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In this paper we revisit the classical problem of nonparametric regression,but impose local differential privacy constraints. Under such constraints, theraw data $(X_1,Y_1),\ldots,(X_n,Y_n)$, taking values in $\mathbb{R}^d \times\mathbb{R}$, cannot be directly observed, and all estimators are functions ofthe randomised output from a suitable privacy mechanism. The statistician isfree to choose the form of the privacy mechanism, and here we add Laplacedistributed 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 asa privatised version of the well-studied partitioning regression estimator. Themain result is that the estimator is strongly universally consistent. Ourmethods and analysis also give rise to a strongly universally consistent binaryclassification rule for locally differentially private data.

Strongly universally consistent nonparametric regression and classification with privatised data