A technique to accelerate the convergence of the Robbins-Monro stochastic approximation algorithm for the multidimensional case is studied. It is based on generalization of Kesten's idea that (in the one-dimensional case) frequent changes of the signs of the differences of subsequent observations indicate that the estimates are close to the real solution and vice versa. The convergence with probability one is proved and the asymptotic normality of the delivered estimates is shown. A Kesten's-like modification of the Ruppert-Polyak algorithm with averaging of trajectories is given. Results of numerical simulations are presented to demonstrate the efficiency of the acceleration procedure.