On the Information Leakage Envelope of the Gaussian Mechanism

We study the pointwise maximal leakage (PML) envelope of the Gaussian mechanism, which characterizes the smallest information leakage bound that holds with high probability under arbitrary post-processing. For the Gaussian mechanism with a Gaussian secret, we derive a closed-form expression for the deterministic PML envelope for sufficiently small failure probabilities. We then extend this result to general unbounded secrets by identifying a sufficient condition under which the envelope coincides with the Gaussian case. In particular, we show that strongly log-concave priors satisfy this condition via an application of the Brascamp-Lieb inequality.