The exploration of the complex composition space for high entropy alloys (HEAs) is extremely challenging and resource intensive using traditional materials discovery approaches. Here, we apply Bayesian optimization in the form of active learning with a neural network model to efficiently explore the vast composition space of HEAs with a focus on predicting their stable phases. For example, by focusing data acquisition on the uncertain regions, we achieved a testing accuracy of 95% with 27% of an experimental HEA dataset, comparable to the accuracy (94.6%) of a random forest model trained with 80% of the full dataset (2198 quinary HEAs). Similarly, with just ∼2% of a CALPHAD dataset we achieved a testing accuracy for phase predictions above 96%, which is comparable to the accuracy (above 97%) achieved when an XGboost model is trained with nearly the full dataset (664,650 quinary HEAs). Thus, our approach greatly facilitates both computational and experimental exploration of HEA design spaces.