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Smooth strongly convex interpolation and exact worst-case performance of first-order methods

Author

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  • Adrien B. TAYLOR
  • Julien M. HENDRICKX
  • François GLINEUR

Abstract

We show that the exact worst-case performance of fixed-step first-order methods for smooth (possibly strongly) convex functions can be obtained by solving convex programs. Finding the worst-case performance of a black-box first-order method is formulated as an optimization problem over a set of smooth (strongly) convex functions and initial conditions. We develop closed-form necessary and sufficient conditions for smooth (strongly) convex interpolation, which provide a finite representation for those functions. This allows us to reformulate the worst-case performance estimation problem as an equivalent finite dimension-independent semidefinite optimization problem, whose exact solution can be recovered up to numerical precision. Optimal solutions to this performance estimation problem provide both worst-case performance bounds and explicit functions matching them, as our smooth (strongly) convex interpolation procedure is constructive. Our works build on those of Drori and Teboulle in [8] who introduced and solved relaxations of the performance estimation problem for smooth convex functions. We apply our approach to different fixed-step first-order methods with several performance criteria, including objective function accuracy and residual gradient norm. We conjecture several numerically supported worst-case bounds on the performance of the gradient, fast gradient and optimized fixed-step methods, both in the smooth convex and the smooth strongly convex cases, and deduce tight estimates of the optimal step size for the gradient method.
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Suggested Citation

  • Adrien B. TAYLOR & Julien M. HENDRICKX & François GLINEUR, 2017. "Smooth strongly convex interpolation and exact worst-case performance of first-order methods," LIDAM Reprints CORE 2813, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:2813
    Note: In : Mathematical Programming Series A, 161, 307-345, 2017
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    1. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2012. "Double smoothing technique for large-scale linearly constrained convex optimization," LIDAM Reprints CORE 2423, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. 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.
    2. 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).
    3. 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.
    4. Vrins, F. & Jeanblanc, M., 2015. "The [phi]-Martingale," LIDAM Discussion Papers CORE 2015022, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Ernest K. Ryu & Bằng Công Vũ, 2020. "Finding the Forward-Douglas–Rachford-Forward Method," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 858-876, March.
    6. Rieger, Janosch & Tam, Matthew K., 2020. "Backward-Forward-Reflected-Backward Splitting for Three Operator Monotone Inclusions," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    7. 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.
    8. Abbaszadehpeivasti, Hadi & de Klerk, Etienne & Zamani, Moslem, 2022. "The exact worst-case convergence rate of the gradient method with fixed step lengths for L-smooth functions," Other publications TiSEM 061688c6-f97c-4024-bb5b-1, Tilburg University, School of Economics and Management.

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