  We study few-shot supervised domain adaptation (DA) for regression problems,
where only a few labeled target domain data and many labeled source domain data
are available. Many of the current DA methods base their transfer assumptions
on either parametrized distribution shift or apparent distribution
similarities, e.g., identical conditionals or small distributional
discrepancies. However, these assumptions may preclude the possibility of
adaptation from intricately shifted and apparently very different
distributions. To overcome this problem, we propose mechanism transfer, a
meta-distributional scenario in which a data generating mechanism is invariant
among domains. This transfer assumption can accommodate nonparametric shifts
resulting in apparently different distributions while providing a solid
statistical basis for DA. We take the structural equations in causal modeling
as an example and propose a novel DA method, which is shown to be useful both
theoretically and experimentally. Our method can be seen as the first attempt
to fully leverage the structural causal models for DA.
