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We consider the problem of multi-fidelity zeroth-order optimization, whereone can evaluate a function $f$ at various approximation levels (of varyingcosts), and the goal is to optimize $f$ with the cheapest evaluations possible.In this paper, we study \emph{certified} algorithms, which are additionallyrequired to output a data-driven upper bound on the optimization error. Wefirst formalize the problem in terms of a min-max game between an algorithm andan evaluation environment. We then propose a certified variant of the MFDOOalgorithm 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 hasa near-optimal cost complexity. We close the paper by addressing the specialcase of noisy (stochastic) evaluations as a direct example.

Certified Multi-Fidelity Zeroth-Order Optimization