Distribution Dynamics in Stochastic Optimization: A Decision-Dependent Formulation

Distribution shifts have long been regarded as troublesome external forces that a decision-maker should counteract or conform to. An intriguing feedback phenomenon termed decision dependence arises when the deployed decision affects the environment and alters the data-generating distribution. In the realm of performative prediction, this is encoded by distribution maps parameterized by decisions due to strategic behaviors. In contrast, we formalize an endogenous distribution shift as a feedback process featuring nonlinear dynamics that couple the evolving distribution with the decision. This formulation provides a fertile ground to examine the various roles played by dynamics in the composite problem structure. To this end, we develop an online algorithm that achieves optimal decision-making by both adapting to and shaping the dynamic distribution. We adopt a distributional perspective and demonstrate how this view facilitates characterizations of distribution dynamics and the optimality of the proposed algorithm. We showcase the theoretical results in a recommender system example of affinity maximization with polarized population dynamics.