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

• URL: http://arxiv.org/abs/2102.01977v5
• Date: Wed, 22 Mar 2023 09:17:12 GMT
• Title: Instance-Dependent Bounds for Zeroth-order Lipschitz Optimization with Error Certificates
• Authors: Fran\c{c}ois Bachoc (IMT, GdR MASCOT-NUM), Tommaso R Cesari (TSE-R), S\'ebastien Gerchinovitz (IMT)
• Abstract summary: We study the problem of zeroth-order (black-box) optimization of a Lipschitz function $f$ defined on a compact subset $mathcal X$ of $mathbb Rd$. We characterize the optimal number of evaluations of any Lipschitz function $f$ to find and certify an approximater of $f$ at accuracy $varepsilon$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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