Impossibility of Differentially Private Universally Optimal Mechanisms

Abstract

The notion of a universally utility-maximizing privacy mechanism was recently introduced by Ghosh, Roughgarden, and Sundararajan [STOC 2009]. These are mechanisms that guarantee optimal utility to a large class of information consumers, simultaneously, while preserving Differential Privacy [Dwork, McSherry, Nissim, and Smith, TCC 2006]. Ghosh et al. have demonstrated, quite surprisingly, a case where such a universally-optimal differentially-private mechanisms exists, when the information consumers are Bayesian. This result was recently extended by Gupte and Sundararajan [PODS 2010] to risk-averse consumers.Both positive results deal with mechanisms (approximately) computing a single count query (i.e., the number of individuals satisfying a specific property in a given population), and the starting point of our work is a trial at extending these results to similar settings, such as sum queries with non-binary individual values, histograms, and two (or more) count queries. We show, however, that universally-optimal mechanisms do not exist for all these queries, both for Bayesian and risk-averse consumers.For the Bayesian case, we go further, and give a characterization of those functions that admit universally-optimal mechanisms, showing that a universally-optimal mechanism exists, essentially, only for a (single) count query. At the heart of our proof is a representation of a query function f by its privacy constraint graph G f whose edges correspond to values resulting by applying f to neighboring databases.

Confidentiality and Differential Privacy in the Dissemination of Frequency Tables

“…More specifically, they show that Laplace with DP(ε) minimizes (a, b) among all DP(ε) mechanisms having the same range, provided (a, b) is nonnegative and nondecreasing in |a − b| for all a, the frequency in the single cell. This was followed by Brenner and Nissim (2010) where it is shown such universality does not extend beyond a single cell and, therefore, does not apply for tables as in this paper. Still, the Laplace perturbation seems to be a very efficient choice, better than the normal perturbations of the next section, in the sense of providing higher utility for a given DP level, as indicated also by our simulations and those of Liu (2017).…”

Broadening the Scope of Differential Privacy Using Metrics

“…On the other hand, a well-known impossibility result [8] states that counting queries are essentially the only ones for which an optimal mechanism exists. This result is based on the concept of the induced graph ∼ f of a query f : V n → Y, defined as:…”

“…Databases differing in a single value have distance at most 1, but the distance can be substantially smaller for small modifications of values, offering higher protection in those cases. The Manhattan metric d 1 on V n and its normalized version d 1 are defined as: 8 …”

“…There are two well known results concerning universal optimality: the first [7] establishes that for counting queries the geometric and the truncated geometric mechanisms are universally optimal. The second [8] says that for any other kind of query no universally optimal mechanism exists.…”

“…Next, we turn our attention to the notion of utility, namely the accuracy of the reported answer, and in particular the Bayesian notion of utility [7,8], which takes into account the prior knowledge of the user. In general mechanisms may provide different degrees of utility for the same level of privacy, and obviously it is desirable to identify the optimal ones.…”