Decision-Dependent Stochastic Optimization: The Role of Distribution Dynamics

2025-03-10

AI Summary

This paper addresses the dynamic evolution of data distributions induced endogenously by decisions in decision-dependent stochastic optimization—challenging the conventional assumption of exogenous distributional shift. We formulate distribution evolution as a decision-driven nonlinear dynamical system, the first such model in the literature. We propose an online optimization framework that jointly optimizes decisions and steers distribution dynamics, designing an adaptive and shaping algorithm operating on the probability simplex to unify optimality and generalization under non-stationarity. We establish theoretical convergence guarantees and derive a generalization error bound. Empirical validation on opinion polarization control and recommender systems demonstrates the method’s ability to actively guide distribution evolution while simultaneously achieving near-optimal decisions. Our core contributions are: (i) a principled endogenous distributional dynamics model grounded in decision feedback, and (ii) the first provably optimal and generalizable decision-distribution co-optimization framework.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Economics and fairness of platforms and recommendation systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Distribution shifts have long been regarded as troublesome external forces that a decision-maker should either 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. Stochastic optimization in this dynamic regime 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. Throughout the paper, we adopt a distributional perspective and demonstrate how this view facilitates characterizations of distribution dynamics and the optimality and generalization performance of the proposed algorithm. We showcase the theoretical results in an opinion dynamics context, where an opportunistic party maximizes the affinity of a dynamic polarized population, and in a recommender system scenario, featuring performance optimization with discrete distributions in the probability simplex.

Problem

• Addresses decision-dependent distribution shifts in stochastic optimization.
• Develops an online algorithm for optimal decision-making in dynamic environments.
• Explores feedback processes coupling decisions with evolving data distributions.

Innovation

• Online algorithm adapts to dynamic distribution shifts
• Nonlinear dynamics couple decisions with evolving distributions
• Stochastic optimization in decision-dependent distribution dynamics