Beyond IID: data-driven decision-making in heterogeneous environments
Omar Besbes, Will Ma, Omar Mouchtaki
TL;DR
This work develops a data-driven decision framework for heterogeneous environments where past data come from distributions within a radius $\epsilon$ of an unknown future distribution, and analyzes the resulting worst-case regret. It introduces a general reduction that upper-bounds asymptotic regret via a distributionally robust optimization (DRO) surrogate with a single past distribution, enabling tractable analysis of a broad policy class including SAA. An integral probability metric (IPM) based methodology links the heterogeneity notion to problem structure through an approximation parameter, yielding problem-specific regret bounds for canonical problems such as Newsvendor and Pricing, and revealing both the strengths and failures of SAA under different distances. To address SAA shortcomings, the paper designs rate-optimal policies, notably a Wasserstein-pricing policy that deflates SAA by $\delta=\sqrt{M\epsilon}$ to achieve tight $O(\sqrt{M\epsilon})$ regret guarantees, and discusses general DRO/RDRO-based approaches for robust, scalable solutions. The results highlight the crucial role of the heterogeneity type and problem structure in guiding the choice of data-driven policies and provide a principled path toward rate-optimal, robust decision-making in non-iid settings.
Abstract
How should one leverage historical data when past observations are not perfectly indicative of the future, e.g., due to the presence of unobserved confounders which one cannot "correct" for? Motivated by this question, we study a data-driven decision-making framework in which historical samples are generated from unknown and different distributions assumed to lie in a heterogeneity ball with known radius and centered around the (also) unknown future (out-of-sample) distribution on which the performance of a decision will be evaluated. This work aims at analyzing the performance of central data-driven policies but also near-optimal ones in these heterogeneous environments and understanding key drivers of performance. We establish a first result which allows to upper bound the asymptotic worst-case regret of a broad class of policies. Leveraging this result, for any integral probability metric, we provide a general analysis of the performance achieved by Sample Average Approximation (SAA) as a function of the radius of the heterogeneity ball. This analysis is centered around the approximation parameter, a notion of complexity we introduce to capture how the interplay between the heterogeneity and the problem structure impacts the performance of SAA. In turn, we illustrate through several widely-studied problems -- e.g., newsvendor, pricing -- how this methodology can be applied and find that the performance of SAA varies considerably depending on the combinations of problem classes and heterogeneity. The failure of SAA for certain instances motivates the design of alternative policies to achieve rate-optimality. We derive problem-dependent policies achieving strong guarantees for the illustrative problems described above and provide initial results towards a principled approach for the design and analysis of general rate-optimal algorithms.