Learning to Optimize with Distributional Shift Constraints: A Novel Framework for Safe Domain Adaptation in Machine Learning

Domain adaptation remains a significant challenge in machine learning, particularly in real-world applications where distributional shifts between training and testing domains can lead to drastic performance degradation. This paper presents a novel framework that integrates explicit constraints for worst-case distributional shifts into the empirical risk minimization process, employing integral probability metrics. By characterizing the optimization problem through optimal transport theory, we derive a mathematically rigorous solution that ensures robustness and safety in the adaptation process. Our approach is scrutinized through comprehensive theoretical analysis, establishing guarantees for the performance of adapted models. Extensive experimental evaluations on synthetic and real-world datasets demonstrate the framework's efficacy, revealing substantial improvements over existing domain adaptation methods in scenarios with severe distributional shifts. We underscore the importance of robust domain adaptation methodologies in fostering trust in AI systems deployed across sensitive domains. The proposed framework not only enhances the reliability of machine learning models but also paves the way for future research addressing the complexities of distributional uncertainty.