Locally Differentially Private Minimum Finding
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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 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((^6N/ϵ^2N)^1/2α) error without knowledge of α and the error rate is near-optimal in the sense that any mechanism incurs Ω((1/ϵ^2N)^1/2α) 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)^1/2α) error adaptively to α.
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