Differentially Private Integer Partitions and their Applications

Given a positive integer N ≥ 0 a partition of N is a non-increasing sequence of numbers x1 ≥ x2 . . . ≥ xN ≥ 0 such that x1 + . . . + xN = N . We say that two partitions x and y are neighbors if the L1 distance between them is at most 1/2. Blocki et al. [BDB16] recently showed that there is a ( , δ)differentially private algorithm which (whp) achieves L1 error O( √ N/ ) and they used their algorithm to publish password frequency data from a password dataset of 70 million Yahoo! users [Bon12]. Their algorithm, which was based on an (approximate) instantiation of the exponential mechanism [MT07], is computationally efficient in 1 , log ( 1 δ ) and N . The applications of the mechanism of Blocki et al. [BDB16] are not limited to passwords. For example, the degree distribution of a social network G is simply a partition of the integer 2 |E(G)|. Thus, the mechanism of [BDB16] could be used to preserve differential privacy when releasing the degree distribution. It is particularly important to understand the performance of the exponential mechanism for integer partitions. We provide a pure -differentially instantiation of the exponential mechanism whenever there is an a priori known upper bound on N . We also upper bound the mean squared error of the exponential mechanism O (√ N log N 2 ) . For comparison, the best known results, due to Hay et al. [HLMJ09], achieved mean squared error O (√ N log N 2 ) . Additionally, we conjecture that the L1 error of the exponential mechanism scales with 1/ √ instead of 1/ . Empirical data from the RockYou password frequency dataset supports this conjecture. The conjecture, if true, could lead to the development of several useful node-differentially private algorithms.