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

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Approximation of Hessian Matrix for Second-order SPSA Algorithm Addressed Toward Parameter Optimization in Non-linear Systems Jorge Ivan Medina Martinez Abstract The research presented in this dissertation has been motivated due to the fact that many algorithms, which are very extended, do not offer sufficient stability in the estimation of a great volume of parameters in non-linear systems or other kinds of systems. They also have a high computational complexity and cost. So that, we have decided to use the simultaneous perturbation stochastic approximation (SPSA) algorithm because it has several important advantages such as low computational complexity and stable convergence. Nevertheless, the typical SPSA algorithm has some difficulties and problems when it is applied to non-linear and complex systems. Therefore, this research proposes a novel extension to the SPSA algorithm based on features and disadvantages shown in the first-order and second-order SPSA (the 1st-SPSA and 2nd-SPSA) algorithms and comparisons made from the perspective of the Hessian loss function. These comparisons are made because at finite iterations, the convergence rate depends on matrix conditioning of the loss function Hessian. It is shown that 2nd-SPSA converges more slowly for a loss function with an ill-conditioned Hessian than the one with a well-conditioned Hessian. On the other hand, the convergence rate of 1st-SPSA is less sensitive to the matrix conditioning of loss function Hessian. The main disadvantages in the 1st-SPSA and 2nd-SPSA algorithms, one that the error for the loss function with an ill-conditioned Hessian is greater than the one with a well-conditioned Hessian. Our proposed modified version of 2nd-SPSA (M2-SPSA) eliminates the error amplification caused by the inversion of an ill-conditioned Hessian at finite iterations, which leads to significant improvements in its convergence rate in problems with an ill-conditioned Hessian matrix and complex systems. Asymptotically, the efficiency analysis shows that our proposed SPSA is also superior to 2nd-SPSA in terms of its convergence rate coefficients. It is iii
ABSTRACT shown that for the same asymptotic convergence rate, the ratio of the mean square errors for our proposed SPSA to 2nd-SPSA is always less than one, except for a perfectly conditioned Hessian. Also, we have proposed to reduce the computational expense by evaluating only a diagonal estimate of the eigenvalues in the Hessian matrix. In this research, a new mapping is suggested for the 2nd-SPSA algorithm in order to eliminate the non-positive definiteness part while preserving key spectral properties of the estimated Hessian using the Fisher information matrix. After defining the M2-SPSA algorithm, we apply this algorithm to parameter estimation. Therefore, using M2-SPSA all parameters are perturbed simultaneously, it is possible to modify parameters with only two measurements of an evaluation function regardless of the dimension of the parameter. A convergence theorem for the proposed algorithm is presented and a simulation result also reveals the feasibility of the identification scheme proposed here. In order to show the efficiency of M2-SPSA, we have proposed three important applications, in which we can see the efficiency of our proposed algorithm for estimating and designing the parameters. In the first application, our proposed algorithm is addressed to control, in this case, the vibration reduction in the model considered here. Therefore, the main objective concerns a vibration control of a one-link flexible arm system. A variable structure system (VSS) non-linear observer has been proposed in order to reduce the oscillation in controlling the angle of the flexible arm. The non-linear observer parameters are optimized using a modified version of SPSA algorithm. This SPSA algorithm is especially useful when the number of parameters to be adjusted is large and makes it possible to estimate them very efficiently. As for the vibration and position control, a model reference sliding-mode control (MR-SMC) has been presented. Our proposed M2-SPSA algorithm obtains the parameters of MR-SMC method. The simulations show that the vibration control of a one-link flexible arm system can be achieved more efficiently using our proposed methods. In the second application, our proposed algorithm is addressed to signal processing, in this case IIR lattice filters. Adaptive infinite impulse response (IIR), or recursive, filters are less attractive mainly because of the stability and the difficulties associated with their adaptive algorithms. Therefore, in this research the adaptive IIR lattice filters are studied in order to devise algorithms that preserve the stability of the corresponding direct form schemes. We analyze the local properties of stationary points, a transformation achieving this goal is suggested, which iv
Page 1 and 2: Approximation of Hessian Matrix for
Page 3 and 4: Copyright 2009 by Jorge Ivan Medina
Page 5: ここで提案するアルゴ
Page 9 and 10: Contents 1. Introduction 1 1.1 Moti
Page 11 and 12: CONTENTS 5.3 Parameter Estimation b
Page 13 and 14: LIST OF FIGURES Fig. 4.1 Block diag
Page 15 and 16: List of Abbreviations SPSA 1st-SPSA
Page 17 and 18: CHAPTER 1. INTRODUCTION the converg
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Page 21 and 22: CHAPTER 1. INTRODUCTION and simulta
Page 23 and 24: CHAPTER 1. INTRODUCTION Typical app
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Page 27 and 28: CHAPTER 1. INTRODUCTION Some of the
Page 29 and 30: CHAPTER 1. INTRODUCTION M − k ( k
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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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154 APPENDIX A
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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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