Sharp Minimax Rates for Highly Smooth Strongly Convex Zeroth-Order Optimization

Abstract

We determine the minimax dimension dependence of noisy zeroth-order optimization for globally strongly convex functions with higher-order smoothness in multilinear operator norm. Each query returns one function value with fresh independent Gaussian noise. For dimension $d$, query budget $T$, and any fixed finite smoothness order $β>2$, the minimax expected objective error satisfies $\mathcal{E}_β(T,d)=Θ_β\left(\min\left\{1,(d^2/T)^{(β-1)/β}\right\}\right)$, with dimension-independent strong-convexity, gradient-smoothness, higher-order-smoothness, noise, and minimizer-radius bounds. Thus, for sufficiently small target error $\varepsilon$, the necessary and sufficient number of noisy values is $Θ_β(d^2\varepsilon^{-β/(β-1)})$, with no logarithmic loss. The new lower bound matches the dimension and budget dependence of the upper bound of Akhavan et al. (2024), and holds for arbitrary randomized adaptive algorithms with unrestricted query points. Its key step localizes a normalized hypercube with one radial cutoff: a query can emphasize one hidden coordinate, but the aggregate information about all coordinates remains small. A self-contained multiscale spherical estimator attains the matching upper bound by cancelling lower-order bias terms. Constants in the rates depend only on the fixed smoothness order under the stated normalization.