On the convergence of linear stochastic approximation procedures

Kouritzin, Michael A.

doi:10.1109/18.508865

Cited by 21 publications

(22 citation statements)

References 20 publications

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“…Assumption G2) is also fairly weak, requiring that the convergence rate of f n g be related to the step size fa n g. In the remarks following the proof of Theorem 2, we indicate how G2) can be relaxed. Note that in the standard case where a n = an 0 , a > 0, 0 < 1, and n = n 0 , > 0, we have c = 0 if < 1, and c = =a if an 10 :…”

Section: G2)mentioning

confidence: 99%

“…Condition N5) is a weighted averaging condition (see [19] and [21]). For convenience and without loss of generality, we assume in N5) that an < 1 for all n. A special case of this condition (with a n = 1=n) is considered in [5] and [10], where the weighted averaging condition reduces to regular arithmetic averaging (i.e., k = 1 for all k).…”

Section: A Noise Conditionsmentioning

confidence: 99%

“…b. Note that the above is a stochastic approximation algorithm of the form (1) with f n = A n x n 0 b n 0 e n (see also [8], [10], [19] for a treatment of convergence for a similar linear case).…”

Section: Background: a Convergence Theoremmentioning

confidence: 99%

“…Assumptions B1) and B2) are standard in results for stochastic approximation algorithms of the type (2); see, for example, [8], [10], [19], [21], where (B1) is expressed as ( n k=1 k A k )=( n k=1 k ) ! A.…”

Section: B2)mentioning

confidence: 99%

“…Note that although f is affine, the form of the stochastic approximation algorithm is not a special case of the usually considered case with f n = f (x n ) 0 e n . Indeed, the form of the algorithm we adopt has been widely studied (e.g., [8], [9], [19]; see also [10] for a survey of references on this form of stochastic approximation algorithms). In our main result (Theorem 2),…”

Section: Introductionmentioning

confidence: 99%

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Noise conditions for prespecified convergence rates of stochastic approximation algorithms

Chong

Wang

Kulkarni

1999

IEEE Trans. Inform. Theory

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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 f n g f n g f n g is a given positive sequence converging monotonically to zero. Consider a stochastic approximation algorithm x n+1 = x n 0 a n (A n x n 0 b n ) + a n e n x n+1 = x n 0 a n (A n x n 0 b n ) + a n e n x n+1 = x n 0 a n (A n x n 0 b n ) + a n e n , where fx n g fx n g fx n g is the iterate sequence, fang fang fang is the step size sequence, feng feng feng is the noise sequence, and x 3 x 3x 3 is the desired zero of the function f (x) = Ax 0 b f (x) = Ax 0 b f (x) = Ax 0 b. Then, under appropriate assumptions, we show that x n 0 x 3 = o( n )x n 0 x 3 = o( n ) x n 0 x 3 = o( n ) if and only if the sequence feng feng feng 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 feng feng feng to achieve a convergence rate of f n g f n g f n g is basically that the sequence fe n = n g fe n = n g fe n = n g 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 a n = a=n a n = a=n a n = a=n and fe n g fe n g fe n g is a martingale difference process with bounded variance, then xn 0 x 3 = o(n xn 0 x 3 = o(n xn 0 x 3 = o(n 01=2 (log (n)) ) (log (n)) ) (log (n)) )for any > > >1=2.Index Terms-Convergence rate, Kiefer-Wolfowitz, necessary and sufficient conditions, noise sequences, Robbins-Monro, stochastic approximation.

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Section: G2)mentioning

confidence: 99%

Section: A Noise Conditionsmentioning

confidence: 99%

Section: Background: a Convergence Theoremmentioning

confidence: 99%

Section: B2)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Noise conditions for prespecified convergence rates of stochastic approximation algorithms

Chong

Wang

Kulkarni

1999

IEEE Trans. Inform. Theory

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An Algorithm for Accurate Distributed Time Synchronization in Mobile Wireless Sensor Networks from Noisy Difference Measurements

Liao¹,

Barooah

2016

Asian Journal of Control

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We propose a distributed algorithm for time synchronization in mobile wireless sensor networks. The problem of time synchronization is formulated as nodes estimating their skews and offsets from noisy difference measurements of offsets and logarithm of skews; the measurements acquired by time-stamped message exchanges between neighbors. The algorithm ensures that the estimation error is mean square convergent (variance converging to 0) under certain conditions. A sequence of scheduled update instants is used to meet the requirement of decreasing time-varying gains that need to be synchronized across nodes with unsynchronized clocks. Moreover, a modification on the algorithm is also presented to improve the initial convergence speed. Simulations indicate that highly accurate global time estimates can be achieved with the proposed algorithm for long time durations, while the errors in competing algorithms increase over time.where the scalar parameters u , u are called its skew (the relative speed of the clock with respect to t) and offset Manuscript

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A sliding-window approximation-based fractional adaptive strategy for Hammerstein nonlinear ARMAX systems

2016

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On the convergence of linear stochastic approximation procedures

Cited by 21 publications

References 20 publications

Noise conditions for prespecified convergence rates of stochastic approximation algorithms

Noise conditions for prespecified convergence rates of stochastic approximation algorithms

An Algorithm for Accurate Distributed Time Synchronization in Mobile Wireless Sensor Networks from Noisy Difference Measurements

A sliding-window approximation-based fractional adaptive strategy for Hammerstein nonlinear ARMAX systems

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