Confirmation Sampling for Exact Nearest Neighbor Search

Abstract: Locality-sensitive hashing (LSH), introduced by Indyk and Motwani in STOC '98, has been an extremely influential framework for nearest neighbor search in high-dimensional data sets. While theoretical work has focused on the approximate nearest neighbor problems, in practice LSH data structures with suitably chosen parameters are used to solve the exact nearest neighbor problem (with some error probability). Sublinear query time is often possible in practice even for exact nearest neighbor search, intuitively because the nearest neighbor tends to be significantly closer than other data points. However, theory offers little advice on how to choose LSH parameters outside of pre-specified worst-case settings. We introduce the technique of confirmation sampling for solving the exact nearest neighbor problem using LSH. First, we give a general reduction that transforms a sequence of data structures that each find the nearest neighbor with a small, unknown probability, into a data structure that returns the nearest neighbor with probability $1-\delta$, using as few queries as possible. Second, we present a new query algorithm for the LSH Forest data structure with $L$ trees that is able to return the exact nearest neighbor of a query point within the same time bound as an LSH Forest of $\Omega(L)$ trees with internal parameters specifically tuned to the query and data.