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A Dynamics Theory of RMSProp-Based Implicit Regularization in Deep Low-Rank Matrix Factorization.

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    This study introduces landscape analysis to explain implicit regularization in deep networks using RMSProp optimization. It shows RMSProp aids saddle point escaping, leading to faster convergence for matrix reconstruction tasks.

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    Area of Science:

    • Machine Learning
    • Deep Learning Theory
    • Optimization Algorithms

    Background:

    • Implicit regularization from gradient optimization aids neural network generalization.
    • Existing theories analyze deep matrix factorization (DMF) and discrete gradient dynamics.
    • Discrete gradient dynamics characterize adaptive gradient methods like RMSProp but are complex for deep networks.

    Purpose of the Study:

    • To theoretically and experimentally explain implicit regularization in RMSProp-based deep networks.
    • To introduce landscape analysis focusing on saddle points and local minima.
    • To investigate the impact of learning rates on saddle point escaping (SPE).

    Main Methods:

    • Developed a discrete gradient dynamics approach using landscape analysis.
    • Analyzed saddle point escaping (SPE) dynamics in deep networks.
    • Proved convergence properties for rank-R matrix reconstruction using DMF and SPE.

    Main Results:

    • Demonstrated that DMF converges to a second-order critical point after R stages of SPE for rank-R matrix reconstruction.
    • Analyzed the time required to escape plateaus during SPE.
    • Experimentally verified findings on low-rank matrix, image, and Hankel matrix reconstruction.

    Conclusions:

    • RMSProp exhibits stronger implicit regularization than Gradient Descent (GD) and AdaGrad, but weaker than Adam.
    • Landscape analysis provides a viable method for understanding implicit regularization in deep networks.
    • The theoretical framework is applicable to GD and Adam, but not AdaGrad.