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Stochastic Block-Coordinate Gradient Projection Algorithms for Submodular Maximization

Author

Listed:
  • Zhigang Li
  • Mingchuan Zhang
  • Junlong Zhu
  • Ruijuan Zheng
  • Qikun Zhang
  • Qingtao Wu

Abstract

We consider a stochastic continuous submodular huge-scale optimization problem, which arises naturally in many applications such as machine learning. Due to high-dimensional data, the computation of the whole gradient vector can become prohibitively expensive. To reduce the complexity and memory requirements, we propose a stochastic block-coordinate gradient projection algorithm for maximizing continuous submodular functions, which chooses a random subset of gradient vector and updates the estimates along the positive gradient direction. We prove that the estimates of all nodes generated by the algorithm converge to some stationary points with probability 1. Moreover, we show that the proposed algorithm achieves the tight approximation guarantee after iterations for DR-submodular functions by choosing appropriate step sizes. Furthermore, we also show that the algorithm achieves the tight approximation guarantee after iterations for weakly DR-submodular functions with parameter by choosing diminishing step sizes.

Suggested Citation

  • Zhigang Li & Mingchuan Zhang & Junlong Zhu & Ruijuan Zheng & Qikun Zhang & Qingtao Wu, 2018. "Stochastic Block-Coordinate Gradient Projection Algorithms for Submodular Maximization," Complexity, Hindawi, vol. 2018, pages 1-11, December.
    • Handle: RePEc:hin:complx:2609471
    DOI: 10.1155/2018/2609471
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    References listed on IDEAS

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    3. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
    4. Fisher, M.L. & Nemhauser, G.L. & Wolsey, L.A., 1978. "An analysis of approximations for maximizing submodular set functions," LIDAM Reprints CORE 341, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Fisher, M.L. & Nemhauser, G.L. & Wolsey, L.A., 1978. "An analysis of approximations for maximizing submodular set functions - 1," LIDAM Reprints CORE 334, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. NESTEROV, Yurii, 2012. "Efficiency of coordinate descent methods on huge-scale optimization problems," LIDAM Reprints CORE 2511, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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