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The contextual combinatorial semi-bandit problem with linear payoff functions is a decision-making problem in which a learner chooses a set of arms with the feature vectors in each round under given constraints so as to maximize the sum of rewards of arms. Several existing algorithms have regret bounds that are optimal with respect to the number of rounds $T$. However, there is a gap of $tilde{O}(max(sqrt{d}, sqrt{k}))$ between the current best upper and lower bounds, where $d$ is the dimension of the feature vectors, $k$ is the number of the chosen arms in a round, and $tilde{O}(cdot)$ ignores the logarithmic factors. The dependence of $k$ and $d$ is of practical importance because $k$ may be larger than $T$ in real-world applications such as recommender systems. In this paper, we fill the gap by improving the upper and lower bounds. More precisely, we show that the C${}^2$UCB algorithm proposed by Qin, Chen, and Zhu (2014) has the optimal regret bound $tilde{O}(dsqrt{kT} + dk)$ for the partition matroid constraints. For general constraints, we propose an algorithm that modifies the reward estimates of arms in the C${}^2$UCB algorithm and demonstrate that it enjoys the optimal regret bound for a more general problem that can take into account other objectives simultaneously. We also show that our technique would be applicable to related problems. Numerical experiments support our theoretical results and considerations.
In this paper we study the adversarial combinatorial bandit with a known non-linear reward function, extending existing work on adversarial linear combinatorial bandit. {The adversarial combinatorial bandit with general non-linear reward is an import
Standard approaches to decision-making under uncertainty focus on sequential exploration of the space of decisions. However, textit{simultaneously} proposing a batch of decisions, which leverages available resources for parallel experimentation, has
We present simple and efficient algorithms for the batched stochastic multi-armed bandit and batched stochastic linear bandit problems. We prove bounds for their expected regrets that improve over the best-known regret bounds for any number of batche
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