Concept learning using complexity regularization

In pattern recognition or, as it has also been called, concept learning, the value of a { 0,1}-valued random variable Y is to be predicted based upon observing an R^d-valued random variable X. We apply the method of complexity regularization to learn concepts from large concept classes. The method is shown to automatically find a good balance between the approximation error and the estimation error. In particular, the error probability of the obtained classifier is shown to decrease as O(√(logn/n)) to the achievable optimum, for large nonparametric classes of distributions, as the sample size n grows. We also show that if the Bayes error probability is zero and the Bayes rule is in a known family of decision rules, the error probability is O(logn/n) for many large families, possibly with infinite VC dimension