Differentially Private Nonparametric Regression Under a Growth Condition

TL;DR

This paper establishes new conditions under which nonparametric regression hypotheses can be learned privately, relaxing previous restrictive assumptions by introducing a novel filtering technique for stable hypothesis output.

Contribution

It proves that if the scaled fat shattering dimension tends to zero, then nonparametric classes are privately learnable, extending private learnability results to broader classes.

Findings

Private learnability under relaxed growth conditions of the fat shattering dimension.

Introduction of a novel filtering procedure for stable hypothesis output.

First nonparametric private learnability guarantee for diverging fat shattering classes.

Abstract

Given a real-valued hypothesis class $\mathcal{H}$, we investigate under what conditions there is a differentially private algorithm which learns an optimal hypothesis from $\mathcal{H}$ given i.i.d. data. Inspired by recent results for the related setting of binary classification (Alon et al., 2019; Bun et al., 2020), where it was shown that online learnability of a binary class is necessary and sufficient for its private learnability, Jung et al. (2020) showed that in the setting of regression, online learnability of $\mathcal{H}$ is necessary for private learnability. Here online learnability of $\mathcal{H}$ is characterized by the finiteness of its $\eta$-sequential fat shattering dimension, ${\rm sfat}_\eta(\mathcal{H})$, for all $\eta > 0$. In terms of sufficient conditions for private learnability, Jung et al. (2020) showed that $\mathcal{H}$ is privately learnable if $\lim_{\eta \downarrow 0} {\rm sfat}_\eta(\mathcal{H})$ is finite, which is a fairly restrictive condition. We show that under the relaxed condition $\lim \inf_{\eta \downarrow 0} \eta \cdot {\rm sfat}_\eta(\mathcal{H}) = 0$, $\mathcal{H}$ is privately learnable, establishing the first nonparametric private learnability guarantee for classes $\mathcal{H}$ with ${\rm sfat}_\eta(\mathcal{H})$ diverging as $\eta \downarrow 0$. Our techniques involve a novel filtering procedure to output stable hypotheses for nonparametric function classes.