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

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.

Figures (13)

• Figure 1: Illustration of the strict hierarchical property of the sequence of TRO ambiguity sets $\{\mathcal{P}'_{N,\theta}\mid 0<\theta_1<\theta_2<1\}$.
• Figure 2: Illustration of $\widehat{\upsilon}_N(\theta)$ when $\mathcal{P}_N$ is convex and $\widehat{\mathbb{P}}_N\in\mathcal{P}_N$. For a given target level of conservatism $\overline{\upsilon}=(1-\lambda)\widehat{\upsilon}_N(0) + \lambda\widehat{\upsilon}_N(1)$ with $\lambda\in(0,1)$, one should pick $\overline{\theta}<\lambda$ in the TRO model.
• Figure 3: Illustration of the SAA bias reduction effect. The solid curve represents the concave function $\mathbb{E}_{\mathbb{P}^N}[\widehat{\upsilon}_N(\theta)]$.
• Figure 4: Optimal solution and optimal value for different values of $\theta$ in the inventory control problem.
• Figure 5: Bias and standard deviation of $\widehat{\upsilon}_N(\theta)$ for different values of $\theta$ with $N=10$ in the inventory control problem.

Theorems & Definitions (60)

• Definition 2.1: Star-Shaped Set
• Definition 2.2: Hierarchical Properties
• Theorem 1
• Proposition 1
• Example 1
• Proposition 2
• Example 2
• Example 3
• Example 4
• Theorem 2