Provably Reduced Sample Cost in Prior-Guided Hyperparameter Optimization

Large-scale hyperparameter optimization (HPO) in automated machine learning (AutoML) consumes substantial computational resources, raising growing concerns about scalability and energy efficiency. Existing methods use prior information heuristically to accelerate both black-box and multi-fidelity settings, but they lack a characterization of how prior informativeness quantitatively reduces sample complexity. In this work, we provide the first distribution-dependent sample complexity bounds for multi-fidelity HPO with priors through the formal lens of fixed-budget best-arm identification. By modeling priors directly over arm means as configuration performance, we derive explicit, distribution-dependent error bounds that quantify the relationship between priors and evaluation budget. Our analysis shows that informative priors, which concentrate probability mass on near-optimal arms, yield reductions in the number of required evaluations, whereas baseline performance is recovered with uninformative or misleading priors. We conduct proof-of-concept experiments on a synthetic benchmark and on LCBench, a common multi-fidelity HPO benchmark for deep learning, to confirm our theoretical results, achieving up to 90