  We study the fundamental problem of transfer learning where a learning
algorithm collects data from some source distribution $P$ but needs to perform
well with respect to a different target distribution $Q$. A standard change of
measure argument implies that transfer learning happens when the density ratio
$dQ/dP$ is bounded. Yet, prior thought-provoking works by Kpotufe and Martinet
(COLT, 2018) and Hanneke and Kpotufe (NeurIPS, 2019) demonstrate cases where
the ratio $dQ/dP$ is unbounded, but transfer learning is possible.
  In this work, we focus on transfer learning over the class of low-degree
polynomial estimators. Our main result is a general transfer inequality over
the domain $\mathbb{R}^n$, proving that non-trivial transfer learning for
low-degree polynomials is possible under very mild assumptions, going well
beyond the classical assumption that $dQ/dP$ is bounded. For instance, it
always applies if $Q$ is a log-concave measure and the inverse ratio $dP/dQ$ is
bounded. To demonstrate the applicability of our inequality, we obtain new
results in the settings of: (1) the classical truncated regression setting,
where $dQ/dP$ equals infinity, and (2) the more recent out-of-distribution
generalization setting for in-context learning linear functions with
transformers. We also provide a discrete analogue of our transfer inequality on
the Boolean Hypercube $\{-1,1\}^n$, and study its connections with the recent
problem of Generalization on the Unseen of Abbe, Bengio, Lotfi and Rizk (ICML,
2023). Our main conceptual contribution is that the maximum influence of the
error of the estimator $\widehat{f}-f^*$ under $Q$,
$\mathrm{I}_{\max}(\widehat{f}-f^*)$, acts as a sufficient condition for
transferability; when $\mathrm{I}_{\max}(\widehat{f}-f^*)$ is appropriately
bounded, transfer is possible over the Boolean domain.
