Near-Optimal Bayesian Ambiguity Sets for Distributionally Robust Optimization

We propose a Bayesian framework for assessing the relative strengths of data-driven ambiguity sets in distributionally robust optimization (DRO) when the underlying distribution is defined by a finite-dimensional parameter. The key idea is to measure the relative size between a candidate ambiguity set and a specific asymptotically optimal set. As the amount of data grows large, this asymptotically optimal set is the smallest convex ambiguity set that satisfies a novel Bayesian robustness guarantee that we introduce. This guarantee is defined with respect to a given class of constraints and is a Bayesian analog of more common frequentist feasibility guarantees from the DRO literature. Using this framework, we prove that many popular existing ambiguity sets are significantly larger than the asymptotically optimal set for constraints that are concave in the ambiguity. By contrast, we construct new ambiguity sets that are tractable, satisfy our Bayesian robustness guarantee, and are at most a small, constant factor larger than the asymptotically optimal set; we call these sets Bayesian near-optimal. We further prove that asymptotically, solutions to DRO models with our Bayesian near-optimal sets enjoy strong frequentist robustness properties, despite their smaller size. Finally, our framework yields guidelines for practitioners selecting between competing ambiguity set proposals in DRO. Computational evidence in portfolio allocation using real and simulated data confirms that our framework, although motivated by asymptotic analysis in a Bayesian setting, provides practical insight into the performance of various DRO models with finite data under frequentist assumptions. This paper was accepted by Yinyu Ye, optimization.