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Discretized-Vapnik-Chervonenkis dimension for analyzing complexity of real function classes.

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    This study introduces the discretized Vapnik-Chervonenkis (VC) dimension for real function complexity. It shows that empirical risk minimization (ERM) is consistent for certain infinite VC dimension function classes, expanding prior knowledge.

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    Area of Science:

    • Machine Learning
    • Computational Learning Theory

    Background:

    • The Vapnik-Chervonenkis (VC) dimension is a key measure of the complexity of function classes.
    • Its classical definition requires the entire output range for the traversal set, posing challenges for real-valued functions.

    Purpose of the Study:

    • Introduce a new metric, the discretized-VC dimension, for real function classes.
    • Analyze the learnability of real function classes and neural networks using this new dimension.
    • Extend the consistency of empirical risk minimization (ERM) to certain infinite-dimensional function classes.

    Main Methods:

    • Propose the discretized-VC dimension using a countable traversal set of rational numbers.
    • Prove that a countable traversal set suffices for achieving the VC dimension.
    • Categorize infinite VC dimension real function classes into TYPE-A and TYPE-B.
    • Analyze the relationship between indicator-output and real-output neural network VC dimensions.
    • Derive a risk bound for TYPE-A infinite VC dimension function classes.

    Main Results:

    • A countable traversal set is sufficient for the VC dimension of real function classes.
    • Discretized-VC dimension simplifies complexity analysis.
    • Infinite VC dimension function classes can be classified into TYPE-A and TYPE-B.
    • The ERM principle is proven to be consistent with overwhelming probability for TYPE-A infinite VC dimension function classes.

    Conclusions:

    • The discretized-VC dimension offers a more practical approach to analyzing real function complexity.
    • This work extends the theoretical understanding of learning theory, particularly the conditions for consistent ERM.
    • The findings have implications for the design and analysis of neural networks and machine learning models.