Adaptive Online Bayesian Estimation of Frequency Distributions with Local Differential Privacy

We propose a novel Bayesian approach for the adaptive and online estimation of the frequency distribution of a finite number of categories under the local differential privacy (LDP) framework. The proposed algorithm performs Bayesian parameter estimation via posterior sampling and adapts the randomization mechanism for LDP based on the obtained posterior samples. We propose a randomized mechanism for LDP which uses a subset of categories as an input and whose performance depends on the selected subset and the true frequency distribution. By using the posterior sample as an estimate of the frequency distribution, the algorithm performs a computationally tractable subset selection step to maximize the utility of the privatized response of the next user. We propose several utility functions related to well-known information metrics, such as (but not limited to) Fisher information matrix, total variation distance, and information entropy. We compare each of these utility metrics in terms of their computational complexity. We employ stochastic gradient Langevin dynamics for posterior sampling, a computationally efficient approximate Markov chain Monte Carlo method. We provide a theoretical analysis showing that (i) the posterior distribution targeted by the algorithm converges to the true parameter even for approximate posterior sampling, and (ii) the algorithm selects the optimal subset with high probability if posterior sampling is performed exactly. We also provide numerical results that empirically demonstrate the estimation accuracy of our algorithm where we compare it with nonadaptive and semi-adaptive approaches under experimental settings with various combinations of privacy parameters and population distribution parameters.