Approximate Differential Privacy of the ${\mathrm{\ell }}_2$ Mechanism

We study the ${\mathrm{\ell }}_2$ mechanism for computing a $d$-dimensional statistic with bounded ${\mathrm{\ell }}_2$ sensitivity under approximate differential privacy. Across a range of privacy parameters, we find that the ${\mathrm{\ell }}_2$ mechanism obtains lower error than the Laplace and Gaussian mechanisms, matching the former at $d=1$ and approaching the latter as $d\to \mathrm{\infty }$.