  In this paper we prove that the sample complexity of properly learning a
class of Littlestone dimension $d$ with approximate differential privacy is
$\tilde O(d^6)$, ignoring privacy and accuracy parameters. This result answers
a question of Bun et al. (FOCS 2020) by improving upon their upper bound of
$2^{O(d)}$ on the sample complexity. Prior to our work, finiteness of the
sample complexity for privately learning a class of finite Littlestone
dimension was only known for improper private learners, and the fact that our
learner is proper answers another question of Bun et al., which was also asked
by Bousquet et al. (NeurIPS 2020). Using machinery developed by Bousquet et
al., we then show that the sample complexity of sanitizing a binary hypothesis
class is at most polynomial in its Littlestone dimension and dual Littlestone
dimension. This implies that a class is sanitizable if and only if it has
finite Littlestone dimension. An important ingredient of our proofs is a new
property of binary hypothesis classes that we call irreducibility, which may be
of independent interest.
