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Thresholding a private noisy count

Example

Assume Countable Choice. Let $U$ be a nonempty record universe and let databases be finite-support count functions $D : U \to N_{0}$ , with neighbors formed by adding or removing one occurrence. Let $P : U \to {0, 1}$ be a predicate satisfied by at least one record, let $f (D)$ count the occurrences satisfying $P$ , let $ε > 0$ , and fix $c \in R$ . Release $X = f (D) + Z$ with $Z \sim Lap (1 / ε)$ , then output $1$ exactly when $X \geq c$ and output $0$ otherwise. The threshold decision is a data-independent deterministic postprocessing and retains the $(ε, 0)$ guarantee of the noisy count.

Example references

The scalar Laplace mechanism

Privacy of a sensitivity-one Laplace release

Randomized and deterministic postprocessing preserves privacy

Dwork and Roth, The Algorithmic Foundations of Differential Privacy, Proposition 2.1 and Example 3.1, printed pp. 19 and 32–33.

Vadhan, The Complexity of Differential Privacy, §1.5, Theorem 1.3, and §2.1, Lemma 2.1.

Facts & Assumptions

Given: A predicate $P$ satisfied by some record $u_{0}$ , $ε > 0$ , and threshold $c \in R$ .

[A1]

Countable Choice is inherited from the Laplace probability law; no stronger choice principle is used.

[F1]

Neighboring databases differ by adding or removing one occurrence.

[F2]

Sensitivity is the attained maximum over neighboring query differences.

[F3]

Laplace privacy theorem gives $(ε, 0)$ -DP for a scalar query of sensitivity one and $ε > 0$ .

[F4]

The Borel sigma-algebra contains closed and open intervals, including $[c, + \infty)$ and $(- \infty, c)$ .

[F5]

Postprocessing theorem preserves the same differential-privacy parameters under any data-independent probability kernel.

Verification

1.1A1F1F2F3givenalgebra

Adding or removing a record $u$ changes $f (D)$ by $P (u) \in {0, 1}$ in absolute value. Every neighboring difference is at most one, and the empty database versus the singleton database containing $u_{0}$ attains one. Thus $Δ_{1} f = 1$ by [F1] and [F2], and [A1, F3] gives the corresponding Laplace law, so the noisy count has noise scale $1 / ε$ and is $(ε, 0)$ -DP.

2.1F4F5step 1.1givenconstruct∎

Define $ϕ (x) = 1_{{x \geq c}}$ . Its preimages of ${1}$ and ${0}$ are $[c, + \infty)$ and $(- \infty, c)$ , respectively, so [F4] makes $ϕ$ Borel measurable. The deterministic kernel $L (x, B) = 1_{B} (ϕ (x))$ on ${0, 1}$ is a probability kernel and is independent of the database. Applying [F5] to the noisy-count release proves that the threshold output retains the same $(ε, 0)$ guarantee.

Depends on

Used by

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Sources

Cynthia Dwork and Aaron Roth, The Algorithmic Foundations of Differential Privacy (standard reference, not scraped)
Salil Vadhan, The Complexity of Differential Privacy, 2017 manuscript (standard reference, not scraped)