With the surge in the number of hyperparameters and training times of modern machine learning models, hyperparameter tuning is becoming increasingly expensive. Although methods have been proposed to speed up tuning via knowledge transfer, they typically require the final performance of hyperparameters and do not focus on low-fidelity information. Nevertheless, this common practice is suboptimal and can incur an unnecessary use of resources. It is more cost-efficient to instead leverage the low-fidelity tuning observations to measure inter-task similarity and transfer knowledge from existing to new tasks accordingly. However, performing multi-fidelity tuning comes with its own challenges in the transfer setting: the noise in the additional observations and the need for performance forecasting. Therefore, we conduct a thorough analysis of the multi-task multi-fidelity Bayesian optimization framework, which leads to the best instantiation-AmorTized Auto-Tuning (AT2). We further present an offline-computed 27-task Hyperparameter Recommendation (HyperRec) database to serve the community. Extensive experiments on HyperRec and other real-world databases illustrate the effectiveness of our AT2 method. Copyright © 2021, The Authors. All rights reserved.