Combinatorial Bandits generalize multi-armed bandits, where k out of n arms are chosen at each round and the sum of the rewards is gained. We address the full-bandit feedback, in which the agent observes only the sum of rewards, in contrast to the semi-bandit feedback, in which the agent observes also the individual arms' rewards. We present the Combinatorial Successive Accepts and Rejects (CSAR) algorithm, which is a generalization of the SAR algorithm ( 2013) for the combinatorial setting. Our main contribution is an efficient sampling scheme that uses Hadamard matrices in order to estimate accurately the individual arms' expected rewards. We discuss two variants of the algorithm, the first minimizes the sample complexity and the second minimizes the regret. For the sample complexity we also prove a matching lower bound that shows it is optimal. For the regret minimization, we prove a lower bound which is tight up to a factor of k. Finally, we run experiments and show that our algorithm outperforms other methods.
The problem of , a.k.a. , was introduced in @cite_6 , and later in @cite_9 , where the goal is to find the best arm using a minimal number of samples. The authors of @cite_6 describe two algorithms for this end, one of them is that in each round estimates all the arms with an increasing level of accuracy, and eliminates the arms which are far from the optimal arm with high confidence. This algorithm is the conceptual basis for a number of algorithms, including the one we describe in this work.
@cite_9 @cite_6
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This paper proposes a new method for the K-armed dueling bandit problem, a variation on the regular K-armed bandit problem that offers only relative feedback about pairs of arms. Our approach extends the Upper Confidence Bound algorithm to the relative setting by using estimates of the pairwise probabilities to select a promising arm and applying Upper Confidence Bound with the winner as a benchmark. We prove a sharp finite-time regret bound of order O(K log T) on a very general class of dueling bandit problems that matches a lower bound proven in (, 2012). In addition, our empirical results using real data from an information retrieval application show that it greatly outperforms the state of the art. Motivated by applications in computational advertising and systems biology, we consider the problem of identifying the best out of several possible soft interventions at a source node V in an acyclic causal directed graph, to maximize the expected value of a target node Y (located downstream of V). Our setting imposes a fixed total budget for sampling under various interventions, along with cost constraints on different types of interventions. We pose this as a best arm identification bandit problem with K arms where each arm is a soft intervention at V, and leverage the information leakage among the arms to provide the first gap dependent error and simple regret bounds for this problem. Our results are a significant improvement over the traditional best arm identification results. We empirically show that our algorithms outperform the state of the art in the Flow Cytometry data-set, and also apply our algorithm for model interpretation of the Inception-v3 deep net that classifies images. We consider the multi-armed bandit problem under the PAC ("probably approximately correct") model. It was shown by Even- (2002) that given n arms, a total of O((n e2)log(1 δ)) trials suffices in order to find an e-optimal arm with probability at least 1-δ. We establish a matching lower bound on the expected number of trials under any sampling policy. We furthermore generalize the lower bound, and show an explicit dependence on the (unknown) statistics of the arms. We also provide a similar bound within a Bayesian setting. The case where the statistics of the arms are known but the identities of the arms are not, is also discussed. For this case, we provide a lower bound of Θ((1 e2)(n+log(1 δ))) on the expected number of trials, as well as a sampling policy with a matching upper bound. If instead of the expected number of trials, we consider the maximum (over all sample paths) number of trials, we establish a matching upper and lower bound of the form Θ((n e2)log(1 δ)). Finally, we derive lower bounds on the expected regret, in the spirit of Lai and Robbins. We present an asymptotically optimal algorithm for the max variant of the k-armed bandit problem. Given a set of k slot machines, each yielding payoff from a fixed (but unknown) distribution, we wish to allocate trials to the machines so as to maximize the expected maximum payoff received over a series of n trials. Subject to certain distributional assumptions, we show that O(k ln(k δ) ln(n)2 e2) trials are sufficient to identify, with probability at least 1 - δ, a machine whose expected maximum payoff is within e of optimal. This result leads to a strategy for solving the problem that is asymptotically optimal in the following sense: the gap between the expected maximum payoff obtained by using our strategy for n trials and that obtained by pulling the single best arm for all n trials approaches zero as n → ∞.