A classic problem in machine learning and data analysis is to partition the vertices of a network in such a way that vertices in the same set are densely connected and vertices in different sets are loosely connected. In practice, the most popular approaches rely on local search algorithms; not only for the ease of implementation and the efficiency, but also because of the accuracy of these methods on many real world graphs. For example, the Louvain algorithm - a local search based algorithm - has quickly become the method of choice for clustering in social networks.

We study a secretary problem which captures the task of ranking in online settings.

We thank all reviewers for giving us the insightful comments. Then we collect all positive samples by a Breadth-First Search algorithm. We will add these results to our paper. We also give the qualitative analysis in Figure 2 (b). About the baseline, our baseline is the KNN method, i.e. directly using the nearst We have compared our algorithm with the KNN algorithm in Sec 4.2.

The lower bound in higher dimensions is attained by an orthogonal subspace argument.

Applying BO with GPs on combinatorial spaces is, therefore, not straightforward.

We consider a sequential learning problem with Gaussian payoffs and side observations: after selecting an action i, the learner receives information about the payoff of every action j in the form of Gaussian observations whose mean is the same as the mean payoff, but the variance depends on the pair (i,j) (and may be infinite). The setup allows a more refined information transfer from one action to another than previous partial monitoring setups, including the recently introduced graph-structured feedback case. For the first time in the literature, we provide non-asymptotic problem-dependent lower bounds on the regret of any algorithm, which recover existing asymptotic problem-dependent lower bounds and finite-time minimax lower bounds available in the literature. We also provide algorithms that achieve the problem-dependent lower bound (up to some universal constant factor) or the minimax lower bounds (up to logarithmic factors).

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