Correlation Clustering with Adaptive Similarity Queries

Abstract

We investigate learning algorithms that use similarity queries to approximately solve correlation clustering problems. The input consists of n objects; each pair of objects has a hidden binary similarity score that we can learn through a query. The goal is to use as few queries as possible to partition the objects into clusters so to achieve the optimal number OPT of disagreements with the scores. Our first set of contributions is algorithmic: we introduce ACC, a simple query-aware variant of an existing algorithm (KwikCluster, with expected error 3OPT but a vacuous O(n 2 ) worst-case bound on the number of queries) for which we prove several desirable properties. First, ACC has expected error 3OPT + O(n 3 /Q) when using Q < n 2 queries, and recovers KwikCluster's bound of 3OPT for Q = n 2 . Second, ACC accurately recovers every adversarially perturbed latent cluster C. Under stronger conditions on C, ACC can even be used to recover exactly all clusters with high probability. Third, we show an efficient variant, ACCESS, with the same expected error as ACC but using significantly less queries on some graphs. We empirically test our algorithms on real-world and synthetic datasets. Our second set of contributions is a nearly complete information-theoretic characterization of the query vs. error trade-off. First, using VC theory, for all Q = Ω(n) we prove the existence of algorithms with expected error at most OPT + n 5/2 / √ Q, and at most O n 3 /Q if OPT = 0. We then show that any randomized algorithm, when using at most Q queries, must output a clustering with expected cost OPT + Ω n 3 /Q , which matches the upper bound for Q = Θ(n). For the special case of OPT = 0 we prove a weaker lower bound of Ω n 2 / √ Q .Preprint. Under review.

Query-Efficient Correlation Clustering

“…Additionally, we perform an experimental study of the algorithm. Some of the results from [5] have been rediscovered several years later (in a weaker form) by Bressan, Cesa-Bianchi, Paudice, and Vitale [9]. They study the problem of query-efficient correlation clustering (Problem 1) in the adaptive setting, and provide a queryefficient algorithm, named ACC.…”

“…In terms of techniques, the only difference between our algorithm QECC and the ACC algorithm from [9] is that the latter adds a check that discards pivots when no neighbor is found after inspecting a random sample of size f (n − 1) = Q/(n − 1). This additional check is unnecessary from a theoretical viewpoint (see Theorem 3.1) and it has the disadvantage that it necessarily results in an adaptive algorithm.…”

“…This additional check is unnecessary from a theoretical viewpoint (see Theorem 3.1) and it has the disadvantage that it necessarily results in an adaptive algorithm. Moreover, the analysis of [9] is significantly more complex than ours, because they need to adapt the proof of the approximation guarantees of the QwickCluster algorithm from [1] to take into account the additional check, whereas we simply take the approximation guarantee as given and argue that stopping QwickCluster after k pivots have been selected only incurs an expected additional cost of n 2 /(2k) (Lemma 3.3).…”