  This paper establishes the generalization error of pooled min-$\ell_2$-norm
interpolation in transfer learning where data from diverse distributions are
available. Min-norm interpolators emerge naturally as implicit regularized
limits of modern machine learning algorithms. Previous work characterized their
out-of-distribution risk when samples from the test distribution are
unavailable during training. However, in many applications, a limited amount of
test data may be available during training, yet properties of min-norm
interpolation in this setting are not well-understood. We address this gap by
characterizing the bias and variance of pooled min-$\ell_2$-norm interpolation
under covariate and model shifts. The pooled interpolator captures both early
fusion and a form of intermediate fusion. Our results have several
implications: under model shift, for low signal-to-noise ratio (SNR), adding
data always hurts. For higher SNR, transfer learning helps as long as the
shift-to-signal (SSR) ratio lies below a threshold that we characterize
explicitly. By consistently estimating these ratios, we provide a data-driven
method to determine: (i) when the pooled interpolator outperforms the
target-based interpolator, and (ii) the optimal number of target samples that
minimizes the generalization error. Under covariate shift, if the source sample
size is small relative to the dimension, heterogeneity between between domains
improves the risk, and vice versa. We establish a novel anisotropic local law
to achieve these characterizations, which may be of independent interest in
random matrix theory. We supplement our theoretical characterizations with
comprehensive simulations that demonstrate the finite-sample efficacy of our
results.
