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

Understand

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.