In this work, we study the problem of answering $k$ queries with $(\epsilon, \delta)$-differential privacy, where each query has sensitivity one. We give an algorithm for this task that achieves an expected $\ell_\infty$ error bound of $O(\frac{ 1}{ \epsilon}\sqrt{ k \log \frac{ 1}{ \delta}})$, which is known to be tight (Steinke and Ullman, 2016). A very recent work by Dagan and Kur (2020) provides a similar result, albeit via a completely different approach. One difference between our work and theirs is that our guarantee holds even when $\delta < 2^{ -\Omega(k/(\log k)^8)}$ whereas theirs does not apply in this case. On the other hand, the algorithm of Dagan and Kur has a remarkable advantage that the $\ell_{ \infty}$ error bound of $O(\frac{ 1}{ \epsilon}\sqrt{ k \log \frac{ 1}{ \delta}})$ holds not only in expectation but always (i.e., with probability one) while we can only get a high probability (or expected) guarantee on the error.
Categories: cs.DS cs.CR cs.LG