A new method to calculate the full training process of a neural network is introduced. No sophisticated methods like the replica trick are used. The results are directly related to the actual number of training steps. Some results are presented here, like the maximal learning rate, an exact description of early stopping, and the necessary number of training steps. Further problems can be addressed with this approach.

A new method to calculate the full training process of a neural network is introduced. No sophisticated methods like the replica trick are used. The results are directly related to the actual number of training steps. Some results are presented here, like the maximal learning rate, an exact description of early stopping, and the necessary number of training steps. Further problems can be addressed with this approach.

In the Bayes approach to statistical inference [Berger, 1985] one assumes that the prior uncertainty about parameters of an unknown data generating mechanism can be encoded in a probability distribution, the so called prior. Using the prior and the likelihood of the data given the parameters, the posterior distribution of the parameters can be derived from Bayes rule. From this posterior, various estimates for functions ofthe parameter, like predictions about unseen data, can be calculated. However, in general, those predictions cannot be realised by specific parameter values, but only by an ensemble average over parameters according to the posterior probability. Hence, exact implementations of Bayes method for neural networks require averages over network parameters which in general can be performed by time consuming 226 M. Opper and O. Winther Monte Carlo procedures.

In the Bayes approach to statistical inference [Berger, 1985] one assumes that the prior uncertainty about parameters of an unknown data generating mechanism can be encoded in a probability distribution, the so called prior. Using the prior and the likelihood of the data given the parameters, the posterior distribution of the parameters can be derived from Bayes rule. From this posterior, various estimates for functions ofthe parameter, like predictions about unseen data, can be calculated. However, in general, those predictions cannot be realised by specific parameter values, but only by an ensemble average over parameters according to the posterior probability. Hence,exact implementations of Bayes method for neural networks require averages over network parameters which in general can be performed by time consuming 226 M.Opper and O. Winther Monte Carlo procedures.

In this paper we investigate an average-case model of concept learning, and give results that place the popular statistical physics and VC dimension theories of learning curve behavior in a common framework.

In this paper we investigate an average-case model of concept learning, and give results that place the popular statistical physics and VC dimension theories of learning curve behavior in a common framework.