The training or fine-tuning of machine learning, vision, and language models is often implemented as a pipeline: a sequence of stages encompassing data preparation, model training and evaluation. In this paper, we exploit pipeline structures to reduce the cost of hyperparameter tuning for model training/fine-tuning, which is particularly valuable for language models given their high costs in GPU-days. We propose a "memoization-aware" Bayesian Optimization (BO) algorithm, EEIPU, that works in tandem with a pipeline caching system, allowing it to evaluate significantly more hyperparameter candidates per GPU-day than other tuning algorithms. The result is better-quality hyperparameters in the same amount of search time, or equivalently, reduced search time to reach the same hyperparameter quality. In our benchmarks on machine learning (model ensembles), vision (convolutional architecture) and language (T5 architecture) pipelines, we compare EEIPU against recent BO algorithms: EEIPU produces an average of $103\%$ more hyperparameter candidates (within the same budget), and increases the validation metric by an average of $108\%$ more than other algorithms (where the increase is measured starting from the end of warm-up iterations).