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

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 slightly improved analysis of the DOO algorithm that we adapt to the certified setting and then link to the above integral. 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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https://hal.archives-ouvertes.fr/hal-03129721
Contributor : Tommaso Cesari Connect in order to contact the contributor
Submitted on : Wednesday, June 9, 2021 - 4:20:58 PM
Last modification on : Tuesday, October 19, 2021 - 11:17:07 PM

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• HAL Id : hal-03129721, version 3
• ARXIV : 2102.01977

Citation

François Bachoc, Tommaso Cesari, Sébastien Gerchinovitz. Instance-Dependent Bounds for Zeroth-order Lipschitz Optimization with Error Certificates. 2021. ⟨hal-03129721v3⟩

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