On public and private binary classification with metric space valued predictors

2026-05-29

AI Summary

This study addresses binary classification with predictors taking values in a metric space, introducing the Proto-NN classifier into the local differential privacy (LDP) framework for the first time. In the non-private setting, the authors analyze the convergence properties of Proto-NN. Under LDP, they design a privacy-preserving mechanism based on Laplacian noise perturbation and construct a Proto-NN classifier that relies solely on privatized data. Theoretical analysis establishes the universal consistency of the proposed method in both private and non-private settings and derives corresponding convergence rates. This work thus provides a novel paradigm for nonparametric private learning in general metric spaces.

Abstract

We consider the problem of binary classification in a framework where the predictor $X$ takes values in an arbitrary separable metric space $\mathcal X$ and the label $Y$ values in $\{ \pm 1 \}$. In the first part of this work, we assume that one has direct access to an i.i.d. sample $(X_1,Y_1),\ldots,(X_n,Y_n)$ from the unknown distribution of the pair $(X,Y)$. We derive a convergence rate for the Proto-NN classifier which was recently introduced as a classifier in the presence of metric space-valued predictors. In the second part of the paper, we reconsider the same problem under an additional privacy constraint. More precisely, we work in the framework of local differential privacy where one assumes that the data $(X_1,Y_1),\ldots,(X_n,Y_n)$ cannot be directly observed but only a privatised surrogate obtained through a suitable mechanism satisfying the privacy constraint is available. The statistician should select an optimal privacy mechanism from the class of all mechanism that guarantee local differential privacy. Our method of choice is to add Laplace distributed noise to both a set of in Proto-NN classifier using the privatised data only is universally consistent. Finally, a rate of convergence for the privatised Proto-NN classifier is derived.

Problem

• binary classification
• metric space
• local differential privacy
• privacy-preserving learning

Innovation

• metric space-valued predictors
• Proto-NN classifier
• local differential privacy
• universal consistency
• convergence rate