Instance-Dependent Bounds for Zeroth-order Lipschitz Optimization with Error Certificates
This solves a foundational open problem in optimization theory, providing instance-dependent bounds for black-box optimization with certification, which is incremental in extending known results from one dimension to higher dimensions.
The paper tackles the problem of zeroth-order Lipschitz optimization with error certificates, deriving an optimal sample complexity nearly proportional to an integral that depends on the function instance, solving an open problem dating back to 1991.
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 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.