  Most differential privacy mechanisms are applied (i.e., composed) numerous
times on sensitive data. We study the design of optimal differential privacy
mechanisms in the limit of a large number of compositions. As a consequence of
the law of large numbers, in this regime the best privacy mechanism is the one
that minimizes the Kullback-Leibler divergence between the conditional output
distributions of the mechanism given two different inputs. We formulate an
optimization problem to minimize this divergence subject to a cost constraint
on the noise. We first prove that additive mechanisms are optimal. Since the
optimization problem is infinite dimensional, it cannot be solved directly;
nevertheless, we quantize the problem to derive near-optimal additive
mechanisms that we call "cactus mechanisms" due to their shape. We show that
our quantization approach can be arbitrarily close to an optimal mechanism.
Surprisingly, for quadratic cost, the Gaussian mechanism is strictly
sub-optimal compared to this cactus mechanism. Finally, we provide numerical
results which indicate that cactus mechanism outperforms the Gaussian mechanism
for a finite number of compositions.
