Certified Multi-Fidelity Zeroth-Order Optimization

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 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. As a direct example, we close the paper by addressing the special case of noisy (stochastic) evaluations, which corresponds to eps-best arm identification in Lipschitz bandits with continuously many arms.