Optimizing a machine learning (ML) pipeline has been an important topic of AI and ML. Despite recent progress, pipeline optimization 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 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 classification benchmarks show that BLDS is superior to competing algorithms. We also combine BLDS with hyperparameter optimization, empirically showing the advantage of BLDS.