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

Abstract

The sequence of recursive estimators for function minimization generated by Spall's (1998) simultaneous perturbation stochastic approximation (SPSA) method, combined with a suitable restarting mechanism is considered. It is proved that this sequence converges under certain conditions with rate O(n/sup -/spl beta//2/) for some /spl beta/>0, the best value being /spl beta/=2/3, where the rate is measured by the L/sub q/-norm of the estimation error for any 1/spl les/q</spl infin/. The authors also present higher order SPSA methods. It is shown that the error exponent /spl 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.