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Optimizing the Efficiency of First-Order Methods for Decreasing the Gradient of Smooth Convex Functions

Author

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  • Donghwan Kim

    (Korea Advanced Institute of Science and Technology (KAIST))

  • Jeffrey A. Fessler

    (University of Michigan)

Abstract

This paper optimizes the step coefficients of first-order methods for smooth convex minimization in terms of the worst-case convergence bound (i.e., efficiency) of the decrease in the gradient norm. This work is based on the performance estimation problem approach. The worst-case gradient bound of the resulting method is optimal up to a constant for large-dimensional smooth convex minimization problems, under the initial bounded condition on the cost function value. This paper then illustrates that the proposed method has a computationally efficient form that is similar to the optimized gradient method.

Suggested Citation

  • Donghwan Kim & Jeffrey A. Fessler, 2021. "Optimizing the Efficiency of First-Order Methods for Decreasing the Gradient of Smooth Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 192-219, January.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:1:d:10.1007_s10957-020-01770-2
    DOI: 10.1007/s10957-020-01770-2
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    References listed on IDEAS

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    1. Taylor, A. & Hendrickx, J. & Glineur, F., 2015. "Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methods," LIDAM Discussion Papers CORE 2015013, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Adrien B. Taylor & Julien M. Hendrickx & François Glineur, 2018. "Exact Worst-Case Convergence Rates of the Proximal Gradient Method for Composite Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 455-476, August.
    3. Donghwan Kim & Jeffrey A. Fessler, 2017. "On the Convergence Analysis of the Optimized Gradient Method," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 187-205, January.
    4. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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