A lower bound on the release of differentially private integer partitions

Tight Bounds for Differentially Private Anonymized Histograms

“…It is simple to see that the DP integer partition problem is equivalent to that of DP anonymized histogram, because an anonymized histogram can be viewed as an integer partition and vice versa 2 while preserving the neighboring relationship. Note that this equivalence is known and is also stated in [BDB16,AS18].…”

“…[HLMJ09,HRMS10] which focus on ℓ 2 2 error. Similar to previous works, we focus on the regime where the universe B is unbounded [AS18] show that any ǫ-DP algorithm for 1/8 > ǫ > Ω( 1 n ) must incur an error of Ω( n/ǫ). Recently, Suresh [Sur19] proved two arguably surprising results.…”

“…Both of which are tight up to constant multiplicative factors. The latter also negatively answers an open question of Aldà and Simon [AS18] who asked whether the bound of Ω( n/ǫ) proved in their paper is tight. We also remark that, while previous works focus their efforts on proving lower bounds for ǫ-DP algorithms, our lower bound holds naturally also for (ǫ, δ)-DP algorithms where δ is sufficiently small (i.e.…”

“…Formally exploiting (ii) however is somewhat complicated and is not needed for our lower bound; thus we will not go into detail here. Now that we have a high level intuition of the algorithm, we briefly note that the lower bounds in [AS18,Sur19] proceed by "encoding" each boolean vector u ∈ {0, 1} m to an integer partition with a property that the distance of any two vectors are preserved (after appropriate scaling); this suffices to prove a lower bound due to a known strong lower bound for privatizing boolean vectors (see Lemma 5). Our proof proceeds similarly but use a new encoding, guided by the above algorithm.…”

“…Our lower bound also holds for (ǫ, δ)-DP for any sufficiently small δ (depending only on ǫ). The lower bounds in the high-privacy regime of our setting requires ǫ ≥ Ω(log n/n) whereas that of [AS18] requires ǫ ≥ Ω(1/n); these are necessary because the error of O(n) is trivial to achieve (by outputting the all-zero vector).…”