We introduce the concurrent shuffle model of differential privacy. In this model we have multiple concurrent shufflers permuting messages from different, possibly overlapping, batches of users. Similarly to the standard (single) shuffle model, the privacy requirement is that the concatenation of all shuffled messages should be differentially private. We study the private continual summation problem (a.k.a. the counter problem) and show that the concurrent shuffle model allows for significantly improved error compared to a standard (single) shuffle model. Specifically, we give a summation algorithm with error $\tilde{O}(n^{1/(2k+1)})$ with $k$ concurrent shufflers on a sequence of length $n$. Furthermore, we prove that this bound is tight for any $k$, even if the algorithm can choose the sizes of the batches adaptively. For $k=\log n$ shufflers, the resulting error is polylogarithmic, much better than $\tilde{\Theta}(n^{1/3})$ which we show is the smallest possible with a single shuffler. We use our online summation algorithm to get algorithms with improved regret bounds for the contextual linear bandit problem. In particular we get optimal $\tilde{O}(\sqrt{n})$ regret with $k= \tilde{\Omega}(\log n)$ concurrent shufflers.

翻译：我们引入差分隐私的并发混洗模型。在该模型中，存在多个并发混洗器，它们对不同（可能重叠）的用户批次中的消息进行置换。与标准（单）混洗模型类似，隐私要求是：所有已混洗消息的拼接结果应满足差分隐私。我们研究了私有持续求和问题（又称计数器问题），并表明与标准（单）混洗模型相比，并发混洗模型能够显著降低误差。具体而言，我们提出了一种求和算法，在长度为$n$的序列上使用$k$个并发混洗器时，误差为$\tilde{O}(n^{1/(2k+1)})$。此外，我们证明对于任意$k$，该界是紧致的，即使算法可以自适应地选择批次大小也是如此。当$k=\log n$个混洗器时，所得误差为多对数级别，远优于我们证明的单个混洗器所能达到的最小可能误差$\tilde{\Theta}(n^{1/3})$。我们利用在线求和算法为上下文线性赌博机问题设计了具有更优遗憾界的算法。特别地，当$k= \tilde{\Omega}(\log n)$个并发混洗器时，我们实现了最优的$\tilde{O}(\sqrt{n})$遗憾界。