  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.
