The bootstrap is a common and powerful statistical tool for numerically computing the standard error of estimators, that is, a calculation of the uncertainty of functions computed on sample data so as to make an inference back to the original population from which the sample was drawn. Understanding uncertainty, and inferential questions, in the context of private data is an increasingly important task within the literature of differential privacy [7, 20, 15]. We show how to construct an implementation of the bootstrap within differential privacy. Most importantly, we show that, for a broad class of functions under zero concentrated differential privacy, the bootstrap can be implemented at no cost. That is, for a given choice of privacy parameter and associated expected error of some query, the bootstrap can be implemented for the exact same privacy guarantee, resulting in the same expected error (or sometimes less) in the desired query, but additionally provide the standard error of that query. In section 2 we provide a brief overview of differential privacy. Then to describe these results on bootstrap inference, in section 3 we describe some foundational results on the aggregation of repeated queries under contrasting privacy and composition definitions. This leads to a tangential result in section 4 on a low-noise Gaussian mechanism for pure differential privacy. Next we provide a brief foundation on the bootstrap algorithm in statistics in section 5, before showing our algorithmic construction of the bootstrap using the mechanisms of differential privacy in section 6. In section 7 we describe how to use the differentially private estimate of the standard error in the construction of confidence intervals and hypothesis tests, and then demonstrate this in section 8 with examples using published Census microdata in the style of privacy sensitive data.
Statistical analysis of network data, while popular in a broad range of fields, can also be highly problematic from a privacy standpoint. In this thesis, we study privacy-preserving inference on network data using the rigorous notion of differential privacy. We propose new methods for differentially private inference using a common class of models known as Exponential Random Graph Models (ERGMs). The goal of our work is to accurately estimate the parameters of an ERGM applied to a network dataset, while offering meaningful privacy guarantees to participants. We propose methods that provably guarantee differential privacy at two different granularities: edge-level privacy, which protects the privacy of any single relationship in the network and node-level privacy, which protects all of the relationships of a participant. Specifically, using the framework of "restricted sensitivity," we take advantage of the sparsity of real-world networks to perturb data much less than prior work while guaranteeing differential privacy. We empirically evaluate the accuracy of inference in a series of experiments on both synthetic networks and a real network dataset. Experimental results suggest that our proposed methods enable accurate inference under meaningful privacy guarantees in settings where current methods do not, moving us closer to the goal of useful differentially private statistical modeling of network data.
The fields of law and computer science incorporate contrasting notions of the privacy risks associated with the analysis and release of statistical data about individuals and groups of individuals. Emerging concepts from the theoretical computer science literature provide formal mathematical models for quantifying and mitigating privacy risks, where the set of risks they take into account is much broader than the privacy risks contemplated by many privacy laws. An example of such a model is differential privacy, which provides a provable guarantee of privacy against a wide range of potential attacks, including types of attacks currently unknown or unforeseen. The subject of much theoretical investigation, new privacy technologies based on formal models have recently been making significant strides towards practical implementation. For these tools to be used with sensitive personal information, it is important to demonstrate that they satisfy relevant legal requirements for privacy protection. However, making such an argument is challenging due to the conceptual gaps between the legal and technical approaches to defining privacy. Notably, information privacy laws are generally subject to interpretation and some degree of flexibility, which creates uncertainty for the implementation of more formal approaches. This Article articulates the gaps between legal and technical approaches to privacy and presents a methodology for rigorously arguing that a technological method for privacy protection satisfies the requirements of a particular law. The proposed methodology has two main components: (i) extraction of a formal mathematical requirement of privacy based on a legal standard found in an information privacy law, and (ii) construction of a rigorous mathematical proof for establishing that a technological privacy solution satisfies the mathematical requirement derived from the law. To handle ambiguities that can lead to different interpretations of a legal standard, the methodology takes a conservative “worst-case” approach and attempts to extract a mathematical requirement that is robust to potential ambiguities. Under this approach, the mathematical proof demonstrates that a technological method satisfies a broad range of reasonable interpretations of a legal standard. The Article demonstrates the application of the proposed methodology with an example bridging between the requirements of the Family Educational Rights and Privacy Act of 1974 and differential privacy.
In the study of differential privacy, composition theorems (starting with the original paper of Dwork, McSherry, Nissim, and Smith (TCC'06)) bound the degradation of privacy when composing several differentially private algorithms. Kairouz, Oh, and Viswanath (ICML'15) showed how to compute the optimal bound for composing k arbitrary (ϵ,δ)-differentially private algorithms. We characterize the optimal composition for the more general case of k arbitrary (ϵ1,δ1),…,(ϵk,δk)-differentially private algorithms where the privacy parameters may differ for each algorithm in the composition. We show that computing the optimal composition in general is #P-complete. Since computing optimal composition exactly is infeasible (unless FP=#P), we give an approximation algorithm that computes the composition to arbitrary accuracy in polynomial time. The algorithm is a modification of Dyer's dynamic programming approach to approximately counting solutions to knapsack problems (STOC'03).
This document is a primer on differential privacy, which is a formal mathematical framework for guaranteeing privacy protection when analyzing or releasing statistical data. Recently emerging from the theoretical computer science literature, differential privacy is now in initial stages of implementation and use in various academic, industry, and government settings. Using intuitive illustrations and limited mathematical formalism, this document provides an introduction to differential privacy for non-technical practitioners, who are increasingly tasked with making decisions with respect to differential privacy as it grows more widespread in use. In particular, the examples in this document illustrate ways in which social scientists can conceptualize the guarantees provided by differential privacy with respect to the decisions they make when managing personal data about research subjects and informing them about the privacy protection they will be afforded.
We study the problem of estimating finite sample confidence intervals of the mean of a normal population under the constraint of differential privacy. We consider both the known and unknown variance cases and construct differentially private algorithms to estimate confidence intervals. Crucially, our algorithms guarantee a finite sample coverage, as opposed to an asymptotic coverage. Unlike most previous differentially private algorithms, we do not require the domain of the samples to be bounded. We also prove lower bounds on the expected size of any differentially private confidence set showing that our the parameters are optimal up to polylogarithmic factors.
This position paper observes how different technical and normative conceptions of privacy have evolved in parallel and describes the practical challenges that these divergent approaches pose. Notably, past technologies relied on intuitive, heuristic understandings of privacy that have since been shown not to satisfy expectations for privacy protection. With computations ubiquitously integrated in almost every aspect of our lives, it is increasingly important to ensure that privacy technologies provide protection that is in line with relevant social norms and normative expectations. Similarly, it is also important to examine social norms and normative expectations with respect to the evolving scientific study of privacy. To this end, we argue for a rigorous analysis of the mapping from normative to technical concepts of privacy and vice versa. We review the landscape of normative and technical definitions of privacy and discuss specific examples of gaps between definitions that are relevant in the context of privacy in statistical computation. We then identify opportunities for overcoming their differences in the design of new approaches to protecting privacy in accordance with both technical and normative standards.
The transparency goals of the open data movement serve important social, economic, and democratic functions in cities like Seattle. At the same time, some municipal datasets about the city and its citizens’ activities carry inherent risks to individual privacy when shared publicly. In 2016, the City of Seattle declared in its Open Data Policy that the city’s data would be “open by preference,” except when doing so may affect individual privacy. To ensure its Open Data program effectively protects individuals, Seattle committed to performing an annual risk assessment and tasked the Future of Privacy Forum (FPF) with creating and deploying an initial privacy risk assessment methodology for open data.
This Draft Report provides tools and guidance to the City of Seattle and other municipalities navigating the complex policy, operational, technical, organizational, and ethical standards that support privacyprotective open data programs. Although there is a growing body of research on open data privacy, open data managers and departmental data owners need to be able to employ a standardized methodology for assessing the privacy risks and benefits of particular datasets internally, without a bevy of expert statisticians, privacy lawyers, or philosophers. By following a flexible, risk-based assessment process, the City of Seattle – and other municipal open data programs – can maximize the utility and openness of civic data while minimizing privacy risks to individuals and community concerns about ethical challenges, fairness, and equity.
This Draft Report first describes inherent privacy risks in an open data landscape, with an emphasis on potential harms related to re-identification, data quality, and fairness. Accompanying this, the Draft Report includes a Model Open Data Benefit Risk Analysis (MODBRA). The model template evaluates the types of data contained in a proposed open dataset, the potential benefits – and concomitant risks – of releasing the dataset publicly, and strategies for effective de-identification and risk mitigation. This holistic assessment guides city officials to determine whether to release the dataset openly, in a limited access environment, or to withhold it from publication (absent countervailing public policy considerations). The Draft Report methodology builds on extensive work done in this field by experts at the National Institute of Standards and Technology, the University of Washington, the Berkman Klein Center for Internet & Society at Harvard University, and others, and adapts existing frameworks to the unique challenges faced by cities as local governments, technological system integrators, and consumer facing service providers.
Ezgi Cicek, Gilles Barthe, Marco Gaboardi, Deepak Garg, and Jan Hoffmann. 1/2017. “Relational Cost Analysis.” Symposium on the Principle of Programming Languages, ACM.Abstract
Establishing quantitative bounds on the execution cost of programs is essential in many areas of computer science such as complexity analysis, compiler optimizations, security and privacy. Techniques based on program analysis, type systems and abstract interpretation are well-studied, but methods for analyzing how the execution costs of two programs compare to each other have not received attention. Naively combining the worst and best case execution costs of the two programs does not work well in many cases because such analysis forgets the similarities between the programs or the inputs. In this work, we propose a relational cost analysis technique that is capable of establishing precise bounds on the difference in the execution cost of two programs by making use of relational properties of programs and inputs. We develop RelCost, a refinement type and effect system for a higher-order functional language with recursion and subtyping. The key novelty of our technique is the combination of relational refinements with two modes of typing—relational typing for reasoning about similar computations/inputs and unary typing for reasoning about unrelated computations/inputs. This combination allows us to analyze the execution cost difference of two programs more precisely than a naive non-relational approach. We prove our type system sound using a semantic model based on step-indexed unary and binary logical relations accounting for non-relational and relational reasoning principles with their respective costs. We demonstrate the precision and generality of our technique through examples.
Program sensitivity measures how robust a program is to small changes in its input, and is a fundamental notion in domains ranging from differential privacy to cyber-physical systems. A natural way to formalize program sensitivity is in terms of metrics on the input and output spaces, requiring that an r-sensitive function map inputs that are at distance d to outputs that are at distance at most r⋅d. Program sensitivity is thus an analogue of Lipschitz continuity for programs. Reed and Pierce introduced Fuzz, a functional language with a linear type system that can express program sensitivity. They show soundness operationally, in the form of a metric preservation property. Inspired by their work, we study program sensitivity and metric preservation from a denotational point of view. In particular, we introduce metric CPOs, a novel semantic structure for reasoning about computation on metric spaces, by endowing CPOs with a compatible notion of distance. This structure is useful for reasoning about metric properties of programs, and specifically about program sensitivity. We demonstrate metric CPOs by giving a model for the deterministic fragment of Fuzz.
We show a new lower bound on the sample complexity of (ε, δ)-differentially private algorithms that accurately answer statistical queries on high-dimensional databases. The novelty of our bound is that it depends optimally on the parameter δ, which loosely corresponds to the probability that the algorithm fails to be private, and is the first to smoothly interpolate between approximate differential privacy (δ > 0) and pure differential privacy (δ = 0).