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We study the problem of private distribution learning with access to publicdata. In this setup, which we refer to as public-private learning, the learneris given public and private samples drawn from an unknown distribution $p$belonging to a class $\mathcal Q$, with the goal of outputting an estimate of$p$ while adhering to privacy constraints (here, pure differential privacy)only with respect to the private samples.  We show that the public-private learnability of a class $\mathcal Q$ isconnected to the existence of a sample compression scheme for $\mathcal Q$, aswell as to an intermediate notion we refer to as list learning. Leveraging thisconnection: (1) approximately recovers previous results on Gaussians over$\mathbb R^d$; and (2) leads to new ones, including sample complexity upperbounds for arbitrary $k$-mixtures of Gaussians over $\mathbb R^d$, results foragnostic and distribution-shift resistant learners, as well as closureproperties for public-private learnability under taking mixtures and productsof distributions. Finally, via the connection to list learning, we show thatfor Gaussians in $\mathbb R^d$, at least $d$ public samples are necessary forprivate learnability, which is close to the known upper bound of $d+1$ publicsamples.

Private Distribution Learning with Public Data: The View from Sample Compression