Polyak’s method based on the stochastic Lyapunov function for justifying the consistency of estimates produced by a stochastic approximation search algorithm under an unknown-but-bounded noise

In 1976–1977, Polyak published in the journal Avtomatica i Telemekhanika (Automation and Remote Control) two remarkable papers on how to study the properties of estimates of iterative pseudogradient algorithms. The first paper published in 1976 considered the general case based on the stochastic Lyapunov function, and the second one considered the linear case. The assumptions formulated in these papers and the estimates obtained in them can still be considered the state-of-the art. In the current paper, Polyak’s approach is applied to the study of the properties of estimates of a (randomized) stochastic approximation search algorithm for the case of unknown-but-bounded noise in observations. The obtained asymptotic estimates were already known earlier, and exact estimates for a finite number of observations are published for the first time.