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A companion for the Kiefer-Wolfowitz-Blum stochastic approximation algorithm

ANNALS OF STATISTICS 2007年第4期35卷 1749-1772页

作者： Mokkadem, Abdelkader Pelletier, Mariane Univ Versailles Dept Matemat F-78035 Versailles France

A stochastic algorithm for the recursive approximation of the location theta of a maximum of a regression function was introduced by Kiefer and Wolfowitz [Ann. Math. Statist. 23 (1952) 462-466] in the univariate framework, and by Blum [Ann. Math. Statist. 25 (1954) 737-744] in the multivariate case. The aim of this paper is to provide a companion algorithm to the Kiefer-Wolfowitz-Blum algorithm, which allows one to simultaneously recursively approximate the size p of the maximum of the regression function. A precise study of the joint weak convergence rate of both algorithms is given;it turns out that, unlike the location of the maximum, the size of the maximum can be approximated by an algorithm which converges at the parametric rate. Moreover, averaging leads to an asymptotically efficient algorithm for the approximation of the couple (theta, mu).

关键词： stochastic approximation algorithm weak convergence rate parametric rate averaging principle

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On the ERM formulation and a stochastic approximation algorithm of the stochastic-R₀ EVLCP

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ANNALS OF OPERATIONS RESEARCH 2014年第1期217卷 513-534页

作者： Wang, Ming-Zheng Ali, M. Montaz Dalian Univ Technol Sch Management Sci & Engn Dalian 116024 Peoples R China Univ Witwatersrand Sch Computat & Appl Math ZA-2050 Johannesburg Johannesburg South Africa

In this paper, a class of stochastic extended vertical linear complementarity problems is studied as an extension of the stochastic linear complementarity problem. The expected residual minimization (ERM) formulation of this stochastic extended vertical complementarity problem is proposed based on an NCP function. We study the corresponding properties of the ERM problem, such as existence of solutions, coercive property and differentiability. Finally, we propose a descent stochastic approximation method for solving this problem. A comprehensive convergence analysis is given. A number of test examples are constructed and the numerical results are presented.

关键词： stochastic programming stochastic R-0-property Existence of solution stochastic approximation algorithm ERM reformulation

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MAXIMUM LIKELIHOOD ESTIMATION FOR SIMPLEX DISTRIBUTION NONLINEAR MIXED MODELS VIA THE stochastic approximation algorithm

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ROCKY MOUNTAIN JOURNAL OF MATHEMATICS 2008年第5期38卷 1863-1875页

作者： Zhang, Wenzhuan Wei, Hongjie Guizhou Coll Finance & Econ Sch Math & Stat Guiyang 550004 Guizhou Peoples R China Chinese Acad Trop Agr Sci Rubber Res Inst Danzhou 571737 Hainan Peoples R China

Longitudinal continuous proportional data. is common in many fields such as biomedical research, psychological research and so on, e.g., the percent decrease in glomerular filtration rate at different follow-up times from the baseline. As shown in Song and Tan [16] such data can be fitted with simplex models. However, the original models of [16] for such longitudinal continuous proportional data assumed a fixed effect for every subject. This paper extends the models of Song and Tan [16] by adding random effects, and proposes simplex distribution nonlinear mixed models which are one kind of nonlinear reproductive dispersion mixed model. By treating random effects in the models as hypothetical missing data and applying the Metropolis-Hastings (M-H) algorithm, this paper develops the stochastic approximation (SA) algorithm with Markov chain Monte-Carlo (MCMC) method for maximum likelihood estimation in the models. Finally, for ease of comparison, the method is illustrated with the same data from an ophthalmology study on the use of intraocular gas in retinal surgeries in [16].

关键词： Simplex distribution nonlinear mixed models maximum likelihood estimates stochastic approximation algorithm M-H algorithm longitudinal continuous proportional data

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An Introduction to Development of Centralized and Distributed stochastic approximation algorithm with Expanding Truncations

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algorithmS 2021年第6期14卷 174页

作者： Zhao, Wenxiao Chinese Acad Sci Acad Math & Syst Sci Key Lab Syst & Control Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China

The stochastic approximation algorithm (SAA), starting from the pioneer work by Robbins and Monro in 1950s, has been successfully applied in systems and control, statistics, machine learning, and so forth. In this paper, we will review the development of SAA in China, to be specific, the stochastic approximation algorithm with expanding truncations (SAAWET) developed by Han-Fu Chen and his colleagues during the past 35 years. We first review the historical development for the centralized algorithm including the probabilistic method (PM) and the ordinary differential equation (ODE) method for SAA and the trajectory-subsequence method for SAAWET. Then, we will give an application example of SAAWET to the recursive principal component analysis. We will also introduce the recent progress on SAAWET in a networked and distributed setting, named the distributed SAAWET (DSAAWET).

关键词： stochastic approximation algorithm expanding truncation technique centralized and distributed algorithm

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A new normalized stochastic approximation algorithm using a time-shift parameter

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ELECTRONICS AND COMMUNICATIONS IN JAPAN PART III-FUNDAMENTAL ELECTRONIC SCIENCE 1995年第8期78卷 41-51页

作者： Okamura, A Kirimoto, T Kondo, M Electro-Optics & Microwave Systems Laboratory Mitsubishi Electric Corporation Kamakura Japan 247 Members

This paper proposes a stochastic approximation adaptive algorithm aiming at a fast convergence speed, where the step size. is normalized by the instantaneous power \\x(k)\\(2) of the input signal and a constant time shift beta is included at the weight update, as alpha/(k + beta)\\x(k)\\(2). If is shown first that the proposed algorithm gives a convergence to the Wiener solution. Then the transient response of the mean error is analyzed. It is shown as a result that beta greater than or equal to alpha/2 is the condition for the time-shift parameter to guarantee the uniform convergence of the algorithm. It is shown also that the speed of convergence is improved greatly compared to the conventional algorithm when a is set large enough so that the product of alpha and the minimum eigenvalue of the autocorrelation matrix exceeds 100 and the time-shift parameter is set as beta similar to alpha/2. The performance is evaluated by a computer simulation. The above result of analysis is verified, and it is shown that the proposed algorithm can realize the same convergence speed as that of the recursive least-squares (RLS) algorithm.

关键词： system identification stochastic approximation algorithm normalized algorithm fast convergence time-varying step size

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Recursive identification of nonlinear nonparametric systems under event-triggered observations

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SYSTEMS & CONTROL LETTERS 2025年 196卷

作者： Ren, Xiaotao Zhao, Wenxiao Zhang, Han Shanxi Vocat Univ Engn Sci & Technol Sch Equipment Engn Jinzhong 030619 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Key Lab Syst & Control Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Shanghai Jiao Tong Univ Dept Automat Shanghai 200240 Peoples R China

In the paper, recursive identification of the nonlinear nonparametric system is considered, where measurements for identification are subject to an event-triggered scheme and the system itself is of a finite impulse response (FIR) type. First, an adaptive event detector is designed and the measurements for identification are transmitted depending whether the value of the detector is 1 or 0. Second, a recursive identification algorithm is proposed with the help of a kernel-based stochastic approximation algorithm with expanding truncations (SAAWET). Third, under some reasonable assumptions, the estimates are proved to be strongly consistent, i.e., converging almost surely to the values of the unknown function in the system at any fixed points, and the transmission rate of the detector is analyzed as well. Finally, numerical examples are given to justify the theoretical results.

关键词： Event-triggered scheme Nonlinear nonparametric system Recursive identification stochastic approximation algorithm

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Asymptotic optimality for consensus-type stochastic approximation algorithms using iterate averaging

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控制理论与应用（英文版） 2013年第1期11卷 1-9页

作者： Gang YIN Le Yi WANG Yu SUN David CASBEER Raymond HOLSAPPLE Derek KINGSTON Department of Mathematics Wayne State University Department of Electrical and Computer Engineering Wayne State University Quantitative Business Analysis Department Siena College Control Science Center of Excellence Air Force Research Laboratory

This paper introduces a post-iteration averaging algorithm to achieve asymptotic optimality in convergence rates of stochastic approximation algorithms for consensus control with structural constraints. The algorithm involves two stages. The first stage is a coarse approximation obtained using a sequence of large stepsizes. Then, the second stage provides a refinement by averaging the iterates from the first stage. We show that the new algorithm is asymptotically efficient and gives the optimal convergence rates in the sense of the best scaling factor and ＇smallest＇ possible asymptotic variance.

关键词： stochastic approximation algorithm Consensus Iterate averaging Asymptotic optimality

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A multi-step Richardson-Romberg extrapolation method for stochastic approximation

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stochastic PROCESSES AND THEIR APPLICATIONS 2015年第11期125卷 4066-4101页

作者： Frikha, N. Huang, L. Univ Paris Diderot LPMA F-75013 Paris France

We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in Frikha (2013). This allows us to extend and develop the Richardson Romberg extrapolation method for Monte Carlo linear estimator (introduced in Talay and Tubaro (1990) and deeply studied in Pages (2007)) to the framework of stochastic optimization by means of stochastic approximation algorithm. We notably apply the method to the estimation of the quantile of diffusion processes. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Euler scheme Weak error Richardson-Romberg extrapolation stochastic approximation algorithm

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Bandwidth selector for nonparametric recursive density estimation for spatial data defined by stochastic approximation method

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2020年第12期49卷 2942-2963页

作者： Bouzebda, Salim Slaoui, Yousri Univ Technol Compiegne LMAC Compiegne France Univ Poitiers Lab Math & Applict Futuroscope France

In this article we propose an automatic selection of the bandwidth of the recursive kernel density estimators for spatial data defined by the stochastic approximation algorithm. We showed that, using the selected bandwidth and the stepsize which minimize the MWISE (Mean Weighted Integrated Squared Error), the recursive estimator will be quite similar to the nonrecursive one in terms of estimation error and much better in terms of computational costs. In addition, we obtain the central limit theorem for the nonparametric recursive density estimator under some mild conditions.

关键词： Bandwidth selection density estimation spatial data stochastic approximation algorithm

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OPTIMAL IMPORTANCE SAMPLING PARAMETER SEARCH FOR LEVY PROCESSES VIA stochastic approximation

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2008年第1期47卷 293-307页

作者： Kawai, Reiichiro Osaka Univ Ctr Study Finance & Insurance Toyonaka Osaka 5608531 Japan

The author proposes stochastic approximation methods of finding the optimal measure change by the exponential tilting for Levy processes in Monte Carlo importance sampling variance reduction. In accordance with the structure of the underlying Levy measure, either a constrained or unconstrained algorithm of the stochastic approximation is chosen. For both cases, the almost sure convergence to a unique stationary point is proved. Numerical examples are presented to illustrate the effectiveness of our method.

关键词： Esscher transform Girsanov theorem Monte Carlo simulation infinitely divisible distribution stochastic approximation algorithm variance reduction

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