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| Foreword | |
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| Preface | |
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| The framework of learning | |
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| Introduction | |
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| A formal setting | |
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| Hypothesis spaces and target functions | |
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| Sample, approximation, and generalization errors | |
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| The bias-variance problem | |
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| The remainder of this book | |
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| References and additional remarks | |
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| Basic hypothesis spaces | |
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| First examples of hypothesis space | |
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| Reminders I | |
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| Hypothesis spaces associated with Sobolev spaces | |
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| Reproducing Kernel Hilbert Spaces | |
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| Some Mercer kernels | |
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| Hypothesis spaces associated with an RKHS | |
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| Reminders II | |
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| On the computation of empirical target functions | |
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| References and additional remarks | |
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| Estimating the sample error | |
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| Exponential inequalities in probability | |
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| Uniform estimates on the defect | |
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| Estimating the sample error | |
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| Convex hypothesis spaces | |
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| References and additional remarks | |
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| Polynomial decay of the approximation error | |
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| Reminders III | |
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| Operators defined by a kernel | |
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| Mercer's theorem | |
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| RKHSs revisited | |
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| Characterizing the approximation error in RKHSs | |
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| An example | |
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| References and additional remarks | |
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| Estimating covering numbers | |
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| Reminders IV | |
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| Covering numbers for Sobolev smooth kernels | |
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| Covering numbers for analytic kernels | |
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| Lower bounds for covering numbers | |
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| On the smoothness of box spline kernels | |
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| References and additional remarks | |
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| Logarithmic decay of the approximation error | |
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| Polynomial decay of the approximation error C[infinity]for kernels | |
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| Measuring the regularity of the kernel | |
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| Estimating the approximation error in RKHSs | |
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| Proof of Theorem 6.1 | |
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| References and additional remarks | |
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| On the bias-variance problem | |
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| A useful lemma | |
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| Proof of Theorem 7.1 | |
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| A concrete example of bias-variance | |
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| References and additional remarks | |
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| Least squares regularization | |
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| Bounds for the regularized error | |
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| On the existence of target functions | |
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| A first estimate for the excess generalization error | |
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| Proof of Theorem 8.1 | |
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| Reminders V | |
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| Compactness and regularization | |
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| References and additional remarks | |
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| Support vector machines for classification | |
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| Binary classifiers | |
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| Regularized classifiers | |
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| Optimal hyperplanes: the separable case | |
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| Support vector machines | |
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| Optimal hyperplanes: the nonseparable case | |
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| Error analysis for separable measures | |
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| Weakly separable measures | |
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| References and additional remarks | |
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| General regularized classifiers | |
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| Bounding the misclassification error in terms of the generalization error | |
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| Projection and error decomposition | |
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| Bounds for the regularized error D([gamma],[pi]of f[subscript gamma] | |
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| Bounds for the sample error term involving f[subscript gamma] | |
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| Bounds for the sample error term involving f[superscript pi][subscript z,gamma] | |
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| Stronger error bounds | |
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| Improving learning rates by imposing noise conditions | |
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| References and additional remarks | |
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| References | |
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| Index | |