An Axiomatic Analysis of Distributionally Robust Optimization with $q$-Norm Ambiguity Sets for Probability Smoothing
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914473023897600 |
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| author | Izunaga, Yoichi Kurihara, Kota Nagano, Hokuto Uchida, Daiki |
| author_facet | Izunaga, Yoichi Kurihara, Kota Nagano, Hokuto Uchida, Daiki |
| contents | 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 $q \in [1, \infty]$, the $q$-DRO estimator satisfies the fundamental axioms of Positivity and Symmetry. For the case of $q \in (1, \infty)$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Axiomatic Analysis of Distributionally Robust Optimization with $q$-Norm Ambiguity Sets for Probability Smoothing Izunaga, Yoichi Kurihara, Kota Nagano, Hokuto Uchida, Daiki Optimization and Control 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 $q \in [1, \infty]$, the $q$-DRO estimator satisfies the fundamental axioms of Positivity and Symmetry. For the case of $q \in (1, \infty)$, 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. |
| title | An Axiomatic Analysis of Distributionally Robust Optimization with $q$-Norm Ambiguity Sets for Probability Smoothing |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2511.18815 |