🤖 AI Summary
This work addresses a long-standing theoretical gap in deterministic derivative-free convex optimization by establishing the first near-tight lower bound on query complexity when only exact function evaluations are available. Specifically, for minimizing convex Lipschitz functions over the $d$-dimensional Euclidean ball, the paper proves a deterministic query complexity lower bound of $\Omega(d^2 / \log(d+1))$, closing the gap that has persisted for nearly three decades up to a logarithmic factor. The result is achieved through a carefully constructed adversarial family of functions, combined with tools from information theory and convex analysis. Furthermore, the authors extend their approach to the mixed-integer setting, yielding a complexity bound of $\widetilde{O}(d^2 \cdot 2^n)$, which significantly improves upon existing estimates.
📝 Abstract
We study the deterministic query complexity of minimizing a convex Lipschitz function over a $d$-dimensional Euclidean ball using only exact function values. At accuracy $Θ(d^{-1/2})$, the previously applicable lower bound was $Ω(d)$, inherited from the stronger full first-order oracle, while an upper bound from Protasov's value-only method requires $O(d^2\log^2 d)$ evaluations. By providing a lower bound of $Ω(\,\frac{d^2}{\log(d+1)})$ on the oracle complexity in this setting, we thereby close this gap dating back to 1996, up to polylogarithmic factors. Furthermore, we are able to lift this result to the mixed-integer setting: Mixed-integer convex optimization with $d$ continuous and $n$ discrete variables using function values requires $\tildeΩ(d^2\cdot 2^n)$ queries.