Fuzzi: A Three-Level Logic for Differential Privacy

Curators of sensitive datasets sometimes need to know whether queries against the data are differentially private [Dwork et al. 2006]. Two sorts of logics have been proposed for checking this property: (1) type systems and other static analyses, which fully automate straightforward reasoning with concepts like "program sensitivity" and "privacy loss," and (2) full-blown program logics such as apRHL (an approximate, probabilistic, relational Hoare logic) [Barthe et al. 2016], which support more flexible reasoning about subtle privacy-preserving algorithmic techniques but offer only minimal automation. We propose a three-level logic for differential privacy in an imperative setting and present a prototype implementation called Fuzzi. Fuzzi's lowest level is a general-purpose logic; its middle level is apRHL; and its top level is a novel sensitivity logic adapted from the linear-logic-inspired type system of Fuzz, a differentially private functional language [Reed and Pierce 2010]. The key novelty is a high degree of integration between the sensitivity logic and the two lower-level logics: the judgments and proofs of the sensitivity logic can be easily translated into apRHL; conversely, privacy properties of key algorithmic building blocks can be proved manually in apRHL and the base logic, then packaged up as typing rules that can be applied by a checker for the sensitivity logic to automatically construct privacy proofs for composite programs of arbitrary size. We demonstrate Fuzzi's utility by implementing four different private machine-learning algorithms and showing that Fuzzi's checker is able to derive tight sensitivity bounds.