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Analysis of Optimization Algorithms via Sum-of-Squares

Author

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  • Sandra S. Y. Tan

    (National University of Singapore)

  • Antonios Varvitsiotis

    (National University of Singapore)

  • Vincent Y. F. Tan

    (National University of Singapore)

Abstract

We introduce a new framework for unifying and systematizing the performance analysis of first-order black-box optimization algorithms for unconstrained convex minimization. The low-cost iteration complexity enjoyed by first-order algorithms renders them particularly relevant for applications in machine learning and large-scale data analysis. Relying on sum-of-squares (SOS) optimization, we introduce a hierarchy of semidefinite programs that give increasingly better convergence bounds for higher levels of the hierarchy. Alluding to the power of the SOS hierarchy, we show that the (dual of the) first level corresponds to the performance estimation problem (PEP) introduced by Drori and Teboulle (Math Program 145(1):451–482, 2014), a powerful framework for determining convergence rates of first-order optimization algorithms. Consequently, many results obtained within the PEP framework can be reinterpreted as degree-1 SOS proofs, and thus, the SOS framework provides a promising new approach for certifying improved rates of convergence by means of higher-order SOS certificates. To determine analytical rate bounds, in this work, we use the first level of the SOS hierarchy and derive new results for noisy gradient descent with inexact line search methods (Armijo, Wolfe, and Goldstein).

Suggested Citation

  • Sandra S. Y. Tan & Antonios Varvitsiotis & Vincent Y. F. Tan, 2021. "Analysis of Optimization Algorithms via Sum-of-Squares," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 56-81, July.
    • Handle: RePEc:spr:joptap:v:190:y:2021:i:1:d:10.1007_s10957-021-01869-0
    DOI: 10.1007/s10957-021-01869-0
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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. TAYLOR, Adrien B. & HENDRICKX, Julien M. & François GLINEUR, 2016. "Exact worst-case performance of first-order methods for composite convex optimization," LIDAM Discussion Papers CORE 2016052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. DE KLERK, Etienne & GLINEUR, François & TAYLOR, Adrien B., 2016. "On the Worst-case Complexity of the Gradient Method with Exact Line Search for Smooth Strongly Convex Functions," LIDAM Discussion Papers CORE 2016027, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    6. David G. Luenberger & Yinyu Ye, 2016. "Linear and Nonlinear Programming," International Series in Operations Research and Management Science, Springer, edition 4, number 978-3-319-18842-3, December.
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