Oracle-Complexity Gap in Derivative-Free Convex Optimization

For deterministically minimizing a convex 1-Lipschitz function on the $d$-dimensional ball using only exact function values, the query complexity sat between $\Omega(d)$ and $O(d^2 \log^2 d)$ since 1996. The paper proves a near-quadratic lower bound $\Omega(d^2 / \log(d+1))$, closing the gap: $Q(d, \sim d^{-1/2}) = \Theta(d^2)$, a polynomial separation from full first-order information.