A Design Framework for Strongly $\chi^2$-Private Data Disclosure

TL;DR

This paper introduces a geometric approach to designing privacy mechanisms that maximize useful data disclosure while satisfying strong $ ext{chi}^2$-privacy constraints, using information geometry to simplify the optimization process.

Contribution

It develops a novel geometric framework for privacy mechanism design under strong $ ext{chi}^2$-privacy, reducing complex optimization to a principal singular vector problem.

Findings

The geometric approach simplifies privacy mechanism optimization.

Optimal mechanisms are characterized by principal singular vectors.

The framework extends to scenarios with noisy adversaries.

Abstract

In this paper, we study a stochastic disclosure control problem using information-theoretic methods. The useful data to be disclosed depend on private data that should be protected. Thus, we design a privacy mechanism to produce new data which maximizes the disclosed information about the useful data under a strong $\chi^2$-privacy criterion. For sufficiently small leakage, the privacy mechanism design problem can be geometrically studied in the space of probability distributions by a local approximation of the mutual information. By using methods from Euclidean information geometry, the original highly challenging optimization problem can be reduced to a problem of finding the principal right-singular vector of a matrix, which characterizes the optimal privacy mechanism. In two extensions we first consider a scenario where an adversary receives a noisy version of the user's message and then we look for a mechanism which finds $U$ based on observing $X$, maximizing the mutual information between $U$ and $Y$ while satisfying the privacy criterion on $U$ and $Z$ under the Markov chain $(Z,Y)-X-U$.