# Contextual Bandit Algorithms with Supervised Learning Guarantees

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Alina Beygelzimer,
John Langford,
Lihong Li,
Lev Reyzin,
Robert Schapire
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Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, PMLR 15:19-26, 2011.

#### 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. 2002] called Exp4.P, which with high probability incurs regret at most O(\sqrtKT\ln N). 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(\sqrtTd\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.

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