  A new line of work, started with Dwork et al., studies the task of answering
statistical queries using a sample and relates the problem to the concept of
differential privacy. By the Hoeffding bound, a sample of size $O(\log
k/\alpha^2)$ suffices to answer $k$ non-adaptive queries within error $\alpha$,
where the answers are computed by evaluating the statistical queries on the
sample. This argument fails when the queries are chosen adaptively (and can
hence depend on the sample). Dwork et al. showed that if the answers are
computed with $(\epsilon,\delta)$-differential privacy then $O(\epsilon)$
accuracy is guaranteed with probability $1-O(\delta^\epsilon)$. Using the
Private Multiplicative Weights mechanism, they concluded that the sample size
can still grow polylogarithmically with the $k$.
  Very recently, Bassily et al. presented an improved bound and showed that (a
variant of) the private multiplicative weights algorithm can answer $k$
adaptively chosen statistical queries using sample complexity that grows
logarithmically in $k$. However, their results no longer hold for every
differentially private algorithm, and require modifying the private
multiplicative weights algorithm in order to obtain their high probability
bounds.
  We greatly simplify the results of Dwork et al. and improve on the bound by
showing that differential privacy guarantees $O(\epsilon)$ accuracy with
probability $1-O(\delta\log(1/\epsilon)/\epsilon)$. It would be tempting to
guess that an $(\epsilon,\delta)$-differentially private computation should
guarantee $O(\epsilon)$ accuracy with probability $1-O(\delta)$. However, we
show that this is not the case, and that our bound is tight (up to logarithmic
factors).
