On Avoiding the Union Bound When Answering Multiple Differentially Private Queries
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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(1/ϵ√(k log1/δ)), 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^-Ω(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(1/ϵ√(k log1/δ)) 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.
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