This chapter introduces a stepwise optimization approach for algorithm selection in Automatic Machine Learning (AutoML). Traditional methods like Combined Algorithm Selection and Hyper-parameter optimization (CASH) often suffer from inefficiency due to the large and redundant search space. To address this, the chapter proposes a cascaded algorithm selection framework, which separates the process into two levels: hyper-parameter optimization for individual algorithms and a resource allocation strategy at the upper level. The upper level is formulated as a multi-armed bandit problem, where each arm represents a hyper-parameter optimization process. The chapter introduces the Extreme-Region Upper Confidence Bound (ER-UCB) strategy, designed to maximize the final feedback by focusing on the arm with the largest extreme region. Theoretical analysis and empirical studies on synthetic and real-world AutoML tasks demonstrate the effectiveness of ER-UCB in improving algorithm selection efficiency and performance. The results highlight the importance of stepwise optimization in reducing redundancy and enhancing AutoML outcomes.