The *turnstile* continual release model of differential privacy captures scenarios where a privacy-preserving real-time analysis is sought for a dataset evolving through additions and deletions. In typical applications of real-time data analysis, both the length of the stream $T$ and the size of the universe $|\mathcal{U}|$ from which data come can be extremely large. This motivates the study of private algorithms in the turnstile setting using space sublinear in both $T$ and $|\mathcal{U}|$. In this paper, we give the first sublinear space differentially private algorithms for the fundamental problems of counting distinct elements in the turnstile streaming model. Our algorithm achieves, on arbitrary streams, $O_{\eta}(T^{1/3})$ space and additive error, and a $(1+\eta)$-relative approximation for all $\eta \in (0,1)$. Our result significantly improves upon the space requirements of the state-of-the-art algorithms for this problem, which is linear, approaching the known $\Omega(T^{1/4})$ additive error lower bound for arbitrary streams. Moreover, when a bound $W$ on the number of times an item appears in the stream is known, our algorithm provides $O_{\eta}(\sqrt{W})$ additive error, using $O_{\eta}(\sqrt{W})$ space. This additive error asymptotically matches that of prior work which required instead linear space. Our results address an open question posed by Jain et al. about designing low-memory mechanisms for this problem. We complement this results with a space lower bound for this problem, which shows that any algorithm that uses similar techniques must use space $\Omega(T^{1/3})$.

In this work we first show that the classical Thompson sampling algorithm for multi-arm bandits is differentially private as-is, without any modification. We provide per-round privacy guarantees as a function of problem parameters and show composition over $T$ rounds; since the algorithm is unchanged, existing $O(\sqrt{NT\log N})$ regret bounds still hold and there is no loss in performance due to privacy. We then show that simple modifications – such as pre-pulling all arms a fixed number of times, increasing the sampling variance – can provide tighter privacy guarantees. We again provide privacy guarantees that now depend on the new parameters introduced in the modification, which allows the analyst to tune the privacy guarantee as desired. We also provide a novel regret analysis for this new algorithm, and show how the new parameters also impact expected regret. Finally, we empirically validate and illustrate our theoretical findings in two parameter regimes and demonstrate that tuning the new parameters substantially improve the privacy-regret tradeoff.

In this work we introduce an alternative model for the design and analysis of strategyproof mechanisms that is motivated by the recent surge of work in "learning-augmented algorithms". Aiming to complement the traditional approach in computer science, which analyzes the performance of algorithms based on worst-case instances, this line of work has focused on the design and analysis of algorithms that are enhanced with machine-learned predictions regarding the optimal solution. The algorithms can use the predictions as a guide to inform their decisions, and the goal is to achieve much stronger performance guarantees when these predictions are accurate (consistency), while also maintaining near-optimal worst-case guarantees, even if these predictions are very inaccurate (robustness). So far, these results have been limited to algorithms, but in this work we argue that another fertile ground for this framework is in mechanism design.