This paper introduces and addresses a wide class of stochastic bandit problems where the function mapping the arm to the corresponding reward exhibits some known structural properties.

Neural Information Processing Systems http://nips.cc/

We address reinforcement learning problems with finite state and action spaces where the underlying MDP has some known structure that could be potentially exploited to minimize the exploration rates of suboptimal (state, action) pairs. For any arbitrary structure, we derive problem-specific regret lower bounds satisfied by any learning algorithm. These lower bounds are made explicit for unstructured MDPs and for those whose transition probabilities and average reward functions are Lipschitz continuous w.r.t. the state and action. For Lipschitz MDPs, the bounds are shown not to scale with the sizes S and A of the state and action spaces, i.e., they are smaller than c log T where T is the time horizon and the constant c only depends on the Lipschitz structure, the span of the bias function, and the minimal action sub-optimality gap. This contrasts with unstructured MDPs where the regret lower bound typically scales as SA log T. We devise DEL (Directed Exploration Learning), an algorithm that matches our regret lower bounds. We further simplify the algorithm for Lipschitz MDPs, and show that the simplified version is still able to efficiently exploit the structure.

We consider the problem of community detection or clustering in the labeled Stochastic Block Model (LSBM) with a finite number K of clusters of sizes linearly growing with the global population of items n. Every pair of items is labeled independently at random, and label l appears with probability p(i, j, l) between two items in clusters indexed by i and j, respectively. The objective is to reconstruct the clusters from the observation of these random labels. Clustering under the SBM and their extensions has attracted much attention recently. Most existing work aimed at characterizing the set of parameters such that it is possible to infer clusters either positively correlated with the true clusters, or with a vanishing proportion of misclassified items, or exactly matching the true clusters. We find the set of parameters such that there exists a clustering algorithm with at most s misclassified items in average under the general LSBM and for any s = o(n), which solves one open problem raised in [2]. We further develop an algorithm, based on simple spectral methods, that achieves this fundamental performance limit within O(npolylog(n)) computations and without the a-priori knowledge of the model parameters.

This paper introduces and addresses a wide class of stochastic bandit problems where the function mapping the arm to the corresponding reward exhibits some known structural properties.