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

Robins et al. (2008) introduced a class of influence functions (IFs) which\ncould be used to obtain doubly robust moment functions for the corresponding\nparameters. However, that class does not include the IF of parameters for which\nthe nuisance functions are solutions to integral equations. Such parameters are\nparticularly important in the field of causal inference, specifically in the\nrecently proposed proximal causal inference framework of Tchetgen Tchetgen et\nal. (2020), which allows for estimating the causal effect in the presence of\nlatent confounders. In this paper, we first extend the class of Robins et al.\nto include doubly robust IFs in which the nuisance functions are solutions to\nintegral equations. Then we demonstrate that the double robustness property of\nthese IFs can be leveraged to construct estimating equations for the nuisance\nfunctions, which enables us to solve the integral equations without resorting\nto parametric models. We frame the estimation of the nuisance functions as a\nminimax optimization problem. We provide convergence rates for the nuisance\nfunctions and conditions required for asymptotic linearity of the estimator of\nthe parameter of interest. The experiment results demonstrate that our proposed\nmethodology leads to robust and high-performance estimators for average causal\neffect in the proximal causal inference framework.\n