Modern systems that rely on Machine Learning (ML) for predictive modelling, may suffer from the cold-start problem: supervised models work well but, initially, there are no labels, which are costly or slow to obtain. This problem is even worse in imbalanced data scenarios, where labels of the positive class take longer to accumulate. We propose an Active Learning (AL) system for datasets with orders of magnitude of class imbalance, in a cold start streaming scenario. We present a computationally efficient Outlier-based Discriminative AL approach (ODAL) and design a novel 3-stage sequence of AL labeling policies where ODAL is used as warm-up. Then, we perform empirical studies in four real world datasets, with various magnitudes of class imbalance. The results show that our method can more quickly reach a high performance model than standard AL policies without ODAL warm-up. Its observed gains over random sampling can reach 80% and be competitive with policies with an unlimited annotation budget or additional historical data (using just 2% to 10% of the labels).