  We study the problem of differentially private query release assisted by
access to public data. In this problem, the goal is to answer a large class
$\mathcal{H}$ of statistical queries with error no more than $\alpha$ using a
combination of public and private samples. The algorithm is required to satisfy
differential privacy only with respect to the private samples. We study the
limits of this task in terms of the private and public sample complexities.
  First, we show that we can solve the problem for any query class
$\mathcal{H}$ of finite VC-dimension using only $d/\alpha$ public samples and
$\sqrt{p}d^{3/2}/\alpha^2$ private samples, where $d$ and $p$ are the
VC-dimension and dual VC-dimension of $\mathcal{H}$, respectively. In
comparison, with only private samples, this problem cannot be solved even for
simple query classes with VC-dimension one, and without any private samples, a
larger public sample of size $d/\alpha^2$ is needed. Next, we give sample
complexity lower bounds that exhibit tight dependence on $p$ and $\alpha$. For
the class of decision stumps, we give a lower bound of $\sqrt{p}/\alpha$ on the
private sample complexity whenever the public sample size is less than
$1/\alpha^2$. Given our upper bounds, this shows that the dependence on
$\sqrt{p}$ is necessary in the private sample complexity. We also give a lower
bound of $1/\alpha$ on the public sample complexity for a broad family of query
classes, which by our upper bound, is tight in $\alpha$.
