Metric Entropy and Minimax Risk in Classi cation

We apply recent results on the minimax risk in density esti mation to the related problem of pattern classi cation The notion of loss we seek to minimize is an information theoretic measure of how well we can predict the classi cation of future examples given the classi cation of previously seen examples We give an asymptotic characterization of the minimax risk in terms of the metric entropy properties of the class of distributions that might be generating the examples We then use these results to characterize the minimax risk in the special case of noisy two valued classi cation problems in terms of the Assouad density and the Vapnik Chervonenkis dimension