Abstract
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 x 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, 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.
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