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    Area of Science:

    • Machine Learning
    • Optimization Theory
    • Deep Neural Networks

    Background:

    • Deep neural networks (DNNs) are widely used in various applications, requiring efficient optimization algorithms.
    • Adaptive-learning-rate optimization algorithms (ALROAs), including Adam and AMSGrad, are popular choices for DNN training due to their practical effectiveness.
    • However, theoretical guarantees for ALROAs in complex optimization settings like constrained nonconvex finite-sum and online problems are crucial.

    Purpose of the Study:

    • To analyze the theoretical performance of ALROAs in constrained nonconvex stochastic finite-sum and online optimization settings relevant to DNNs.
    • To establish that ALROAs function as ϵ-approximations for these challenging optimization problems.
    • To provide concrete guidance on the parameters required to achieve these ϵ-approximations.

    Main Methods:

    • Theoretical analysis of ALROAs applied to constrained nonconvex stochastic finite-sum and online optimization.
    • Derivation of convergence bounds and approximation guarantees.
    • Identification of the relationship between learning rates, mini-batch sizes, iteration counts, and stochastic gradient complexity for achieving ϵ-accuracy.

    Main Results:

    • Demonstration that ALROAs provide ϵ-approximations for the considered optimization problems.
    • Quantification of the learning rates, mini-batch sizes, and number of iterations necessary to achieve a specified ϵ-approximation.
    • Characterization of the stochastic gradient complexity required for convergence.

    Conclusions:

    • ALROAs offer a theoretically sound approach to optimizing deep neural networks under complex constraints.
    • The findings provide practical insights for tuning ALROAs to achieve desired levels of approximation accuracy.
    • This work bridges the gap between the practical success and theoretical understanding of ALROAs in deep learning.