Maximum Likelihood Estimation Is All You Need for Well-Specified Covariate Shift

Abstract

A key challenge of modern machine learning systems is to achieve Out-of-Distribution (OOD) generalization---generalizing to target data whose distribution differs from that of source data. Despite its significant importance, the fundamental question of ``what are the most effective algorithms for OOD generalization'' remains open even under the standard setting of covariate shift. This paper addresses this fundamental question by proving that, surprisingly, classical Maximum Likelihood Estimation (MLE) purely using source data (without any modification) achieves the *minimax* optimality for covariate shift under the *well-specified* setting. This result holds for a very large class of parametric models, including but not limited to linear regression, logistic regression, and phase retrieval, and does not require any boundedness condition on the density ratio. This paper further complement the study by proving that for the *misspecified setting*, MLE can perform poorly, and the Maximum Weighted Likelihood Estimator (MWLE) emerges as minimax optimal in specific scenarios, outperforming MLE.