Extremal Mechanisms for Local Differential Privacy

TL;DR

This paper introduces extremal staircase mechanisms for local differential privacy, characterizes their optimality for various utilities, and identifies simple mechanisms that are optimal in extreme privacy regimes.

Contribution

It develops a family of extremal privatization mechanisms called staircase mechanisms and proves their optimality for a broad class of utility functions under local differential privacy constraints.

Findings

Staircase mechanisms are optimal for mutual information and $f$-divergences.

Optimal privacy-utility trade-offs can be computed via linear programming.

Binary and randomized response mechanisms are optimal in low and high privacy regimes.

Abstract

Local differential privacy has recently surfaced as a strong measure of privacy in contexts where personal information remains private even from data analysts. Working in a setting where both the data providers and data analysts want to maximize the utility of statistical analyses performed on the released data, we study the fundamental trade-off between local differential privacy and utility. This trade-off is formulated as a constrained optimization problem: maximize utility subject to local differential privacy constraints. We introduce a combinatorial family of extremal privatization mechanisms, which we call staircase mechanisms, and show that it contains the optimal privatization mechanisms for a broad class of information theoretic utilities such as mutual information and $f$-divergences. We further prove that for any utility function and any privacy level, solving the privacy-utility maximization problem is equivalent to solving a finite-dimensional linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has a number of variables that is exponential in the size of the alphabet the data lives in. To account for this, we show that two simple privatization mechanisms, the binary and randomized response mechanisms, are universally optimal in the low and high privacy regimes, and well approximate the intermediate regime.