A Bayesian Risk Approach to Data-driven Stochastic Optimization: Formulations and Asymptotics

TL;DR

This paper introduces a Bayesian risk optimization framework for stochastic programs with unknown distributions, providing theoretical insights into its consistency, asymptotics, and interpretation as a weighted combination of posterior mean and confidence interval half-width.

Contribution

It proposes a novel Bayesian risk approach for parametric stochastic optimization, analyzing its asymptotic properties and offering a new interpretation of its objectives.

Findings

Establishes the consistency of the Bayesian risk optimization solutions.

Proves the asymptotic normality of the objective functions and optimal values.

Reveals that BRO objectives approximate a weighted sum of posterior mean and confidence interval half-width.

Abstract

A large class of stochastic programs involve optimizing an expectation taken with respect to an underlying distribution that is unknown in practice. One popular approach to addressing the distributional uncertainty, known as the distributionally robust optimization (DRO), is to hedge against the worst case over an uncertainty set of candidate distributions. However, it has been observed that inappropriate construction of the uncertainty set can sometimes result in over-conservative solutions. To explore the middle ground between optimistically ignoring the distributional uncertainty and pessimistically fixating on the worst-case scenario, we propose a Bayesian risk optimization (BRO) framework for parametric underlying distributions, which is to optimize a risk functional applied to the posterior distribution of an unknown distribution parameter. Of our particular interest are four risk functionals: mean, mean-variance, value-at-risk, and conditional value-at-risk. To unravel the implication of BRO, we establish the consistency of objective functions and optimal solutions, as well as the asymptotic normality of objective functions and optimal values. More importantly, our analysis reveals a hidden interpretation: the objectives of BRO can be approximately viewed as a weighted sum of posterior mean objective and the (squared) half-width of the true objective's confidence interval.