### Abstract

We consider the non-stochastic Multi-Armed Bandit problem in a setting where there is a fixed and known metric on the action space that determines a cost for switching between any pair of actions. The loss of the online learner has two components: the first is the usual loss of the selected actions, and the second is an additional loss due to switching between actions. Our main contribution gives a tight characterization of the expected minimax regret in this setting, in terms of a complexity measure C of the underlying metric which depends on its covering numbers. In finite metric spaces with k actions, we give an efficient algorithm that achieves regret of the form Õ(max{C^{1/3}T^{2/3}, √kT}), and show that this is the best possible. Our regret bound generalizes previous known regret bounds for some special cases: (i) the unit-switching cost regret ⊙(max{k^{1}/^{3}T^{2/3}, √kT}) where C = ⊙(k), and (ii) the interval metric with regret ⊙(max{T^{2}/^{3}, √kT}) where C = ⊙(1). For infinitemetrics spaces with Lipschitz loss functions, we derive a tight regret bound of ⊙(T d+1/d+2) where d ≥ 1 is the Minkowski dimension of the space, which is known to be tight even when there are no switching costs.

Original language | English (US) |
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Pages (from-to) | 4120-4129 |

Number of pages | 10 |

Journal | Advances in Neural Information Processing Systems |

Volume | 2017-December |

State | Published - Jan 1 2017 |

Event | 31st Annual Conference on Neural Information Processing Systems, NIPS 2017 - Long Beach, United States Duration: Dec 4 2017 → Dec 9 2017 |

### All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Information Systems
- Signal Processing

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## Cite this

*Advances in Neural Information Processing Systems*,

*2017-December*, 4120-4129.