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Accelerated first-order methods for large-scale convex optimization: nearly optimal complexity under strong convexity

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Abstract

We introduce four accelerated (sub)gradient algorithms (ASGA) for solving several classes of convex optimization problems. More specifically, we propose two estimation sequences majorizing the objective function and develop two iterative schemes for each of them. In both cases, the first scheme requires the smoothness parameter and a Hölder constant, while the second scheme is parameter-free (except for the strong convexity parameter which we set zero if it is not available) at the price of applying a finitely terminated backtracking line search. The proposed algorithms attain the optimal complexity for smooth problems with Lipschitz continuous gradients, nonsmooth problems with bounded variation of subgradients, and weakly smooth problems with Hölder continuous gradients. Further, for strongly convex problems, they are optimal for smooth problems while nearly optimal for nonsmooth and weakly smooth problems. Finally, numerical results for some applications in sparse optimization and machine learning are reported, which confirm the theoretical foundations.

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Acknowledgements

I would like to thank Arnold Neumaier for his useful comments on this paper. I am really grateful of anonymous referees and the associated editor for their constructive comments and suggestions that improved the quality of this paper.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Masoud Ahookhosh

  2. Department of Electrical Engineering (ESAT-STADIUS) – KU Leuven, Kasteelpark Arenberg 10, 3001, Leuven, Belgium

    Masoud Ahookhosh

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  1. Masoud Ahookhosh
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Correspondence to Masoud Ahookhosh.

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Appendix

Appendix

The numerical results for the elastic net minimization and the support vector machine are given in Tables 3 and 4, respectively.

Table 3 Numerical results of NSDSG, NESCO, NESUN, ASGA-1, ASGA-2, ASGA-3, and ASGA-4 for the elastic net minimization problems (54) and (55)
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Table 4 Numerical results of NSDSG, NESUN, ASGA-1, ASGA-2, ASGA-3, and ASGA-4 for the binary classification with linear support vector machines (57)
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Ahookhosh, M. Accelerated first-order methods for large-scale convex optimization: nearly optimal complexity under strong convexity. Math Meth Oper Res 89, 319–353 (2019). https://doi.org/10.1007/s00186-019-00674-w

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  • DOI: https://doi.org/10.1007/s00186-019-00674-w

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