Locally Private Hypothesis Selection

We initiate the study of hypothesis selection under local differential privacy. Given samples from an unknown probability distribution $p$ and a set of $k$ probability distributions $mathcal{Q}$, we aim to output, under the constraints of $varepsilon$-local differential privacy, a distribution from $mathcal{Q}$ whose total variation distance to $p$ is comparable to the best such distribution. This is a generalization of the classic problem of $k$-wise simple hypothesis testing, which corresponds to when $p in mathcal{Q}$, and we wish to identify $p$. Absent privacy constraints, this problem requires $O(log k)$ samples from $p$, and it was recently shown that the same complexity is achievable under (central) differential privacy. However, the naive approach to this problem under local differential privacy would require $tilde O(k^2)$ samples. We first show that the constraint of local differential privacy incurs an exponential increase in cost: any algorithm for this problem requires at least $Omega(k)$ samples. Second, for the special case of $k$-wise simple hypothesis testing, we provide a non-interactive algorithm which nearly matches this bound, requiring $tilde O(k)$ samples. Finally, we provide sequentially interactive algorithms for the general case, requiring $tilde O(k)$ samples and only $O(log log k)$ rounds of interactivity. Our algorithms are achieved through a reduction to maximum selection with adversarial comparators, a problem of independent interest for which we initiate study in the parallel setting. For this problem, we provide a family of algorithms for each number of allowed rounds of interaction $t$, as well as lower bounds showing that they are near-optimal for every $t$. Notably, our algorithms result in exponential improvements on the round complexity of previous methods.