On the Trade-Off Between Distributional Belief and Ambiguity: Conservatism, Finite-Sample Guarantees, and Asymptotic Properties

TL;DR

This paper introduces a new data-driven trade-off (TRO) approach for modeling uncertainty, balancing optimism and robustness, with theoretical guarantees and practical applications in inventory and portfolio management.

Contribution

It develops a TRO ambiguity set with a star-shaped shape parameter, providing hierarchical structure, and analyzes its properties including conservatism, bias, and asymptotic behavior.

Findings

TRO ambiguity set guarantees hierarchical structure.

TRO model can produce unbiased estimates of true optimal value.

Almost-sure convergence of solutions to true counterparts.

Abstract

We propose and analyze a new data-driven trade-off (TRO) approach for modeling uncertainty that serves as a middle ground between the optimistic approach, which adopts a distributional belief, and the pessimistic distributionally robust optimization approach, which hedges against distributional ambiguity. We equip the TRO model with a TRO ambiguity set characterized by a size parameter controlling the level of optimism and a shape parameter representing distributional ambiguity. We first show that constructing the TRO ambiguity set using a general star-shaped shape parameter with the empirical distribution as its star center is necessary and sufficient to guarantee the hierarchical structure of the sequence of TRO ambiguity sets. Then, we analyze the properties of the TRO model, including quantifying conservatism, quantifying bias and generalization error, and establishing asymptotic properties. Specifically, we show that the TRO model could generate a spectrum of decisions, ranging from optimistic to conservative decisions. Additionally, we show that it could produce an unbiased estimator of the true optimal value. Furthermore, we establish the almost-sure convergence of the optimal value and the set of optimal solutions of the TRO model to their true counterparts. We exemplify our theoretical results using an inventory control problem and a portfolio optimization problem.