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.

翻译：我们研究定义在$\mathbb R^d$中紧子集$\mathcal X$上的Lipschitz函数$f$的零阶（黑箱）优化问题，并附加算法必须认证其推荐结果准确性的约束条件。我们刻画了任意Lipschitz函数$f$在精度$\varepsilon$下找到并认证其近似最大值点所需的最优函数评估次数。在关于$\mathcal X$的弱假设下，这一最优样本复杂度被证明近似正比于积分$\int_{\mathcal X} \mathrm{d}\boldsymbol x/( \max(f) - f(\boldsymbol x) + \varepsilon )^d$。这一结果仅在$d=1$维情况下（部分）已知，解决了可追溯至1991年的公开问题。在技术方面，我们的上界依赖于Bouttier等人(2020)针对Piyavskii-Shubert算法建立的打包界，并将其与上述积分相关联。我们还证明，计算可处理的DOO算法的认证版本能够达到这些打包界和积分界。与Lipschitz设置中传统的 Worst-case 下界不同，我们的实例依赖下界基于局部最坏情况分析，该分析方法可能对其他学习任务具有实用价值。