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# Contextual Bandit Algorithms with Supervised Learning Guarantees

*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., called Exp4.P, which with high probability incurs regret at most $O(\sqrt{KT\ln N})$. 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 $O(\sqrt{Td\ln T})$ 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.