Deterministic learning rules for boltzmann machines

Nonmonotonic generalization bias of Gaussian mixture models

Theories of learning and generalization hold that the generalization bias, defined as the difference between the training error and the generalization error, increases on average with the number of adaptive parameters. This article, however, shows that this general tendency is violated for a gaussian mixture model. For temperatures just below the first symmetry breaking point, the effective number of adaptive parameters increases and the generalization bias decreases. We compute the dependence of the neural information criterion on temperature around the symmetry breaking. Our results are confirmed by numerical cross-validation experiments.

Efficient Learning in Boltzmann Machines Using Linear Response Theory

The learning process in Boltzmann machines is computationally very expensive. The computational complexity of the exact algorithm is exponential in the number of neurons. We present a new approximate learning algorithm for Boltzmann machines, based on mean-field theory and the linear response theorem. The computational complexity of the algorithm is cubic in the number of neurons.

In the absence of hidden units, we show how the weights can be directly computed from the fixed-point equation of the learning rules. Thus, in this case we do not need to use a gradient descent procedure for the learning process. We show that the solutions of this method are close to the optimal solutions and give a significant improvement when correlations play a significant role. Finally, we apply the method to a pattern completion task and show good performance for networks up to 100 neurons.

Symmetry breaking and training from incomplete data with Radial Basis Boltzmann Machines

A Radial Basis Boltzmann Machine (RBBM) is a specialized Boltzmann Machine architecture that combines feed-forward mapping with probability estimation in the input space, and for which very efficient learning rules exist. The hidden representation of the network displays symmetry breaking as a function of the noise in the dynamics. Thus, generalization can be studied as a function of the noise in the neuron dynamics instead of as a function of the number of hidden units. We show that the RBBM can be seen as an elegant alternative of k-nearest neighbor, leading to comparable performance without the need to store all data. We show that the RBBM has good classification performance compared to the MLP. The main advantage of the RBBM is that simultaneously with the input-output mapping, a model of the input space is obtained which can be used for learning with missing values. We derive learning rules for the case of incomplete data, and show that they perform better on incomplete data than the traditional learning rules on a 'repaired' data set.