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An adaptive accelerated proximal gradient method and its homotopy continuation for sparse optimization

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  • Qihang Lin
  • Lin Xiao

Abstract

We consider optimization problems with an objective function that is the sum of two convex terms: one is smooth and given by a black-box oracle, and the other is general but with a simple, known structure. We first present an accelerated proximal gradient (APG) method for problems where the smooth part of the objective function is also strongly convex. This method incorporates an efficient line-search procedure, and achieves the optimal iteration complexity for such composite optimization problems. In case the strong convexity parameter is unknown, we also develop an adaptive scheme that can automatically estimate it on the fly, at the cost of a slightly worse iteration complexity. Then we focus on the special case of solving the $$\ell _1$$ ℓ 1 -regularized least-squares problem in the high-dimensional setting. In such a context, the smooth part of the objective (least-squares) is not strongly convex over the entire domain. Nevertheless, we can exploit its restricted strong convexity over sparse vectors using the adaptive APG method combined with a homotopy continuation scheme. We show that such a combination leads to a global geometric rate of convergence, and the overall iteration complexity has a weaker dependency on the restricted condition number than previous work. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Qihang Lin & Lin Xiao, 2015. "An adaptive accelerated proximal gradient method and its homotopy continuation for sparse optimization," Computational Optimization and Applications, Springer, vol. 60(3), pages 633-674, April.
    • Handle: RePEc:spr:coopap:v:60:y:2015:i:3:p:633-674
    DOI: 10.1007/s10589-014-9694-4
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2007. "Gradient methods for minimizing composite objective function," LIDAM Discussion Papers CORE 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Olivier Fercoq & Zheng Qu, 2020. "Restarting the accelerated coordinate descent method with a rough strong convexity estimate," Computational Optimization and Applications, Springer, vol. 75(1), pages 63-91, January.
    2. Qihang Lin & Runchao Ma & Yangyang Xu, 2022. "Complexity of an inexact proximal-point penalty method for constrained smooth non-convex optimization," Computational Optimization and Applications, Springer, vol. 82(1), pages 175-224, May.
    3. Masaru Ito & Mituhiro Fukuda, 2021. "Nearly Optimal First-Order Methods for Convex Optimization under Gradient Norm Measure: an Adaptive Regularization Approach," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 770-804, March.

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