This paper proposes an invariant causal predictor that is robust to
distribution shift across domains and maximally reserves the transferable
invariant information. Based on a disentangled causal factorization, we
formulate the distribution shift as soft interventions in the system, which
covers a wide range of cases for distribution shift as we do not make prior
specifications on the causal structure or the intervened variables. Instead of
imposing regularizations to constrain the invariance of the predictor, we
propose to predict by the intervened conditional expectation based on the
do-operator and then prove that it is invariant across domains. More
importantly, we prove that the proposed predictor is the robust predictor that
minimizes the worst-case quadratic loss among the distributions of all domains.
For empirical learning, we propose an intuitive and flexible estimating method
based on data regeneration and present a local causal discovery procedure to
guide the regeneration step. The key idea is to regenerate data such that the
regenerated distribution is compatible with the intervened graph, which allows
us to incorporate standard supervised learning methods with the regenerated
data. Experimental results on both synthetic and real data demonstrate the
efficacy of our predictor in improving the predictive accuracy and robustness
across domains