On the Robustness of Information-Theoretic Privacy Measures and Mechanisms

Abstract

Consider a data publishing setting for a dataset composed by both private and non-private features. The publisher uses an empirical distribution, estimated from n i.i.d. samples, to design a privacy mechanism which is applied to new fresh samples afterward. In this paper, we study the discrepancy between the privacy-utility guarantees for the empirical distribution, used to design the privacy mechanism, and those for the true distribution, experienced by the privacy mechanism in practice. We first show that, for any privacy mechanism, these discrepancies vanish at speed O(1/ √ n) with high probability. These bounds follow from our main technical results regarding the Lipschitz continuity of the considered information leakage measures. Then we prove that the optimal privacy mechanisms for the empirical distribution approach the corresponding mechanisms for the true distribution as the sample size n increases, thereby establishing the statistical consistency of the optimal privacy mechanisms. Finally, we introduce and study uniform privacy mechanisms which, by construction, provide privacy to all the distributions within a neighborhood of the estimated distribution and, thereby, guarantee privacy for the true distribution with high probability.

Bottleneck Problems: An Information and Estimation-Theoretic View

“…Following this observation, we introduced a large family of optimization problems, which we call bottleneck problems, by replacing mutual information in IB and PF with Arimoto's mutual information [21] or f -information [22]. Invoking results from [31], [32], we also demonstrated that these information measures are in general easier to estimate from data than mutual information. Similar to IB and PF, the bottleneck problems were shown to be fully characterized by boundaries of a two-dimensional convex set parameterized by two real-valued non-negative functions Φ and Ψ.…”

“…These bounds can then be used to shed light on the de facto guarantee of the bottleneck problems. Relying on [ 34 ] (Theorem 1), one can obtain that the gaps between the measures , , and computed on empirical distributions and the true one scale as where n is the number of samples. This is in contrast with mutual information for which the similar upper bound scales as as shown in [ 33 ].…”

“…Finally, when is unknown, mutual information is known to be notoriously difficult to estimate. Nevertheless, Arimoto’s mutual information and f -information are easier to estimate: While mutual information can be estimated with estimation error that scales as [ 33 ], Diaz et al [ 34 ] showed that this estimation error for Arimoto’s mutual information and f -information is . We also generalize our computation technique that enables us to analytically compute these bottleneck problems.…”

The Roadmap to 6G Security and Privacy

“…Quantifying this information leakage is important in order to limit this. Different notions of privacy leakage have been proposed to capture the capacity of adversaries to estimate private information, for example, Shannon's mutual information, differential privacy, among others [159], as well as different leakage measures. In that sense, privacy can, under careful control, tolerate some leakage to get some utility.…”

Finite-Sample Concentration of the Multinomial in Relative Entropy

“…To apply this test in practice an upper bound on ε α is needed, so to maximize the power of a provably correct finite-sample test we seek upper bounds on Pr[V ≥ ε] which are meaningful (less than 1) for ε as small as possible. Equivalently, tight control on ε reduces the number of samples needed to obtain a given level of significance, which is of importance in areas as disparate as high-dimensional statistics [4], combinatorial constructions in complexity theory [5], and private machine learning [6].…”