We propose an end-to-end deep learning method that learns to estimate emphysema extent from proportions of the diseased tissue. These proportions were visually estimated by experts using a standard grading system, in which grades correspond to intervals (label example: 1–5% of diseased tissue). The proposed architecture encodes the knowledge that the labels represent a volumetric proportion. A custom loss is designed to learn with intervals. Thus, during training, our network learns to segment the diseased tissue such that its proportions fit the ground truth intervals. Our architecture and loss combined improve the performance substantially (8% ICC) compared to a more conventional regression network. We outperform traditional lung densitometry and two recently published methods for emphysema quantification by a large margin (at least 7% AUC and 15% ICC), and achieve near-human-level performance. Moreover, our method generates emphysema segmentations that predict the spatial distribution of emphysema at human level.