**Abstract:**Estimating the quantiles of a large dataset is a fundamental problem in both the streaming algorithms literature and the differential privacy literature. However, all existing private mechanisms for distribution-independent quantile computation require space at least linear in the input size $n$. In this work, we devise a differentially private algorithm for the quantile estimation problem, with strongly sublinear space complexity, in the one-shot and continual observation settings. Our basic mechanism estimates any $\alpha$-approximate quantile of a length-$n$ stream over a data universe $\mathcal{X}$ with probability $1-\beta$ using $O\left( \frac{\log (|\mathcal{X}|/\beta) \log (\alpha \epsilon n)}{\alpha \epsilon} \right)$ space while satisfying $\epsilon$-differential privacy at a single time point. Our approach builds upon deterministic streaming algorithms for non-private quantile estimation instantiating the exponential mechanism using a utility function defined on sketch items, while (privately) sampling from intervals defined by the sketch. We also present another algorithm based on histograms that is especially well-suited to the multiple quantiles case. We implement our algorithms and experimentally evaluate them on synthetic and real-world datasets.

**Submission Length:**Regular submission (no more than 12 pages of main content)

**Changes Since Last Submission:**We updated Section 3.2 to reflect that the universe need not be finite; explicitly defined $Q_{\mathcal{D}}^q$ in def'n 3.5. We clarified our use of the phrase "confidence intervals" and the $\ell_1$ metric. We disambiguated the use of $\Delta$ for both the sketch representation and the sensitivity. We clarified what we meant by "neighboring databases" and updated our notational use of $S$ and $R$. We also added additional experiments. Generalized requirements for sketching algorithms we build upon.

**Assigned Action Editor:**~Gautam_Kamath1

**Submission Number:**866

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