  Real-world machine learning applications often involve deploying neural
networks to domains that are not seen in the training time. Hence, we need to
understand the extrapolation of nonlinear models -- under what conditions on
the distributions and function class, models can be guaranteed to extrapolate
to new test distributions. The question is very challenging because even
two-layer neural networks cannot be guaranteed to extrapolate outside the
support of the training distribution without further assumptions on the domain
shift. This paper makes some initial steps toward analyzing the extrapolation
of nonlinear models for structured domain shift. We primarily consider settings
where the marginal distribution of each coordinate of the data (or subset of
coordinates) does not shift significantly across the training and test
distributions, but the joint distribution may have a much bigger shift. We
prove that the family of nonlinear models of the form $f(x)=\sum f_i(x_i)$,
where $f_i$ is an arbitrary function on the subset of features $x_i$, can
extrapolate to unseen distributions, if the covariance of the features is
well-conditioned. To the best of our knowledge, this is the first result that
goes beyond linear models and the bounded density ratio assumption, even though
the assumptions on the distribution shift and function class are stylized.
