An Axiomatic Analysis of Distributionally Robust Optimization with q-Norm Ambiguity Sets for Probability Smoothing

We analyze the axiomatic properties of a class of probability estimators derived from Distributionally Robust Optimization (DRO) with q -norm ambiguity sets ( q -DRO), a principled approach to the zero-frequency problem. While classical estimators such as Laplace smoothing are characterized by strong linearity axioms like Ratio Preservation, we show that q -DRO provides a flexible alternative that satisfies other desirable properties. We first prove that for any q ∈ [1, ∞], the q -DRO estimator satisfies the fundamental axioms of Positivity and Symmetry. For the case of q ∈ (1, ∞), we then prove that it also satisfies Order Preservation. Our analysis of the optimality conditions also reveals that the q -DRO formulation is equivalent to the regularized empirical loss minimization.