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    Area of Science:

    • Optimization algorithms
    • Machine learning theory

    Background:

    • The limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm is widely used in machine learning and optimization.
    • Previous work demonstrated linear convergence for stochastic L-BFGS (sL-BFGS) with variance-reduced stochastic gradients.

    Purpose of the Study:

    • To propose a novel sL-BFGS algorithm incorporating momentum.
    • To analyze the convergence rate of the proposed algorithm.

    Main Methods:

    • Development of a new sL-BFGS algorithm with momentum.
    • Theoretical analysis to prove convergence rates.
    • Empirical validation on diverse datasets.

    Main Results:

    • The proposed sL-BFGS algorithm with momentum achieves an accelerated linear convergence rate.
    • Theoretical conditions for this accelerated convergence are established.
    • Experimental results confirm the performance advantage over existing methods.

    Conclusions:

    • The integration of momentum significantly enhances the convergence speed of sL-BFGS.
    • The new algorithm offers a more efficient optimization approach for machine learning applications.