Asymptotic Optimality in Data-Driven Decision Making

TL;DR

This paper develops a framework for constructing statistically optimal decisions in stochastic optimization, accounting for complex data structures and balancing risk decay rates with decision consistency.

Contribution

It introduces a novel decision-making approach based on a multi-objective optimization that generalizes large deviation principles and classical data-driven methods.

Findings

The approach achieves exponential decay of shifted regret.

It generalizes distributionally robust optimization.

Effective in heterogeneous data scenarios.

Abstract

Given data generated by an observable 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 randomly evolving data-generating mechanisms. We propose a decision-making approach that identifies optimal decisions for which a specific notion of risk of shifted regret decays to zero at a prescribed exponential rate. This optimal decision arises as the solution to a multi-objective optimization problem, which reflects asymptotic behavior properties of the data-generating process. Central to our framework is a rate function that characterizes this behavior via a Laplace principle, thereby generalizing standard concepts from large deviation theory. Our general formulation enables our approach to account for data from uncertain distributions and recovers classical results in data-driven decision making under uncertainty as special cases, including distributionally robust optimization. Moreover, our method 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 illustrative examples from operations research, such as the newsvendor problem, under aleatoric uncertainty induced by heterogeneous data sources.