Privately Answering Counting Queries with Generalized Gaussian Mechanisms
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We consider the problem of answering k counting (i.e. sensitivity-1) queries about a database with (ϵ, δ)-differential privacy. We give a mechanism such that if the true answers to the queries are the vector d, the mechanism outputs answers d̃ with the ℓ_∞-error guarantee: ℰ[||d̃ - d||_∞] = O(√(k logloglog k log(1/δ))/ϵ). This reduces the multiplicative gap between the best known upper and lower bounds on ℓ_∞-error from O(√(loglog k)) to O(√(logloglog k)). Our main technical contribution is an analysis of the family of mechanisms of the following form for answering counting queries: Sample x from a Generalized Gaussian, i.e. with probability proportional to (-(||x||_p/σ)^p), and output d̃ = d + x. This family of mechanisms offers a tradeoff between ℓ_1 and ℓ_∞-error guarantees and may be of independent interest. For p = O(loglog k), this mechanism already matches the previous best known ℓ_∞-error bound. We arrive at our main result by composing this mechanism for p = O(logloglog k) with the sparse vector mechanism, generalizing a technique of Steinke and Ullman.
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