C++ Neural Networks and Fuzzy Logic: Preface

Learning and Stability

Learning, convergence, and stability are matters of much interest. As learning is taking place, you want to

know if the process is going to halt at some appropriate point, which is a question of convergence. Is what is

learned stable, or will the network have to learn all over again, as each new event occurs? These questions

have their answers within a mathematical model with differential equations developed to describe a learning

algorithm. Proofs showing stability are part of the model inventor’s task. One particular tool that aids in the

process of showing convergence is the idea of state energy, or cost, to describe whether the direction the

process is taking can lead to convergence.

The Lyapunov function, discussed later in this chapter, is found to provide the right energy function, which

can be minimized during the operation of the neural network. This function has the property that the value

gets smaller with every change in the state of the system, thus assuring that a minimum will be reached

eventually. The Lyapunov function is discussed further because of its significant utility for neural network

models, but briefly because of the high level of mathematics involved. Fortunately, simple forms are derived

and put into learning algorithms for neural networks. The high−level mathematics is used in making the

proofs to show the viability of the models.

Alternatively, temperature relationships can be used, as in the case of the Boltzmann machine, or any other

well−suited cost function such as a function of distances used in the formulation of the Traveling Salesman

Problem, in which the total distance for the tour of the traveling salesman is to be minimized, can be

employed. The Traveling Salesman Problem is important and well−known. A set of cities is to be visited by

C++ Neural Networks and Fuzzy Logic:Preface

Associative Memory Models and One−Shot Learning

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the salesman, each only once, and the aim is to devise a tour that minimizes the total distance traveled. The

search continues for an efficient algorithm for this problem. Some algorithms solve the problem in a large

number but not all of the situations. A neural network formulation can also work for the Traveling Salesman

Problem. You will see more about this in Chapter 15.