Betweenness centrality is a popular centrality measure with applications in several domains, and whose exact computation is impractical for modern-sized networks. We present SILVAN, a novel, efficient algorithm to compute, with high probability, accurate estimates of the betweenness centrality of all nodes of a graph and a high-quality approximation of the top-k betweenness centralities. SILVAN follows a progressive sampling approach, and builds on novel bounds based on Monte-Carlo Empirical Rademacher Averages, a powerful and flexible tool from statistical learning theory. SILVAN relies on a novel estimation scheme providing non-uniform bounds on the deviation of the estimates of the betweenness centrality of all the nodes from their true values, and a refined characterisation of the number of samples required to obtain a high-quality approximation. Our extensive experimental evaluation shows that SILVAN extracts high-quality approximations while outperforming, in terms of number of samples and accuracy, the state-of-the-art approximation algorithm with comparable quality guarantees.