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An Optimal stochastic approximation for Estimating the Effective Window of a Control Factor

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Journal of Systems Science & Complexity 2015年第6期28卷 1258-1270页

作者： WANG Dianpeng TIAN Yubin XU Ying School of Mathematics and Statistics Beijing Institute of Technology

For control processes with one control factor and ternary response, it is important to find the effective settings of the control factor. Especially, it is often of scientific and practical interest tofind the window of the control factor that is necessary to cause the probability of normal response larger than a specified p. This article proposes an optimal stochastic approximation to estimate the window of interest. Simulation study shows that the proposed method gives an effcient estimation with small sample size.

关键词： Control factor Robbins-Monro procedure stochastic approximation ternary response

Revisiting the ODE Method for Recursive Algorithms:Fast Convergence Using Quasi stochastic approximation

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Journal of Systems Science & Complexity 2021年第5期34卷 1681-1702页

作者： CHEN Shuhang DEVRAJ Adithya BERSTEIN Andrey MEYN Sean Department of Mathematics University of FloridaGainesville FL32605USA Stanford University Stanford CA94305USA NREL Golden CO80401USA University of Florida Gainesville FL32605USA

Several decades ago,*** Meyn and Lei Guo were postdoctoral fellows at ANU,where they shared interest in recursive *** seems fitting to celebrate Lei Guo’s 60 th birthday with a review of the ODE Method and its recent evolution,with focus on the following themes:The method has been regarded as a technique for algorithm *** is argued that this viewpoint is backwards:The original stochastic approximation method was surely motivated by an ODE,and tools for analysis came much later(based on establishing robustness of Euler approximations).The paper presents a brief survey of recent research in machine learning that shows the power of algorithm design in continuous time,following by careful approximation to obtain a practical recursive *** these methods are usually presented in a stochastic setting,this is not a *** fact,recent theory shows that rates of convergence can be dramatically accelerated by applying techniques inspired by quasi *** to conditions,the optimal rate of convergence can be obtained by applying the averaging technique of Polyak and *** conditions are not universal,but theory suggests alternatives to achieve *** theory is illustrated with applications to gradient-free optimization,and policy gradient algorithms for reinforcement learning.

关键词： Learning and adaptive systems in artificial intelligence reinforcement learning stochastic approximation

Nearly optimal stochastic approximation for online principal subspace estimation

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Science China Mathematics 2023年第5期66卷 1087-1122页

作者： Xin Liang Zhen-Chen Guo Li Wang Ren-Cang Li Wen-Wei Lin Yau Mathematical Sciences Center Tsinghua UniversityBeijing 100084China Yanqi Lake Beijing Institute of Mathematical Sciences and Applications Beijing 101408China Department of Mathematics Nanjing UniversityNanjing 210093China Department of Mathematics University of Texas at ArlingtonArlingtonTX 76019USA Department of Mathematics Hong Kong Baptist UniversityHong KongChina Nanjing Center for Applied Mathematics Nanjing 211135China Department of Applied Mathematics Yang Ming Chiao Tung UniversityHsinchu 300China

Principal component analysis(PCA) has been widely used in analyzing high-dimensional data. It converts a set of observed data points of possibly correlated variables into a set of linearly uncorrelated variables via an orthogonal transformation. To handle streaming data and reduce the complexities of PCA,(subspace)online PCA iterations were proposed to iteratively update the orthogonal transformation by taking one observed data point at a time. Existing works on the convergence of(subspace) online PCA iterations mostly focus on the case where the samples are almost surely uniformly bounded. In this paper, we analyze the convergence of a subspace online PCA iteration under more practical assumptions and obtain a nearly optimal finite-sample error bound. Our convergence rate almost matches the minimax information lower bound. We prove that the convergence is nearly global in the sense that the subspace online PCA iteration is convergent with high probability for random initial guesses. This work also leads to a simpler proof of the recent work on analyzing online PCA for the first principal component only.

关键词： principal component analysis principal component subspace stochastic approximation high-dimensional data online algorithm nite-sample analysis

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A stochastic approximation Frame Algorithm with Adaptive Directions

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Numerical Mathematics(Theory,Methods and Applications) 2008年第4期1卷 460-474页

作者： Zi Xu Yu-Hong Dai Department of Mathematics College of Sciences Shanghai University Shang-hai 200444 PR. China. State Key Laboratory of Scientific and Engineering Computing Institute of Com-putational Mathematics and Scientific/Engineering Computing The Academy ofMathematics and Systems Science Chinese Academy of Sciences P.O.Box 2719Beijing 100190 P.R. China.

stochastic approximation problem is to find some root or extremum of a nonlinear function for which only noisy measurements of the function are available. The classical algorithm for stochastic approximation problem is the Robbins-Monro （RM） algorithm, which uses the noisy evaluation of the negative gradient direction as the iterative direction. In order to accelerate the RM algorithm, this paper gives a flame algorithm using adaptive iterative directions. At each iteration, the new algorithm goes towards either the noisy evaluation of the negative gradient direction or some other directions under some switch criterions. Two feasible choices of the criterions are proposed and two corresponding frame algorithms are formed. Different choices of the directions under the same given switch criterion in the frame can also form different algorithms. We also proposed the simultanous perturbation difference forms for the two frame algorithms. The almost surely convergence of the new algorithms are all established. The numerical experiments show that the new algorithms are promising.

关键词： stochastic approximation conjugate gradient adaptive directions.

GEOMETRIC STRUCTURE IN stochastic approximation

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Acta Mathematicae Applicatae Sinica 2001年第1期17卷 53-59页

作者：程代展杜宏陈翰馥 he Academy of Mathematics and Systems Sciences Institute of Systems Science Beijing China he Chinese Academy of Sciences Institute of Systems Science Beijing China

Let J be the zero set of the gradient fx of a function f:Rn→R. Under fairly general conditions the stochastic approximation algorithm ensures d(f(xk),f(J))→0, as k→∞. First of all, the paper considers this problem: Under what conditions the convergence d(f(xk),f(J)) → 0 implies k →∞ d(xk,J)→O. It is shown that such implication takes place if fx is continuous and f(J) is nowhere dense. Secondly, an intensified version of Sard's theorem has been proved, which itself is interesting. As a particular case, it provides two independent sufficient conditions as answers to the previous question: If f is a C1 function and either i) J is a compact set or ii) for any bounded set B, f-1(B)is bounded, then f(J) is nowhere dense. Finally, some tools in algebraic geometry are used to prove that j(J) is a finite set if f is a polynomial. Hence f(J) is nowhere dense in the polynomial case.

关键词： stochastic approximation regular value intensified 'Sard's theorem irreducible algebraic variety

stochastic approximation with Non-Additive Measurement Noise

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Stochastic Approximation with Non-Additive Measurement Noise

On Efficiency of Simultaneous Perturbation stochastic approximation Method for Implementation of an Adaptive Filter

1997年中国控制会议

作者： Han-Fu CHEN Laboratory of Systems & Control, Institute of Systems Science, The Chinese Academy of Sciences, Beijing 100080, China.

＜正＞The Robbins-Monro algorithm with randomly varying truncations for measurements with non-additive noise is considered. Assuming that the function under observation is locally Lipschitz-continuous in its first argument and the noise is a （?）-mixing process, strong consistency of the estimate is shown. Neither growth rate restriction on the function, nor the decreasing rate of the mixing coefficients are required.

关键词： stochastic approximation non-additive noise randomly varying truncation

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Computer Technology and Application 2011年第12期2卷 948-962页

作者： Hong Son Hoang Remy Baraille SHO M/HOM Toulouse 31057 France

In this paper, the simultaneous perturbation stochastic approximation （SPSA） algorithm is used for seeking optimal parameters in an adaptive filter developed for assimilating observations in the very high dimensional dynamical systems. The main results show that the SPSA is capable of yielding the high filter performance similar to that produced by classical optimization algorithms, with better performance for non-linear filtering problems as more and more observations are assimilated. The advantage of the SPSA is that at each iteration it requires only two measurements of the objective function to approximate the gradient vector regardless of the dimension of the control vector （or maximally, three measurements if second-order optimization algorithms are used）. The SPSA approach is thus free from the need to develop a discrete adjoint of tangent linear model as it is required up to now for solving optimization problems in very high dimensional systems. This technique offers promising perspectives on developing optimal assimilation systems encountered in the field of data assimilation in meteorology and oceanography.

关键词： stochastic approximation dominant Schur vectors minimum prediction error adaptive filter stability.

Adaptive tracking for NARX system based on stochastic approximation algorithm

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Adaptive tracking for NARX system based on stochastic approx...

Convergence Rate of stochastic approximation Algorithms in the Degenerate Case

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作者： Wenhui Feng Han-Fu Chen Shijiazhuang Tiedao University Academy of Mathematics and Systems Science Chinese Academy of Sciences

In this paper we study the adaptive tracking control for nonlinear ARX（NARX） systems with additive observation noise. Compared to the existing works the dynamical function in the model is allowed to process arbitrary growth rate with respect to the input argument. The adaptive control is designed on the basis of stochastic approximation algorithm with expanding truncations（SAAWET）, and it is pointed out that there is a trade-off between the growth rate of the dynamical function with respect to control argument and that of the truncation bound to be chosen. We proved that the tracking purpose is achieved and the numerical simulation given in the paper justifies the theoretical assertions.

关键词： Nonlinear ARX system adaptive tracking adaptive control stochastic approximation

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Convergence Rate of Stochastic Approximation Algorithms in t...