We study the almost sure asymptotic behaviour of stochastic approximation algorithms for the search of zero of a real function. The quadratic strong law of large numbers is extended to the powers greater than one. In ...
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We study the almost sure asymptotic behaviour of stochastic approximation algorithms for the search of zero of a real function. The quadratic strong law of large numbers is extended to the powers greater than one. In other words, the convergence of moments in the almost sure central limit theorem (ASCLT) is established. As a by-product of this convergence, one gets another proof of ASCLT for stochastic approximation algorithms. The convergence result is applied to several examples as estimation of quantiles and recursive estimation of the mean.
We present a sufficient and necessary condition for the convergence of stochastic approximation algorithms, which were proposed 50 years ago, have been widely applied to various areas and intensively investigated in t...
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We present a sufficient and necessary condition for the convergence of stochastic approximation algorithms, which were proposed 50 years ago, have been widely applied to various areas and intensively investigated in theory. In the literature, only various sufficient conditions are known. The obtained condition is simple and has a clear physical meaning. (C) 2005 Elsevier B.V. All rights reserved.
We prove an almost sure central limit theorem for some multidimensional stochasticalgorithms used for the search of zeros of a function and known to satisfy a central limit theorem. The almost sure version of the cen...
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We prove an almost sure central limit theorem for some multidimensional stochasticalgorithms used for the search of zeros of a function and known to satisfy a central limit theorem. The almost sure version of the central limit theorem requires either a logarithmic empirical mean (in the same way as in the ease of independent identically distributed variables) or another scale, depending on the choice of the algorithm gains. (C) 1999 Academic Press.
This paper gives a robustness analysis of the stochastic approximation algorithms in the situations: when the regression function does not exactly equal zero at the sought-for χo, when the Liapunov function is not ze...
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This paper gives a robustness analysis of the stochastic approximation algorithms in the situations: when the regression function does not exactly equal zero at the sought-for χo, when the Liapunov function is not zero at χoand when anΣni=1X;i+1differs from zero where {aig} are the weighting coefficients of the algorithm and {Xi} are the measurement errors. It is shown that the estimation error is small if the abovementioned differences are small.
We consider a continuous-time Robbins-Monro-type stochasticapproximation procedure for a system described by a (multidimensional) stochastic differential equation driven by a general Levy process, and we find suffici...
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We consider a continuous-time Robbins-Monro-type stochasticapproximation procedure for a system described by a (multidimensional) stochastic differential equation driven by a general Levy process, and we find sufficient conditions for its convergence in terms of Lyapunov functions. While the jump part of the noise may spoil convergence to the root of the drift in some cases, we show that by a suitable choice of noise coefficients we obtain convergence under hypotheses on the drift weaker than those used in the diffusion case or convergence to a selected root in the case of multiple roots of the drift.
The authors develop a two-timescale simultaneous perturbation stochasticapproximation algorithm for simulation-based parameter optimization over discrete sets. This algorithm is applicable in cases where the cost to ...
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The authors develop a two-timescale simultaneous perturbation stochasticapproximation algorithm for simulation-based parameter optimization over discrete sets. This algorithm is applicable in cases where the cost to be optimized is in itself the long-run average of certain cost functions whose noisy estimates are obtained via simulation. The authors present the convergence analysis of their algorithm. Next, they study applications of their algorithm to the problem of admission control in communication networks. They study this problem under two different experimental settings and consider appropriate continuous time queuing models in both settings. Their algorithm finds optimal threshold-type policies within suitable parameterized classes of these. They show results of several experiments for different network parameters and rejection cost. The authors also study the sensitivity of their algorithm with respect to its parameters and step sizes. The results obtained are along expected lines.
The problem of the definition and estimation of generative models based on deformable templates from raw data is of particular importance for modeling non-aligned data affected by various types of geometric variabilit...
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The problem of the definition and estimation of generative models based on deformable templates from raw data is of particular importance for modeling non-aligned data affected by various types of geometric variability. This is especially true in shape modeling in the computer vision community or in probabilistic atlas building in computational anatomy. A first coherent statistical framework modeling geometric variability as hidden variables was described in Allassonniere. Amit and Trouve [J. R. Stat. Soc. Ser: B Stat. Methodol. 69 (2007) 3-29]. The present paper gives a theoretical proof of convergence of effective stochasticapproximation expectation strategies to estimate such models and shows the robustness of this approach against noise through numerical experiments in the context of handwritten digit modeling.
This paper considers a Markov-modulated duplication-deletion random graph where at each time instant, one node can either join or leave the network;the probabilities of joining or leaving evolve according to the reali...
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This paper considers a Markov-modulated duplication-deletion random graph where at each time instant, one node can either join or leave the network;the probabilities of joining or leaving evolve according to the realization of a finite state Markov chain. Two results are presented. First, motivated by social network applications, the asymptotic behavior of the degree distribution is analyzed. Second, a stochasticapproximation algorithm is presented to track empirical degree distribution as it evolves over time. The tracking performance of the algorithm is analyzed in terms of mean square error and a functional central limit theorem is presented for the asymptotic tracking error. Also, a Hilbert-space-valued stochasticapproximation algorithm that tracks a Markov-modulated probability mass function with support on the set of nonnegative integers is analyzed.
We consider the optimal sensor scheduling problem formulated as a partially observed Markov decision process (POMDP). Due to operational constraints, at each time instant, the scheduler can dynamically select one out ...
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We consider the optimal sensor scheduling problem formulated as a partially observed Markov decision process (POMDP). Due to operational constraints, at each time instant, the scheduler can dynamically select one out of a finite number of sensors and record a noisy measurement of an underlying Markov chain. The aim is to compute the optimal measurement scheduling policy, so as to minimize a cost function comprising of estimation errors and measurement costs. The formulation results in a nonstandard POMDP that is nonlinear in the information state. We give sufficient conditions on the cost function, dynamics of the Markov chain and observation probabilities so that the optimal scheduling policy has a threshold structure with respect to a monotone likelihood ratio (MLR) ordering. As a result, the computational complexity of implementing the optimal scheduling policy is inexpensive. We then present stochastic approximation algorithms for estimating the best linear MLR order threshold policy.
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