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    Area of Science:

    • Optimization Theory
    • Machine Learning Algorithms
    • Convex Analysis

    Background:

    • Adaptive stochastic optimization methods are crucial for machine learning.
    • Mirror maps capture geometric properties in optimization.
    • Adaptive moment estimation (Adam)-type algorithms are widely used but lack theoretical grounding for hyperparameter choices.

    Purpose of the Study:

    • To present a family of adaptive stochastic optimization methods based on mirror maps.
    • To theoretically analyze the convergence rates of these methods for various function types.
    • To provide explanations for hyperparameter selection in Adam-type algorithms.

    Main Methods:

    • Developing adaptive stochastic optimization algorithms incorporating mirror maps.
    • Analyzing average regret convergence rates for convex and strongly convex objective functions.
    • Investigating convergence for smooth, non-convex functions using properties of strongly convex differentiable mirror maps.

    Main Results:

    • Achieved convergence rates for convex objective functions under standard assumptions.
    • Improved convergence rates for strongly convex objective functions.
    • Demonstrated convergence rates of order up to a logarithmic term for smooth objective functions, aligning with practical Adam-type algorithm usage.

    Conclusions:

    • The proposed family of adaptive stochastic optimization methods offers theoretical guarantees.
    • The study provides insights into the effectiveness and hyperparameter choices of Adam-type algorithms.
    • This work bridges the gap between theoretical analysis and practical application in adaptive optimization.