Observations on the bias of nonnegative mechanisms for differential privacy

Aisling McGlinchey; Oliver Mason

doi:10.3934/fods.2020020

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2020, Volume 2, Issue 4: 429-442. Doi: 10.3934/fods.2020020

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Observations on the bias of nonnegative mechanisms for differential privacy

Aisling McGlinchey and
Oliver Mason^,

Dept. of Mathematics and Statistics, Maynooth University, Co. Kildare, Ireland & Lero, the Science Foundation Ireland Research Centre for Software

^* Corresponding author: Oliver Mason
^* Corresponding author: Oliver Mason

Received: July 31, 2020

Revised: November 30, 2020

Early access: December 2020

Published: December 2020

This work is supported by SFI grant 13/RC/2094

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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.

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Mathematics Subject Classification: Primary: 68P27; Secondary: 62E99.

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