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A Two-Step Fixed-Point Proximity Algorithm for a Class of Non-differentiable Optimization Models in Machine Learning

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Abstract

Sparse learning models are popular in many application areas. Objective functions in sparse learning models are usually non-smooth, which makes it difficult to solve them numerically. We develop a fast and convergent two-step iteration scheme for solving a class of non-differentiable optimization models motivated from sparse learning. To overcome the difficulty of the non-differentiability of the models, we first present characterizations of their solutions as fixed-points of mappings involving the proximity operators of the functions appearing in the objective functions. We then introduce a two-step fixed-point algorithm to compute the solutions. We establish convergence results of the proposed two-step iteration scheme and compare it with the alternating direction method of multipliers (ADMM). In particular, we derive specific two-step iteration algorithms for three models in machine learning: \(\ell ^1\)-SVM classification, \(\ell ^1\)-SVM regression, and the SVM classification with the group LASSO regularizer. Numerical experiments with some synthetic datasets and some benchmark datasets show that the proposed algorithm outperforms ADMM and the linear programming method in computational time and memory storage costs.

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Authors and Affiliations

  1. School of Mathematics, and Guangdong Province Key Lab of Computational Science, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Zheng Li

  2. Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA, 23529, USA

    Guohui Song & Yuesheng Xu

Authors
  1. Zheng Li
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  2. Guohui Song
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  3. Yuesheng Xu
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Correspondence to Yuesheng Xu.

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This research was supported in part by the Ministry of Science and Technology of China under Grant 2016YFB0200602, by the Natural Science Foundation of China under Grants 11471013, 11771464, and by the US National Science Foundation under Grants DMS-1521661, DMS-1522332, DMS-1912958 and DMS-1939203.

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Li, Z., Song, G. & Xu, Y. A Two-Step Fixed-Point Proximity Algorithm for a Class of Non-differentiable Optimization Models in Machine Learning. J Sci Comput 81, 923–940 (2019). https://doi.org/10.1007/s10915-019-01045-7

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  • DOI: https://doi.org/10.1007/s10915-019-01045-7

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