Generative models for decision-making under distributional shift
Authors: Xiuyuan Cheng, Yunqin Zhu, Yao Xie
Organizations: Department of Mathematics, Duke University, Durham, NC, USA · H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA
Many data-driven decision problems are formulated using a nominal distribution estimated from historical data, while 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.
Figures & tables
Type of shift
Why P=Q
Distribution needed
Regime shift
Deployment occurs in a new region, season, market, or operating condition
Corrected deployment distribution, often Q=T#P .
Stress shift
The task requires adverse but plausible scenarios beyond typical historical behavior
Stressed or least-favorable distribution.
Contextual shift
Uncertainty changes with observed covariates such as weather, demand signals, or network state
Conditional distribution Q=P(Y∣X=x) .
Partial observation
New noisy measurements reveal information about an unobserved state
Posterior distribution Q=P(⋅∣y) .
Equilibrium shift
The distribution is generated by interacting agents’ collective behavior
Endogenous equilibrium distribution.
Table 1: Common reasons why the deployment distribution Q differs from the nominal distribution P .
Figure 1: Roadmap for Section 3. Particle dynamics induce density-evolution equations; the Fokker–Planck equation can be rewritten as a continuity equation; Benamou–Brenier adds an optimization principle over continuity-constrained paths.
Model class
Mechanism
Object learned
Diffusion models
Add noise forward, then learn reverse-time denoising dynamics
Score field or reverse dynamics mapping noise back to data.
Normalizing flows
Compose invertible maps or integrate an ODE
Transport map between a reference and target distribution.
Flow matching
Match velocities along interpolation paths
Velocity field whose continuity equation transports distributions.
Consistency models
Distill diffusion dynamics into few-step generation
Direct map from noisy states to clean samples.
Table 2: Generative-model classes discussed in Section 4.
Distribution-construction task
Example OR situation
Role of the deployment distribution Q
Distribution-to-distribution transport
Transfer a demand, traffic, or outage model from one region, season, or operating regime to another
Q is a target-regime distribution obtained by transporting samples from a nominal source distribution P .
Worst-case generation and robust construction
Stress testing a supply chain, portfolio, power grid, or service system under adverse but plausible conditions
Q is a stressed or least-favorable distribution constructed from P to expose vulnerabilities of a decision.
Conditional generation with observed context
Generate demand, renewable supply, travel time, or outage scenarios given weather, calendar, market, or network covariates
Q is a context-dependent conditional distribution, such as P(Y∣X=x) .
Posterior sampling and inverse problems
Update uncertainty about a latent state after partial observations, sensor readings, or noisy measurements
Q is a posterior distribution obtained by combining a prior model with an observation or likelihood model.
Mean-field game formulations
Model fleet repositioning, traffic routing, ride-sharing, or congestion systems with many interacting agents
Q is an endogenous equilibrium population distribution induced by agents’ best responses.
Table 3: Roadmap for the distribution-construction tasks in Section 5.
Task
Source of distribution shift
Representative guarantee
Data imitation
Transport from data distribution P toward reference distribution Q
Contraction / forward–reverse convergence in Wasserstein space
Robust generation
Adversarial perturbation of a nominal distribution
First-order convergence of min–max optimization in transport-map space
Posterior updating
Partial observation through a prior–likelihood update
Error-transfer bounds from prior approximation and sampling error to posterior error
Table 4: Representative theory templates for different distribution-construction tasks.
Figure 2: Empirical and generated marginal outage distributions for representative counties (Fulton, DeKalb, Clayton, and Rockdale), shown via empirical cumulative distribution functions of outage counts. The four counties illustrate both high-volume and sparse outage regimes.
Figure 3: Empirical and generated cross-county correlation matrices in the transformed space.
Figure 4: Marginal distributions of nominal and generated worst-case returns for the six assets, shown for different values of the regularization parameter. Larger perturbations produce more pronounced shifts toward adverse return regions.
Figure 5: Out-of-sample cumulative wealth of the nominal portfolio and portfolios optimized under generated worst-case distributions, evaluated on a non-stress test setting (left) and a stress test setting (right). Curves correspond to different values of the regularization parameter. These examples illustrate that γ=0.1 achieves the best performance among the choices.