Nonparametric Instrumental Variable Regression through Stochastic Approximate Gradients

Causal inference from observational data presents unique challenges, primarily due to the potential for confounding variables that can affect both outcomes and variables of interest. The unconfoundedness assumption, crucial in this context, posits that all confounding variables are observed and properly accounted for, allowing for an unbiased estimation of causal effects. However, in many real-world scenarios, this assumption is difficult to satisfy. When this is the case, approaches that rely on instrumental variables (IVs) -- quantities that are correlated with the variable of interest (relevance condition), do not affect the outcome in any other way (exclusion condition) and are independent of the unobservable confounders -- offer a way to identify causal effects even when unobserved confounders exist. Moreover, as traditional parametric models often require assumptions about the relationship between variables that may not hold in practice, nonparametric IV (NPIV) models can adapt to the intrinsic structure of the data, allowing for a more nuanced understanding of causal relationships. There has been a recent boost of new algorithms applied to the NPIV estimation problem and its theoretical properties. The challenge is that NPIV estimation is an ill-posed inverse problem (Newey and Powell, 2003; Carrasco et al., 2007; Cavalier, 2011), and recent methods aim to incorporate developments from predictive models, e.g., deep learning, while also accounting for the particular structure of the inverse problem at hand. In this work, we present a novel framework for NPIV estimation that relies on stochastic approximate gradients, allowing it to seamlessly incorporate a variety of machine learning methods, such as those based on Reproducing Kernel Hilbert Space (RKHS) and deep learning. Moreover, we demonstrate, under minimal assumptions, finite sample guarantees for the projected populational risk for both continuous and binary responses.