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 consistent mechanism 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 α 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((ln⁶ N/ϵ²N)¹/2^α) error without knowledge of α and the error rate is near-optimal in the sense that any mechanism incurs Ω((1/ϵ²N)¹/2^α) 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 Õ((1/N)¹/2^α) error adaptively to α.