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# Linear Queries Estimation with Local Differential Privacy

*Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics*, PMLR 89:721-729, 2019.

#### Abstract

We study the problem of estimating a set of d linear queries with respect to some unknown distribution p over a domain $[J]$ based on a sensitive data set of n individuals under the constraint of local differential privacy. This problem subsumes a wide range of estimation tasks, e.g., distribution estimation and d-dimensional mean estimation. We provide new algorithms for both the offline (non-adaptive) and adaptive versions of this problem. In the offline setting, the set of queries are fixed before the algorithm starts. In the regime where $n < d^2/\log(J)$, our algorithms attain $L_2$ estimation error that is independent of d. For the special case of distribution estimation, we show that projecting the output estimate of an algorithm due to [Acharya et al. 2018] on the probability simplex yields an $L_2$ error that depends only sub-logarithmically on $J$ in the regime where $n < J^2/\log(J)$. Our bounds are within a factor of at most $(\log(J))^{1/4}$ from the optimal $L_2$ error. These results show the possibility of accurate estimation of linear queries in the high-dimensional settings under the $L_2$ error criterion. In the adaptive setting, the queries are generated over d rounds; one query at a time. In each round, a query can be chosen adaptively based on all the history of previous queries and answers. We give an algorithm for this problem with optimal $L_{\infty}$ estimation error (worst error in the estimated values for the queries w.r.t. the data distribution). Our bound matches a lower bound on the $L_{\infty}$ error in the offline version of this problem [Duchi et al. 2013].