Sharp Minimax Rates for Bounded Lipschitz Gaussian Functionals

We study recovery of a bounded scalar Lipschitz functional from point evaluations when prediction error is measured under a known infinite-dimensional Gaussian law. Suppose its covariance eigenvalues satisfy a two-sided polynomial envelope, $c_{\lambda }{j}^{-\tau }\le {\lambda }_j\le C_{\lambda }{j}^{-\tau }$, with fixed $\tau >1$. For the common class of functions with global sup norm and Lipschitz constant at most one, we establish the sharp expected $L^2$ error $\Theta ([\log (m+2){]}^{-(\tau +1)/2})$. The expected squared error has order $[\log (m+2){]}^{-(\tau +1)}$. These rates hold for passive regression with unit Gaussian noise, adaptive noisy point queries, and adaptive exact point queries. The lower bound assigns independent signs to small Gaussian cells; an adaptive query can reveal at most one sign. The upper bound estimates a finite collection of Hermite coefficients from passive samples. For passive regression, the same sharp rates also hold when the covariance and its eigenbasis are unknown, using only the original random samples and known spectral-envelope constants. The result identifies the sharp rate for this uniform function class and shows that removing label noise and choosing queries adaptively do not change its asymptotic order. Classical geometric partitions and Gaussian spectral approximation provide the principal proof ingredients.