Learning in Motion: Online Optimization under Distributional Drift via Wasserstein Geometry

Abstract

In dynamic learning environments, the assumption of a fixed data distribution often fails, especially in applications such as online recommendation systems, adaptive control, and financial forecasting. This paper presents a novel framework for online optimization in the presence of distributional drift, where the underlying data distribution evolves over time. By leveraging the geometry of the Wasserstein space, we introduce a principled approach to quantify and adapt to these shifts. We propose a Wasserstein-Proximal learning algorithm that adjusts to the evolving landscape using transport-based regularization, and we establish tight regret bounds under various smoothness and convexity assumptions. Empirical results on both synthetic and real-world data confirm that incorporating Wasserstein drift leads to significantly improved performance in non-stationary environments. Our findings bridge the gap between dynamic regret minimization and distributionally robust optimization, offering new insights for adaptive learning under uncertainty.