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# Sleeping Experts and Bandits with Stochastic Action Availability and Adversarial Rewards

*Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics*, PMLR 5:272-279, 2009.

#### Abstract

We consider algorithms for selecting actions in order to maximize rewards chosen by an adversary, where the set of actions available on any given round is selected stochastically. We present the first polynomial-time no-regret algorithms for this setting. In the full-observation (experts) version of the problem, we present an exponential-weights algorithm that achieves regret $\mathcal{O}(\sqrt{T \log n})$, which is the best possible. For the bandit setting (where the algorithm only observes the reward of the action selected), we present a no-regret algorithm based on follow-the-perturbed-leader. This algorithm runs in polynomial time, unlike the EXP4 algorithm which can also be applied to this setting. Our algorithm has the interesting interpretation of solving a geometric experts problem where the embedding in which rewards are linear is never explicitly constructed. We argue that this adversarial-reward, stochastic availability formulation is important in practice, as assuming stationary stochastic rewards is unrealistic in many domains.