### Abstract

We address the problem of competing with any large set of N policies in the non-stochastic bandit setting, where the learner must repeatedly select among K actions but observes only the reward of the chosen action. We present a modification of the Exp4 algorithm of Auer et al. [2], called Exp4.P, which with high probability incurs regret at most O(√KT lnN). Such a bound does not hold for Exp4 due to the large variance of the importance-weighted estimates used in the algorithm. The new algorithm is tested empirically in a large-scale, real-world dataset. For the stochastic version of the problem, we can use Exp4.P as a subroutine to compete with a possibly infinite set of policies of VC-dimension d while incurring regret at most O(√Td ln T) with high probability. These guarantees improve on those of all previous algorithms, whether in a stochastic or adversarial environment, and bring us closer to providing guarantees for this setting that are comparable to those in standard supervised learning.

Original language | English (US) |
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Pages (from-to) | 19-26 |

Number of pages | 8 |

Journal | Journal of Machine Learning Research |

Volume | 15 |

State | Published - Dec 1 2011 |

Event | 14th International Conference on Artificial Intelligence and Statistics, AISTATS 2011 - Fort Lauderdale, FL, United States Duration: Apr 11 2011 → Apr 13 2011 |

### All Science Journal Classification (ASJC) codes

- Software
- Control and Systems Engineering
- Statistics and Probability
- Artificial Intelligence

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## Cite this

*Journal of Machine Learning Research*,

*15*, 19-26.