  Posterior sampling, i.e., exponential mechanism to sample from the posterior
distribution, provides $\varepsilon$-pure differential privacy (DP) guarantees
and does not suffer from potentially unbounded privacy breach introduced by
$(\varepsilon,\delta)$-approximate DP. In practice, however, one needs to apply
approximate sampling methods such as Markov chain Monte Carlo (MCMC), thus
re-introducing the unappealing $\delta$-approximation error into the privacy
guarantees. To bridge this gap, we propose the Approximate SAample Perturbation
(abbr. ASAP) algorithm which perturbs an MCMC sample with noise proportional to
its Wasserstein-infinity ($W_\infty$) distance from a reference distribution
that satisfies pure DP or pure Gaussian DP (i.e., $\delta=0$). We then leverage
a Metropolis-Hastings algorithm to generate the sample and prove that the
algorithm converges in $W_\infty$ distance. We show that by combining our new
techniques with a localization step, we obtain the first nearly linear-time
algorithm that achieves the optimal rates in the DP-ERM problem with strongly
convex and smooth losses.
