Dimension-sharp bound for the finite-difference bias second moment

Establish a general bound of order [?] O(L^2\mu^2d^2) for the finite-difference bias second moment \(\Sigma_b^2\), in place of the current generic \(O(L^2\mu^2d^3)\) bound, in order to move the admissible sketch step size from \(\Theta(1/d^2)\) to \(\Theta(1/d)\) and make the lowest-deployment operating point free in function evaluations.

Background

The convergence and deployment analysis distinguishes between the mean finite-difference bias and its second moment. For Gaussian and spherical probes, the paper derives a sharper bound for the weighted bias vector, which permits a sketch step size of order $1/d$. However, the generic second-moment estimate for the scalar finite-difference bias yields a stricter cap of order $1/d^2$.

The authors note that examples suggest the generic bound is loose by a factor of $d$: a bound of order $L^2\mu^2d^2$ would shift the first step-size cap to order $1/d$, allowing the deployment-optimal operating point to be reached without increasing the total number of function evaluations. This bound is explicitly left unresolved.