A scheme that publishes aggregate information about sensitive data must resolve the trade-off between utility to information consumers and privacy of the database participants. Differential privacy is a well-established definition of privacy--this is a universal guarantee against all attackers, whatever their side-information or intent. In this paper, we present a universal treatment of utility based on the standard minimax rule from decision theory (in contrast to the utility model in, which is Bayesian). In our model, information consumers are minimax (risk-averse) agents, each possessing some side-information about the query, and each endowed with a loss-function which models their tolerance to inaccuracies. Further, information consumers are rational in the sense that they actively combine information from the mechanism with their side-information in a way that minimizes their loss. Under this assumption of rational behavior, we show that for every fixed count query, a certain geometric mechanism is universally optimal for all minimax information consumers. Additionally, our solution makes it possible to release query results at multiple levels of privacy in a collusion-resistant manner.
A recent thorough survey of the state of the field of differential privacy is given in @cite_14 . Dinur and Nissim @cite_23 , @cite_19 establish upper-bounds on the number of queries that can be answered with reasonable accuracy. Most of the differential privacy literature circumvents these impossibility results by focusing on interactive models where a mechanism supplies answers to only a sub-linear (in @math ) number of queries. Count queries (e.g. @cite_23 @cite_15 ) and more general queries (e.g. @cite_8 @cite_10 ) have been studied from this perspective.
@cite_14 @cite_8 @cite_19 @cite_23 @cite_15 @cite_10
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We continue a line of research initiated in [10, 11] on privacy-preserving statistical databases. Consider a trusted server that holds a database of sensitive information. Given a query function f mapping databases to reals, the so-called true answer is the result of applying f to the database. To protect privacy, the true answer is perturbed by the addition of random noise generated according to a carefully chosen distribution, and this response, the true answer plus noise, is returned to the user. Previous work focused on the case of noisy sums, in which f = Σ i g(x i ), where x i denotes the ith row of the database and g maps database rows to [0,1]. We extend the study to general functions f, proving that privacy can be preserved by calibrating the standard deviation of the noise according to the sensitivity of the function f. Roughly speaking, this is the amount that any single argument to f can change its output. The new analysis shows that for several particular applications substantially less noise is needed than was previously understood to be the case. The first step is a very clean characterization of privacy in terms of indistinguishability of transcripts. Additionally, we obtain separation results showing the increased value of interactive sanitization mechanisms over non-interactive. This work is at theintersection of two lines of research. One line, initiated by Dinurand Nissim, investigates the price, in accuracy, of protecting privacy in a statistical database. The second, growing from an extensive literature on compressed sensing (see in particular the work of Donoho and collaborators [4,7,13,11])and explicitly connected to error-correcting codes by Candes and Tao ([4]; see also [5,3]), is in the use of linearprogramming for error correction. Our principal result is the discovery of a sharp threshhold ρ*∠ 0.239, so that if ρ In the context of privacy-preserving datamining our results say thatany privacy mechanism, interactive or non-interactive, providingreasonably accurate answers to a 0.761 fraction of randomly generated weighted subset sum queries, and arbitrary answers on the remaining 0.239 fraction, is blatantly non-private. We examine the tradeoff between privacy and usability of statistical databases. We model a statistical database by an n-bit string d 1 ,..,d n , with a query being a subset q ⊆ [n] to be answered by Σ ieq d i . Our main result is a polynomial reconstruction algorithm of data from noisy (perturbed) subset sums. Applying this reconstruction algorithm to statistical databases we show that in order to achieve privacy one has to add perturbation of magnitude (Ω√n). That is, smaller perturbation always results in a strong violation of privacy. We show that this result is tight by exemplifying access algorithms for statistical databases that preserve privacy while adding perturbation of magnitude O(√n).For time-T bounded adversaries we demonstrate a privacypreserving access algorithm whose perturbation magnitude is ≈ √T. In a recent paper Dinur and Nissim considered a statistical database in which a trusted database administrator monitors queries and introduces noise to the responses with the goal of maintaining data privacy [5]. Under a rigorous definition of breach of privacy, Dinur and Nissim proved that unless the total number of queries is sub-linear in the size of the database, a substantial amount of noise is required to avoid a breach, rendering the database almost useless. We introduce a new, generic framework for private data analysis.The goal of private data analysis is to release aggregate information about a data set while protecting the privacy of the individuals whose information the data set contains.Our framework allows one to release functions f of the data withinstance-based additive noise. That is, the noise magnitude is determined not only by the function we want to release, but also bythe database itself. One of the challenges is to ensure that the noise magnitude does not leak information about the database. To address that, we calibrate the noise magnitude to the smoothsensitivity of f on the database x --- a measure of variabilityof f in the neighborhood of the instance x. The new frameworkgreatly expands the applicability of output perturbation, a technique for protecting individuals' privacy by adding a smallamount of random noise to the released statistics. To our knowledge, this is the first formal analysis of the effect of instance-basednoise in the context of data privacy. Our framework raises many interesting algorithmic questions. Namely,to apply the framework one must compute or approximate the smoothsensitivity of f on x. We show how to do this efficiently for several different functions, including the median and the cost ofthe minimum spanning tree. We also give a generic procedure based on sampling that allows one to release f(x) accurately on manydatabases x. This procedure is applicable even when no efficient algorithm for approximating smooth sensitivity of f is known orwhen f is given as a black box. We illustrate the procedure by applying it to k-SED (k-means) clustering and learning mixtures of Gaussians.