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

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2.6 FISHER INFORMATION MATRIX under the same rate of convergence. Under this circumstance, there is no superiority of either one of M2-SPSA and 2nd-SPSA to the other in terms of the efficiency or the rate of convergence. The superiority of our proposed SPSA algorithm to 2nd-SPSA indicated by (2.25) only shows an improvement in the multiplier for the convergence rate ( R 0 ) when the common convergence rate is sub-optimal. In [25] is showed that by setting α = 1 and γ = 1 / 6 asymptotically optimal MSE can be achieved with a maximum rate of convergence for the MSE of − / 3 θˆ of k β = k − 2 in both 1st-SPSA and 2nd-SPSA. We have already shown that in order to k avoid the violation of the condition min i ( λ i / λ ) ≥ β / 2 the setting of α = 1 (with β ≈ 2 / 3 ) is often not allowed in our proposed SPSA algorithm. Neither is it possible to choose a different set of α and m γ to yield m β m = 2 / 3 when γ = 1 / 6 an . Under this circumstance, the / 3 maximum rate of convergence of − 2 for MSE cannot be achieved by our proposed SPSA. It is noted that the mapping k f k such as the one proposed in Sec. 2.3 will leave the asymptotic H k unchanged (when we set Λˆ = Λ ) as k → ∞ . On the other hand, our proposed SPSA k k algorithm changes H k when its Λ is replaced by Λ . k k 2.6 -Fisher Information Matrix 2.6.1 -Introduction to Fisher Information Matrix In this section, we presented a relatively simple MCNR method for obtaining the FIM that is used in order to estimate the Hessian matrix efficiently. So that, the resampling-based method relies on an efficient technique for estimating the Hessian matrix. The FIM plays a central role in the practice and theory of identification and estimation. This matrix provides a summary of the amount of information in the data relative to the quantities of interest [22]. Suppose that the i-th measurement of a process is z i and that a stacked vector of n such measurement vectors is n T T T [ z z z ] T z ≡ ,..., 1 , 2 n . Let us assume that the general form for the joint probability density or probability mass function for zn is known, but that this function depends on an unknown vector θ . Let the probability density/mass function for z be z( ζ θ ) where ζ (“zeta”) is a n p f 31
CHAPTER 2. PROPOSED SPSA ALGORITHM dummy vector representing the possible outcomes for z n (in p f z( ζ θ) ), the index n on z n is being suppressed for notational convenience). The corresponding likelihood function, say l ( θ ζ ) = z( ζ θ ). (2.27) p f With the definition of the likelihood function in (2.27), we are now in a position to present the Fisher information matrix. The expectations below are with respect to the dataset z n . The p xp f f information matrix F n(θ ) for a differentiable log-likelihood function is given by [22] ⎛ ∂ log l ∂ logl ⎞ F n ( θ ) ≡ E⎜ ⋅ θ ⎟. T ⎝ ∂θ ∂θ ⎠ (2.28) In the case where the underlying data { z z ,..., } , 2 1 are independent (and even in many cases where the data may be dependent), the magnitude of F n(θ ) will grow at a rate proportional to n since logl( ⋅) will represent a sum of n random terms. Then, the bounded quantity F n (θ ) / n is employed as an average information matrix over all measurements. Except for relatively simple problems, however, the form in (2.28) is generally not useful in the practical calculation of the information matrix. Computing the expectation of a product of multivariate non-linear functions is usually a hopeless task. A well-known equivalent form follows by assuming that logl ( ⋅) is twice differentiable in θ . The following Hessian matrix z n H ( θ ζ ) ≡ ∂ 2 log l ( θ ζ ) ∂ θ ∂ θ T is assumed to exist [22]. One of these conditions is that the set { ζ : l ( θ ζ ) > 0} does not depend on θ . A fundamental implication of the regularity for the likelihood is that the necessary interchanges of differentiation and integration are valid. Then, the information matrix is related to the Hessian matrix of l through: [ H ( θ Z θ ] F θ ) = − E ) . (2.29) n ( n The form in (2.29) is usually more amenable to calculating the matrix than the product-based 32
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
Page 17 and 18: CHAPTER 1. INTRODUCTION the converg
Page 19 and 20: CHAPTER 1.INTRODUCTION approximatio
Page 21 and 22: CHAPTER 1. INTRODUCTION and simulta
Page 23 and 24: CHAPTER 1. INTRODUCTION Typical app
Page 25 and 26: CHAPTER 1. INTRODUCTION 1.4--Featur
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 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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