Weighted averages of Kiefer-Wolfowitz-type procedures, which are driven by larger step lengths than usual, can achieve the optimal rate of convergence. A priori knowledge of a lower bound on the smallest eigenvalue of the Hessian matrix is avoided. The asymptotic mean squared error of the weighted averaging algorithm is the same as would emerge using a Newton-type adaptive algorithm. Several different gradient estimates are considered; one of them leads to a vanishing asymptotic bias. This gradient estimate applied with the weighted averaging algorithm usually yields a better asymptotic mean squared error than applied with the standard algorithm.