LNAI 2842 On Ordinal VC Dimension and Some Notions of Complexity 1st Edition by Eric Martin, Arun Sharma, Frank Stephan ISBN 9783540200574 354020057X

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Product details:
ISBN 10: 354020057X
ISBN 13: 9783540200574
Author: Eric Martin, Arun Sharma, Frank Stephan

We generalize the classical notion of VC-dimension to ordinal VC-dimension, in the context of logical learning paradigms. Logical learning paradigms encompass the numerical learning paradigms commonly studied in Inductive inference. A logical learning paradigm is defined as a set mathcal{W} of structures over some vocabulary, and a set mathcal{D} of first-order formulas that represent data. The sets of models of ϕ in mathcal{W}, where ϕ varies over mathcal{D}, generate a natural topology mathbb{W} over mathcal{W}.

LNAI 2842 On Ordinal VC Dimension and Some Notions of Complexity 1st Edition Table of contents:

• Introduction

  • Overview of the Book
  • Motivation and Objectives
  • Key Concepts in Complexity Theory
  • Ordinal VC Dimension: A New Approach

• Preliminaries

  • Mathematical Background
  • Set Theory and Logic
  • Basic Notions in Complexity Theory
  • Classical VC Dimension

• Ordinal VC Dimension

  • Definition and Motivation
  • Relationship to Classical VC Dimension
  • Key Theorems and Results
  • Applications to Machine Learning and Data Analysis

• Complexity Measures in Learning Theory

  • Computational Complexity and Its Implications
  • Complexity of Hypothesis Classes
  • Ordinal Complexity of Learning Algorithms
  • Ordinal VC Dimension in Computational Learning Theory

• Properties of Ordinal VC Dimension

  • Generalization and Bounds
  • Trade-offs Between VC Dimension and Sample Complexity
  • Comparative Analysis with Other Complexity Measures

• Applications and Examples

  • Ordinal VC Dimension in Supervised Learning
  • The Role in Neural Networks and Deep Learning
  • Case Studies and Examples in Practical Machine Learning

• Advanced Topics

  • Ordinal VC Dimension in Non-Standard Settings
  • Extensions to Infinite and Continuous Spaces
  • Connections with Other Theories in Computational Complexity

• Conclusion

  • Summary of Key Findings
  • Open Problems and Future Research Directions
  • The Impact of Ordinal VC Dimension on Learning Theory and Beyond

• Appendices

  • Mathematical Proofs and Derivations
  • Additional Resources
  • Glossary of Terms

• References

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