  We study the sample complexity of learning threshold functions under the
constraint of differential privacy. It is assumed that each labeled example in
the training data is the information of one individual and we would like to
come up with a generalizing hypothesis $h$ while guaranteeing differential
privacy for the individuals. Intuitively, this means that any single labeled
example in the training data should not have a significant effect on the choice
of the hypothesis. This problem has received much attention recently; unlike
the non-private case, where the sample complexity is independent of the domain
size and just depends on the desired accuracy and confidence, for private
learning the sample complexity must depend on the domain size $X$ (even for
approximate differential privacy). Alon et al. (STOC 2019) showed a lower bound
of $\Omega(\log^*|X|)$ on the sample complexity and Bun et al. (FOCS 2015)
presented an approximate-private learner with sample complexity
$\tilde{O}\left(2^{\log^*|X|}\right)$. In this work we reduce this gap
significantly, almost settling the sample complexity. We first present a new
upper bound (algorithm) of $\tilde{O}\left(\left(\log^*|X|\right)^2\right)$ on
the sample complexity and then present an improved version with sample
complexity $\tilde{O}\left(\left(\log^*|X|\right)^{1.5}\right)$.
  Our algorithm is constructed for the related interior point problem, where
the goal is to find a point between the largest and smallest input elements. It
is based on selecting an input-dependent hash function and using it to embed
the database into a domain whose size is reduced logarithmically; this results
in a new database, an interior point of which can be used to generate an
interior point in the original database in a differentially private manner.
