Stochastic approximation on a discrete set and the multi— armed

Let R be a real funccion on a discrete set Z∊R. A Robbins-Moniotype stochastic approximation procedure is considered for solving the equation R(z) = 0, or for finding the point of the minimal absolute value of R, if there is no solution of R(z) =0 in Z, The procedure is modified so as to be also asymptotically optimal as a strategy for the multi-armed bandit problem. The Kiefer-Wolfowitz-type situation and the multi-dimensional generalizations are also Considered.