  We study the multi-armed bandit problem where the rewards are realizations of
general non-stationary stochastic processes, a setting that generalizes many
existing lines of work and analyses. In particular, we present a theoretical
analysis and derive regret guarantees for rested bandits in which the reward
distribution of each arm changes only when we pull that arm. Remarkably, our
regret bounds are logarithmic in the number of rounds under several natural
conditions. We introduce a new algorithm based on classical UCB ideas combined
with the notion of weighted discrepancy, a useful tool for measuring the
non-stationarity of a stochastic process. We show that the notion of
discrepancy can be used to design very general algorithms and a unified
framework for the analysis of multi-armed rested bandit problems with
non-stationary rewards. In particular, we show that we can recover the regret
guarantees of many specific instances of bandit problems with non-stationary
rewards that have been studied in the literature. We also provide experiments
demonstrating that our algorithms can enjoy a significant improvement in
practice compared to standard benchmarks.
