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    Local linear regression models improve function learning by minimizing empirical risk. Smaller data point discrepancy enhances estimation accuracy, suggesting low-discrepancy sequences are beneficial for nonparametric regression.

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    Area of Science:

    • Machine Learning
    • Nonparametric Statistics
    • Data Analysis

    Background:

    • Local linear regression models offer nonparametric approaches for estimating target functions.
    • Empirical Risk Minimization (ERM) is a key framework for function learning.
    • Geometric properties of data significantly influence regression model performance.

    Purpose of the Study:

    • To analyze local linear regression within the ERM framework for function learning.
    • To investigate the impact of data point discrepancy on estimation consistency and approximation capabilities.
    • To explore the utility of low-discrepancy sequences for improving nonparametric regression.

    Main Methods:

    • Nonparametric function estimation using local linear regression.
    • Empirical Risk Minimization (ERM) principles applied to function learning.
    • Analysis of geometric data properties and sample discrepancy.
    • Theoretical convergence analysis for ERM procedures.
    • Evaluation of low-discrepancy sequences for sampling.

    Main Results:

    • Established conditions for the consistency and convergence of the ERM procedure in local linear regression.
    • Demonstrated that reduced data point discrepancy enhances estimation accuracy.
    • Identified low-discrepancy sequences as a promising method for improving nonparametric regression performance.
    • Simulation results validated the theoretical findings in practical function learning scenarios.

    Conclusions:

    • The study provides theoretical guarantees for the convergence of ERM in local linear regression.
    • Data point discrepancy is a critical factor influencing the effectiveness of nonparametric regression.
    • Low-discrepancy sequences offer a practical approach to enhance function learning accuracy.