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Stochastic approximation (SA) has long been applied for problems of minimizing loss functions or root finding with noisy input information. As with all stochastic search algorithms, there are adjustable algorithm coefficients that must be specified, and that can have a profound effect on algorithm performance. It is known that choosing these coefficients according to an SA analog of the deterministic Newton-Raphson algorithm provides an optimal or near-optimal form of the algorithm. However, directly determining the required Hessian matrix (or Jacobian matrix for root finding) to achieve this algorithm form has often been difficult or impossible in practice. The paper presents a general adaptive SA algorithm that is based on a simple method for estimating the Hessian matrix, while concurrently estimating the primary parameters of interest. The approach applies in both the gradient-free optimization (Kiefer-Wolfowitz) and root-finding/stochastic gradient-based (Robbins-Monro) settings, and is based on the "simultaneous perturbation (SP)" idea introduced previously. The algorithm requires only a small number of loss function or gradient measurements per iteration-independent of the problem dimension-to adaptively estimate the Hessian and parameters of primary interest. Aside from introducing the adaptive SP approach, the paper presents practical implementation guidance, asymptotic theory, and a nontrivial numerical evaluation. Also included is a discussion and numerical analysis comparing the adaptive SP approach with the iterate-averaging approach to accelerated SA.