Query-Efficient Locally Private Hypothesis Selection via the Scheffe Graph

We propose an algorithm with improved query-complexity for the problem of hypothesis selection under local differential privacy constraints. Given a set of $k$ probability distributions $Q$, we describe an algorithm that satisfies local differential privacy, performs $\tilde{O}(k^{3/2})$ non-adaptive queries to individuals who each have samples from a probability distribution $p$, and outputs a probability distribution from the set $Q$ which is nearly the closest to $p$. Previous algorithms required either $\Omega(k^2)$ queries or many rounds of interactive queries. Technically, we introduce a new object we dub the Scheff\'e graph, which captures structure of the differences between distributions in $Q$, and may be of more broad interest for hypothesis selection tasks.