  We examine machine learning models in a setup where individuals have the
choice to share optional personal information with a decision-making system, as
seen in modern insurance pricing models. Some users consent to their data being
used whereas others object and keep their data undisclosed. In this work, we
show that the decision not to share data can be considered as information in
itself that should be protected to respect users' privacy. This observation
raises the overlooked problem of how to ensure that users who protect their
personal data do not suffer any disadvantages as a result. To address this
problem, we formalize protection requirements for models which only use the
information for which active user consent was obtained. This excludes implicit
information contained in the decision to share data or not. We offer the first
solution to this problem by proposing the notion of Protected User Consent
(PUC), which we prove to be loss-optimal under our protection requirement. We
observe that privacy and performance are not fundamentally at odds with each
other and that it is possible for a decision maker to benefit from additional
data while respecting users' consent. To learn PUC-compliant models, we devise
a model-agnostic data augmentation strategy with finite sample convergence
guarantees. Finally, we analyze the implications of PUC on challenging real
datasets, tasks, and models.
