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Accelerated gradient sliding for structured convex optimization

Author

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  • Guanghui Lan

    (Georgia Institute of Technology)

  • Yuyuan Ouyang

    (Clemson University)

Abstract

Our main goal in this paper is to show that one can skip gradient computations for gradient descent type methods applied to certain structured convex programming (CP) problems. To this end, we first present an accelerated gradient sliding (AGS) method for minimizing the summation of two smooth convex functions with different Lipschitz constants. We show that the AGS method can skip the gradient computation for one of these smooth components without slowing down the overall optimal rate of convergence. This result is much sharper than the classic black-box CP complexity results especially when the difference between the two Lipschitz constants associated with these components is large. We then consider an important class of bilinear saddle point problem whose objective function is given by the summation of a smooth component and a nonsmooth one with a bilinear saddle point structure. Using the aforementioned AGS method for smooth composite optimization and Nesterov’s smoothing technique, we show that one only needs $${{\mathcal{O}}}(1/\sqrt{\varepsilon })$$ O ( 1 / ε ) gradient computations for the smooth component while still preserving the optimal $${{\mathcal{O}}}(1/\varepsilon )$$ O ( 1 / ε ) overall iteration complexity for solving these saddle point problems. We demonstrate that even more significant savings on gradient computations can be obtained for strongly convex smooth and bilinear saddle point problems.

Suggested Citation

  • Guanghui Lan & Yuyuan Ouyang, 2022. "Accelerated gradient sliding for structured convex optimization," Computational Optimization and Applications, Springer, vol. 82(2), pages 361-394, June.
    • Handle: RePEc:spr:coopap:v:82:y:2022:i:2:d:10.1007_s10589-022-00365-z
    DOI: 10.1007/s10589-022-00365-z
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2005. "Excessive gap technique in nonsmooth convex minimization," LIDAM Reprints CORE 1818, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Samid Hoda & Andrew Gilpin & Javier Peña & Tuomas Sandholm, 2010. "Smoothing Techniques for Computing Nash Equilibria of Sequential Games," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 494-512, May.
    3. 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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