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) shuffler 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) shuffler 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.

We give an $(\varepsilon,\delta)$-differentially private algorithm for the Multi-Armed Bandit (MAB) problem in the shuffle model with a distribution-dependent regret of $O\left(\left(\sum_{a:\Delta_a>0}\frac{\log T}{\Delta_a}\right)+\frac{k\sqrt{\log\frac{1}{\delta}}\log T}{\varepsilon}\right)$, and a distribution-independent regret of $O\left(\sqrt{kT\log T}+\frac{k\sqrt{\log\frac{1}{\delta}}\log T}{\varepsilon}\right)$, 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.

Locality Sensitive Hashing (LSH) is an effective method of indexing a set of items to support efficient nearest neighbors queries in high-dimensional spaces. The basic idea of LSH is that similar items should produce hash collisions with higher probability than dissimilar items. We study LSH for (not necessarily convex) polygons, and use it to give efficient data structures for similar shape retrieval. Arkin et al. [Arkin et al., 1991] represent polygons by their "turning function" - a function which follows the angle between the polygon’s tangent and the x-axis while traversing the perimeter of the polygon. They define the distance between polygons to be variations of the L_p (for p = 1,2) distance between their turning functions. This metric is invariant under translation, rotation and scaling (and the selection of the initial point on the perimeter) and therefore models well the intuitive notion of shape resemblance. We develop and analyze LSH near neighbor data structures for several variations of the L_p distance for functions (for p = 1,2). By applying our schemes to the turning functions of a collection of polygons we obtain efficient near neighbor LSH-based structures for polygons. To tune our structures to turning functions of polygons, we prove some new properties of these turning functions that may be of independent interest. As part of our analysis, we address the following problem which is of independent interest. Find the vertical translation of a function f that is closest in L₁ distance to a function g. We prove tight bounds on the approximation guarantee obtained by the translation which is equal to the difference between the averages of g and f.