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We consider in this work Edge Computing (EC) in a multi-tenant environment: the resource owner, i.e., the Network Operator (NO), virtualizes the resources and lets third party Service Providers (SPs - tenants) run their services, which can be diverse and with heterogeneous requirements. Due to confidentiality guarantees, the NO cannot observe the nature of the traffic of SPs, which is encrypted. This makes resource allocation decisions challenging, since they must be taken based solely on observed monitoring information. We focus on one specific resource, i.e., cache space, deployed in some edge node, e.g., a base station. We study the decision of the NO about how to partition cache among several SPs in order to minimize the upstream traffic. Our goal is to optimize cache allocation using purely data-driven, model-free Reinforcement Learning (RL). Differently from most applications of RL, in which the decision policy is learned offline on a simulator, we assume no previous knowledge is available to build such a simulator. We thus apply RL in an \emph{online} fashion, i.e., the policy is learned by directly perturbing the actual system and monitoring how its performance changes. Since perturbations generate spurious traffic, we also limit them. We show in simulation that our method rapidly converges toward the theoretical optimum, we study its fairness, its sensitivity to several scenario characteristics and compare it with a method from the state-of-the-art.