4. MATHEMATICAL THEORY OF

NEURAL LEARNING

This chapter deals with theoretical aspects of learning in artificial neural networks. It investigates mathematically, the nature and stability of the asymptotic solutions obtained using the basic supervised, Hebbian and reinforcement learning rules, which were introduced in the previous chapter. Formal analysis is also given for simple competitive learning and self-organizing feature map learning.

A unifying framework for the characterization of various learning rules is presented. This framework is based on the notion that learning in general neural networks can be viewed as search, in a multidimensional space, for a solution which optimizes a prespecified criterion function, with or without constraints. Under this framework, a continuous-time learning rule is viewed as a first-order, stochastic differential equation/dynamical system, whereby the state of the system evolves so as to minimize an associated instantaneous criterion function. Approximation techniques are employed to determine, in an average sense, the nature of the asymptotic solutions of the stochastic system. This approximation leads to an "average learning equation" which, in most cases, can be cast as a globally, asymptotically stable gradient system whose stable equilibria are minimizers of a well-defined average criterion function. Finally, and subject to certain assumptions, these stable equilibria can be taken as the possible limits (attractor states) of the stochastic learning equation.

The chapter also treats two important issues associated with learning in a general feedforward neural network. These are learning generalization and learning complexity. The section on generalization presents a theoretical method for calculating the asymptotic probability of correct generalization of a neural network as a function of the training set size and the number of free parameters in the network. Here, generalization in deterministic and stochastic nets are investigated. The chapter concludes by reviewing some significant results on the complexity of learning in neural networks.

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