In this paper, we present an extensive study of the cutting-plane algorithm (CPA) applied to structural kernels for advanced text classification on large datasets. In particular, we carry out a comprehensive experimentation on two interesting natural language tasks, e.g. predicate argument extraction and question answering. Our results show that (i) CPA applied to train a non-linear model with different tree kernels fully matches the accuracy of the conventional SVM algorithm while being ten times faster; (ii) by using smaller sampling sizes to approximate subgradients in CPA we can trade off accuracy for speed, yet the optimal parameters and kernels found remain optimal for the exact SVM. These results open numerous research perspectives, e.g. in natural language processing, as they show that complex structural kernels can be efficiently used in real-world applications. For example, for the first time, we could carry out extensive tests of several tree kernels on millions of training instances. As a direct benefit, we could experiment with a variant of the partial tree kernel, which we also propose in this paper.