  We study the behavior of optimal ridge regularization and optimal ridge risk
for out-of-distribution prediction, where the test distribution deviates
arbitrarily from the train distribution. We establish general conditions that
determine the sign of the optimal regularization level under covariate and
regression shifts. These conditions capture the alignment between the
covariance and signal structures in the train and test data and reveal stark
differences compared to the in-distribution setting. For example, a negative
regularization level can be optimal under covariate shift or regression shift,
even when the training features are isotropic or the design is
underparameterized. Furthermore, we prove that the optimally-tuned risk is
monotonic in the data aspect ratio, even in the out-of-distribution setting and
when optimizing over negative regularization levels. In general, our results do
not make any modeling assumptions for the train or the test distributions,
except for moment bounds, and allow for arbitrary shifts and the widest
possible range of (negative) regularization levels.
