A stochastic approximation algorithm with multiplicative step size adaptation

An algorithm of searching a zero of an unknown undimensional function is considered, measured at a point x with some error. The step sizes are random positive values and are calculated according to the rule: if two consecutive iterations are in same direction step is multiplied by u>1, otherwise, it is multiplied by 0<d<1. The function may have one or more zeros; the random values are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on the conditions on the two parameters u and d almost sure convergence of the sequence as well as under some conditions is guaranteed almost sure divergence. In particular, if the error distribuition as median 0 and zero probability for particular poinst then it is established that for ud<1, convergence takes place, and for ud>1, divergence. Due to the multiplicative rule of updating of the step, it is natural to expect that the sequence converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with, but instead, approximates one of zeros of the function. By adjusting the parameters u and d, one can reach necessary precision of approximation; higher precision is obtained at the expense of lower convergence rate.