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    Area of Science:

    • Optimization
    • Machine Learning
    • Data Science

    Background:

    • Stochastic algorithms are vital for big data processing.
    • Nonsmooth stochastic Difference-of-Convex (DC) programs are a key class of nonconvex optimization problems with broad applications, especially in machine learning.
    • Existing methods may not be optimal for continuously generated, streaming data.

    Purpose of the Study:

    • To develop novel online stochastic algorithms for nonsmooth DC programs.
    • To adapt the powerful DC Algorithm (DCA) for the online, streaming data context.
    • To provide rigorous convergence analysis for the proposed algorithms.

    Main Methods:

    • The study proposes new online stochastic algorithms.
    • These algorithms are based on the Difference-of-Convex (DC) Algorithm (DCA) framework.
    • Stochastic approximations (SAs) are employed, replacing deterministic DCA quantities with noisy estimators from new data samples.

    Main Results:

    • New online stochastic algorithms for nonsmooth DC programs are presented.
    • Convergence analysis is rigorously established using convex analysis and martingale theory.
    • The algorithms' efficacy is demonstrated on the expected principal component analysis (PCA) problem in machine learning.

    Conclusions:

    • The developed algorithms offer an effective approach for nonconvex stochastic optimization with streaming data.
    • The theoretical analysis provides a strong foundation for the algorithms' performance.
    • The application to PCA highlights practical utility in machine learning.