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Convergence Rates for Lasserre's Upper-Bound Hierarchy on ℝd with Gaussian Measure

  • Sander Gribling
  • Igor Klep
  • Poornendu Kumar
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Abstract

We establish quantitative convergence rates for Lasserre's measure-based upper-bound hierarchy on ℝd using the standard Gaussian measure as the reference measure. For any polynomial f attaining its global infimum fmin, we prove that the order-r upper bounds satisfy λr−fmin=O(1/r), 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=x2, while demonstrating that, without attainment of the infimum, convergence can be arbitrarily slow: for every m≥0, we construct a polynomial whose upper bounds converge at rate Θ(r−1/(m+2)).

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Cite

BibTeXRIS
BibTeX
@misc{gribling2026convergence,
  title = {{Convergence Rates for Lasserre's Upper-Bound Hierarchy on $\mathbbR^d$ with Gaussian Measure}},
  author = {Sander Gribling and Igor Klep and Poornendu Kumar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22030},
  eprint = {2610.22030},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22030},
}
RIS
TY  - UNPB
TI  - Convergence Rates for Lasserre's Upper-Bound Hierarchy on ℝ^d with Gaussian Measure
AU  - Sander Gribling
AU  - Igor Klep
AU  - Poornendu Kumar
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22030
UR  - https://arxiv.org/abs/2610.22030
AB  - We establish quantitative convergence rates for Lasserre's measure-based upper-bound hierarchy on 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(1/r), 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≥0, we construct a polynomial whose upper bounds converge at rate Θ(r^−1/(m+2)).
ER  -