{
  "abstract": "We establish quantitative convergence rates for Lasserre's measure-based upper-bound hierarchy on $\\mathbb{R}^d$ using the standard Gaussian measure as the reference measure. For any polynomial $f$ attaining its global infimum $f_{\\min}$, we prove that the order-$r$ upper bounds satisfy $$ λ_r-f_{\\min}=O\\Big(\\frac{1}{r}\\Big), $$ yielding, to the best of our knowledge, the first explicit convergence rate for this hierarchy on a noncompact domain. We further show that the rate is sharp by proving an $Ω(1/r)$ rate for $f=x^2$, while demonstrating that, without attainment of the infimum, convergence can be arbitrarily slow: for every $m\\ge0$, we construct a polynomial whose upper bounds converge at rate $Θ(r^{-1/(m+2)})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22030",
  "author_total": 3,
  "authors": [
    {
      "name": "Sander Gribling"
    },
    {
      "name": "Igor Klep"
    },
    {
      "name": "Poornendu Kumar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0wv15w3akb1nx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22030",
  "fulltext_url": null,
  "id": "0wv15w3akb1nx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22030",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7222654463",
  "pdf_url": "https://arxiv.org/pdf/2610.22030",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0wv15w3akb1nx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22030",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22030"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Convergence Rates for Lasserre's Upper-Bound Hierarchy on $\\mathbb{R}^d$ with Gaussian Measure",
  "title_text": "Convergence Rates for Lasserre's Upper-Bound Hierarchy on ℝ^d with Gaussian Measure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/0wv15w3akb1nx",
  "venue": "arXiv",
  "versions": []
}