Instrumental Variable Regression via Kernel Maximum Moment Loss
This work addresses instrumental variable regression for causal inference, offering a novel method with theoretical guarantees and practical algorithms, though it appears incremental in the context of existing kernel-based approaches.
The authors tackled nonlinear instrumental variable regression by proposing a kernel maximum moment loss objective, which simplifies the problem to empirical risk minimization and provides consistency and asymptotic normality guarantees. They demonstrated the effectiveness of their algorithms on synthetic and real-world data.
We investigate a simple objective for nonlinear instrumental variable (IV) regression based on a kernelized conditional moment restriction (CMR) known as a maximum moment restriction (MMR). The MMR objective is formulated by maximizing the interaction between the residual and the instruments belonging to a unit ball in a reproducing kernel Hilbert space (RKHS). First, it allows us to simplify the IV regression as an empirical risk minimization problem, where the risk functional depends on the reproducing kernel on the instrument and can be estimated by a U-statistic or V-statistic. Second, based on this simplification, we are able to provide the consistency and asymptotic normality results in both parametric and nonparametric settings. Lastly, we provide easy-to-use IV regression algorithms with an efficient hyper-parameter selection procedure. We demonstrate the effectiveness of our algorithms using experiments on both synthetic and real-world data.