The Asymptotic Behaviour of Information Leakage Metrics

Information theoretic leakage metrics quantify the amount of information about a private random variable $X$ that is leaked through a correlated revealed variable $Y$. They can be used to evaluate the privacy of a system in which an adversary, from whom we want to keep $X$ private, is given access to $Y$. Global information theoretic leakage metrics quantify the overall amount of information leaked upon observing $Y$, whilst their pointwise counterparts define leakage as a function of the particular realisation $Y=y$ that the adversary sees, and thus can be viewed as random variables. We consider an adversary who observes a large number of independent identically distributed realisations of $Y$. We formalise the essential asymptotic behaviour of an information theoretic leakage metric, considering in turn what this means for pointwise and global metrics. With the resulting requirements in mind, we take an axiomatic approach to defining a set of pointwise leakage metrics, as well as a set of global leakage metrics that are constructed from them. The global set encompasses many known measures including mutual information, Sibson mutual information, Arimoto mutual information, maximal leakage, min entropy leakage, $f$-divergence metrics, and g-leakage. We prove that both sets follow the desired asymptotic behaviour. Finally, we derive composition theorems which quantify the rate of privacy degradation as an adversary is given access to a large number of independent observations of $Y$. It is found that, for both pointwise and global metrics, privacy degrades exponentially with increasing observations for the adversary, at a rate governed by the minimum Chernoff information between distinct conditional channel distributions. This extends the work of Wu et al. (2024), who have previously found this to be true for certain known metrics, including some that fall into our more general set.