{
  "abstract": "The probability smoothing problem seeks estimators that assign positive probability to all categories, including those unobserved in a finite sample. In our previous work, we established that the estimator derived from distributionally robust optimization (DRO) with a $q$-norm ambiguity set ($q$-DRO) satisfies the axioms of Positivity, Symmetry, and Order Preservation, while relaxing the rigid Ratio Preservation axiom characterizing Laplace smoothing. In this paper, we observe that Symmetry itself imposes a strong constraint: it forces all unobserved categories to receive identical probability regardless of their relation to observed categories. Motivated by this observation, we study DRO with an optimal transport ambiguity set (OT-DRO), which incorporates inter-category dissimilarities through a cost matrix and thereby enables structure-respecting smoothing. We derive a tractable dual reformulation of OT-DRO as an empirical cross-entropy minimization with a Lipschitz-type penalty on the log-probability vector, and we identify an explicit threshold on the robustness radius at which the optimal estimator reduces to the uniform distribution, paralleling the threshold established for $q$-DRO. We then show, through Sion's minimax theorem, that DRO for probability smoothing over any compact convex ambiguity set that contains a strictly positive distribution is equivalent to entropy maximization over that set. This unifying perspective explains the parallel threshold structure in $q$-DRO and OT-DRO.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25684",
  "author_total": 3,
  "authors": [
    {
      "name": "Kota Kurihara"
    },
    {
      "name": "Yuya Yamakawa"
    },
    {
      "name": "Yoichi Izunaga"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6d3pvh1bmcrfd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25684",
  "fulltext_url": null,
  "id": "6d3pvh1bmcrfd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25684",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1803|Risk and Portfolio Optimization"
    ]
  ],
  "openalex": "W7223026266",
  "pdf_url": "https://arxiv.org/pdf/2610.25684",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6d3pvh1bmcrfd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25684",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25684"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A Unifying View of Distributionally Robust Optimization for Probability Smoothing via Maximum Entropy",
  "title_text": "A Unifying View of Distributionally Robust Optimization for Probability Smoothing via Maximum Entropy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/6d3pvh1bmcrfd",
  "venue": "arXiv",
  "versions": []
}