Locally Differentially-Private Randomized Response for Discrete Distribution Learning

We consider a setup in which confidential i.i.d. samples X1, . . . , Xn from an unknown finite-support distribution p are passed through n copies of a discrete privatization chan- nel (a.k.a. mechanism) producing outputs Y1, . . . , Yn. 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 out- puts Y1, . . . , Yn 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.