We consider the problem of multi-fidelity zeroth-order optimization, where one can evaluate a function $f$ at various approximation levels (of varying costs), and the goal is to optimize $f$ with the cheapest evaluations possible. In this paper, we study \emph{certified} algorithms, which are additionally required to output a data-driven upper bound on the optimization error. We first formalize the problem in terms of a min-max game between an algorithm and an evaluation environment. We then propose a certified variant of the MFDOO algorithm and derive a bound on its cost complexity for any Lipschitz function $f$. We also prove an $f$-dependent lower bound showing that this algorithm has a near-optimal cost complexity. We close the paper by addressing the special case of noisy (stochastic) evaluations as a direct example.

翻译：我们考虑多保真度零阶优化问题，其中可以在不同近似精度（具有不同成本）下评估函数$f$，目标是以最经济的评估方式优化$f$。本文研究*认证*算法，该算法额外要求输出一个数据驱动的优化误差上界。我们首先通过算法与评估环境之间的极小极大博弈形式化该问题。随后提出MFDOO算法的认证变体，并针对任意Lipschitz函数$f$推导其成本复杂度的界。我们还证明了$f$相关的下界，表明该算法具有近乎最优的成本复杂度。最后通过有噪声（随机）评估这一特例作为直接示例进行讨论。