In many natural language processing applications, identifying predictive text can be as important as the predictions themselves. When predicting medical diagnoses, for example, identifying predictive content in clinical notes not only enhances interpretability, but also allows unknown, descriptive (i.e., text-based) risk factors to be identified. We here formalize this problem as predictive extraction and address it using a simple mechanism based on linear attention. Our method preserves differentiability, allowing scalable inference via stochastic gradient descent. Further, the model decomposes predictions into a sum of contributions of distinct text spans. Importantly, we require only document labels, not ground-truth spans. Results show that our model identifies semantically-cohesive spans and assigns them scores that agree with human ratings, while preserving classification performance.