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    Random features (RFs) methods offer theoretical guarantees but often assume target functions are within kernel space. This study proves RFs achieve statistical optimality even outside kernel space, especially with data-dependent sampling.

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    Area of Science:

    • Nonparametric statistical learning
    • Machine learning theory

    Background:

    • Random features (RFs) methods are popular for their theoretical guarantees and flexibility.
    • Existing RFs research assumes target functions reside within the associated kernel space, limiting practical applications.

    Purpose of the Study:

    • To investigate the effectiveness of RFs in an agnostic setting where target functions may lie outside the kernel space.
    • To prove that RFs can still achieve capacity-dependent statistical optimality in such agnostic scenarios.

    Main Methods:

    • Developed a finer-grained estimate for the capacity of the hypothesis space.
    • Conducted a refined analysis of error terms using concise error decomposition.

    Main Results:

    • RFs with uniform sampling guarantee optimality in 50% of agnostic situations.
    • RFs with data-dependent sampling achieve optimal rates across all agnostic settings.
    • Data-dependent sampling reduces the number of RFs required and enhances applicability.

    Conclusions:

    • RFs are effective even when target functions are outside the kernel space.
    • Data-dependent sampling significantly improves the performance and applicability of RFs in agnostic settings.
    • Experimental results validate the theoretical findings on real-world datasets.