6 - Stochastic Feedforward Neural Networks: Universal Approximation

Summary

We take a look at the universal approximation question for stochastic feedforward neural networks. In contrast with deterministic networks, which represent mappings from inputs to outputs, stochastic networks represent mappings from inputs to probability distributions over outputs. Even if the sets of inputs and outputs are finite, the set of stochastic mappings is continuous. Moreover, the values of the output variables may be correlated, which requires that their values are computed jointly. A prominent class of stochastic feedforward networks are deep belief networks. We discuss the representational power in terms of compositions of Markov kernels expressed by the layers of the network. We investigate different types of shallow and deep architectures, and the minimal number of layers and units that are necessary and sufficient in order for the network to be able to approximate any stochastic mapping arbitrarily well. The discussion builds on notions of probability sharing, focusing on the case of binary variables and sigmoid units. After reviewing existing results, we present a detailed analysis of shallow networks and a unified analysis for a variety of deep networks.