Optimal Ridge Regularization for Out-of-Distribution Prediction

Oregon Convention Center 

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 alignment between the covariance and signal structures in the train and test data and reveal stark differences compared to the in-distribution setting (where the test and train distributions agree); for example, a negative regularization level can be optimal under covariate shift, even when the training features are isotropic. Furthermore, we prove that the optimally-tuned risk is monotonic in the data aspect ratio, even in the out-of-distribution setting. 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 level.

Ridge regression

Optimal regularization

Distribution shift

Covariate shift

Regression shift

Risk monotonicity 

IMS