Missing Mass for Differentially Private Domain Discovery

Abstract

We study several problems in differentially private domain discovery, where each user holds a subset of items from a shared but unknown domain, and the goal is to output an informative subset of items. For set union, we show that the simple baseline Weighted Gaussian Mechanism (WGM) has a near-optimal $\ell_1$ missing mass guarantee on Zipfian data as well as a distribution-free $\ell_\infty$ missing mass guarantee. We then apply the WGM as a domain-discovery precursor for existing known-domain algorithms for private top-$k$ and $k$-hitting set and obtain new utility guarantees for their unknown domain variants. Finally, experiments demonstrate that all of our WGM-based methods are competitive with or outperform existing baselines for all three problems.

Figures (13)

• Figure 1: Set Union MM as a function of $\Delta_0$. Note that lower is better.
• Figure 2: Top-$k$ MM as a function of $k$, using $\Delta_0 = 100$.
• Figure 3: Number of missed users as a function of $k$, using $\Delta_0 = 100$.
• Figure 4: Log-log plot of frequency vs. rank for large datasets
• Figure 5: Log-log plot of frequency vs. rank for small datasets

Theorems & Definitions (46)

• Definition 2.1: DMNS06
• Definition 2.2
• Definition 3.1: $(C, s)$-Zipfian
• Lemma 3.1
• Theorem 3.2: Theorem 5.1 gopi2020differentially
• Theorem 3.3
• Corollary 3.4
• Theorem 3.5
• Theorem 3.6
• Definition 4.1