Certified Multi-Fidelity Zeroth-Order Optimization
This addresses optimization efficiency for scenarios with costly function evaluations, such as in engineering or machine learning, but is incremental as it builds on existing multi-fidelity methods.
The paper tackles the problem of multi-fidelity zeroth-order optimization, where functions are evaluated at varying costs to minimize optimization expense, by proposing a certified algorithm that outputs an error bound and proving it has near-optimal cost complexity for Lipschitz functions.
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.