Near-Optimal Lower Bounds for Exact Zeroth-Order Convex Optimization

The fundamental oracle limits of exact function value access remain poorly understood in zeroth-order optimization: even for canonical convex problems, the optimal joint dependence on dimension $d$ and accuracy $\epsilon$ has remained unresolved. We resolve this question for arbitrary adaptive randomized algorithms minimizing nonsmooth $L_0$-Lipschitz convex functions over the $d$-dimensional Euclidean unit ball, where each query returns only the exact scalar value $f(\mathbf x)$ of a fixed objective. With universal $L_0>0$, let $T_\epsilon$ denote the minimum number of queries required to return an $\epsilon$-suboptimal point with probability at least $1/2$, uniformly over the function class. We prove that $$T_\epsilon \ge c\, \frac{ d\min\{d,\epsilon^{-2}\} }{ \log\!\bigl(e\min\{d,\epsilon^{-2}\}\bigr) },$$ for all $d\ge d_0$ and $0<\epsilon\le\epsilon_0$, for universal constants $c,\epsilon_0>0$ and $d_0\in\mathbb N$. In the low-accuracy regime $\epsilon\ge d^{-1/2}$, this matches the $O(d\epsilon^{-2})$ two-point exact value upper bound up to a logarithmic factor. In the high-accuracy regime $\epsilon\le d^{-1/2}$, the lower bound saturates at $\Omega(d^2/\log(ed))$, independently of $\epsilon$, matching the $\widetilde O(d^2)$ evaluation oracle upper bound up to polylogarithmic factors. The proof uses a random support function hard family and develops a posterior mean energy method for adaptive exact max observations, in place of first-order zero chain constructions and noise based transcript inequalities.