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Efficiency of Stochastic Coordinate Proximal Gradient Methods on Nonseparable Composite Optimization

Author

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  • Ion Necoara

    (Automatic Control and Systems Engineering Department, University Politehnica Bucharest, 060042 Bucharest, Romania; and Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania)

  • Flavia Chorobura

    (Automatic Control and Systems Engineering Department, University Politehnica Bucharest, 060042 Bucharest, Romania)

Abstract

This paper deals with composite optimization problems having the objective function formed as the sum of two terms; one has a Lipschitz continuous gradient along random subspaces and may be nonconvex, and the second term is simple and differentiable but possibly nonconvex and nonseparable. Under these settings, we design a stochastic coordinate proximal gradient method that takes into account the nonseparable composite form of the objective function. This algorithm achieves scalability by constructing at each iteration a local approximation model of the whole nonseparable objective function along a random subspace with user-determined dimension. We outline efficient techniques for selecting the random subspace, yielding an implementation that has low cost per iteration, also achieving fast convergence rates. We present a probabilistic worst case complexity analysis for our stochastic coordinate proximal gradient method in convex and nonconvex settings; in particular, we prove high-probability bounds on the number of iterations before a given optimality is achieved. Extensive numerical results also confirm the efficiency of our algorithm.

Suggested Citation

  • Ion Necoara & Flavia Chorobura, 2025. "Efficiency of Stochastic Coordinate Proximal Gradient Methods on Nonseparable Composite Optimization," Mathematics of Operations Research, INFORMS, vol. 50(2), pages 993-1018, May.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:2:p:993-1018
    DOI: 10.1287/moor.2023.0044
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    References listed on IDEAS

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    1. Haihao Lu & Robert M. Freund & Yurii Nesterov, 2018. "Relatively smooth convex optimization by first-order methods, and applications," LIDAM Reprints CORE 2965, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. 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).
    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. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Dmitry Grishchenko & Franck Iutzeler & Jérôme Malick, 2021. "Proximal Gradient Methods with Adaptive Subspace Sampling," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1303-1323, November.
    6. David Kozak & Stephen Becker & Alireza Doostan & Luis Tenorio, 2021. "A stochastic subspace approach to gradient-free optimization in high dimensions," Computational Optimization and Applications, Springer, vol. 79(2), pages 339-368, June.
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