Robust Estimation of Discrete Distributions under Local Differential Privacy

• URL: http://arxiv.org/abs/2202.06825v1
• Date: Mon, 14 Feb 2022 15:59:02 GMT
• Title: Robust Estimation of Discrete Distributions under Local Differential Privacy
• Authors: Julien Chhor and Flore Sentenac
• Abstract summary: We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. We show that combining the two constraints leads to a minimax estimation rate of $epsilonsqrtd/alpha2 k+sqrtd2/alpha2 kn$ up to a $sqrtlog (1/epsilon)$ factor.
• Score: 1.52292571922932
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is an almost unexplored topic. We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. A fraction $1-\epsilon$ of the batches contain $k$ i.i.d. samples drawn from a discrete distribution $p$ over $d$ elements. To protect the users' privacy, each of the samples is privatized using an $\alpha$-locally differentially private mechanism. The remaining $\epsilon n $ batches are an adversarial contamination. The minimax rate of estimation under contamination alone, with no privacy, is known to be $\epsilon/\sqrt{k}+\sqrt{d/kn}$, up to a $\sqrt{\log(1/\epsilon)}$ factor. Under the privacy constraint alone, the minimax rate of estimation is $\sqrt{d^2/\alpha^2 kn}$. We show that combining the two constraints leads to a minimax estimation rate of $\epsilon\sqrt{d/\alpha^2 k}+\sqrt{d^2/\alpha^2 kn}$ up to a $\sqrt{\log(1/\epsilon)}$ factor, larger than the sum of the two separate rates. We provide a polynomial-time algorithm achieving this bound, as well as a matching information theoretic lower bound.

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