Strongly universally consistent nonparametric regression and classification with privatised data
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In this paper we revisit the classical problem of nonparametric regression, but impose local differential privacy constraints. Under such constraints, the raw data (X_1,Y_1),…,(X_n,Y_n), taking values in ℝ^d ×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 X_i and to the value of its response variable Y_i. 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. Our methods and analysis also give rise to a strongly universally consistent binary classification rule for locally differentially private data.
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