Motivated by a problem of targeted advertising in social networks, we introduce and study a new model of online learning on labeled graphs where the graph is initially unknown and the algorithm is free to choose which vertex to predict next. After observing that natural nonadaptive exploration/prediction strategies, like depth-first with majority vote, do not behave satisfactorily on simple binary labeled graphs, we introduce an adaptive strategy that performs well under the hypothesis that the vertices of the unknown graph (i.e., members of the social network) can be partitioned into a few well-separated clusters within which labels (i.e., members' preferences) are roughly constant. Our algorithm is efficiently implementable and provably competitive against the best of these partitions.
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.
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