Tight Bounds for Differentially Private Anonymized Histograms

In this note, we consider the problem of differentially privately (DP) computing an anonymized histogram, which is defined as the multiset of counts of the input dataset (without bucket labels). In the low-privacy regime epsilon >= 1, we give an epsilon-DP algorithm with an expected l(1)-error bound of O(root n/e(epsilon)). In the high-privacy regime epsilon < 1, we give an Omega(root n log(1/epsilon)/epsilon) lower bound on the expected l(1) error. In both cases, our bounds asymptotically match the previously known lower/upper bounds due to [Sur19].