Generalized Rainbow Differential Privacy

TL;DR

This paper introduces rainbow differential privacy, a novel framework using graph colorings to design optimal DP mechanisms with a closed-form solution, extending previous work to more colors and $(,)$-DP.

Contribution

It establishes the existence and uniqueness of optimal DP mechanisms under rainbow boundary conditions and provides a closed-form expression, generalizing prior results.

Findings

Optimal DP mechanisms can be characterized by boundary conditions.

The framework applies to any finite number of colors and $(,)$-DP.

Homogeneous boundary conditions are necessary for optimality.

Abstract

We study a new framework for designing differentially private (DP) mechanisms via randomized graph colorings, called rainbow differential privacy. In this framework, datasets are nodes in a graph, and two neighboring datasets are connected by an edge. Each dataset in the graph has a preferential ordering for the possible outputs of the mechanism, and these orderings are called rainbows. Different rainbows partition the graph of connected datasets into different regions. We show that if a DP mechanism at the boundary of such regions is fixed and it behaves identically for all same-rainbow boundary datasets, then a unique optimal $(\epsilon,\delta)$-DP mechanism exists (as long as the boundary condition is valid) and can be expressed in closed-form. Our proof technique is based on an interesting relationship between dominance ordering and DP, which applies to any finite number of colors and for $(\epsilon,\delta)$-DP, improving upon previous results that only apply to at most three colors and for $\epsilon$-DP. We justify the homogeneous boundary condition assumption by giving an example with non-homogeneous boundary condition, for which there exists no optimal DP mechanism.