Stochastic searching on the line and its applications to parameter learning in nonlinear optimization

We consider the problem of a learning mechanism (for example, a robot) locating a point on a line when it is interacting with a random environment which essentially informs it, possibly erroneously, which may it should move. In this paper we present a novel scheme by which the point can be learned using some recently devised learning principles. The heart of the strategy involves discretizing the space and performing a controlled random walk on this space. The scheme is shown to be epsilon-optimal and to converge with probability 1. Although the problem is solved in its generality, its application in nonlinear optimization has also been suggested. Typically, an optimization process involves working one's way toward the maximum (minimum) using the local information that is available. However, the crucial issue in these strategies is that of determining the parameter to be used in the optimization itself. If the parameter is too small the convergence is sluggish. On the other hand, if the parameter is too large, the system could erroneously converge or even oscillate. Our strategy can be used to determine the best parameter to be used in the optimization.