Locally Differentially Private Minimum Finding

We investigate a problem of finding the minimum, in which each user has a real value and we want to estimate the minimum of these values under the local differential privacy constraint. We reveal that this problem is fundamentally difficult, and we cannot construct a mechanism that is consistent in the worst case. Instead of considering the worst case, we aim to construct a private mechanism whose error rate is adaptive to the easiness of estimation of the minimum. As a measure of easiness, we introduce a parameter $\alpha$ that characterizes the fatness of the minimum-side tail of the user data distribution. As a result, we reveal that the mechanism can achieve $O((\ln^6N/\epsilon^2N)^{1/2\alpha})$ error without knowledge of $\alpha$ and the error rate is near-optimal in the sense that any mechanism incurs $\Omega((1/\epsilon^2N)^{1/2\alpha})$ error. Furthermore, we demonstrate that our mechanism outperforms a naive mechanism by empirical evaluations on synthetic datasets. Also, we conducted experiments on the MovieLens dataset and a purchase history dataset and demonstrate that our algorithm achieves $\tilde{O}((1/N)^{1/2\alpha})$ error adaptively to $\alpha$.