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Iteratively reweighted $$\ell _1$$ ℓ 1 algorithms with extrapolation

Author

Listed:
  • Peiran Yu

    (The Hong Kong Polytechnic University)

  • Ting Kei Pong

    (The Hong Kong Polytechnic University)

Abstract

Iteratively reweighted $$\ell _1$$ ℓ 1 algorithm is a popular algorithm for solving a large class of optimization problems whose objective is the sum of a Lipschitz differentiable loss function and a possibly nonconvex sparsity inducing regularizer. In this paper, motivated by the success of extrapolation techniques in accelerating first-order methods, we study how widely used extrapolation techniques such as those in Auslender and Teboulle (SIAM J Optim 16:697–725, 2006), Beck and Teboulle (SIAM J Imaging Sci 2:183–202, 2009), Lan et al. (Math Program 126:1–29, 2011) and Nesterov (Math Program 140:125–161, 2013) can be incorporated to possibly accelerate the iteratively reweighted $$\ell _1$$ ℓ 1 algorithm. We consider three versions of such algorithms. For each version, we exhibit an explicitly checkable condition on the extrapolation parameters so that the sequence generated provably clusters at a stationary point of the optimization problem. We also investigate global convergence under additional Kurdyka–Łojasiewicz assumptions on certain potential functions. Our numerical experiments show that our algorithms usually outperform the general iterative shrinkage and thresholding algorithm in Gong et al. (Proc Int Conf Mach Learn 28:37–45, 2013) and an adaptation of the iteratively reweighted $$\ell _1$$ ℓ 1 algorithm in Lu (Math Program 147:277–307, 2014, Algorithm 7) with nonmonotone line-search for solving random instances of log penalty regularized least squares problems in terms of both CPU time and solution quality.

Suggested Citation

  • Peiran Yu & Ting Kei Pong, 2019. "Iteratively reweighted $$\ell _1$$ ℓ 1 algorithms with extrapolation," Computational Optimization and Applications, Springer, vol. 73(2), pages 353-386, June.
    • Handle: RePEc:spr:coopap:v:73:y:2019:i:2:d:10.1007_s10589-019-00081-1
    DOI: 10.1007/s10589-019-00081-1
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. Xiaojun Chen & Weijun Zhou, 2014. "Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2 –ℓ p minimization," Computational Optimization and Applications, Springer, vol. 59(1), pages 47-61, October.
    3. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
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    6. Bo Wen & Xiaojun Chen & Ting Kei Pong, 2018. "A proximal difference-of-convex algorithm with extrapolation," Computational Optimization and Applications, Springer, vol. 69(2), pages 297-324, March.
    7. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
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    Cited by:

    1. Shuqin Sun & Ting Kei Pong, 2023. "Doubly iteratively reweighted algorithm for constrained compressed sensing models," Computational Optimization and Applications, Springer, vol. 85(2), pages 583-619, June.

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