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E-raamat: Learning Kernel Classifiers – Theory and Algorithms: Theory and Algorithms

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An overview of the theory and application of kernel classification methods.

Linear classifiers in kernel spaces have emerged as a major topic within the field of machine learning. The kernel technique takes the linear classifier—a limited, but well-established and comprehensively studied model—and extends its applicability to a wide range of nonlinear pattern-recognition tasks such as natural language processing, machine vision, and biological sequence analysis. This book provides the first comprehensive overview of both the theory and algorithms of kernel classifiers, including the most recent developments. It begins by describing the major algorithmic advances: kernel perceptron learning, kernel Fisher discriminants, support vector machines, relevance vector machines, Gaussian processes, and Bayes point machines. Then follows a detailed introduction to learning theory, including VC and PAC-Bayesian theory, data-dependent structural risk minimization, and compression bounds. Throughout, the book emphasizes the interaction between theory and algorithms: how learning algorithms work and why. The book includes many examples, complete pseudo code of the algorithms presented, and an extensive source code library.
Series Foreword xv
Preface xvii
Introduction
1(16)
The Learning Problem and (Statistical) Inference
1(7)
Supervised Learning
3(3)
Unsupervised Learning
6(1)
Reinforcement Learning
7(1)
Learning Kernel Classifiers
8(3)
The Purposes of Learning Theory
11(6)
I LEARNING ALGORITHMS
Kernel Classifiers from a Machine Learning Perspective
17(56)
The Basic Setting
17(7)
Learning by Risk Minimization
24(6)
The (Primal) Perceptron Algorithm
26(1)
Regularized Risk Functionals
27(3)
Kernels and Linear Classifiers
30(19)
The Kernel Technique
33(3)
Kernel Families
36(11)
The Representer Theorem
47(2)
Support Vector Classification Learning
49(12)
Maximizing the Margin
49(4)
Soft Margins---Learning with Training Error
53(3)
Geometrical Viewpoints on Margin Maximization
56(2)
The v-Trick and Other Variants
58(3)
Adaptive Margin Machines
61(7)
Assessment of Learning Algorithms
61(2)
Leave-One-Out Machines
63(1)
Pitfalls of Minimizing a Leave-One-Out Bound
64(2)
Adaptive Margin Machines
66(2)
Bibliographical Remarks
68(5)
Kernel Classifiers from a Bayesian Perspective
73(42)
The Bayesian Framework
73(8)
The Power of Conditioning on Data
79(2)
Gaussian Processes
81(11)
Bayesian Linear Regression
82(5)
From Regression to Classification
87(5)
The Relevance Vector Machine
92(5)
Bayes Point Machines
97(6)
Estimating the Bayes Point
100(3)
Fisher Discriminants
103(7)
Bibliographical Remarks
110(5)
II LEARNING THEORY
Mathematical Models of Learning
115(48)
Generative vs. Discriminative Models
116(5)
PAC and VC Frameworks
121(13)
Classical PAC and VC Analysis
123(4)
Growth Function and VC Dimension
127(4)
Structural Risk Minimization
131(3)
The Luckiness Framework
134(6)
PAC and VC Frameworks for Real-Valued Classifiers
140(18)
VC Dimensions for Real-Valued Function Classes
146(4)
The PAC Margin Bound
150(1)
Robust Margin Bounds
151(7)
Bibliographical Remarks
158(5)
Bounds for Specific Algorithms
163(168)
The PAC-Bayesian Framework
164(11)
PAC-Bayesian Bounds for Bayesian Algorithms
164(8)
A PAC-Bayesian Margin Bound
172(3)
Compression Bounds
175(10)
Compression Schemes and Generalization Error
176(6)
On-line Learning and Compression Schemes
182(3)
Algorithmic Stability Bounds
185(8)
Algorithmic Stability for Regression
185(5)
Algorithmic Stability for Classification
190(3)
Bibliographical Remarks
193(6)
III APPENDICES
A Theoretical Background and Basic Inequalities
199(54)
A.1 Notation
199(1)
A.2 Probability Theory
200(3)
A.2.1 Some Results for Random Variables
203(4)
A.2.2 Families of Probability Measures
207(8)
A.3 Functional Analysis and Linear Algebra
215(5)
A.3.1 Covering, Packing and Entropy Numbers
220(2)
A.3.2 Matrix Algebra
222(17)
A.4 Ill-Posed Problems
239(1)
A.5 Basic Inequalities
240(1)
A.5.1 General (In)equalities
240(3)
A.5.2 Large Deviation Bounds
243(10)
B Proofs and Derivations---Part I
253(28)
B.1 Functions of Kernels
253(1)
B.2 Efficient Computation of String Kernels
254(1)
B.2.1 Efficient Computation of the Substring Kernel
255(1)
B.2.2 Efficient Computation of the Subsequence Kernel
255(2)
B.3 Representer Theorem
257(1)
B.4 Convergence of the Perceptron
258(1)
B.5 Convex Optimization Problems of Support Vector Machines
259(1)
B.5.1 Hard Margin SVM
260(1)
B.5.2 Linear Soft Margin Loss SVM
260(1)
B.5.3 Quadratic Soft Margin Loss SVM
261(1)
B.5.4 v-Linear Margin Loss SVM
262(1)
B.6 Leave-One-Out Bound for Kernel Classifiers
263(2)
B.7 Laplace Approximation for Gaussian Processes
265(1)
B.7.1 Maximization of ftm+1&barver;X=x,Xm=x
266(2)
B.7.2 Computation of Σ
268(1)
B.7.3 Stabilized Gaussian Process Classification
269(2)
B.8 Relevance Vector Machines
271(1)
B.8.1 Derivative of the Evidence w.r.t. &thetas;
271(2)
B.8.2 Derivative of the Evidence w.r.t. σ
273(1)
B.8.3 Update Algorithms for Maximizing the Evidence
274(1)
B.8.4 Computing the Log-Evidence
275(1)
B.8.5 Maximization of fW&barver;Zm=z
276(1)
B.9 A Derivation of the Operation Åμ
277(1)
B.10 Fisher Linear Discriminant
278(3)
C Proofs and Derivations---Part II
281(40)
C.1 VC and PAC Generalization Error Bounds
281(1)
C.1.1 Basic Lemmas
281(3)
C.1.2 Proof of Theorem 4.7
284(3)
C.2 Bound on the Growth Function
287(2)
C.3 Luckiness Bound
289(3)
C.4 Empirical VC Dimension Luckiness
292(4)
C.5 Bound on the Fat Shattering Dimension
296(2)
C.6 Margin Distribution Bound
298(2)
C.7 The Quantifier Reversal Lemma
300(2)
C.8 A PAC-Bayesian Marin Bound
302(1)
C.8.1 Balls in Version Space
303(3)
C.8.2 Volume Ratio Theorem
306(2)
C.8.3 A Volume Ratio Bound
308(3)
C.8.4Bollmann's Lemma
311(3)
C.9 Algorithmic Stability Bounds
314(1)
C.9.1 Uniform Stability of Functions Minimizing a Regularized Risk
315(1)
C.9.2 Algorithmic Stability Bounds
316(5)
D Pseudocodes
321(10)
D.1 Perceptron Algorithm
321(2)
D.2 Support Vector and Adaptive Margin Machines
323(1)
D.2.1 Standard Support Vector Machines
323(1)
D.2.2-Support Vector Machines
324(1)
D.2.3 Adaptive Margin Machines
324(1)
D.3 Gaussian Processes
325(1)
D.4 Relevance Vector Machines
325(4)
D.5 Fisher Discriminants
329(1)
D.6 Bayes Point Machines
330(1)
List of Symbols 331(8)
References 339(18)
Index 357