Hypothesis Testing Interpretations and Renyi Differential Privacy

Differential privacy is a de facto standard in data privacy, with applications in the public and private sectors. A way to explain differential privacy, which is particularly appealing to statistician and social scientists is by means of its statistical hypothesis testing interpretation. Informally, one cannot effectively test whether a specific individual has contributed her data by observing the output of a private mechanism---any test cannot have both high significance and high power. In this paper, we identify some conditions under which a privacy definition given in terms of a statistical divergence satisfies a similar interpretation. These conditions are useful to analyze the distinguishability power of divergences and we use them to study the hypothesis testing interpretation of some relaxations of differential privacy based on Renyi divergence. This analysis also results in an improved conversion rule between these definitions and differential privacy

Synthesizing Probabilistic Invariants via Doob's Decomposition

When analyzing probabilistic computations, a powerful approach is to first find a martingale---an expression on the program variables whose expectation remains invariant---and then apply the optional stopping theorem in order to infer properties at termination time. One of the main challenges, then, is to systematically find martingales. We propose a novel procedure to synthesize martingale expressions from an arbitrary initial expression. Contrary to state-of-the-art approaches, we do not rely on constraint solving. Instead, we use a symbolic construction based on Doob's decomposition. This procedure can produce very complex martingales, expressed in terms of conditional expectations. We show how to automatically generate and simplify these martingales, as well as how to apply the optional stopping theorem to infer properties at termination time. This last step typically involves some simplification steps, and is usually done manually in current approaches. We implement our techniques in a prototype tool and demonstrate our process on several classical examples. Some of them go beyond the capability of current semi-automatic approaches

Proving uniformity and independence by self-composition and coupling

Proof by coupling is a classical proof technique for establishing probabilistic properties of two probabilistic processes, like stochastic dominance and rapid mixing of Markov chains. More recently, couplings have been investigated as a useful abstraction for formal reasoning about relational properties of probabilistic programs, in particular for modeling reduction-based cryptographic proofs and for verifying differential privacy. In this paper, we demonstrate that probabilistic couplings can be used for verifying non-relational probabilistic properties. Specifically, we show that the program logic pRHL---whose proofs are formal versions of proofs by coupling---can be used for formalizing uniformity and probabilistic independence. We formally verify our main examples using the EasyCrypt proof assistant

A Relational Logic for Higher-Order Programs

Relational program verification is a variant of program verification where one can reason about two programs and as a special case about two executions of a single program on different inputs. Relational program verification can be used for reasoning about a broad range of properties, including equivalence and refinement, and specialized notions such as continuity, information flow security or relative cost. In a higher-order setting, relational program verification can be achieved using relational refinement type systems, a form of refinement types where assertions have a relational interpretation. Relational refinement type systems excel at relating structurally equivalent terms but provide limited support for relating terms with very different structures. We present a logic, called Relational Higher Order Logic (RHOL), for proving relational properties of a simply typed $\lambda$-calculus with inductive types and recursive definitions. RHOL retains the type-directed flavour of relational refinement type systems but achieves greater expressivity through rules which simultaneously reason about the two terms as well as rules which only contemplate one of the two terms. We show that RHOL has strong foundations, by proving an equivalence with higher-order logic (HOL), and leverage this equivalence to derive key meta-theoretical properties: subject reduction, admissibility of a transitivity rule and set-theoretical soundness. Moreover, we define sound embeddings for several existing relational type systems such as relational refinement types and type systems for dependency analysis and relative cost, and we verify examples that were out of reach of prior work.Comment: Submitted to ICFP 201