Multi-armed bandit problem with known trend

We consider a variant of the multi-armed bandit model, which we call multi-armed bandit problem with known trend, where the gambler knows the shape of the reward function of each arm but not its distribution. This new problem is motivated by different on-line problems like active learning, music and interface recommendation applications, where when an arm is sampled by the model the received reward change according to a known trend. By adapting the standard multi-armed bandit algorithm UCB1 to take advantage of this setting, we propose the new algorithm named Adjusted Upper Confidence Bound (A-UCB) that assumes a stochastic model. We provide upper bounds of the regret which compare favorably with the ones of UCB1. We also confirm that experimentally with different simulations.