  Differential privacy provides a rigorous framework to quantify data privacy,
and has received considerable interest recently. A randomized mechanism
satisfying $(\epsilon, \delta)$-differential privacy (DP) roughly means that,
except with a small probability $\delta$, altering a record in a dataset cannot
change the probability that an output is seen by more than a multiplicative
factor $e^{\epsilon} $. A well-known solution to $(\epsilon, \delta)$-DP is the
Gaussian mechanism initiated by Dwork et al. [1] in 2006 with an improvement by
Dwork and Roth [2] in 2014, where a Gaussian noise amount $\sqrt{2\ln
\frac{2}{\delta}} \times \frac{\Delta}{\epsilon}$ of [1] or $\sqrt{2\ln
\frac{1.25}{\delta}} \times \frac{\Delta}{\epsilon}$ of [2] is added
independently to each dimension of the query result, for a query with
$\ell_2$-sensitivity $\Delta$. Although both classical Gaussian mechanisms
[1,2] assume $0 < \epsilon \leq 1$, our review finds that many studies in the
literature have used the classical Gaussian mechanisms under values of
$\epsilon$ and $\delta$ where the added noise amounts of [1,2] do not achieve
$(\epsilon,\delta)$-DP. We obtain such result by analyzing the optimal noise
amount $\sigma_{DP-OPT}$ for $(\epsilon,\delta)$-DP and identifying $\epsilon$
and $\delta$ where the noise amounts of classical mechanisms are even less than
$\sigma_{DP-OPT}$.
  Since $\sigma_{DP-OPT}$ has no closed-form expression and needs to be
approximated in an iterative manner, we propose Gaussian mechanisms by deriving
closed-form upper bounds for $\sigma_{DP-OPT}$. Our mechanisms achieve
$(\epsilon,\delta)$-DP for any $\epsilon$, while the classical mechanisms [1,2]
do not achieve $(\epsilon,\delta)$-DP for large $\epsilon$ given $\delta$.
Moreover, the utilities of our mechanisms improve those of [1,2] and are close
to that of the optimal yet more computationally expensive Gaussian mechanism.
