We develop deterministic necessary and sufficient conditions on individual noise sequences of a stochastic approximation algorithm for the error of the iterates to converge at a given rate. Specifically, suppose {ρn} is a given positive sequence converging monotonically to zero. Consider a stochastic approximation algorithm x n+1=xn-an(Anxn-b n)+anen, where {xn} is the iterate sequence, {an} is the step size sequence, {en } is the noise sequence, and x* is the desired zero of the function f(x)=Ax-b. Then, under appropriate assumptions, we show that x n-x*=o(ρn) if and only if the sequence {en} satisfies one of five equivalent conditions. These conditions are based on well-known formulas for noise sequences: Kushner and Clark's (1978) condition, Chen's (see Proc. IFAC World Congr., p.375-80, 1996) condition, Kulkarni and Horn's (see IEEE Trails Automat. Contr., vol.41, p.419-24, 1996) condition, a decomposition condition, and a weighted averaging condition. Our necessary and sufficient condition on {en} to achieve a convergence rate of {ρn} is basically that the sequence {enn} satisfies any one of the above five well-known conditions. We provide examples to illustrate our result. In particular, we easily recover the familiar result that if an=a/n and {en} is a martingale difference process with bounded variance, then xn-x*=o(n-1/2(log(n))β ) for any β>1/2

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Information Theory, IEEE Transactions on  (Volume:45 ,  Issue: 2 )

Mar 1999