Testing the Lipschitz Property over Product Distributions with Applications to Data Privacy

Abstract

Abstract. In the past few years, the focus of research in the area of statistical data privacy has been in designing algorithms for various problems which satisfy some rigorous notions of privacy. However, not much effort has gone into designing techniques to computationally verify if a given algorithm satisfies some predefined notion of privacy. In this work, we address the following question: Can we design algorithms which tests if a given algorithm satisfies some specific rigorous notion of privacy (e.g., differential privacy)?We design algorithms to test privacy guarantees of a given algorithm A when run on a dataset x containing potentially sensitive information about the individuals. More formally, we design a computationally efficient algorithm Tpriv that verifies whether A satisfies differential privacy on typical datasets (DPTD) guarantee in time sublinear in the size of the domain of the datasets. DPTD, a similar notion to generalized differential privacy first proposed by [3], is a distributional relaxation of the popular notion of differential privacy [14].To design algorithm Tpriv, we show a formal connection between the testing of privacy guarantee for an algorithm and the testing of the Lipschitz property of a related function. More specifically, we show that an efficient algorithm for testing of Lipschitz property can be used as a subroutine in Tpriv that tests if an algorithm satisfies differential privacy on typical datasets.Apart from formalizing the connection between the testing of privacy guarantee and testing of the Lipschitz property, we generalize the work of [21] to the setting of property testing under product distribution. More precisely, we design an efficient Lipschitz tester for the case where the domain points are drawn from hypercube according to some fixed but unknown product distribution instead of the uniform distribution.

L p -testing

“…The c-Lipschitz properties are a subject of a recent wave of investigation [21,2,8,5,7], with a focus on hypergrid domains, due to their applications to differential privacy [10]. Definition 1.5.…”

“…We can get a reduction that works for adaptive algorithms by viewing L1-testing monotonicity as a multi-input concatenation problem[16]. This reduction preserves the query complexity for the special class of proximityoblivious testers[8], but incurs a loss of O 1 ε in general and, specifically, when applied to our adaptive tester. As this approach would not improve our results, we focus on reductions for nonadaptive algorithms.…”

Property Testing For Differential Privacy

“…We focus on the two extreme settings: when we are given no information about the black-box (except the domain and range), and the full information setting where we have an untrusted full description of the algorithm A. A similar formulation of DP in the property testing framework was first introduced in Dixit et al (2013), who consider testing for DP given oracle access to the probability density functions on outputs. Dixit et al (2013) reduce this version of the problem to testing the Lipschitz property of functions and make progress on this more general problem.…”

Property Testing on Product Distributions: Optimal Testers for Bounded Derivative Properties

“…A distribution on databases is given by independent priors on each individual. The goal in [DJRT13] is to distinguish private mechanisms from those that aren't private on 'typical' databases. Random testing of hardware: Given an actual silicon implementation of a circuit, it is standard practice for engineers to test it on a set of random instances.…”

“…The study of Lipschitz continuity in property testing is more recent [JR11, AJMR12, CS13a, DJRT13,BRY14b]. With the exception of [HK07,HK08b,AC06,DJRT13], all the previous works are in the uniform distribution setting, for which the story is mostly clear: there is an O(ε −1 d log n)-query tester for both properties [CS13a], and this is optimal for monotonicity [CS13b]. For Lipschitz continuity, an Ω(d log n) non-adaptive lower bound has been proved [BRY14b] recently.…”