Optimizing a machine learning (ML) pipeline has been an important topic of AI and ML. Despite recent progress, this topic remains a challenging problem, due to potentially many combinations to consider as well as slow training and validation. We present the BLDS algorithm for optimized algorithm selection (ML operations) in a fixed ML pipeline structure. BLDS performs multi-fidelity optimization for selecting ML algorithms trained with smaller computational overhead, while controlling its pipeline search based on multi-armed bandit and limited discrepancy search. Our experiments on well-known benchmarks show that BLDS is superior to competing algorithms.