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# Pseudo-reward Algorithms for Contextual Bandits with Linear Payoff Functions

*Proceedings of the Sixth Asian Conference on Machine Learning*, PMLR 39:344-359, 2015.

#### Abstract

We study the contextual bandit problem with linear payoff functions, which is a generalization of the traditional multi-armed bandit problem. In the contextual bandit problem, the learner needs to iteratively select an action based on an observed context, and receives a linear score on only the selected action as the reward feedback. Motivated by the observation that better performance is achievable if the other rewards on the non-selected actions can also be revealed to the learner, we propose a new framework that feeds the learner with pseudo-rewards, which are estimates of the rewards on the non-selected actions. We argue that the pseudo-rewards should better contain over-estimates of the true rewards, and propose a forgetting mechanism to decrease the negative influence of the over-estimation in the long run. Then, we couple the two key ideas above with the linear upper confidence bound (LinUCB) algorithm to design a novel algorithm called linear pseudo-reward upper confidence bound (LinPRUCB). We prove that LinPRUCB shares the same order of regret bound to LinUCB, while enjoying the practical observation of faster reward-gathering in the earlier iterations. Experiments on artificial and real-world data sets justify that LinPRUCB is competitive to and sometimes even better than LinUCB. Furthermore, we couple LinPRUCB with a special parameter to formalize a new algorithm that yields faster computation in updating the internal models while keeping the promising practical performance. The two properties match the real-world needs of the contextual bandit problem and make the new algorithm a favorable choice in practice.