Differentially Private Two-Stage Gradient Descent for Instrumental Variable Regression

Instrumental variable regression (IV aR) is a key tool in causal inference, designed to recover structural parameters when standard estimators fail due to endogeneity. In many observational settings, covariates are influenced by unobserved confounders, causing naive methods (such as the ordinary least squares (OLS) in the context of linear regression) to produce biased and inconsistent estimates. IV aR circumvents this by leveraging instruments, which are variables that are predictive of the endogenous regressors but independent of hidden confounders, to enable consistent estimation of causal effects [Hausman, 2001, Wooldridge, 2010, Angrist and Krueger, 2001]. This perspective is increasingly important in machine learning, for example in recommendation systems where user exposure is confounded by prior preferences [Si et al., 2022], or in reinforcement learning where actions and rewards are jointly influenced by unobserved context [Xu et al., 2023]. In such settings, IV aR provides a principled way to disentangle causal effects from spurious correlations, enabling more reliable decision making. However, many applications of IV aR involve sensitive data, such as individual health records, financial transactions, or user interactions, where protecting privacy is of paramount importance.