Convergence Rates for Lasserre's Upper-Bound Hierarchy on ${\mathbb{R}}^d$ with Gaussian Measure

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 $${\lambda }_r-f_{\min }=O\left(\frac{1}{r}\right),$$ 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 $\Omega (1/r)$ rate for $f=x^2$, while demonstrating that, without attainment of the infimum, convergence can be arbitrarily slow: for every $m\ge 0$, we construct a polynomial whose upper bounds converge at rate $\Theta (r^{-1/(m+2)})$.

20 pages, 2 figures