In this paper, we propose a learning approach to adaptive performance tuning of database applications. The objective is to validate the opportunity to devise a tuning strategy that does not need prior knowledge of a cost model. Instead, the cost model is learned through reinforcement learning. We instantiate our approach to the use case of index tuning. We model the execution of queries and updates as a Markov decision process whose states are database configurations, actions are configuration changes, and rewards are functions of the cost of configuration change and query and update evaluation. During the reinforcement learning process, we face two important challenges: not only the unavailability of a cost model, but also the size of the state space. To address the latter, we devise strategies to prune the state space, both in the general case and for the use case of index tuning. We empirically and comparatively evaluate our approach on a standard OLTP dataset. We show that our approach is competitive with state-of-the-art adaptive index tuning, which is dependent on a cost model.