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Register a new userProblem Dependent View on Structured Thresholding Bandit Problems
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We investigate the problem dependent regime in the stochastic Thresholding Bandit problem (TBP) under several shape constraints. In the TBP, the objective of the learner is to output, at the end of a sequential game, the set of arms whose means are above a given threshold. The vanilla, unstructured, case is already well studied in the literature. Taking $K$ as the number of arms, we consider the case where (i) the sequence of arms means $(mu_k)_{k=1}^K$ is monotonically increasing (MTBP) and (ii) the case where $(mu_k)_{k=1}^K$ is concave (CTBP). We consider both cases in the problem dependent regime and study the probability of error - i.e. the probability to mis-classify at least one arm. In the fixed budget setting, we provide upper and lower bounds for the probability of error in both the concave and monotone settings, as well as associated algorithms. In both settings the bounds match in the problem dependent regime up to universal constants in the exponential.
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The Thresholding Bandit Problem (TBP) aims to find the set of arms with mean rewards greater than a given threshold. We consider a new setting of TBP, where in addition to pulling arms, one can also emph{duel} two arms and get the arm with a greater mean. In our motivating application from crowdsourcing, dueling two arms can be more cost-effective and time-efficient than direct pulls. We refer to this problem as TBP with Dueling Choices (TBP-DC). This paper provides an algorithm called Rank-Search (RS) for solving TBP-DC by alternating between ranking and binary search. We prove theoretical guarantees for RS, and also give lower bounds to show the optimality of it. Experiments show that RS outperforms previous baseline algorithms that only use pulls or duels.
The stochastic multi-armed bandit (MAB) problem is a common model for sequential decision problems. In the standard setup, a decision maker has to choose at every instant between several competing arms, each of them provides a scalar random variable, referred to as a reward. Nearly all research on this topic considers the total cumulative reward as the criterion of interest. This work focuses on other natural objectives that cannot be cast as a sum over rewards, but rather more involved functions of the reward stream. Unlike the case of cumulative criteria, in the problems we study here the oracle policy, that knows the problem parameters a priori and is used to center the regret, is not trivial. We provide a systematic approach to such problems, and derive general conditions under which the oracle policy is sufficiently tractable to facilitate the design of optimism-based (upper confidence bound) learning policies. These conditions elucidate an interesting interplay between the arm reward distributions and the performance metric. Our main findings are illustrated for several commonly used objectives such as conditional value-at-risk, mean-variance trade-offs, Sharpe-ratio, and more.
We study the $K$-armed dueling bandit problem, a variation of the standard stochastic bandit problem where the feedback is limited to relative comparisons of a pair of arms. We introduce a tight asymptotic regret lower bound that is based on the information divergence. An algorithm that is inspired by the Deterministic Minimum Empirical Divergence algorithm (Honda and Takemura, 2010) is proposed, and its regret is analyzed. The proposed algorithm is found to be the first one with a regret upper bound that matches the lower bound. Experimental comparisons of dueling bandit algorithms show that the proposed algorithm significantly outperforms existing ones.
We study the K-armed dueling bandit problem, a variation of the standard stochastic bandit problem where the feedback is limited to relative comparisons of a pair of arms. The hardness of recommending Copeland winners, the arms that beat the greatest number of other arms, is characterized by deriving an asymptotic regret bound. We propose Copeland Winners Relative Minimum Empirical Divergence (CW-RMED) and derive an asymptotically optimal regret bound for it. However, it is not known whether the algorithm can be efficiently computed or not. To address this issue, we devise an efficient version (ECW-RMED) and derive its asymptotic regret bound. Experimental comparisons of dueling bandit algorithms show that ECW-RMED significantly outperforms existing ones.
We discuss a multiple-play multi-armed bandit (MAB) problem in which several arms are selected at each round. Recently, Thompson sampling (TS), a randomized algorithm with a Bayesian spirit, has attracted much attention for its empirically excellent performance, and it is revealed to have an optimal regret bound in the standard single-play MAB problem. In this paper, we propose the multiple-play Thompson sampling (MP-TS) algorithm, an extension of TS to the multiple-play MAB problem, and discuss its regret analysis. We prove that MP-TS for binary rewards has the optimal regret upper bound that matches the regret lower bound provided by Anantharam et al. (1987). Therefore, MP-TS is the first computationally efficient algorithm with optimal regret. A set of computer simulations was also conducted, which compared MP-TS with state-of-the-art algorithms. We also propose a modification of MP-TS, which is shown to have better empirical performance.
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