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We utilize a differential privacy mechanism (Laplace noise, with scale $b$) to provide the privacy of records stored within a dataset. Each individual is associated with two, and at most $k$, records. These records are represented as $(\textsf{Username}, O_i)$, where $O_i$ denotes the $i$th noisy value (generated by the mechanism) corresponding to an individual, namely $\textsf{Username}$. How do we calculate the privacy loss for an individual, taking into account that two or at most $k$ records are associated with the individual?

For example, considering user Alice, the following records are assigned to her in the dataset: $(\textsf{Alice}, O_1=1.25), (\textsf{Alice}, O_2=1.89), (\textsf{Alice}, O_3=2.14)$.

I am wondering how much privacy she loses due to these records.

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The straightforward way to analyze this is via the notion of group privacy. Recall the standard notion of differential privacy.

A mechanism $M$ is said to be $(\epsilon,\delta)$ differentially private if, for all adjacent databases $D\sim D'$, for all sets $S$, one has that $$\Pr[M(D)\in S] \leq \exp(\epsilon) \Pr[M(D')] + \delta$$

Here, the adjacency notion is that $D, D'$ differ in a single record. In other words, when formatted appropriately, one has that $\lVert D-D'\rVert_0 = 1$, where $\lVert x\rVert_0 = |\{i\mid x_i\neq 0\}|$ is the Hamming "metric".

Group privacy deals with the related question for adjacent databases $D\sim_k D'$, where now this means that $\lVert D-D'\rVert_0 \leq k$. The following is well-known (see the footnote of page 20 of the DP book)

If $M$ is $(\epsilon,\delta)$-differentially private for databases $D\sim D'$, then for any $k\in\mathbb{N}$ it is $(k\epsilon, k\exp((k-1)\epsilon)\delta)$-differentially private for databases $D\sim_k D'$.

You're essentially asking about the privacy loss of the "alice group", which may have size at most $k$ members. The easy thing to say is that group privacy implies the resulting scheme is $(k\epsilon, k\exp((k-1)\epsilon)\delta)$-DP.

Note that if you care about approximate-DP schemes, group privacy is somewhat easier to reason about in terms of concentrated DP. Here, a $\rho$-CDP scheme becomes a $k^2\rho$-CDP scheme for groups of size $k$.

One thing I don't know is the following. Let $U$ be your set of users, and for user $u\in U$ let $k_u$ be the number of records your database contains associated with that user. Does the privacy loss of user $u$ scale with $k_u$ or $k:=\max_u k_u$? When reasoning about user $u$, if the quantity $k_u$ is publicly known the privacy loss should scale with $k_u$, but if $k_u$ is a sensitive quantity itself then I am not sure what happens.

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  • $\begingroup$ To Mark Schultz-Wu: For example, let's assume there are three users. If they have $k_1$, $k_2$, and $k_3$ records associated with them respectively, then we should consider epsilon as $Max(k_1, k_2, k_3) \times \epsilon$. Am I correct? $\endgroup$ Commented Mar 27 at 9:44
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    $\begingroup$ @AmirhosseinAdavoudi yes $\endgroup$
    – Mark Schultz-Wu
    Commented Mar 27 at 17:54

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