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 $\epsilon \geq 1$, we give an $\epsilon$-DP algorithm with an expected $\ell_1$-error bound of $O(\sqrt{n} / e^\epsilon)$. In the high-privacy regime $\epsilon < 1$, we give an $\Omega(\sqrt{n \log(1/\epsilon) / \epsilon})$ lower bound on the expected $\ell_1$ error. In both cases, our bounds asymptotically match the previously known lower/upper bounds due to [Suresh, NeurIPS 2019].
翻译:在本说明中,我们考虑了以不同方式私下(DP)计算匿名直方图的问题,该直方图被定义为输入数据集(无桶标签)的数组数数。在低原始系统 $\ epsilon\ geq 1$, 我们给低原始系统 $\ epsilon\ $\ $1$- error 算法, 预期其值为$( lsqrt{n} / e ⁇ esilon) 。 在高原始系统 $\ epsilon < 1$ 中, 我们给$\ omega (sqrt{n\ log (1/\ epsilon) /\ epslon}) $( $\ el_ 1美元) 。 在这两种情况下, 我们的界限都与先前已知的因[ Suresh, NurIPS 20199] 而导致的低/上下限值匹配。