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The Informational Complexity of Learning: Perspectives on Neural Networks and Generative Grammar - Softcover
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Inhaltsangabe
1. Introduction.- 1.1 The Components of a Learning Paradigm.- 1.1.1 Concepts, Hypotheses, and Learners.- 1.1.2 Generalization, Learnability, Successful learning.- 1.1.3 Informational Complexity.- 1.2 Parametric Hypothesis Spaces.- 1.3 Technical Contents and Major Contributions.- 1.3.1 A Final Word.- 2. Generalization Error For Neural Nets.- 2.1 Introduction.- 2.2 Definitions and Statement of the Problem.- 2.2.1 Random Variables and Probability Distributions.- 2.2.2 Learning from Examples and Estimators.- 2.2.3 The Expected Risk and the Regression Function.- 2.2.4 The Empirical Risk.- 2.2.5 The Problem.- 2.2.6 Bounding the Generalization Error.- 2.2.7 A Note on Models and Model Complexity.- 2.3 Stating the Problem for Radial Basis Functions.- 2.4 Main Result.- 2.5 Remarks.- 2.5.1 Observations on the Main Result.- 2.5.2 Extensions.- 2.5.3 Connections with Other Results.- 2.6 Implications of the Theorem in Practice: Putting In the Numbers.- 2.6.1 Rate of Growth of n for Guaranteed Convergence.- 2.6.2 Optimal Choice of n.- 2.6.3 Experiments.- 2.7 Conclusion.- 2-A Notations.- 2-B A Useful Decomposition of the Expected Risk.- 2-C A Useful Inequality.- 2-D Proof of the Main Theorem.- 2-D.1 Bounding the approximation error.- 2-D.2 Bounding the estimation error.- 2-D.3 Bounding the generalization error.- 3. Active Learning.- 3.1 A General Framework For Active Approximation.- 3.1.1 Preliminaries.- 3.1.2 The Problem of Collecting Examples.- 3.1.3 In Context.- 3.2 Example 1: A Class of Monotonically Increasing Bounded Functions.- 3.2.1 Lower Bound for Passive Learning.- 3.2.2 Active Learning Algorithms.- 3.2.2.1 Derivation of an optimal sampling strategy.- 3.2.3 Empirical Simulations, and other Investigations.- 3.2.3.1 Distribution of Points selected.- 3.2.3.2 Classical Optimal Recovery.- 3.2.3.3 Error Rates and Sample Complexities for some Arbitrary Functions: Some Simulations.- 3.3 Example 2: A Class of Functions with Bounded First Derivative.- 3.3.1 Lower Bounds.- 3.3.2 Active Learning Algorithms.- 3.3.2.1 Derivation of an optimal sampling strategy.- 3.3.3 Some Simulations.- 3.3.3.1 Distribution of points selected.- 3.3.3.2 Error Rates:.- 3.4 Conclusions, Extensions, and Open Problems.- 3.5 A Simple Example.- 3.6 Generalizations.- 3.6.1 Localized Function Classes.- 3.6.2 The General ?-focusing strategy;.- 3.6.3 Generalizations and Open Problems.- 4. Language Learning.- 4.1 Language Learning and The Poverty of Stimulus.- 4.2 Constrained Grammars-Principles and Parameters.- 4.2.1 Example: A 3-parameter System from Syntax.- 4.2.2 Example: Parameterized Metrical Stress in Phonology.- 4.3 Learning in the Principles and Parameters Framework.- 4.4 Formal Analysis of the Triggering Learning Algorithm.- 4.4.1 Background.- 4.4.2 The Markov formulation.- 4.4.2.1 Parameterized Grammars and their Corresponding Markov Chains.- 4.4.2.2 Markov Chain Criteria for Learnability.- 4.4.2.3 The Markov chain for the 3-parameter Example.- 4.4.3 Derivation of the transition probabilities for the Markov TLA structure.- 4.4.3.1 Formalization.- 4.4.3.2 Additional Properties of the Learning System.- 4.5 Characterizing Convergence Times for the Markov Chain Model.- 4.5.1 Some Transition Matrices and Their Convergence Curves.- 4.5.2 Absorption Times.- 4.5.3 Eigenvalue Rates of Convergence.- 4.5.3.1 Eigenvalues and Eigenvectors.- 4.5.3.2 Representation of Tk.- 4.5.3.3 Initial Conditions and Limiting Distributions.- 4.5.3.4 Rate of Convergence.- 4.5.3.5 Transition Matrix Recipes:.- 4.6 Exploring Other Points.- 4.6.1 Changing the Algorithm.- 4.6.2 Distributional Assumptions.- 4.6.3 Natural Distributions-CHILDES COORPUS.- 4.7 Batch Learning Upper and Lower Bounds: An Aside.- 4.8 Conclusions, Open Questions, and Future Directions.- 4-A Unembedded Sentences For Parametric Grammars.- 4-B Memoryless Algorithms and Markov Chains.- 4-C Proof of Learnability Theorem.- 4-C.1 Markov state terminology.- 4-C.2 Canonical Decomposition.- 4-D Formal Proof.- 5. Language Change.- 5.1 I
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