Learning Theory: An Approximation Theory

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.

Foreword	p. ix
Preface	p. x
The framework of learning	p. 1
Introduction	p. 1
A formal setting	p. 5
Hypothesis spaces and target functions	p. 9
Sample, approximation, and generalization errors	p. 11
The bias-variance problem	p. 13
The remainder of this book	p. 14
References and additional remarks	p. 15
Basic hypothesis spaces	p. 17
First examples of hypothesis space	p. 17
Reminders I	p. 18
Hypothesis spaces associated with Sobolev spaces	p. 21
Reproducing Kernel Hilbert Spaces	p. 22
Some Mercer kernels	p. 24
Hypothesis spaces associated with an RKHS	p. 31
Reminders II	p. 33
On the computation of empirical target functions	p. 34
References and additional remarks	p. 35
Estimating the sample error	p. 37
Exponential inequalities in probability	p. 37
Uniform estimates on the defect	p. 43
Estimating the sample error	p. 44
Convex hypothesis spaces	p. 46
References and additional remarks	p. 49
Polynomial decay of the approximation error	p. 54
Reminders III	p. 55
Operators defined by a kernel	p. 56
Mercer's theorem	p. 59
RKHSs revisited	p. 61
Characterizing the approximation error in RKHSs	p. 63
An example	p. 68
References and additional remarks	p. 69
Estimating covering numbers	p. 72
Reminders IV	p. 73
Covering numbers for Sobolev smooth kernels	p. 76
Covering numbers for analytic kernels	p. 83
Lower bounds for covering numbers	p. 101
On the smoothness of box spline kernels	p. 106
References and additional remarks	p. 108
Logarithmic decay of the approximation error	p. 109
Polynomial decay of the approximation error C[infinity]for kernels	p. 110
Measuring the regularity of the kernel	p. 112
Estimating the approximation error in RKHSs	p. 117
Proof of Theorem 6.1	p. 125
References and additional remarks	p. 125
On the bias-variance problem	p. 127
A useful lemma	p. 128
Proof of Theorem 7.1	p. 129
A concrete example of bias-variance	p. 132
References and additional remarks	p. 133
Least squares regularization	p. 134
Bounds for the regularized error	p. 135
On the existence of target functions	p. 139
A first estimate for the excess generalization error	p. 140
Proof of Theorem 8.1	p. 148
Reminders V	p. 151
Compactness and regularization	p. 151
References and additional remarks	p. 155
Support vector machines for classification	p. 157
Binary classifiers	p. 159
Regularized classifiers	p. 161
Optimal hyperplanes: the separable case	p. 166
Support vector machines	p. 169
Optimal hyperplanes: the nonseparable case	p. 171
Error analysis for separable measures	p. 173
Weakly separable measures	p. 182
References and additional remarks	p. 185
General regularized classifiers	p. 187
Bounding the misclassification error in terms of the generalization error	p. 189
Projection and error decomposition	p. 194
Bounds for the regularized error D([gamma],[pi]of f[subscript gamma]	p. 196
Bounds for the sample error term involving f[subscript gamma]	p. 198
Bounds for the sample error term involving f[superscript pi][subscript z,gamma]	p. 201
Stronger error bounds	p. 204
Improving learning rates by imposing noise conditions	p. 210
References and additional remarks	p. 211
References	p. 214
Index	p. 111
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Learning Theory: An Approximation Theory Viewpoint

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