The Laplace Mechanism has optimal utility for differential privacy over continuous queries

Differential Privacy protects individuals' data when statistical queries are published from aggregated databases: applying "obfuscating" mechanisms to the query results makes the released information less specific but, unavoidably, also decreases its utility. Yet it has been shown that for discrete data (e.g. counting queries), a mandated degree of privacy and a reasonable interpretation of loss of utility, the Geometric obfuscating mechanism is optimal: it loses as little utility as possible. For continuous query results however (e.g. real numbers) the optimality result does not hold. Our contribution here is to show that optimality is regained by using the Laplace mechanism for the obfuscation. The technical apparatus involved includes the earlier discrete result by Ghosh et al., recent work on abstract channels and their geometric representation as hyper-distributions, and the dual interpretations of distance between distributions provided by the Kantorovich-Rubinstein Theorem.