Learning Theory ：An Approximation Theory Viewpoint ( Cambridge Monographs on Applied and Computational Mathematics )

Publication subTitle ：An Approximation Theory Viewpoint

Publication series ：Cambridge Monographs on Applied and Computational Mathematics

Author： Felipe Cucker; Ding Xuan Zhou  

Publisher： Cambridge University Press‎

Publication year： 2007

E-ISBN: 9780511271663

P-ISBN(Paperback): 9780521865593

Subject： TP181 automatic reasoning, machine learning

Keyword： 一般性问题

Language： ENG

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Learning Theory

Description

The goal of learning theory is to approximate a function from sample values. To attain this goal learning theory draws on a variety of diverse subjects, specifically statistics, approximation theory, and algorithmics. Ideas from all these areas blended to form a subject whose many successful applications have triggered a rapid growth during the last two decades. This is the first book to give a general overview of the theoretical foundations of the subject emphasizing the approximation theory, while still giving a balanced overview. It is based on courses taught by the authors, and is reasonably self-contained so will appeal to a broad spectrum of researchers in learning theory and adjacent fields. It will also serve as an introduction for graduate students and others entering the field, who wish to see how the problems raised in learning theory relate to other disciplines.

Chapter

Cover

Half-title

Series-title

Title

Copyright

Contents

Foreword

Preface

1 The framework of learning

1.1 Introduction

1.2 A formal setting

1.3 Hypothesis spaces and target functions

1.4 Sample, approximation, and generalization errors

1.5 The bias–variance problem

1.6 The remainder of this book

1.7 References and additional remarks

2 Basic hypothesis spaces

2.1 First examples of hypothesis space

2.2 Reminders I

2.3 Hypothesis spaces associated with Sobolev spaces

2.4 Reproducing Kernel Hilbert Spaces

2.5 Some Mercer kernels

2.6 Hypothesis spaces associated with an RKHS

2.7 Reminders II

2.8 On the computation of empirical target functions

2.9 References and additional remarks

3 Estimating the sample error

3.1 Exponential inequalities in probability

3.2 Uniform estimates on the defect

3.3 Estimating the sample error

3.4 Convex hypothesis spaces

3.5 References and additional remarks

4 Polynomial decay of the approximation error

4.1 Reminders III

4.2 Operators defined by a kernel

4.3 Mercer's theorem

4.4 RKHSs revisited

4.5 Characterizing the approximation error in RKHSs

4.6 An example

4.7 References and additional remarks

5 Estimating covering numbers

5.1 Reminders IV

5.2 Covering numbers for Sobolev smooth kernels

5.3 Covering numbers for analytic kernels

5.4 Lower bounds for covering numbers

5.5 On the smoothness of box spline kernels

5.6 References and additional remarks

6 Logarithmic decay of the approximation error

6.1 Polynomial decay of the approximation error for … kernels

6.2 Measuring the regularity of the kernel

6.3 Estimating the approximation error in RKHSs

6.4 Proof of Theorem 6.1

6.5 References and additional remarks

7 On the bias–variance problem

7.1 A useful lemma

7.2 Proof of Theorem 7.1

7.3 A concrete example of bias–variance

7.4 References and additional remarks

8 Least squares regularization

8.1 Bounds for the regularized error

8.2 On the existence of target functions

8.3 A first estimate for the excess generalization error

8.4 Proof of Theorem 8.1

8.5 Reminders V

8.6 Compactness and regularization

8.7 References and additional remarks

9 Support vector machines for classification

9.1 Binary classifiers

9.2 Regularized classifiers

9.3 Optimal hyperplanes: the separable case

9.4 Support vector machines

9.5 Optimal hyperplanes: the nonseparable case

9.6 Error analysis for separable measures

9.7 Weakly separable measures

9.8 References and additional remarks

10 General regularized classifiers

10.1 Bounding the misclassification error in terms of the generalization error

10.2 Projection and error decomposition

10.3 Bounds for the regularized error D(Gamma,Phi) of fGamma

10.4 Bounds for the sample error term involving fGamma

10.5 Bounds for the sample error term involving fPhiz,Gamma

10.6 Stronger error bounds

10.7 Improving learning rates by imposing noise conditions

10.8 References and additional remarks

References

Index

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