Privacy measure used to define \(\epsilon(\alpha)\)-Rényi differential privacy.

In the following proof definition, \(d\) corresponds to an RDP curve when also quantified over all adjacent datasets. That is, an RDP curve \(\epsilon(\alpha)\) is no smaller than \(d(\alpha)\) for any possible choices of \(\alpha\), and 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

renyi_divergence()

Value

Measure

Details

Proof Definition:

For any two distributions \(Y, Y'\) and any curve \(d\), \(Y, Y'\) are \(d\)-close under the Rényi divergence measure if, for any given \(\alpha \in (1, \infty)\),

\(D_\alpha(Y, Y') = \frac{1}{1 - \alpha} \mathbb{E}_{x \sim Y'} \Big[\ln \left( \dfrac{\Pr[Y = x]}{\Pr[Y' = x]} \right)^\alpha \Big] \leq d(\alpha)\)

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