Locally Private Set-valued Data Analyses: Distribution and Heavy Hitters Estimation

In many mobile applications, user-generated data are presented as set-valued data. To tackle potential privacy threats in analyzing these valuable data, local differential privacy has been attracting substantial attention. However, existing approaches only provide sub-optimal utility and are expensive in computation and communication for set-valued data distribution estimation and heavy-hitter identification. In this paper, we propose a utility-optimal and efficient set-valued data publication method (i.e., Wheel mechanism ). On the user side, the computational complexity is only $O(\min \lbrace m\log m, m e^\epsilon \rbrace )$ and communication costs are $O(\epsilon +\log m)$ bits, where $m$ is the number of items, $d$ is the domain size and $\epsilon$ is the privacy budget, while existing approaches usually depend on $O(d)$ or $O(\log d)$ ( $d \gg m$ ). Our theoretical analyses reveal the estimation errors have been reduced from the previously known $O(\frac{m^{2} d}{n\epsilon ^{2}})$ to the optimal rate $O(\frac{m d}{n\epsilon ^{2}})$ . Additionally, for heavy-hitter identification, we present a variant of the Wheel mechanism as an efficient frequency oracle, entailing only $O(\sqrt{n})$ computational complexity. This heavy-hitter protocol achieves an identification bar of $\tilde{O}(\frac{1}{\epsilon }\sqrt{\frac{m}{n} \log d})$ , reducing by a factor of $\sqrt{m}$ relative to existing protocols. Extensive experiments demonstrate our methods are 3-100x faster than existing approaches and have optimized statistical efficiency.