Advanced Probabilistic Couplings for Differential Privacy

Differential privacy is a promising formal approach to data privacy, which provides a quantitative bound on the privacy cost of an algorithm that operates on sensitive information. Several tools have been developed for the formal verification of differentially private algorithms, including program logics and type systems. However, these tools do not capture fundamental techniques that have emerged in recent years, and cannot be used for reasoning about cutting-edge differentially private algorithms. Existing techniques fail to handle three broad classes of algorithms: 1) algorithms where privacy depends accuracy guarantees, 2) algorithms that are analyzed with the advanced composition theorem, which shows slower growth in the privacy cost, 3) algorithms that interactively accept adaptive inputs. We address these limitations with a new formalism extending apRHL, a relational program logic that has been used for proving differential privacy of non-interactive algorithms, and incorporating aHL, a (non-relational) program logic for accuracy properties. We illustrate our approach through a single running example, which exemplifies the three classes of algorithms and explores new variants of the Sparse Vector technique, a well-studied algorithm from the privacy literature. We implement our logic in EasyCrypt, and formally verify privacy. We also introduce a novel coupling technique called \emph{optimal subset coupling} that may be of independent interest

Probabilistic Couplings For Probabilistic Reasoning

This thesis explores proofs by coupling from the perspective of formal verification. Long employed in probability theory and theoretical computer science, these proofs construct couplings between the output distributions of two probabilistic processes. Couplings can imply various probabilistic relational properties, guarantees that compare two runs of a probabilistic computation. To give a formal account of this clean proof technique, we first show that proofs in the program logic pRHL (probabilistic Relational Hoare Logic) describe couplings. We formalize couplings that establish various probabilistic properties, including distribution equivalence, convergence, and stochastic domination. Then we deepen the connection between couplings and pRHL by giving a proofs-as-programs interpretation: a coupling proof encodes a probabilistic product program, whose properties imply relational properties of the original two programs. We design the logic xpRHL (product pRHL) to build the product program, with extensions to model more advanced constructions including shift coupling and path coupling. We then develop an approximate version of probabilistic coupling, based on approximate liftings. It is known that the existence of an approximate lifting implies differential privacy, a relational notion of statistical privacy. We propose a corresponding proof technique---proof by approximate coupling---inspired by the logic apRHL, a version of pRHL for building approximate liftings. Drawing on ideas from existing privacy proofs, we extend apRHL with novel proof rules for constructing new approximate couplings. We give approximate coupling proofs of privacy for the Report-noisy-max and Sparse Vector mechanisms, well-known algorithms from the privacy literature with notoriously subtle privacy proofs, and produce the first formalized proof of privacy for these algorithms in apRHL. Finally, we enrich the theory of approximate couplings with several more sophisticated constructions: a principle for showing accuracy-dependent privacy, a generalization of the advanced composition theorem from differential privacy, and an optimal approximate coupling relating two subsets of samples. We also show equivalences between approximate couplings and other existing definitions. These ingredients support the first formalized proof of privacy for the Between Thresholds mechanism, an extension of the Sparse Vector mechanism

*-Liftings for Differential Privacy

Recent developments in formal verification have identified approximate liftings (also known as approximate couplings) as a clean, compositional abstraction for proving differential privacy. There are two styles of definitions for this construction. Earlier definitions require the existence of one or more witness distributions, while a recent definition by Sato uses universal quantification over all sets of samples. These notions have different strengths and weaknesses: the universal version is more general than the existential ones, but the existential versions enjoy more precise composition principles. We propose a novel, existential version of approximate lifting, called *-lifting, and show that it is equivalent to Sato\u27s construction for discrete probability measures. Our work unifies all known notions of approximate lifting, giving cleaner properties, more general constructions, and more precise composition theorems for both styles of lifting, enabling richer proofs of differential privacy. We also clarify the relation between existing definitions of approximate lifting, and generalize our constructions to approximate liftings based on f-divergences