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.
翻译:根据真实价值的假设等级 $\ mathcal{H},我们调查在什么条件下有不同的私人算法,从 $\ mathcal{H} 获得 i.d. d. 数据。受相关二进制分类最近结果的启发(Alon 等人, 2019; Bun 等人, 2020) 显示二进制班的在线学习能力对于其私人学习能力是必要和足够的, Jung 等人(202020) 显示在回归设置中, $\ macal{ H} $的在线学习能力对于私人学习是必需的。 $\ mathcal{ H} 的在线学习能力是美元- 美元- 后期脂肪的有限性( 美元; bun等人) 显示,对于所有 美元/ eta> 0 美元, 在私人学习的充足条件中, Jung等人(2020) 显示, 美元/ madcal{ h} 美元是私人学习的 美元,如果 美元- detatamaral_ lexlexal rodeal rodeal) a rodeal rodeal rode rodeal.