Generalization performance of Bayes optimal classification algorithm for learning a perceptron

The generalization error of the Bayes optimal classification algorithm when learning a perceptron from noise-free random training examples is calculated exactly using methods of statistical mechanics. It is shown that if an assumption of replica symmetry is made, then, in the thermodynamic limit, the error of the Bayes optimal algorithm is less than the error of a canonical stochastic learning algorithm, by a factor approaching 2 as the ratio of the number of training examples to perceptron weights grows. In addition, it is shown that approximations to the generalization error of the Bayes optimal algorithm can be achieved by learning algorithms that use a two-layer neutral net to learn a perceptron.