
Locally private nonasymptotic testing of discrete distributions is faster using interactive mechanisms
We find separation rates for testing multinomial or more general discret...
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Interactive versus noninteractive locally, differentially private estimation: Two elbows for the quadratic functional
Local differential privacy has recently received increasing attention fr...
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Differentially Private Algorithms for Clustering with Stability Assumptions
We study the problem of differentially private clustering under inputst...
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Learning discrete distributions: user vs itemlevel privacy
Much of the literature on differential privacy focuses on itemlevel pri...
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Sharp phase transitions for exact support recovery under local differential privacy
We address the problem of variable selection in the Gaussian mean model ...
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Test without Trust: Optimal Locally Private Distribution Testing
We study the problem of distribution testing when the samples can only b...
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Local Differential Privacy for Physical Sensor Data and Sparse Recovery
In this work we explore the utility of locally differentially private th...
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Locally differentially private estimation of nonlinear functionals of discrete distributions
We study the problem of estimating nonlinear functionals of discrete distributions in the context of local differential privacy. The initial data x_1,…,x_n ∈ [K] are supposed i.i.d. and distributed according to an unknown discrete distribution p = (p_1,…,p_K). Only αlocally differentially private (LDP) samples z_1,...,z_n are publicly available, where the term 'local' means that each z_i is produced using one individual attribute x_i. We exhibit privacy mechanisms (PM) that are interactive (i.e. they are allowed to use already published confidential data) or noninteractive. We describe the behavior of the quadratic risk for estimating the power sum functional F_γ = ∑_k=1^K p_k^γ, γ >0 as a function of K, n and α. In the noninteractive case, we study two plugin type estimators of F_γ, 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 twostep procedure which attains the faster parametric rate (n α^2)^1/2 when γ≥ 2. We give lower bounds results over all αLDP mechanisms and all estimators using the private samples.
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