It has been recently shown that, under the margin (or low noise) assumption, there exist classifiers attaining fast rates of convergence of the excess Bayes risk, that is, rates faster than n−1/2. The work on this subject has suggested the following two conjectures: (i) the best achievable fast rate is of the order n−1, and (ii) the plug-in classifiers generally converge more slowly than the classifiers based on empirical risk minimization. We show that both conjectures are not correct. In particular, we construct plug-in classifiers that can achieve not only fast, but also super-fast rates, that is, rates faster than n−1. We establish minimax lower bounds showing that the obtained rates cannot be improved.
"Fast learning rates for plug-in classifiers." Ann. Statist. 35 (2) 608 - 633, April 2007. https://doi.org/10.1214/009053606000001217