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We propose a novel framework for analyzing the dynamics of distribution shiftin real-world systems that captures the feedback loop between learningalgorithms and the distributions on which they are deployed. Prior work largelymodels feedback-induced distribution shift as adversarial or via an overlysimplistic distribution-shift structure. In contrast, we propose a coupledpartial differential equation model that captures fine-grained changes in thedistribution over time by accounting for complex dynamics that arise due tostrategic responses to algorithmic decision-making, non-local endogenouspopulation interactions, and other exogenous sources of distribution shift. Weconsider two common settings in machine learning: cooperative settings withinformation asymmetries, and competitive settings where a learner facesstrategic users. For both of these settings, when the algorithm retrains viagradient descent, we prove asymptotic convergence of the retraining procedureto a steady-state, both in finite and in infinite dimensions, obtainingexplicit rates in terms of the model parameters. To do so we derive new resultson the convergence of coupled PDEs that extends what is known on multi-speciessystems. Empirically, we show that our approach captures well-documented formsof distribution shifts like polarization and disparate impacts that simplermodels cannot capture.

Strategic Distribution Shift of Interacting Agents via Coupled Gradient Flows