  Transfer learning is a critical part of real-world machine learning
deployments and has been extensively studied in experimental works with
overparameterized neural networks. However, even in the simplest setting of
linear regression a notable gap still exists in the theoretical understanding
of transfer learning. In-distribution research on high-dimensional linear
regression has led to the identification of a phenomenon known as
\textit{benign overfitting}, in which linear interpolators overfit to noisy
training labels and yet still generalize well. This behavior occurs under
specific conditions on the source covariance matrix and input data dimension.
Therefore, it is natural to wonder how such high-dimensional linear models
behave under transfer learning. We prove the first non-asymptotic excess risk
bounds for benignly-overfit linear interpolators in the transfer learning
setting. From our analysis, we propose a taxonomy of \textit{beneficial} and
\textit{malignant} covariate shifts based on the degree of
overparameterization. We follow our analysis with empirical studies that show
these beneficial and malignant covariate shifts for linear interpolators on
real image data, and for fully-connected neural networks in settings where the
input data dimension is larger than the training sample size.
