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 careful 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.

提出了一种使用后验抽样和近似采样方法相结合的算法，通过引入噪声来保持隐私性，并结合局部化步骤，在DP-ERM问题中实现了最佳的速率。

基于纯差分和高斯差分隐私的可处理的MCMC私密学习