the other nuisance function. We frame this idea as a minimax optimization problem and use RKHSes as the function spaces. We provide convergence rates for the nuisance functions and conditions required for asymptotic linearity of the estimator of the functional of interest. The experiment results demonstrate that our proposed methodology leads to robust and high-performance estimators for average causal effect in the proximal inference framework.