to transform a learner in the realizable setting to a learner in the agnostic setting[^6]. We note that formally the transformation in [@alon2020closure] is stated for a constant ${\varepsilon}=O(1)$. Taking ${\varepsilon}=O(1)$ is without loss of generality as a standard “secrecy-of-the-sample” argument can be used to convert this learner into one which is $({\varepsilon}, \delta)$-differentially private by increasing the sample size by a factor of roughly $1/{\varepsilon}$ and running the algorithm on a random subsample. See [@KasiviswanathanLNRS11; @Vadhan17] for further details.

\[thm:equivalence\] The following statements are equivalent for a class ${\mathcal{H}}\subseteq \{\pm 1\}^X$:

1.  ${\mathcal{H}}$ is online learnable.

2.  ${\mathcal{H}}$ is approximate differentially-privately PAC learnable.

Theorem \[thm:equivalence\] is a corollary of Theorem \[thm:main\] (which gives $1\to 2$) and the result by Alon et al. [@AlonLMM19] (which gives $2\to 1$). We comment that a quantitative relation between the learning and regret rates is also implied: it is known that the optimal regret bound for ${\mathcal{H}}$ is $\tilde \Theta_d(\sqrt{T})$, where the $\tilde \Theta_d$ conceals a constant which depends on the Littlestone dimension of ${\mathcal{H}}$ [@Bendavid09agnostic]. Similarly, we get that the optimal sample complexity of agnostically privately learning ${\mathcal{H}}$ is $\Theta_d(\frac{\log({1}/(\beta\delta))}{\alpha^2{\varepsilon}})$.

We remark however that the above equivalence is mostly interesting from a theoretical 