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810 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 2, MARCH 1999 Noise Conditions for Prespecified Convergence
 

Summary: 810 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 2, MARCH 1999
Noise Conditions for Prespecified Convergence
Rates of Stochastic Approximation Algorithms
Edwin K. P. Chong, Senior Member, IEEE,
I-Jeng Wang, Member, IEEE, and Sanjeev R. Kulkarni
Abstract-- 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 fngfngfng is a given positive sequence converging
monotonically to zero. Consider a stochastic approximation algorithm
xn+1 = xn 0an(Anxn 0bn) + anenxn+1 = xn 0an(Anxn 0bn) + anenxn+1 = xn 0an(Anxn 0bn) + anen, where fxngfxngfxng is the iterate
sequence, fangfangfang is the step size sequence, fengfengfeng is the noise sequence,
and x3x3x3 is the desired zero of the function f(x) = Ax 0bf(x) = Ax 0bf(x) = Ax 0b. Then, under
appropriate assumptions, we show that xn 0x3 = o(n)xn 0x3 = o(n)xn 0x3 = o(n) if and only
if the sequence fengfengfeng satisfies one of five equivalent conditions. These
conditions are based on well-known formulas for noise sequences:
Kushner and Clark's condition, Chen's condition, Kulkarni and
Horn's condition, a decomposition condition, and a weighted averaging
condition. Our necessary and sufficient condition on fengfengfeng to achieve
a convergence rate of fngfngfng is basically that the sequence fen=ngfen=ngfen=ng

  

Source: Amir, Yair - Department of Computer Science, Johns Hopkins University

 

Collections: Computer Technologies and Information Sciences