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We consider a remote contextual multi-armed bandit (CMAB) problem, in which the decision-maker observes the context and the reward, but must communicate the actions to be taken by the agents over a rate-limited communication channel. This can model, for example, a personalized ad placement application, where the content owner observes the individual visitors to its website, and hence has the context information, but must convey the ads that must be shown to each visitor to a separate entity that manages the marketing content. In this remote CMAB (R-CMAB) problem, the constraint on the communication rate between the decision-maker and the agents imposes a trade-off between the number of bits sent per agent and the acquired average reward. We are particularly interested in characterizing the rate required to achieve sub-linear regret. Consequently, this can be considered as a policy compression problem, where the distortion metric is induced by the learning objectives. We first study the fundamental information theoretic limits of this problem by letting the number of agents go to infinity, and study the regret achieved when Thompson sampling strategy is adopted. In particular, we identify two distinct rate regions resulting in linear and sub-linear regret behavior, respectively. Then, we provide upper bounds on the achievable regret when the decision-maker can reliably transmit the policy without distortion.