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Robins et al. (2008) introduced a class of influence functions (IFs) whichcould be used to obtain doubly robust moment functions for the correspondingparameters. However, that class does not include the IF of parameters for whichthe nuisance functions are solutions to integral equations. Such parameters areparticularly important in the field of causal inference, specifically in therecently proposed proximal causal inference framework of Tchetgen Tchetgen etal. (2020), which allows for estimating the causal effect in the presence oflatent confounders. In this paper, we first extend the class of Robins et al.to include doubly robust IFs in which the nuisance functions are solutions tointegral equations. Then we demonstrate that the double robustness property ofthese IFs can be leveraged to construct estimating equations for the nuisancefunctions, which enables us to solve the integral equations without resortingto parametric models. We frame the estimation of the nuisance functions as aminimax optimization problem. We provide convergence rates for the nuisancefunctions and conditions required for asymptotic linearity of the estimator ofthe parameter of interest. The experiment results demonstrate that our proposedmethodology leads to robust and high-performance estimators for average causaleffect in the proximal causal inference framework.

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