Privacy measure used to define \((\epsilon, \delta)\)-approximate differential privacy.

In the following definition, \(d\) corresponds to \((\epsilon, \delta)\) when also quantified over all adjacent datasets. That is, \((\epsilon, \delta)\) is no smaller than \(d\) (by product ordering), over all pairs of adjacent datasets \(x, x'\) where \(Y \sim M(x)\), \(Y' \sim M(x')\). \(M(\cdot)\) is a measurement (commonly known as a mechanism). The measurement's input metric defines the notion of adjacency, and the measurement's input domain defines the set of possible datasets.

Usage

fixed_smoothed_max_divergence()

Value

Measure

Details

Proof Definition:

For any two distributions \(Y, Y'\) and any 2-tuple \(d\) of non-negative numbers \(\epsilon\) and \(\delta\), \(Y, Y'\) are \(d\)-close under the fixed smoothed max divergence measure whenever

\(D_\infty^\delta(Y, Y') = \max_{S \subseteq \textrm{Supp}(Y)} \Big[\ln \dfrac{\Pr[Y \in S] + \delta}{\Pr[Y' \in S]} \Big] \leq \epsilon\).

Note that this \(\epsilon\) and \(\delta\) are not privacy parameters \(\epsilon\) and \(\delta\) until quantified over all adjacent datasets, as is done in the definition of a measurement.