Convergence rate of moments in stochastic approximation with simultaneous perturbation gradient approximation and resetting

The sequence of recursive estimators for function minimization generated by Spall's simultaneous perturbation stochastic approximation (SPSA) method, presented in [25], combined with a suitable restarting mechanism is considered. It is proved that this sequence converges under certain conditions with rate O(n(-beta/2)) for some beta>0, the best value being beta = 2/3, where the rate is measured by the L(q)-norm of the estimation error for any 1 less than or equal to q < infinity, The authors also present higher order SPSA methods. It is shown that the error exponent beta/2 can be arbitrarily close to 1/2 if the Hessian matrix of the cost function at the minimizing point has all its eigenvalues to the right of 1/2, the cost function is sufficiently smooth, and a sufficiently high-order approximation of the derivative is used.