Special issue on the theory and practice of differential privacy

This special issue presents papers based on contributions to the first international workshop on the “Theory and Practice of Differential Privacy” (TPDP) held in London, UK, 18 April 2015, as part of the European joint conference on Theory And Practice of Software (ETAPS). Differential privacy is a mathematically rigorous definition of the privacy protection provided by a data release mechanism: it offers a strong guaranteed bound on what can be learned about a user as a result of participating in a differentially private data analysis. Researchers in differential privacy come from several areas of computer science, including algorithms, programming languages, security, databases and machine learning, as well as from several areas of statistics and data analysis. The workshop was intended to be an occasion for researchers from these different research areas to discuss the recent developments in the theory and practice of differential privacy. The program of the workshop included 10 contributed talks, 1 invited speaker and 1 joint invited speaker with the workshop “Hot Issues in Security Principles and Trust” (HotSpot 2016). Participants at the workshop were invited to submit papers to this special issue. Six papers were accepted, most of which directly reflect talks presented at the workshop

Formal verification of higher-order probabilistic programs

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on semantical issues. In this paper, we take a step further and we develop a set of program logics, named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous distributions and operators for conditioning distributions by real-valued functions. Pleasingly, our program logics retain the comfortable reasoning style of informal proofs thanks to carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show the expressiveness of our logics by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) TT-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs

Really Natural Linear Indexed Type Checking

Recent works have shown the power of linear indexed type systems for enforcing complex program properties. These systems combine linear types with a language of type-level indices, allowing more fine-grained analyses. Such systems have been fruitfully applied in diverse domains, including implicit complexity and differential privacy. A natural way to enhance the expressiveness of this approach is by allowing the indices to depend on runtime information, in the spirit of dependent types. This approach is used in DFuzz, a language for differential privacy. The DFuzz type system relies on an index language supporting real and natural number arithmetic over constants and variables. Moreover, DFuzz uses a subtyping mechanism to make types more flexible. By themselves, linearity, dependency, and subtyping each require delicate handling when performing type checking or type inference; their combination increases this challenge substantially, as the features can interact in non-trivial ways. In this paper, we study the type-checking problem for DFuzz. We show how we can reduce type checking for (a simple extension of) DFuzz to constraint solving over a first-order theory of naturals and real numbers which, although undecidable, can often be handled in practice by standard numeric solvers

A Theory AB Toolbox

Randomized algorithms are a staple of the theoretical computer science literature. By careful use of randomness, algorithms can achieve properties that are simply not possible with deterministic algorithms. Today, these properties are proved on paper, by theoretical computer scientists; we investigate formally verifying these proofs. The main challenges are two: proofs about algorithms can be quite complex, using various facts from probability theory; and proofs are highly customized - two proofs of the same property for two algorithms can be completely different. To overcome these challenges, we propose taking inspiration from paper proofs, by building common tools - abstractions, reasoning principles, perhaps even notations - into a formal verification toolbox. To give an idea of our approach, we consider three common patterns in paper proofs: the union bound, concentration bounds, and martingale arguments

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