Observations on the Bias of Nonnegative Mechanisms for Differential Privacy

TL;DR

This paper analyzes the bias introduced by nonnegative mechanisms in differential privacy, extending existing methods and characterizing optimal functions to minimize bias, revealing inherent limitations of these approaches.

Contribution

It introduces a generalized boundary inflated truncation method using translated ramp functions and characterizes the optimal function for worst-case bias in nonnegative differential privacy mechanisms.

Findings

Boundary inflated truncation causes positive bias in nonnegative queries.

Any square-integrable post-processing function increases maximal bias.

Multiplicative mechanisms can lead to infinite bias without restrictions.

Abstract

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.