A Method for Stopping Nonconvergent Stochastic Approximation Processes

When a stochastic approximation process satisfies the conditions for convergence there are well-established methods to terminate the iterative process in a manner that allows approximate statistics to be calculated on the final result. Many of these use the asymptotic properties of convergent stochastic approximation. However, such methods converge slowly due to step size restrictions, so in practical application it is common to use a step size that violates the conditions for convergence in order to obtain an answer more quickly. Constant gain stochastic approximation is a special case of this practice. In these cases stopping rules based on asymptotic methods are no longer analytically supportable, and other techniques must be found. This paper presents one such method based on the use of a surrogate process to calculate the stopping condition. A discussion of this approach to stopping stochastic approximation is offered in the context of a simple example, including some empirical results.