Differential Privacy with Multiple Selections

TL;DR

This paper introduces a multi-selection approach for differentially private recommendations, characterizes the optimal mechanism for one-dimensional features, and demonstrates that Laplace noise is optimal with error decreasing as more results are returned.

Contribution

It provides a precise characterization of the optimal differentially private mechanism for multi-selection recommendations in one-dimensional settings, including the optimal noise distribution and response algorithm.

Findings

Laplace noise distribution is proven optimal.

Optimal mechanism's error decreases inversely with number of results.

Explicit characterization of the optimal mechanism for one-dimensional features.

Abstract

We consider the setting where a user with sensitive features wishes to obtain a recommendation from a server in a differentially private fashion. We propose a ``multi-selection'' architecture where the server can send back multiple recommendations and the user chooses one from these that matches best with their private features. When the user feature is one-dimensional -- on an infinite line -- and the accuracy measure is defined w.r.t some increasing function $\mathfrak{h}(.)$ of the distance on the line, we precisely characterize the optimal mechanism that satisfies differential privacy. The specification of the optimal mechanism includes both the distribution of the noise that the user adds to its private value, and the algorithm used by the server to determine the set of results to send back as a response and further show that Laplace is an optimal noise distribution. We further show that this optimal mechanism results in an error that is inversely proportional to the number of results returned when the function $\mathfrak{h}(.)$ is the identity function.