Near-Optimal Regret Bounds for Contextual Combinatorial Semi-Bandits with Linear Payoff Functions

Authors

  • Kei Takemura NEC Corporation
  • Shinji Ito NEC Corporation
  • Daisuke Hatano RIKEN AIP
  • Hanna Sumita Tokyo Institute of Technology
  • Takuro Fukunaga Chuo University JST PRESTO RIKEN AIP
  • Naonori Kakimura Keio University
  • Ken-ichi Kawarabayashi National Institute of Informatics

Keywords:

Online Learning & Bandits

Abstract

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 Õ(max(√d, √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 Õ(·) 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 C2UCB algorithm proposed by Qin, Chen, and Zhu (2014) has the optimal regret bound Õ(d√kT + dk) for the partition matroid constraints. For general constraints, we propose an algorithm that modifies the reward estimates of arms in the C2UCB 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.

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Published

2021-05-18

How to Cite

Takemura, K., Ito, S., Hatano, D., Sumita, H., Fukunaga, T., Kakimura, N., & Kawarabayashi, K.- ichi. (2021). Near-Optimal Regret Bounds for Contextual Combinatorial Semi-Bandits with Linear Payoff Functions. Proceedings of the AAAI Conference on Artificial Intelligence, 35(11), 9791-9798. Retrieved from https://ojs.aaai.org/index.php/AAAI/article/view/17177

Issue

Section

AAAI Technical Track on Machine Learning IV