Generative Models for Decision Making Under Distributional Shift
Abstract
Many data-driven decision problems are formulated using a nominal distribution estimated from historical data, whereas performance is ultimately determined by a deployment distribution that may be shifted, context dependent, partially observed, or stress induced. This tutorial presents modern generative models, particularly flow- and score-based methods, as mathematical tools for constructing decision-relevant distributions. From an operations research perspective, their primary value lies not in unconstrained sample synthesis but in representing and transforming distributions through transport maps, velocity fields, score fields, and guided stochastic dynamics. We present a unified framework based on pushforward maps, continuity, Fokker-Planck equations, Wasserstein geometry, and optimization in probability space. Within this framework, generative models can be used to learn nominal uncertainty, construct stressed or least-favorable distributions for robustness, and produce conditional or posterior distributions under side information and partial observation. We also highlight representative theoretical guarantees, including forward-reverse convergence for iterative flow models, first-order minimax analysis in transport-map space, and error-transfer bounds for posterior sampling with generative priors. The tutorial provides a principled introduction to using generative models for scenario generation, robust decision making, uncertainty quantification, and related problems under distributional shift.
Funding: The work of Y. Zhu and Y. Xie was partially supported by the National Science Foundation (NSF) [Grant CMMI-2112533] and the Coca-Cola Foundation. Y. Xie and X. Cheng were partially supported by the NSF [Grant DMS-2134037]. X. Cheng was also partially supported by the NSF [Grants DMS-2237842 and DMS-2031849] and the Simons Foundation [Grant MPS-MODL-00814643].
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