Stochastic Approximation Via Averaging: The Polyak’s Approach Revisited

Recursive stochastic optimization algorithms are considered in this work. A class of multistage procedures is developed. We analyze essentially the same kind of procedures as proposed in Polyak’s recent work. A quite different approach is taken and correlated noise processes are dealt with. In lieu of evaluating the second moments, the methods of weak convergence are employed and the asymptotic properties are obtained by examining a suitably scaled sequence. Under rather weak conditions, we show that the algorithm via averaging is an efficient approach in that it provides us with the optimal convergence speed. In addition, no prewhitening filters are needed.