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    Convergence Analysis of Gradient Descent for Eigenvector Computation

    Convergence analysis of gradient descent for top-k eigenspace computation

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    Type
    Conference Paper
    Authors
    Xu, Zhiqiang
    Cao, Xin
    Gao, Xin cc
    KAUST Department
    Computational Bioscience Research Center (CBRC)
    Computer Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2018-07-05
    Online Publication Date
    2018-07-05
    Print Publication Date
    2018-07
    Permanent link to this record
    http://hdl.handle.net/10754/628358
    
    Metadata
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    Abstract
    We present a novel, simple and systematic convergence analysis of gradient descent for eigenvector computation. As a popular, practical, and provable approach to numerous machine learning problems, gradient descent has found successful applications to eigenvector computation as well. However, surprisingly, it lacks a thorough theoretical analysis for the underlying geodesically non-convex problem. In this work, the convergence of the gradient descent solver for the leading eigenvector computation is shown to be at a global rate O(min{ (lambda_1/Delta_p)^2 log(1/epsilon), 1/epsilon }), where Delta_p=lambda_p-lambda_p+1>0 represents the generalized positive eigengap and always exists without loss of generality with lambda_i being the i-th largest eigenvalue of the given real symmetric matrix and p being the multiplicity of lambda_1. The rate is linear at (lambda_1/Delta_p)^2 log(1/epsilon) if (lambda_1/Delta_p)^2=O(1), otherwise sub-linear at O(1/epsilon). We also show that the convergence only logarithmically instead of quadratically depends on the initial iterate. Particularly, this is the first time the linear convergence for the case that the conventionally considered eigengap Delta_1= lambda_1 - lambda_2=0 but the generalized eigengap Delta_p satisfies (lambda_1/Delta_p)^2=O(1), as well as the logarithmic dependence on the initial iterate are established for the gradient descent solver. We are also the first to leverage for analysis the log principal angle between the iterate and the space of globally optimal solutions. Theoretical properties are verified in experiments.
    Citation
    Xu Z, Cao X, Gao X (2018) Convergence Analysis of Gradient Descent for Eigenvector Computation. Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence. Available: http://dx.doi.org/10.24963/ijcai.2018/407.
    Sponsors
    This research is supported in part by the funding from King Abdullah University of Science and Technology (KAUST).
    Publisher
    International Joint Conferences on Artificial Intelligence
    Journal
    Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence
    Conference/Event name
    The 27th International Joint Conference on Artificial Intelligence and The 23rd European Conference on Artificial Intelligence (IJCAI-ECAI-18)
    DOI
    10.24963/ijcai.2018/407
    Additional Links
    https://www.ijcai.org/proceedings/2018/407
    ae974a485f413a2113503eed53cd6c53
    10.24963/ijcai.2018/407
    Scopus Count
    Collections
    Conference Papers; Computer Science Program; Computational Bioscience Research Center (CBRC); Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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