Optimal Tree-Based Mechanisms for Differentially Private Approximate CDFs

TL;DR

This paper develops optimal level-uniform tree-based mechanisms for differentially private approximate CDFs, minimizing error by choosing optimal tree structures and privacy budgets, and proposes strategies to enhance estimate accuracy.

Contribution

It extends binary tree mechanisms to level-uniform trees, identifying optimal structures and parameters for DP CDF release, including both real-valued and integer branching factors.

Findings

Optimal tree structures depend on parameters and privacy budgets.

Equal branching factors and privacy budgets yield near-optimal mechanisms.

Strategies for combining estimates and post-processing improve accuracy.

Abstract

This paper considers the $\varepsilon$-differentially private (DP) release of an approximate cumulative distribution function (CDF) of the samples in a dataset. We assume that the true (approximate) CDF is obtained after lumping the data samples into a fixed number $K$ of bins. In this work, we extend the well-known binary tree mechanism to the class of \emph{level-uniform tree-based} mechanisms and identify $\varepsilon$-DP mechanisms that have a small $\ell_2$-error. We identify optimal or close-to-optimal tree structures when either of the parameters, which are the branching factors or the privacy budgets at each tree level, are given, and when the algorithm designer is free to choose both sets of parameters. Interestingly, when we allow the branching factors to take on real values, under certain mild restrictions, the optimal level-uniform tree-based mechanism is obtained by choosing equal branching factors \emph{independent} of $K$, and equal privacy budgets at all levels. Furthermore, for selected $K$ values, we explicitly identify the optimal \emph{integer} branching factors and tree height, assuming equal privacy budgets at all levels. Finally, we describe general strategies for improving the private CDF estimates further, by combining multiple noisy estimates and by post-processing the estimates for consistency.