Abstract

We address the problem of competing with any large set of N policies in the nonstochastic 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 VCdimension d while incurring regret at most O(√TdlnT) 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.