Must the dimension and accuracy costs of zeroth-order optimization multiply? Global upper bounds for smooth stochastic nonconvex optimization scale as $O(dε^{-4})$, with both factors separately necessary. We show that their product is unavoidable. We establish an $Ω(dε^{-4})$ lower bound for smooth stochastic nonconvex optimization, yielding an $Ω(dδ^{-1}ε^{-3})$ bound for nonsmooth Goldstein stationarity. Both match the best-known full-space upper bounds, establishing joint minimax optimality in dimension and accuracy. These bounds hold against fully adaptive randomized algorithms with unrestricted queries and outputs, even under arbitrary sample reuse. As a byproduct, we establish noiseless exact-value lower bounds for both smooth and Goldstein stationarity, even against randomized algorithms.

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