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

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1.3 INTRODUCTION TO SPSA ALGORITHM One properly generated simultaneous random change of all p variables in the problem contains as much information for optimization as a full set of p one at time changes of each variable [13]. Further, SPSA—like other stochastic approximation methods—formally accommodates noisy measurements of the objective function. This is an important practical concern in a wide variety of problems involving Monte Carlo simulations, physical experiments, feedback systems, or incomplete knowledge. The need for solving multivariate optimization problems is pervasive in engineering and the physical and social sciences. The SPSA algorithm has already attracted considerable attention for challenging optimization problems where it is difficult or impossible to directly obtain a gradient of the objective function, not on measurement of the gradient objective function. As we mentioned above, the gradient approximation is based on only two functions measurements (regardless of the dimension of the gradient vector). Therefore, contrasts with standard finite-difference approaches, which require a number of function measurements proportional to the dimension of the gradient vector. The SPSA is generally used in non-linear problems having many variables where the objective function gradient is difficult or impossible to obtain. As a SA algorithm, SPSA may be rigorously applied when noisy measurements of the objective function are all that are available. There have also been many successful applications of SPSA in settings where perfect measurements of the loss function are available. Fig. 1.2. Performance of SPSA algorithm (two measurements). 9
CHAPTER 1. INTRODUCTION 1.4--Features of SPSA 1. SPSA allows for the input the algorithm to be measurement of the objective function corrupted by noise. For example, this is ideal for the case where Monte Carlo simulations are being used because each simulation run provides one noisy estimate of the performance measure. This is especially relelvant in practice as a very large number of scenarios often need to be evaluated, and it will not be possible to run a large number of simulations at each scenario (to average out noise). So, an algorithm explicitly designed to handle noise is needed. 2. The algorithm is appropiate for high-dimensional problems where many terms are being determined in the optimization process. Many practical applications have a significant. 3. Performance guarantees for SPSA exist in the form of an extentive convergence theory. The algorithm has desirable properties for the both global and local optimization in the sense that the gradient approximation is sufficiently noisy to allow for escape from local minima while being informative about the slope for the function to faciliate local convergence. This may avoid the cumbersome need in many global optimization problems to manually switch from a global to a local algorithm. However, we concentrate in the optimal area, so that we omite the local mimina problem. 4. Implementation of SPSA may be easier than other stochastic optimization methods since there are fewer algorithm coefficients that need to be specfied, and there are some published guidelines [12] proving insight into how to pick th coefficients in practical applications. 5. While the original SPSA method is designed for conitnuos optimization problems, there have been recent extensions to discrete optimization problems. This may be revelant to certain design problems, for example, where one wants to find the best number of items to use in a particular application. 10
Page 1 and 2: Approximation of Hessian Matrix for
Page 3 and 4: Copyright 2009 by Jorge Ivan Medina
Page 5 and 6: ここで提案するアルゴ
Page 7 and 8: ABSTRACT shown that for the same as
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
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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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