Strongly universally consistent nonparametric regression and classification with privatised data

In this paper we revisit the classical problem of nonparametric regression,\nbut impose local differential privacy constraints. Under such constraints, the\nraw data $(X_1,Y_1),\\ldots,(X_n,Y_n)$, taking values in $\\mathbb{R}^d \\times\n\\mathbb{R}$, cannot be directly observed, and all estimators are functions of\nthe randomised output from a suitable privacy mechanism. The statistician is\nfree to choose the form of the privacy mechanism, and here we add Laplace\ndistributed noise to a discretisation of the location of a feature vector $X_i$\nand to the value of its response variable $Y_i$. Based on this randomised data,\nwe design a novel estimator of the regression function, which can be viewed as\na privatised version of the well-studied partitioning regression estimator. The\nmain result is that the estimator is strongly universally consistent. Our\nmethods and analysis also give rise to a strongly universally consistent binary\nclassification rule for locally differentially private data.\n