Multi-class classification with a very large number of classes, or extreme classification, is a challenging problem from both statistical and computational perspectives. Most of the classical approaches to multi-class classification, including one-vs-rest or multi-class support vector machines, require the exact estimation of the classifier's margin, at both the training and the prediction steps making them intractable in extreme classification scenarios. In this paper, we study the impact of computing an approximate margin using nearest neighbor (ANN) search structures combined with locality-sensitive hashing (LSH). This approximation allows to dramatically reduce both the training and the prediction time without a significant loss in performance. We theoretically prove that this approximation does not lead to a significant loss of the risk of the model and provide empirical evidence over five publicly available large scale datasets, showing that the proposed approach is highly competitive with respect to state-of-the-art approaches on time, memory and performance measures.