Information Leakage Envelopes

TL;DR

This paper introduces the PML envelope, a new privacy guarantee measure that remains robust under post-processing and bounds leakage failure probability, addressing limitations of previous definitions.

Contribution

The paper defines the PML envelope, proving its properties and analyzing its behavior for common privacy mechanisms, advancing the understanding of robust privacy guarantees.

Findings

PML envelope satisfies both post-processing robustness and failure probability bounds.

Structural properties like monotonicity are established for the PML envelope.

Analysis of PML envelope for extremal mechanisms and randomized response demonstrates its practical relevance.

Abstract

We study privacy guarantees in the framework of pointwise maximal leakage (PML) that satisfy two requirements: they are robust under post-processing and upper bound the failure probability, i.e., the probability that the information leakage exceeds a given threshold. We first examine two candidate definitions inspired by (approximate) differential privacy and show that neither one satisfies both requirements simultaneously. We then introduce the notion of the PML envelope, which quantifies the largest amount of information leakage about a secret after arbitrary post-processing of a mechanism's output. By construction, the PML envelope satisfies both requirements. We discuss basic structural properties of the envelope, such as monotonicity, and derive general upper and lower bounds. We further analyze the envelope for two widely used privacy mechanisms: the PML-extremal mechanisms in the high-privacy regime and randomized response. Overall, this work establishes the PML envelope as a natural and operationally meaningful definition for providing privacy guarantees that are preserved under arbitrary downstream transformations.