  The shuffle model of Differential Privacy (DP) has gained significant
attention in privacy-preserving data analysis due to its remarkable tradeoff
between privacy and utility. It is characterized by adding a shuffling
procedure after each user's locally differentially private perturbation, which
leads to a privacy amplification effect, meaning that the privacy guarantee of
a small level of noise, say $\epsilon_0$, can be enhanced to
$O(\epsilon_0/\sqrt{n})$ (the smaller, the more private) after shuffling all
$n$ users' perturbed data. Most studies in the shuffle DP focus on proving a
tighter privacy guarantee of privacy amplification. However, the current
results assume that the local privacy budget $\epsilon_0$ is within a limited
range. In addition, there remains a gap between the tightest lower bound and
the known upper bound of the privacy amplification. In this work, we push
forward the state-of-the-art by making the following contributions. Firstly, we
present the first asymptotically optimal analysis of Renyi Differential Privacy
(RDP) in the shuffle model without constraints on $\epsilon_0$. Secondly, we
introduce hypothesis testing for privacy amplification through shuffling,
offering a distinct analysis technique and a tighter upper bound. Furthermore,
we propose a DP-SGD algorithm based on RDP. Experiments demonstrate that our
approach outperforms existing methods significantly at the same privacy level.
