Open Access
2021 Strongly universally consistent nonparametric regression and classification with privatised data
Thomas B. Berrett, László Györfi, Harro Walk
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Electron. J. Statist. 15(1): 2430-2453 (2021). DOI: 10.1214/21-EJS1845

Abstract

In this paper we revisit the classical problem of nonparametric regression, but impose local differential privacy constraints. Under such constraints, the raw data (X1,Y1),...,(Xn,Yn), taking values in Rd×R, cannot be directly observed, and all estimators are functions of the randomised output from a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and here we add Laplace distributed noise to a discretisation of the location of a feature vector Xi and to the value of its response variable Yi. Based on this randomised data, we design a novel estimator of the regression function, which can be viewed as a privatised version of the well-studied partitioning regression estimator. The main result is that the estimator is strongly universally consistent, and we further establish an upper bound on the rate of convergence. Our methods and analysis also give rise to a strongly universally consistent binary classification rule for locally differentially private data.

Citation

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Thomas B. Berrett. László Györfi. Harro Walk. "Strongly universally consistent nonparametric regression and classification with privatised data." Electron. J. Statist. 15 (1) 2430 - 2453, 2021. https://doi.org/10.1214/21-EJS1845

Information

Received: 1 November 2020; Published: 2021
First available in Project Euclid: 28 April 2021

Digital Object Identifier: 10.1214/21-EJS1845

Keywords: 62G08 , 62G20 , 68P27 , ‎classification‎ , Local differential privacy , regression estimate , universal consistency

Vol.15 • No. 1 • 2021
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