On Avoiding the Union Bound When Answering Multiple Differentially Private Queries

In this work, we study the problem of answering k queries with (ε, δ)-differential privacy, where each query has sensitivity one. We give an algorithm for this task that achieves an expected ℓ_∞ error bound of O(frac1ε√k log frac1δ), 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 δ < 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 ℓ∞ error bound of O(frac1ε√k log frac1δ) 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.