  It has become increasingly common nowadays to collect observations of feature
and response pairs from different environments. As a consequence, one has to
apply learned predictors to data with a different distribution due to
distribution shifts. One principled approach is to adopt the structural causal
models to describe training and test models, following the invariance principle
which says that the conditional distribution of the response given its
predictors remains the same across environments. However, this principle might
be violated in practical settings when the response is intervened. A natural
question is whether it is still possible to identify other forms of invariance
to facilitate prediction in unseen environments. To shed light on this
challenging scenario, we focus on linear structural causal models (SCMs) and
introduce invariant matching property (IMP), an explicit relation to capture
interventions through an additional feature, leading to an alternative form of
invariance that enables a unified treatment of general interventions on the
response as well as the predictors. We analyze the asymptotic generalization
errors of our method under both the discrete and continuous environment
settings, where the continuous case is handled by relating it to the
semiparametric varying coefficient models. We present algorithms that show
competitive performance compared to existing methods over various experimental
settings including a COVID dataset.
