  Federated Domain Adaptation (FDA) describes the federated learning (FL)
setting where source clients and a server work collaboratively to improve the
performance of a target client where limited data is available. The domain
shift between the source and target domains, coupled with limited data of the
target client, makes FDA a challenging problem, e.g., common techniques such as
federated averaging and fine-tuning fail due to domain shift and data scarcity.
To theoretically understand the problem, we introduce new metrics that
characterize the FDA setting and a theoretical framework with novel theorems
for analyzing the performance of server aggregation rules. Further, we propose
a novel lightweight aggregation rule, Federated Gradient Projection
($\texttt{FedGP}$), which significantly improves the target performance with
domain shift and data scarcity. Moreover, our theory suggests an
$\textit{auto-weighting scheme}$ that finds the optimal combinations of the
source and target gradients. This scheme improves both $\texttt{FedGP}$ and a
simpler heuristic aggregation rule. Extensive experiments verify the
theoretical insights and illustrate the effectiveness of the proposed methods
in practice.
