Interpretability is a pressing issue for machine learning. Common approaches to interpretable machine learning constrain interactions between features of the input, sacrificing model complexity in order to render more comprehensible the effects of those features on the model's output. We approach interpretability from a new angle: constrain the information about the features without restricting the complexity of the model. We use the Distributed Information Bottleneck to optimally compress each feature so as to maximally preserve information about the output. The learned information allocation, by feature and by feature value, provides rich opportunities for interpretation, particularly in problems with many features and complex feature interactions. The central object of analysis is not a single trained model, but rather a spectrum of models serving as approximations that leverage variable amounts of information about the inputs. Information is allocated to features by their relevance to the output, thereby solving the problem of feature selection by constructing a learned continuum of feature inclusion-to-exclusion. The optimal compression of each feature---at every stage of approximation---allows fine-grained inspection of the distinctions among feature values that are most impactful for prediction. We develop a framework for extracting insight from the spectrum of approximate models and demonstrate its utility on a range of tabular datasets.