Instance-Dependent Bounds for Zeroth-order Lipschitz Optimization with Error Certificates

We study the problem of zeroth-order (black-box) optimization of a Lipschitz function $f$ defined on a compact subset $\mathcal{X}$ of $\mathbb{R}^d$, with the additional constraint that algorithms must certify the accuracy of their recommendations. We characterize the optimal number of evaluations of any Lipschitz function $f$ to find and certify an approximate maximizer of $f$ at accuracy $\varepsilon$. Under a weak assumption on $\mathcal{X}$, this optimal sample complexity is shown to be nearly proportional to the integral $\int_{\mathcal{X}} \mathrm{d}\boldsymbol{x}/( \max(f) - f(\boldsymbol{x}) + \varepsilon )^d$. This result, which was only (and partially) known in dimension $d=1$, solves an open problem dating back to 1991. In terms of techniques, our upper bound relies on a packing bound by Bouttier et al. (2020) for the Piyavskii-Shubert algorithm that we link to the above integral. We also show that a certified version of the computationally tractable DOO algorithm matches these packing and integral bounds. Our instance-dependent lower bound differs from traditional worst-case lower bounds in the Lipschitz setting and relies on a local worst-case analysis that could likely prove useful for other learning tasks.