BEHAVIOR OF STOCHASTIC APPROXIMATIONALGORITHM IN ROOT SET OF REGRESSIONFUNCTION

1.IntroductionStochasticapproximation(SA)iswidelyappliedinoptimization,systemindelltification,adaptivecontrolandsignalprocessing.ThebasicproblemofSAistoseektherootsoreXtremaofafunctionf(.)calledregressionfunctionwhichcanbeestimateattimekandletthe(k 1)thobservationi,e,Yk 1~f(Xk) ek 1,(1)whereek 1istheobservationnoise.RobbinsandMonroll]in1951proposedthefollowingalgorithmxk l=xk akyk 1,(2)toestimatetherootsofj(.),whereahisthestepsize.ThisalgorithmisnowcalledRobbinsMonro(RM)algorithm.Denoteb…

BEHAVIOR OF STOCHASTIC APPROXIMATION ALGORITHM IN ROOT SET OF REGRESSION FUNCTION

By convergence of a stochastic approximation (SA) algorithm one usually means that the distance d(xn, J) between the estimate xn given by the algorithm and the root set J of the regression function tends to zero. This paper characterizes the properties of the limit points of xn. An illustrative example is given to show that xn may move about in J even thought d(xn, J) 0. Finally, the paper gives conditions guaranteeing not only d(xn, J)0 but also the convergence of {xn} itself in the one-dimensional case.