Higher-Order Expansion and Bartlett Correctability of Distributionally Robust Optimization

Distributionally robust optimization (DRO) is a worst-case framework for stochastic optimization under uncertainty that has drawn fast-growing studies in recent years. When the underlying probability distribution is unknown and observed from data, DRO suggests computing the worst-case distribution within a so-called uncertainty set that captures the involved statistical uncertainty. In particular, DRO with uncertainty set constructed as a statistical divergence neighborhood ball has been shown to provide a tool for constructing valid confidence intervals for nonparametric functionals and bears a duality with the empirical likelihood (EL). In this paper, we show how adjusting the ball size of such type of DRO can reduce higher-order coverage errors similar to the so-called Bartlett correction. Our correction, which applies to general von Mises differentiable functionals, is more general than the existing EL literature that only focuses on smooth function models or M-estimation. Moreover, we demonstrate a higher-order “self-normalizing” property of DRO regardless of the choice of divergence. Our approach builds on the development of a higher-order expansion of DRO, which is obtained through an asymptotic analysis on a fixed-point equation arising from the Karush-Kuhn-Tucker conditions. Funding: This work was supported by the National Science Foundation, Division of Information and Intelligent Systems [Grant IIS-1849280] and the Division of Civil, Mechanical and Manufacturing Innovation [Grant CAREER CMMI-1834710].