# Locally Differentially-Private Randomized Response for Discrete Distribution Learning

We consider a setup in which confidential i.i.d. samples X_1,,X_n from an unknown finite-support distribution p are passed through n copies of a discrete privatization channel (a.k.a. mechanism) producing outputs Y_1,,Y_n. The channel law guarantees a local differential privacy of ϵ. Subject to a prescribed privacy level ϵ, the optimal channel should be designed such that an estimate of the source distribution based on the channel outputs Y_1,,Y_n converges as fast as possible to the exact value p. For this purpose we study the convergence to zero of three distribution distance metrics: f-divergence, mean-squared error and total variation. We derive the respective normalized first-order terms of convergence (as n→∞), which for a given target privacy ϵ represent a rule-of-thumb factor by which the sample size must be augmented so as to achieve the same estimation accuracy as that of a non-randomizing channel. We formulate the privacy-fidelity trade-off problem as being that of minimizing said first-order term under a privacy constraint ϵ. We further identify a scalar quantity that captures the essence of this trade-off, and prove bounds and data-processing inequalities on this quantity. For some specific instances of the privacy-fidelity trade-off problem, we derive inner and outer bounds on the optimal trade-off curve.

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