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Learning and generalisation : with applications to neural networks / M. Vidyasagar

Main Author Vidyasagar, Mathukumalli, 1947- Country Reino Unido. Publication London : Springer, cop. 2003 Description XXI, 488 p. ; 24 cm Series Communications and control engineering ISBN 1-85233-373-1 CDU 519.7
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Holdings
Item type Current location Call number Status Date due Barcode Item holds
Monografia Biblioteca Geral da Universidade do Minho
BGUM 519.7 - V Available 324152
Total holds: 0

Enhanced descriptions from Syndetics:

How does a machine learn a new concept on the basis of examples? This second edition takes account of important new developments in the field. It also deals extensively with the theory of learning control systems, now comparably mature to learning of neural networks.

Table of contents provided by Syndetics

  • Preface to the Second Edition (p. xiii)
  • Preface to the First Edition (p. xvii)
  • 1 Introduction (p. 1)
  • 2 Preliminaries (p. 13)
  • 2.1 Pseudometric Spaces, Packing and Covering Numbers (p. 13)
  • 2.1.1 Pseudometric Spaces (p. 13)
  • 2.1.2 Packing and Covering Numbers (p. 14)
  • 2.1.3 Compact and Totally Bounded Sets (p. 16)
  • 2.2 Probability Measures (p. 17)
  • 2.2.1 Definition of a Probability Space (p. 17)
  • 2.2.2 A Pseudometric Induced by a Probability Measure (p. 18)
  • 2.2.3 A Metric on the Set of Probability Measures (p. 19)
  • 2.2.4 Random Variables (p. 21)
  • 2.2.5 Conditional Expectations (p. 23)
  • 2.3 Large Deviation Type Inequalities (p. 24)
  • 2.3.1 Chernoff Bounds (p. 24)
  • 2.3.2 Chernoff-Okamoto Bound (p. 26)
  • 2.3.3 Hoeffding's Inequality (p. 26)
  • 2.4 Stochastic Processes, Almost Sure Convergence (p. 29)
  • 2.4.1 Probability Measures on Infinite Cartesian Products (p. 29)
  • 2.4.2 Stochastic Processes (p. 29)
  • 2.4.3 The Borel-Cantelli Lemma and Almost Sure Convergence (p. 30)
  • 2.5 Mixing Properties of Stochastic Processes (p. 33)
  • 2.5.1 Definitions of Various Kinds of Mixing Coefficients (p. 34)
  • 2.5.2 Inequalities for Mixing Processes (p. 36)
  • 3 Problem Formulations (p. 43)
  • 3.1 Uniform Convergence of Empirical Means (p. 43)
  • 3.1.1 The UCEM Property (p. 43)
  • 3.1.2 The UCEMUP Property (p. 52)
  • 3.1.3 Extension to Dependent Input Sequences (p. 54)
  • 3.2 Learning Concepts and Functions (p. 55)
  • 3.2.1 Concept Learning (p. 55)
  • 3.2.2 Function Learning (p. 64)
  • 3.2.3 Extension to Dependent Input Sequences (p. 65)
  • 3.2.4 Assumptions Underlying the Model of Learning (p. 66)
  • 3.2.5 Alternate Notions of Learnability (p. 70)
  • 3.3 Model-Free Learning (p. 76)
  • 3.3.1 Problem Formulation (p. 76)
  • 3.3.2 Relationship to the Uniform Convergence of Empirical Means (p. 81)
  • 3.4 Preservation of UCEMUP and PAC Properties (p. 83)
  • 3.4.1 Preservation of UCEMUP Property with Beta-Mixing Inputs (p. 84)
  • 3.4.2 Law of Large Numbers Under Alpha-Mixing Inputs (p. 89)
  • 3.4.3 Preservation of PAC Learning Property with Beta-Mixing Inputs (p. 94)
  • 3.4.4 Preservation of PAC Learning Property with Beta-Mixing Inputs: Continued (p. 95)
  • 3.4.5 Replacing {{\cal P}} by its Closure (p. 97)
  • 3.5 Markov Chains and Beta-Mixing (p. 100)
  • 3.5.1 Geometric Ergodicity and Beta-Mixing (p. 100)
  • 3.5.2 Beta-Mixing Properties of Markov Sequences (p. 105)
  • 3.5.3 Mixing Properties of Hidden Markov Models (p. 110)
  • 4 Vapnik-Chervonenkis, Pseudo- and Fat-Shattering Dimen-sions (p. 115)
  • 4.1 Definitions (p. 115)
  • 4.1.1 The Vapnik-Chervonenkis Dimension (p. 115)
  • 4.1.2 The Pseudo-Dimension (p. 120)
  • 4.1.3 The Fat-Shattering Dimension (p. 122)
  • 4.2 Bounds on Growth Functions (p. 123)
  • 4.2.1 Growth Functions of Collections of Sets (p. 123)
  • 4.2.2 Bounds on Covering Numbers Based on the Pseudo-Dimension (p. 128)
  • 4.2.3 Metric Entropy Bounds for Families of Functions (p. 132)
  • 4.2.4 Bounds on Covering Numbers Based on the Fat-Shattering Dimension (p. 139)
  • 4.3 Growth Functions of Iterated Families (p. 141)
  • 5 Uniform Convergence of Empirical Means (p. 149)
  • 5.1 Restatement of the Problems Under Study (p. 149)
  • 5.2 Equivalence of the UCEM and ASCEM Properties (p. 153)
  • 5.3 Main Theorems (p. 155)
  • 5.4 Preliminary Lemmas (p. 161)
  • 5.5 Theorem 5.1: Proof of Necessity (p. 173)
  • 5.6 Theorem 5.1: Proof of Sufficiency (p. 178)
  • 5.7 Proofs of the Remaining Theorems (p. 190)
  • 5.8 Uniform Convergence Properties of Iterated Families (p. 194)
  • 5.8.1 Boolean Operations on Collections of Sets (p. 195)
  • 5.8.2 Uniformly Continuous Mappings on Families of Functions (p. 196)
  • 5.8.3 Families of Loss Functions (p. 200)
  • 6 Learning Under a Fixed Probability Measure (p. 207)
  • 6.1 Introduction (p. 207)
  • 6.2 UCEM Property Implies ASEC Learnability (p. 209)
  • 6.3 Finite Metric Entropy Implies Learnability (p. 216)
  • 6.4 Consistent Learnability (p. 224)
  • 6.4.1 Consistent PAC Learnability (p. 224)
  • 6.4.2 Consistent PUAC Learnability (p. 226)
  • 6.5 Examples (p. 230)
  • 6.6 Learnable Concept Classes Have Finite Metric Entropy (p. 236)
  • 6.7 Model-Free Learning (p. 242)
  • 6.7.1 A Sufficient Condition for Learnability (p. 244)
  • 6.7.2 A Necessary Condition (p. 248)
  • 6.8 Dependent Inputs (p. 250)
  • 6.8.1 Finite Metric Entropy and Alpha-Mixing Input Sequences (p. 250)
  • 6.8.2 Consistent Learnability and Beta-Mixing Input Sequences (p. 251)
  • 7 Distribution-Free Learning (p. 255)
  • 7.1 Uniform Convergence of Empirical Means (p. 255)
  • 7.1.1 Function Classes (p. 256)
  • 7.1.2 Concept Classes (p. 258)
  • 7.1.3 Loss Functions (p. 261)
  • 7.2 Function Learning (p. 263)
  • 7.2.1 Finite P-Dimension Implies PAC and PUAC Learnability (p. 264)
  • 7.2.2 Finite P-Dimension is not Necessary for PAC Learnability (p. 267)
  • 7.3 Concept Learning (p. 269)
  • 7.3.1 Improved Upper Bound for the Sample Complexity (p. 269)
  • 7.3.2 A Universal Lower Bound for the Sample Complexity (p. 273)
  • 7.3.3 Learnability Implies Finite VC-Dimension (p. 278)
  • 7.4 Learnability of Functions with a Finite Range (p. 280)
  • 8 Learning Under an Intermediate Family of Probabilities (p. 285)
  • 8.1 General Families of Probabilities (p. 287)
  • 8.1.1 Uniform Convergence of Empirical Means (p. 287)
  • 8.1.2 Function Learning (p. 288)
  • 8.1.3 Concept Learning (p. 292)
  • 8.2 Totally Bounded Families of Probabilities (p. 297)
  • 8.3 Families of Probabilities with a Nonempty Interior (p. 308)
  • 9 Alternate Models of Learning (p. 311)
  • 9.1 Efficient Learning (p. 312)
  • 9.1.1 Definition of Efficient Learnability (p. 313)
  • 9.1.2 The Complexity of Finding a Consistent Hypothesis (p. 317)
  • 9.2 Active Learning (p. 326)
  • 9.2.1 Fixed-Distribution Learning (p. 329)
  • 9.2.2 Distribution-Free Learning (p. 332)
  • 9.3 Learning with Prior Information: Necessary and Sufficient Conditions (p. 335)
  • 9.3.1 Definition of Learnability with Prior Information (p. 335)
  • 9.3.2 Some Simple Sufficient Conditions (p. 337)
  • 9.3.3 Dispersability of Function Classes (p. 341)
  • 9.3.4 Connections Between Dispersability and Learnability WPI (p. 344)
  • 9.3.5 Distribution-Free Learning with Prior Information (p. 348)
  • 9.4 Learning with Prior Information: Bounds on Learning Rates (p. 352)
  • 10 Applications to Neural Networks (p. 365)
  • 10.1 What is a Neural Network? (p. 366)
  • 10.2 Learning in Neural Networks (p. 369)
  • 10.2.1 Problem Formulation (p. 369)
  • 10.2.2 Reprise of Sample Complexity Estimates (p. 372)
  • 10.2.3 Complexity-Theoretic Limits to Learnability (p. 377)
  • 10.3 Estimates of VC-Dimensions of Families of Networks (p. 381)
  • 10.3.1 Multi-Layer Perceptron Networks (p. 382)
  • 10.3.2 A Network with Infinite VC-Dimension (p. 388)
  • 10.3.3 Neural Networks as Verifiers of Formulas (p. 390)
  • 10.3.4 Neural Networks with Piecewise-Polynomial Activation Functions (p. 396)
  • 10.3.5 A General Approach (p. 402)
  • 10.3.6 An Improved Bound (p. 406)
  • 10.3.7 Networks with Pfaffian Activation Functions (p. 410)
  • 10.3.8 Results Based on Order-Minimality (p. 413)
  • 10.4 Structural Risk Minimization (p. 415)
  • 11 Applications to Control Systems (p. 421)
  • 11.1 Randomized Algorithms for Robustness Analysis (p. 421)
  • 11.1.1 Introduction to Robust Control (p. 421)
  • 11.1.2 Some NP-Hard Problems in Robust Control (p. 424)
  • 11.1.3 Randomized Algorithms for Robustness Analysis (p. 426)
  • 11.2 Randomized Algorithms for Robust Controller Synthesis: Gen-eral Approach (p. 429)
  • 11.2.1 Paradigm of Robust Controller Synthesis Problem (p. 429)
  • 11.2.2 Various Types of "Near" Minima (p. 432)
  • 11.2.3 A General Approach to Randomized Algorithms (p. 435)
  • 11.2.4 Two Algorithms for Finding Probably Approximate Near Minima (p. 436)
  • 11.3 VC-Dimension Estimates for Problems in Robust Controller Synthesis (p. 438)
  • 11.3.1 A General Result (p. 438)
  • 11.3.2 Robust Stabilization (p. 438)
  • 11.3.3 Weighted H ∞ -Norm Minimization (p. 441)
  • 11.3.4 Weighted H 2 -Norm Minimization (p. 444)
  • 11.3.5 Sample Complexity Considerations (p. 445)
  • 11.3.6 Robust Controller Design Using Randomized Algorithms: An Example (p. 449)
  • 11.4 A Learning Theory Approach to System Identification (p. 453)
  • 11.4.1 Problem Formulation (p. 453)
  • 11.4.2 A General Result (p. 455)
  • 11.4.3 Sufficient Conditions for the UCEM Property (p. 458)
  • 11.4.4 Bounds on the P-Dimension (p. 461)
  • 12 Some Open Problems (p. 465)

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