Data minimization is a legal obligation defined in the European Union's General Data Protection Regulation (GDPR) as the responsibility to process an adequate, relevant, and limited amount of personal data in relation to a processing purpose. However, unlike fairness or transparency, the principle has not seen wide adoption for machine learning systems due to a lack of computational interpretation. In this paper, we build on literature in machine learning and law to propose the first learning framework for limiting data collection based on an interpretation that ties the data collection purpose to system performance. We formalize a data minimization criterion based on performance curve derivatives and provide an effective and interpretable piecewise power law technique that models distinct stages of an algorithm's performance throughout data collection. Results from our empirical investigation offer deeper insights into the relevant considerations when designing a data minimization framework, including the choice of feature acquisition algorithm, initialization conditions, as well as impacts on individuals that hint at tensions between data minimization and fairness.