Discrete Distribution Estimation under User-level Local Differential Privacy

TL;DR

This paper investigates the problem of estimating discrete distributions under user-level local differential privacy, revealing phase transitions and equivalences that inform privacy-utility trade-offs and connecting to shuffled differential privacy.

Contribution

It provides tight bounds for user-level LDP distribution estimation, demonstrating equivalences between multiple samples per user and more users, and links to shuffled DP for improved guarantees.

Findings

More samples per user can be simulated by more users with fewer samples each.

Phase transitions depend on the number of samples, privacy level, and estimation risk.

Algorithms achieve near-optimal error bounds, verified by simulations.

Abstract

We study discrete distribution estimation under user-level local differential privacy (LDP). In user-level $\varepsilon$-LDP, each user has $m\ge1$ samples and the privacy of all $m$ samples must be preserved simultaneously. We resolve the following dilemma: While on the one hand having more samples per user should provide more information about the underlying distribution, on the other hand, guaranteeing the privacy of all $m$ samples should make the estimation task more difficult. We obtain tight bounds for this problem under almost all parameter regimes. Perhaps surprisingly, we show that in suitable parameter regimes, having $m$ samples per user is equivalent to having $m$ times more users, each with only one sample. Our results demonstrate interesting phase transitions for $m$ and the privacy parameter $\varepsilon$ in the estimation risk. Finally, connecting with recent results on shuffled DP, we show that combined with random shuffling, our algorithm leads to optimal error guarantees (up to logarithmic factors) under the central model of user-level DP in certain parameter regimes. We provide several simulations to verify our theoretical findings.