## Abstract

Top-k Combinatorial Bandits generalize multi-armed bandits, where at each round any subset of k out of n arms may be chosen 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 generalizes SAR (Bubeck et al., 2013) for top-k combinatorial bandits. 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. We also prove a lower bound on sample complexity, which is tight for k = O(1). Finally, we run experiments and show that our algorithm outperforms other methods.

Original language | English |
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Pages (from-to) | 752-776 |

Number of pages | 25 |

Journal | Proceedings of Machine Learning Research |

Volume | 117 |

State | Published - 2020 |

Event | 31st International Conference on Algorithmic Learning Theory, ALT 2020 - San Diego, United States Duration: 8 Feb 2020 → 11 Feb 2020 |

### Funding

Funders | Funder number |
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Tel Aviv University Yandex Initiative in Machine Learning | |

Israel Science Foundation |

## Keywords

- Combinatorial Bandits
- Experimental Design
- Hadamard Matrix
- Multi-Armed Bandits
- Regret Minimization
- Sample Complexity
- Top-k Bandits