  An interactive mechanism is an algorithm that stores a data set and answers
adaptively chosen queries to it. The mechanism is called differentially
private, if any adversary cannot distinguish whether a specific individual is
in the data set by interacting with the mechanism. We study composition
properties of differential privacy in concurrent compositions. In this setting,
an adversary interacts with k interactive mechanisms in parallel and can
interleave its queries to the mechanisms arbitrarily. Previously, Vadhan and
Wang [2021] proved an optimal concurrent composition theorem for
pure-differential privacy. We significantly generalize and extend their
results. Namely, we prove optimal parallel composition properties for several
major notions of differential privacy in the literature, including approximate
DP, R\'enyi DP, and zero-concentrated DP. Our results demonstrate that the
adversary gains no advantage by interleaving its queries to independently
running mechanisms. Hence, interactivity is a feature that differential privacy
grants us for free.
  Concurrently and independently of our work, Vadhan and Zhang [2022] proved an
optimal concurrent composition theorem for f-DP [Dong et al., 2022], which
implies our result for the approximate DP case.
