Locally differentially private estimation of nonlinear functionals of discrete distributions

We study the problem of estimating non-linear functionals of discrete distributions in the context of local differential privacy. The initial data x1,ldots,xn ∈ [K] are supposed i.i.d. and distributed according to an unknown discrete distribution p = (p1,ldots,pK). Only α-locally differentially private (LDP) samples z1,...,zn are publicly available, where the term 'local' means that each zi is produced using one individual attribute xi. We exhibit privacy mechanisms (PM) that are interactive (i.e. they are allowed to use already published confidential data) or non-interactive. We describe the behavior of the quadratic risk for estimating the power sum functional Fγ = sumk=1K pkγ, γ >0 as a function of K, , n and α. In the non-interactive case, we study two plug-in type estimators of Fγ, for all γ >0, that are similar to the MLE analyzed by Jiao et al. (2017) in the multinomial model. However, due to the privacy constraint the rates we attain are slower and similar to those obtained in the Gaussian model by Collier et al. (2020). In the interactive case, we introduce for all γ >1 a two-step procedure which attains the faster parametric rate (n α2)-1/2 when γ ≥ 2. We give lower bounds results over all α-LDP mechanisms and all estimators using the private samples.