Stochastic approximation in finite samples using surrogate processes

The practical application of stochastic approximation methods require a reliable means to stop the iterative process when the estimate is close to the optimal value or when further improvement of the estimate is doubtful. Conventional ideas on stopping stochastic algorithms employ probabilistic criteria based on the asymptotic distribution of the stochastic approximation process, often with the parameters of the distribution determined by sequential estimation. Difficulties may arise when this approach is applied to small (finite) samples. We propose a different approach that uses the notion of a surrogate process as a proxy for the stochastic approximation. A discussion of this approach to stopping stochastic approximation is offered in the context of a simple example, including some empirical results.