ERM learning algorithm for multi-class classification

The multi-class classification problem is considered by an empirical risk minimization (ERM) approach. The hypothesis space for the learning algorithm is taken to be a ball of a Banach space of continuous functions. When the regression function lies in some interpolation space, satisfactory learning rates for the excess misclassification error are provided in terms of covering numbers of the unit ball of the Banach space. A comparison theorem is proved and is used to bound the excess misclassification error by means of the excess generalization error.