Decision Making under the Exponential Family: Distributionally Robust Optimisation with Bayesian Ambiguity Sets

TL;DR

This paper introduces a novel distributionally robust optimization framework leveraging Bayesian ambiguity sets, improving decision-making under uncertainty by accounting for model ambiguity with efficient solutions and better performance.

Contribution

It proposes DRO-BAS, a new approach combining Bayesian inference with distributionally robust optimization, applicable to exponential family models, with proven dual formulations and improved computational efficiency.

Findings

Outperforms existing Bayesian DRO on Newsvendor problem

Achieves faster solve times with comparable robustness on Portfolio problem

Provides strong dual formulations for efficient stochastic programming

Abstract

Decision making under uncertainty is challenging as the data-generating process (DGP) is often unknown. Bayesian inference proceeds by estimating the DGP through posterior beliefs on the model's parameters. However, minimising the expected risk under these beliefs can lead to suboptimal decisions due to model uncertainty or limited, noisy observations. To address this, we introduce Distributionally Robust Optimisation with Bayesian Ambiguity Sets (DRO-BAS) which hedges against model uncertainty by optimising the worst-case risk over a posterior-informed ambiguity set. We provide two such sets, based on posterior expectations (DRO-BAS(PE)) or posterior predictives (DRO-BAS(PP)) and prove that both admit, under conditions, strong dual formulations leading to efficient single-stage stochastic programs which are solved with a sample average approximation. For DRO-BAS(PE) this covers all conjugate exponential family members while for DRO-BAS(PP) this is shown under conditions on the predictive's moment generating function. Our DRO-BAS formulations outperform existing Bayesian DRO on the Newsvendor problem and achieve faster solve times with comparable robustness on the Portfolio problem.