Approximation of Hessian Matrix for Second-order SPSA Algorithm ...

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ABSTRACT gives algorithms that can be efficiently implemented. Application to the Steiglitz-McBride (SM) and Simple Hyperstable Adaptive Recursive Filter (SHARF) algorithms is presented. Also, our proposed M2-SPSA algorithm is presented in order to get the coefficients in a lattice form more efficiently and with a lower computational cost and complexity. The results are compared with previous lattice versions of these algorithms. These previous lattice versions may fail to preserve the stability of stationary points. Finally, the M2-SPSA algorithm is addressed to the problem of estimation of unknown static parameters in non-linear state-space models. The M2-SPSA algorithm can generate maximum likelihood estimates efficiently. The performance of the proposed algorithm is assessed through simulation. Here, the M2-SPSA algorithm is compared with the finite difference stochastic approximation (FDSA) in order to show its efficiency. Therefore, in this dissertation, we have proposed a modification to SPSA algorithm where the main objective is to estimate the parameters in complex systems, improve the convergence and reduce the computational cost. Then, this modification to the simultaneous perturbation seems particularly useful when there are number of parameters to be identified is very large or when the observed values for what is to be identified can only be obtained via an unknown observation system. Finally, this dissertation is organized as follows. In Chapter 1, we describe an introduction to SPSA, so that we explain the mean concepts, advantages, disadvantages, recursions, formulation and implementation of SPSA. Our proposed SPSA algorithm is analyzed in detail in Chap. 2. The asymptotic normality, the Hessian estimation and the efficiency between M2-SPSA and the previous versions of SPSA are shown. In addition, we show how the M2-SPSA algorithm is applied to parameter estimation, and prove its efficiency in several simple numerical simulations. The first important application of M2-SPSA algorithm is described in Chap. 3, in this case, in the control area; M2-SPSA is applied to parameter estimation of some methods for controlling the vibration in the proposed system. Other application for M2-SPSA algorithm is described in Chap. 4. In this application, our proposed algorithm is applied to signal processing, here M2-SPSA calculates the coefficients in some adaptive algorithms. In the final application, M2-SPSA algorithm is addressed to the problem of estimation of unknown static parameters in non-linear state-space models, this is described in Chap. 5. Finally, the conclusions and future work are given in Chap. 6. v
Contents 1. Introduction 1 1.1 Motivation and Background 1 -1.1.1 Motivation 1 -1.1.2 Background 2 1.2 Overview of Stochastic Algorithms 5 1.3 Introduction to SPSA Algorithm 7 1.4 Features of SPSA 10 1.5 Applications Areas 11 1.6 Formulation of SPSA Algorithm 12 1.7 Basic Assumptions of SPSA Algorithm 14 1.8 Versions of SPSA Algorithms 15 2. Proposed SPSA Algorithm 19 2.1 Overview of Modified 2nd-SPSA Algorithm 19 2.2 SPSA Algorithm Recursions 20 2.3 Proposed Mapping 22 2.4 Description of Proposed SPSA Algorithm 26 2.5 Asymptotic Normality 27 2.6 Fisher Information Matrix 31 -2.6.1 Introduction to Fisher Information Matrix 31 -2.6.2 Two Key Properties of the Information Matrix: Connections to --Covariance Matrix of Parameter Estimates 33 -2.6.3 Estimation of F (θ n ) 34 2.7 Efficiency Between 1st-SPSA, 2nd-SPSA and 2M-SPSA 40 2.8 Implementation Aspects 41 2.9 Strong Convergence 44 2.10 Asymptotic Distribution and Efficiency Analysis 50 2.11 Perturbation Distribution for M2-SPSA 54 2.12 Parameter Estimation 57 2.12.1 Introduction 57 2.12.2 System to be Applied 64 2.12.3 Convergence Theorem 69 vii
Page 1 and 2: Approximation of Hessian Matrix for
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CHAPTER 2. PROPOSED SPSA ALGORITHM
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CHAPTER 3. APPLICATION USING M2-SPS
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CHAPTER 6. CONCLUSIONS AND FUTURE W
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REFERENCE [10] S. A. Billings, G. N
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REFERENCE [29] M. Metivier and P. P
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REFERENCE [51] D. Parikh, N. Ahmed
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REFERENCE [72] N. J. Gordon, D. J S
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APPENDIX A that this random vector
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APPENDIX A Part 2: To show that ~
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APPENDIX A Proof of Theorem 2a (M2-
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APPENDIX A 1 ⎡~ ~ ~ ~ ( ( ) ( ))
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APPENDIX A ˆ θ * −α * k+ 1 −
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APPENDIX A results. Here, zk+n+ 1 i
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APPENDIX A Because the second eleme
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APPENDIX B The Wei [48] approach is
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158 APPENDIX B
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LIST OF THE PUBLICATIONS AND INTERN
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LIST OF THE PUBLICATIONS AND INTERN
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Author Biography Jorge Ivan Medina
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Approximation of Hessian Matrix for Second-order SPSA Algorithm ...

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