Linear dependent types for differential privacy

Abstract

Differential privacy offers a way to answer queries about sensitive information while providing strong, provable privacy guarantees, ensuring that the presence or absence of a single individual in the database has a negligible statistical effect on the query's result. Proving that a given query has this property involves establishing a bound on the query's sensitivity-how much its result can change when a single record is added or removed.A variety of tools have been developed for certifying that a given query is differentially private. In one approach, Reed and Pierce [34] proposed a functional programming language, Fuzz, for writing differentially private queries. Fuzz uses linear types to track sensitivity and a probability monad to express randomized computation; it guarantees that any program with a certain type is differentially private. Fuzz can successfully verify many useful queries. However, it fails when the sensitivity analysis depends on values that are not known statically.We present DFuzz, an extension of Fuzz with a combination of linear indexed types and lightweight dependent types. This combination allows a richer sensitivity analysis that is able to certify a larger class of queries as differentially private, including ones whose sensitivity depends on runtime information. As in Fuzz, the differential privacy guarantee follows directly from the soundness theorem of the type system. We demonstrate the enhanced expressivity of DFuzz by certifying differential privacy for a broad class of iterative algorithms that could not be typed previously.

Relational cost analysis for functional-imperative programs

“…[2010] present a differential static analysis to find code defects looking at two pieces of code relationally. Probabilistic relational verification has seen many applications in cryptography [Barthe et al 2014] and differential privacy [Barthe et al 2015;Gaboardi et al 2013]. The indexed types used by Gaboardi et al [2013] are similar in spirit to ours.…”

Duet: an expressive higher-order language and linear type system for statically enforcing differential privacy

“…Type-system-based solutions to proving that a program adheres to differential privacy began with Reed and Pierce's Fuzz language [44], which is based on linear typing. Fuzz, as well as subsequent work based on linear types aided by SMT solvers [29], supports type inference of privacy bounds with type-level dependency and higher-order composition of programs. However, these systems only support the original and most basic variant of differential privacy called ϵ-differential privacy.…”

“…Higher-order Relational Type Systems. Following the initial work on linear typing for differential privacy [44], a parallel line of work [6,8] leverages relational refinement types aided by SMT solvers in order to support type-level dependency of privacy parameters (à la DFuzz [29]) in addition to more powerful variants of differential privacy such as (ϵ, δ )-differential privacy. These approaches support (ϵ, δ )-differential privacy, but did not support usable type inference until a recently proposed heuristic bi-directional type system [19].…”

“…Duet consists of two languages: one for tracking sensitivities (typeset in green), and one for tracking privacy cost (typeset in red). The sensitivity language is similar to that of DFuzz [29]; its typing rules track the sensitivity of each variable by annotating the context. For example, the expression x + x is 2-sensitive in x; the typing rules in Figure 4 allow us to conclude:…”

“…Duet's sensitivity language is similar to that of DFuzz [29], except that we extend it with significant new tools for machine learning in Section 4. We do not present standard linear logic connectives such as sums, additive products and multiplicative products (a la Fuzz), or symbolic type-level expressions (a la DFuzz), although each are implemented in our tool and described formally in the extended version of this paper.…”