On the Information Leakage Envelope of the Gaussian Mechanism

Abstract: 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.