Answering Private Linear Queries Adaptively Using the Common Mechanism

When analyzing confidential data through a privacy filter, a data scientist often needs to decide which queries will best support their intended analysis. For example, an analyst may wish to study noisy two-way marginals in a dataset produced by a mechanism M 1 . But, if the data are relatively sparse, the analyst may choose to examine noisy one-way marginals, produced by a mechanism M 2 , instead. Since the choice of whether to use M 1 or M 2 is data-dependent, a typical differentially private workflow is to first split the privacy loss budget ρ into two parts: ρ 1 and ρ 2 , then use the first part ρ 1 to determine which mechanism to use, and the remainder ρ 2 to obtain noisy answers from the chosen mechanism. In a sense, the first step seems wasteful because it takes away part of the privacy loss budget that could have been used to make the query answers more accurate. In this paper, we consider the question of whether the choice between M 1 and M 2 can be performed without wasting any privacy loss budget. For linear queries, we propose a method for decomposing M 1 and M 2 into three parts: (1) a mechanism M * that captures their shared information, (2) a mechanism M′1 that captures information that is specific to M 1 , (3) a mechanism M′2 that captures information that is specific to M 2 . Running M * and M′ 1 together is completely equivalent to running M 1 (both in terms of query answer accuracy and total privacy cost ρ ). Similarly, running M * and M′ 2 together is completely equivalent to running M 2 . Since M * will be used no matter what, the analyst can use its output to decide whether to subsequently run M ′ 1 (thus recreating the analysis supported by M 1 )or M′ 2 (recreating the analysis supported by M 2 ), without wasting privacy loss budget.