Iterative Constructions and Private Data Release

Abstract

In this paper we study the problem of approximately releasing the cut function of a graph while preserving differential privacy, and give new algorithms (and new analyses of existing algorithms) in both the interactive and non-interactive settings.Our algorithms in the interactive setting are achieved by revisiting the problem of releasing differentially private, approximate answers to a large number of queries on a database. We show that several algorithms for this problem fall into the same basic framework, and are based on the existence of objects which we call iterative database construction algorithms. We give a new generic framework in which new (efficient) IDC algorithms give rise to new (efficient) interactive private query release mechanisms. Our modular analysis simplifies and tightens the analysis of previous algorithms, leading to improved bounds. We then give a new IDC algorithm (and therefore a new private, interactive query release mechanism) based on the Frieze/Kannan low-rank matrix decomposition. This new release mechanism gives an improvement on prior work in a range of parameters where the size of the database is comparable to the size of the data universe (such as releasing all cut queries on dense graphs).We also give a non-interactive algorithm for efficiently releasing private synthetic data for graph cuts with error O(|V | 1.5 ). Our algorithm is based on randomized response and a nonprivate implementation of the SDP-based, constant-factor approximation algorithm for cut-norm due to Alon and Naor. Finally, we give a reduction based on the IDC framework showing that an efficient, private algorithm for computing sufficiently accurate rank-1 matrix approximations would lead to an improved efficient algorithm for releasing private synthetic data for graph cuts. We leave finding such an algorithm as our main open problem.

Distributed Private Heavy Hitters

“…[GRU12]). Let X i , i ∈ [n] be IID random variables drawn from the Lap(b) (the Laplace distribution with parameter b) and let X = n i=1 X i .…”

The Privacy of the Analyst and the Power of the State

“…Our algorithm is based on a nested version of the Private Multiplicative Weights (PMW) algorithm of Hardt and Rothblum [15], which achieves the best parameters of any known stateful differentially private algorithm (using its analysis from [24]; a simpler proof appears in [25]):…”

Linear dependent types for differential privacy

“…Each algorithm is taken from a different paper from the differential privacy literature (specifically, [4,18,19,27]). The first three algorithms rely on the following new feature that is enabled by DFuzz's dependent types:…”

“…A k-means program that performs a different number of iterations would have a different type. Worse, in Section 6, we will see several examples of practical algorithms whose types cannot be expressed in Fuzz, e.g., the IDC algorithm [19], in which the desired sensitivity is itself a parameter. This means that we cannot even add such operations to Fuzz as primitives.…”

“…Most importantly, our type system (like that of Fuzz) natively captures the concept of differential privacy, since DFuzz enjoys an extension of the metric preservation property of [33,34]. To demonstrate the capabilities of DFuzz, we show that it can certify several examples from the differential privacy literature quite naturally, including Iterative Database Construction [19], k-means [4], k-medians [18], and the Cumulative Distribution Function [27].…”