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Accelerated Randomized Mirror Descent Algorithms for Composite Non-strongly Convex Optimization

Author

Listed:
  • Le Thi Khanh Hien

    (National University of Singapore)

  • Cuong V. Nguyen

    (University of Cambridge)

  • Huan Xu

    (Georgia Institute of Technology)

  • Canyi Lu

    (Carnegie Mellon University)

  • Jiashi Feng

    (National University of Singapore)

Abstract

We consider the problem of minimizing the sum of an average function of a large number of smooth convex components and a general, possibly non-differentiable, convex function. Although many methods have been proposed to solve this problem with the assumption that the sum is strongly convex, few methods support the non-strongly convex case. Adding a small quadratic regularization is a common devise used to tackle non-strongly convex problems; however, it may cause loss of sparsity of solutions or weaken the performance of the algorithms. Avoiding this devise, we propose an accelerated randomized mirror descent method for solving this problem without the strongly convex assumption. Our method extends the deterministic accelerated proximal gradient methods of Paul Tseng and can be applied, even when proximal points are computed inexactly. We also propose a scheme for solving the problem, when the component functions are non-smooth.

Suggested Citation

  • Le Thi Khanh Hien & Cuong V. Nguyen & Huan Xu & Canyi Lu & Jiashi Feng, 2019. "Accelerated Randomized Mirror Descent Algorithms for Composite Non-strongly Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 541-566, May.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:2:d:10.1007_s10957-018-01469-5
    DOI: 10.1007/s10957-018-01469-5
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    References listed on IDEAS

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    1. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2011. "First-order methods of smooth convex optimization with inexact oracle," LIDAM Discussion Papers CORE 2011002, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    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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    Cited by:

    1. Le Thi Khanh Hien & Renbo Zhao & William B. Haskell, 2024. "An Inexact Primal-Dual Smoothing Framework for Large-Scale Non-Bilinear Saddle Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 34-67, January.

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