The Randomness Complexity of Differential Privacy

Clément L. Canonne, Francis E. Su, Salil P. Vadhan

Abstract: We initiate the study of the randomness complexity of differential privacy, i.e., how many random bits an algorithm needs in order to generate accurate differentially private releases. As a test case, we focus on the task of releasing the results of d counting queries, or equivalently all one-way marginals on a d-dimensional dataset with boolean attributes. While standard differentially private mechanisms for this task have randomness complexity that grows linearly with d, we show that, surprisingly, only log₂ d+O(1) random bits (in expectation) suffice to achieve an error that depends polynomially on d (and is independent of the size n of the dataset), and furthermore this is possible with pure, unbounded differential privacy and privacy-loss parameter ε = 1/poly(d). Conversely, we show that at least log₂ d-O(1) random bits are also necessary for nontrivial accuracy, even with approximate, bounded DP, provided the privacy-loss parameters satisfy ε,δ ≤ 1/poly(d). We obtain our results by establishing a close connection between the randomness complexity of differentially private mechanisms and the geometric notion of "deterministic rounding schemes" recently introduced and studied by Vander Woude et al. (2022, 2023).