Optimal Ridge Regularization for Out-of-Distribution Prediction

TL;DR

This paper investigates the behavior of optimal ridge regularization in out-of-distribution prediction, revealing conditions under which negative regularization can be optimal and how risk varies with data aspect ratio, without strong modeling assumptions.

Contribution

It provides a theoretical analysis of optimal ridge regularization under arbitrary distribution shifts, including conditions for negative regularization and monotonic risk behavior.

Findings

Negative regularization can be optimal under certain shifts.

Optimal risk is monotonic in data aspect ratio.

Results hold under minimal assumptions, allowing arbitrary shifts.

Abstract

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.