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Sample-Optimal Locally Private Hypothesis Selection and the Provable Benefits of Interactivity
Proceedings of Thirty Seventh Conference on Learning Theory, PMLR 247:4240-4275, 2024.
Abstract
We study the problem of hypothesis selection under the constraint of local differential privacy. Given a class F of k distributions and a set of i.i.d. samples from an unknown distribution h, the goal of hypothesis selection is to pick a distribution ˆf whose total variation distance to h is comparable with the best distribution in F (with high probability). We devise an ε-locally-differentially-private (ε-LDP) algorithm that uses Θ(kα2min samples to guarantee that d_{TV}(h,\hat{f})\leq \alpha + 9 \min_{f\in \mathcal{F}}d_{TV}(h,f) with high probability. This sample complexity is optimal for varepsilon<1, matching the lower bound of Gopi et al. (2020). All previously known algorithms for this problem required \Omega\left(\frac{k\log k}{\alpha^2\min \{\varepsilon^2 ,1\}} \right) samples to work. Moreover, our result demonstrates the power of interaction for \varepsilon-LDP hypothesis selection. Namely, it breaks the known lower bound of \Omega\left(\frac{k\log k}{\alpha^2 \varepsilon^2} \right) for the sample complexity of non-interactive hypothesis selection. Our algorithm achieves this using only \Theta(\log \log k) rounds of interaction. To prove our results, we define the notion of \emph{critical queries} for a Statistical Query Algorithm (SQA) which may be of independent interest. Informally, an SQA is said to use a small number of critical queries if its success relies on the accuracy of only a small number of queries it asks. We then design an LDP algorithm that uses a smaller number of critical queries.