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

Abstract

A moment function is called doubly robust if it is comprised of two nuisance functions and the estimator based on it is a consistent estimator of the target parameter even if one of the nuisance functions is misspecified. In this paper, we consider a class of doubly robust moment functions originally introduced in (Robins et al., 2008). We demonstrate that this moment function can be used to construct estimating equations for the nuisance functions. 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 implement this idea as a minimax optimization problem. We then provide conditions required for asymptotic linearity of the estimator of the parameter of interest, which are based on the convergence rate of the product of the errors of the nuisance functions, as well as the local ill-posedness of a conditional expectation operator. The convergence rates of the nuisance functions are analyzed using the modern techniques in statistical learning theory based on the Rademacher complexity of the function spaces. We specifically focus on the case that the function spaces are reproducing kernel Hilbert spaces, which enables us to use its spectral properties to analyze the convergence rates. As an application of the proposed methodology, we consider the parameter of average causal effect both in presence and absence of latent confounders. For the case of presence of latent confounders, we use the recently proposed proximal causal inference framework of Tchetgen Tchetgen et al., 2020), and hence our results lead to a robust non-parametric estimator for average causal effect in this framework.

Proximal mediation analysis

“…A complication in the proximal learning framework is that the nuisances are defined as the solutions to integral equations. Progress in this direction (in the context of reproducing kernel Hilbert spaces) is described in Ghassami et al (2021) and Kallus et al (2021). It would thus be useful to extend the work of Ghassami et al (2021) for nonparametric estimation of bridge functions in proximal mediation analysis.…”

Identification and Estimation of Causal Peer Effects Using Double Negative Controls for Unmeasured Network Confounding

“…This will be able to extend previous studies that consider double negative control adjustment of unmeasured confounding in i.i.d. or panel data settings (Deaner, 2018;Shi et al, 2020;Cui et al, 2020;Tchetgen Tchetgen et al, 2020a;Ghassami et al, 2021) and methods that examine network effects without unmeasured network confounding (van der Laan, 2014; Ogburn et al, 2017;Forastiere et al, 2020;Ogburn et al, 2020;Tchetgen Tchetgen et al, 2020b). Another interesting open question is identification and estimation of causal peer effects in complex longitudinal studies with time-varying treatments (e.g., Robins et al, 2000;Tchetgen Tchetgen et al, 2020a).…”

A Finite Sample Theorem for Longitudinal Causal Inference with Machine Learning: Long Term, Dynamic, and Mediated Effects

“…Despite the additional complexity of the longitudinal setting, I recover equally strong results. In particular, [54,52,14,28,101,35,31,51,29,2] study static local parameters and [81,50,40,29] study static proximal parameters. Like the latter, I arrive at mean square rates and projected mean square rates of confounding bridges (also called nonparametric instrumental variable regressions).…”