A projected stochastic approximation algorithm

It is noted that classical stochastic approximation algorithms often diverge because of boundedness problems. The standard approach to preventing this is to project the sequence generated by the algorithm onto a predetermined compact set K. However, in the typical application, the approximate location of the solution is not known. To minimize the probability that the solution lies outside the set K , it is therefore necessary to let K be large. This can seriously curtail the efficiency of the algorithm. The author proposes a stochastic approximation algorithm which bounds the sequence of estimates of the solution to an increasing sequence of sets. This eliminates the possibility of bounding the algorithm to a set which does not contain the solution. Furthermore, it is possible to let the initial set be small, which can result in improved empirical performance