Boltzmann Machines

To understand how the brain works, we need to understand how it figures out how to adjust the connection strengths between neurons so as to build a model of the world. Neurons are complicated devices, but we may be able to gain insight into the principles used for adjusting connection strengths in the brain by making some gross simplifications. Consider a system composed of binary neurons that can be active or inactive with weighted pairwise couplings between pairs of neurons, including long range couplings. If the neurons represent pixels in a binary image, we can store a set of binary training images by adjusting the coupling weights so that the images are local minima of a Hopfield-like energy function which is minus the sum over all pairs of active neurons of their coupling weights. But this energy function can only capture pairwise correlations. It cannot represent the kinds of complicated higher-order correlations that occur in images. Now suppose that in addition to the "visible" neurons that represent the pixel intensities, we also have a large set of hidden neurons that have weighted couplings with each other and with the visible neurons. Suppose also that all of the neurons are asynchronous and stochastic: They adopt the active state with a log probability that is equal to the difference in the energy function when the neuron is inactive versus active. Given a set of training images, is there a simple way to set the weights on all of the couplings so that the training images are local minima of the free energy function obtained by integrating out the states of the hidden neurons? The Boltzmann machine learning algorithm solved this problem in an elegant way. It was proof of principle that learning in neural networks with hidden neurons was possible using only locally available information, contrary to what was generally believed at the time.