{
  "abstract": "Quantifying regret typically involves repeatedly re-solving a stochastic optimization problem across simulated scenarios. Such an approach is costly: the cost scales with the number of scenarios and the cost of each solve. We propose that regret can instead be measured using the joint moments of the cost vector $c$ and the optimal decision $π^*(c)$. Working with the optimal concave and Lipschitz under only a feasible region value function $v(c) = \\min_{z\\in Z} c^\\top z$, we show that regret always decomposes as $\\mathrm{Regret} = -\\mathrm{Cov}(c,π^*(c)) - R(c)$ for a residual $R(c)$. We further show that $R(c)=0$ when $π^*$ is affine in $c$, which holds for unconstrained and equality-constrained (e.g., budget-constrained Markowitz) quadratic programs and many, but not all, linear programs. For all cases, we give a distribution-free bound $\\mathrm{Regret} \\le D\\sqrt{\\mathrm{tr}(Σ_c)}$ that depends only on the radius of the feasible region and the cost covariance. We also derive concentration and asymptotic-normality results for the empirical regret, the latter under an explicit and checkable non-degeneracy condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.14019",
  "author_total": 1,
  "authors": [
    {
      "name": "Irene Aldridge"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/27j0vj7bpyzyq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.14019",
  "fulltext_url": null,
  "id": "27j0vj7bpyzyq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.14019",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "econ.EM"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.CO"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1803|Risk and Portfolio Optimization"
    ]
  ],
  "openalex": "W7161237416",
  "pdf_url": "https://arxiv.org/pdf/2605.14019",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/27j0vj7bpyzyq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2605.14019",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.14019"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Regret Equals Covariance: A Closed-Form Characterization for Stochastic Optimization",
  "title_text": "Regret Equals Covariance: A Closed-Form Characterization for Stochastic Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/27j0vj7bpyzyq",
  "venue": "arXiv",
  "versions": []
}