  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.
