Jay Tenenbaum
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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.
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We give an $(\eps,\delta)$-differentially private algorithm for the Multi-Armed Bandit (MAB) problem in the shuffle model with a distribution-dependent regret of $O\left(\left(\sum_{a:\Delta_a>0}\frac{\log T}{\Delta_a}\right)+\frac{k\sqrt{\log\frac{1}{\delta}}\log T}{\eps}\right)$, and a distribution-independent regret of $O\left(\sqrt{kT\log T}+\frac{k\sqrt{\log\frac{1}{\delta}}\log T}{\eps}\right)$, where $T$ is the number of rounds, $\Delta_a$ is the suboptimality gap of the action $a$, and $k$ is the total number of actions.
Our upper bound almost matches the regret of the best known algorithms for the centralized model, and significantly outperforms the best known algorithm in the local model.
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