Asymptotic Optimality in Data-driven Decision Making

Abstract: Given data generated by an abstract stochastic process, we study how to construct statistically optimal decisions for general stochastic optimization problems. Our setting encompasses non-standard data structures, including data originating from heterogeneous sources or from processes with randomly evolving data-generating mechanisms. We propose a decision-making approach that identifies optimal decisions for which a specific notion of shifted regret risk, evaluated with respect to the underlying stochastic process, decays to zero at a prescribed exponential rate under a minimal model perturbation. This optimal decision arises as the solution to a multi-objective optimization problem, where the trade-offs are governed by the statistical properties of the data-generating process. Central to our framework is a rate function, generalizing the classical concept from large deviation theory. Our general formulation recovers classical results in decision-making under uncertainty, such as those from distributionally robust optimization, as special cases. However, it goes beyond them: it enables decision-makers to systematically balance a desired rate of asymptotic risk decay against a potential loss in statistical consistency of the resulting data-driven decision. We demonstrate the effectiveness of the proposed approach through two illustrative examples from operations research: the classical newsvendor problem and a portfolio optimization problem under aleatoric uncertainty induced by heterogeneous data sources.