| ### Chapter 10 - Classical Optimization Techniques
Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks
Russell D. Reed and Robert J. Marks II
Copyright © 1999 Massachusetts Institute of Technology

## 10.1 The Objective Function

To treat network training as an optimization problem, an objective function (or cost function) must be defined that provides an unambiguous numerical rating of system performance. The cost function reduces all the various good and bad aspects of a possibly complex system down to a single number, a scalar value, which allows candidate solutions to be ranked and compared. In short, it provides the working definition of optimal for the search algorithm, telling it what kinds of solutions to look for. It is important, therefore, that the function faithfully represent our design goals. If we choose a poor error function and obtain unsatisfactory results, the fault is ours for badly specifying the goal of the search.

Selection of the objective function can be a problem in itself since it is not always easy to develop a function that measures exactly what we want when goals are vague. It is often necessary to compromise between what we want, what we can measure, and what we can optimize efficiently. A few basic functions are very commonly used. The mean squared error is popular for function approximation (regression) problems because of its convenience in mathematical analysis. The cross-entropy error function is often used for classification problems when outputs are interpreted as probabilities of membership in an indicated class. In real-world applications, it may be necessary to complicate the function with additional terms to balance conflicting subgoals or to introduce heuristics favoring preferred classes of solutions.