Observations on the Bias of Nonnegative Mechanisms for Differential Privacy
This work is significant for researchers and practitioners in differential privacy, as it highlights inherent bias issues in common mechanisms when dealing with nonnegative data, potentially affecting the accuracy of privacy-preserving analyses.
This paper investigates the bias of differentially private mechanisms, specifically Laplace and multiplicative mechanisms, when applied to nonnegative queries on bounded data. It demonstrates that standard methods like boundary inflated truncation (BIT) and truncation, as well as any square-integrable post-processing of the Laplace mechanism, result in strictly positive bias. The study also shows that multiplicative mechanisms can lead to infinite bias without further restrictions.
We study two methods for differentially private analysis of bounded data and extend these to nonnegative queries. We first recall that for the Laplace mechanism, boundary inflated truncation (BIT) applied to nonnegative queries and truncation both lead to strictly positive bias. We then consider a generalization of BIT using translated ramp functions. We explicitly characterise the optimal function in this class for worst case bias. We show that applying any square-integrable post-processing function to a Laplace mechanism leads to a strictly positive maximal absolute bias. A corresponding result is also shown for a generalisation of truncation, which we refer to as restriction. We also briefly consider an alternative approach based on multiplicative mechanisms for positive data and show that, without additional restrictions, these mechanisms can lead to infinite bias.