## Abstract

We study the Thompson sampling algorithm in an adversarial setting, specifically, for adversarial bit prediction. We characterize the bit sequences with the smallest and largest expected regret. Among sequences of length T with k < ^{T}_{2} zeros, the sequences of largest regret consist of alternating zeros and ones followed by the remaining ones, and the sequence of smallest regret consists of ones followed_by zeros. We also bound the regret of those sequences, the worst case sequences have regret O(^{√}T) and the best case sequence have regret O(1). We extend our results to a model where false positive and false negative errors have different weights. We characterize the sequences with largest expected regret in this generalized setting, and derive their regret bounds. We also show that there are sequences with O(1) regret.

Original language | English |
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Pages (from-to) | 518-553 |

Number of pages | 36 |

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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Yandex Initiative in Machine Learning | |

Israel Science Foundation |

## Keywords

- Adversarial setting
- Bit prediction
- Multi-armed bandits
- Regret
- Thompson sampling