  We derive the optimal $(0, \delta)$-differentially private query-output
independent noise-adding mechanism for single real-valued query function under
a general cost-minimization framework. Under a mild technical condition, we
show that the optimal noise probability distribution is a uniform distribution
with a probability mass at the origin. We explicitly derive the optimal noise
distribution for general $\ell^p$ cost functions, including $\ell^1$ (for noise
magnitude) and $\ell^2$ (for noise power) cost functions, and show that the
probability concentration on the origin occurs when $\delta > \frac{p}{p+1}$.
Our result demonstrates an improvement over the existing Gaussian mechanisms by
a factor of two and three for $(0,\delta)$-differential privacy in the high
privacy regime in the context of minimizing the noise magnitude and noise
power, and the gain is more pronounced in the low privacy regime. Our result is
consistent with the existing result for $(0,\delta)$-differential privacy in
the discrete setting, and identifies a probability concentration phenomenon in
the continuous setting.
