When Is Low Fidelity Worth Buying? Exact Certificates and Empirical Safeguards for Multi-Fidelity Bayesian Optimization

Multi-fidelity Bayesian optimization can reduce the cost of expensive high-fidelity evaluations, but low-fidelity information is valuable only when its benefit justifies the additional evaluation cost. Existing approaches often model low-fidelity sources or select among fidelities without explicitly accounting for this procurement decision. We introduce a cost-aware framework that treats each high-fidelity query and its optional low-fidelity purchase as a single prepaid macro-round. The framework combines an exact finite-family certificate for reliable procurement with an empirical safeguard that probes low-fidelity usefulness before activation and falls back to high-fidelity-only optimization when the evidence is insufficient. We prove expected and high-probability bounds on cost-adjusted simple regret, a finite-horizon comparison bound against matched high-fidelity-only procurement, and an exact bounded-cost fallback property. Experiments show that the method actively exploits informative low-fidelity sources, rejects unsupported sources, and achieves a substantial improvement on a standard multi-fidelity benchmark while preserving the declared cost accounting. These results demonstrate how certification and empirical safeguards can turn low-fidelity evaluation into a controlled procurement option.