Convergence properties of multi-environment causal regularization

Causal regularization was introduced as a stable causal inference strategy in a two-environment setting in Kania and Wit [2022] for linear structural equations models (SEMs). We start with observing that causal regularizer can be extended to several shifted environments and non-linear SEMs. We derive the multi-environment casual regularizer in the population setting. We propose its plug-in estimator, and study its concentration in measure behavior. Although the variance of the plug-in estimator is not well-defined in general, we instead study its conditional variance both with respect to a natural filtration of the empirical as well as conditioning with respect to certain events. We also study generalizations where we consider conditional expectations of higher central absolute moments of the estimator. The results presented here are also new in the prior setting of Kania and Wit [2022] as well as in Rothenhausler et al. [2021].