Differentially Private Set Representations

TL;DR

This paper introduces new differentially private mechanisms for representing large sets efficiently, achieving optimal privacy-utility trade-offs with novel embedding techniques and matching lower bounds.

Contribution

It presents two new DP set representation algorithms with optimal space and error bounds, and introduces a novel embedding approach deviating from prior noise-injection methods.

Findings

Achieves near-optimal space complexity for DP set representations.

Provides matching lower bounds for privacy-utility trade-offs.

Introduces a new embedding technique for sets into random linear systems.

Abstract

We study the problem of differentially private (DP) mechanisms for representing sets of size $k$ from a large universe. Our first construction creates $(\epsilon,\delta)$-DP representations with error probability of $1/(e^\epsilon + 1)$ using space at most $1.05 k \epsilon \cdot \log(e)$ bits where the time to construct a representation is $O(k \log(1/\delta))$ while decoding time is $O(\log(1/\delta))$. We also present a second algorithm for pure $\epsilon$-DP representations with the same error using space at most $k \epsilon \cdot \log(e)$ bits, but requiring large decoding times. Our algorithms match our lower bounds on privacy-utility trade-offs (including constants but ignoring $\delta$ factors) and we also present a new space lower bound matching our constructions up to small constant factors. To obtain our results, we design a new approach embedding sets into random linear systems deviating from most prior approaches that inject noise into non-private solutions.