  In real-world applications, it is important and desirable to learn a model
that performs well on out-of-distribution (OOD) data. Recently, causality has
become a powerful tool to tackle the OOD generalization problem, with the idea
resting on the causal mechanism that is invariant across domains of interest.
To leverage the generally unknown causal mechanism, existing works assume a
linear form of causal feature or require sufficiently many and diverse training
domains, which are usually restrictive in practice. In this work, we obviate
these assumptions and tackle the OOD problem without explicitly recovering the
causal feature. Our approach is based on transformations that modify the
non-causal feature but leave the causal part unchanged, which can be either
obtained from prior knowledge or learned from the training data in the
multi-domain scenario. Under the setting of invariant causal mechanism, we
theoretically show that if all such transformations are available, then we can
learn a minimax optimal model across the domains using only single domain data.
Noticing that knowing a complete set of these causal invariant transformations
may be impractical, we further show that it suffices to know only a subset of
these transformations. Based on the theoretical findings, a regularized
training procedure is proposed to improve the OOD generalization capability.
Extensive experimental results on both synthetic and real datasets verify the
effectiveness of the proposed algorithm, even with only a few causal invariant
transformations.
