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On the Efficiency of Random Permutation for ADMM and Coordinate Descent

Author

Listed:
  • Ruoyu Sun

    (Department of Industrial and Enterprise Systems Engineering and Coordinated Science Laboratory, University of Illinois Urbana–Champaign, Champaign, Illinois 61801;)

  • Zhi-Quan Luo

    (Chinese University of Hong Kong, 518172 Shenzhen, China; Shenzhen Research Institute of Big Data, 518172 Shenzhen, China;)

  • Yinyu Ye

    (Department of Management Science and Engineering, Stanford University, Palo Alto, California 94305)

Abstract

Random permutation is observed to be powerful for optimization algorithms: for multiblock ADMM (alternating direction method of multipliers), whereas the classical cyclic version diverges, the randomly permuted version converges in practice; for BCD (block coordinate descent), the randomly permuted version is typically faster than other versions. In this paper we provide strong theoretical evidence that random permutation has positive effects on ADMM and BCD, by analyzing randomly permuted ADMM (RP-ADMM) for solving linear systems of equations, and randomly permuted BCD (RP-BCD) for solving unconstrained quadratic problems. First, we prove that RP-ADMM converges in expectation for solving systems of linear equations. The key technical result is that the spectrum of the expected update matrix of RP-BCD lies in (−1/3, 1), instead of the typical range (−1, 1). Second, we establish expected convergence rates of RP-ADMM for solving linear systems and RP-BCD for solving unconstrained quadratic problems. This expected rate of RP-BCD is O ( n ) times better than the worst-case rate of cyclic BCD, thus establishing a gap of at least O ( n ) between RP-BCD and cyclic BCD. To analyze RP-BCD, we propose a conjecture of a new matrix algebraic mean-geometric mean inequality and prove a weaker version of it.

Suggested Citation

  • Ruoyu Sun & Zhi-Quan Luo & Yinyu Ye, 2020. "On the Efficiency of Random Permutation for ADMM and Coordinate Descent," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 233-271, February.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:1:p:233-271
    DOI: 10.1287/moor.2019.0990
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    References listed on IDEAS

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    1. Min Li & Defeng Sun & Kim-Chuan Toh, 2015. "A Convergent 3-Block Semi-Proximal ADMM for Convex Minimization Problems with One Strongly Convex Block," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 32(04), pages 1-19.
    2. P. Tseng, 2001. "Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization," Journal of Optimization Theory and Applications, Springer, vol. 109(3), pages 475-494, June.
    3. Bingsheng He & Min Tao & Xiaoming Yuan, 2017. "Convergence Rate Analysis for the Alternating Direction Method of Multipliers with a Substitution Procedure for Separable Convex Programming," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 662-691, August.
    4. D. Leventhal & A. S. Lewis, 2010. "Randomized Methods for Linear Constraints: Convergence Rates and Conditioning," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 641-654, August.
    5. Caihua Chen & Yuan Shen & Yanfei You, 2013. "On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, October.
    6. Deren Han & Xiaoming Yuan, 2012. "A Note on the Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 227-238, October.
    7. 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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    1. Merrick, James H. & Bistline, John E.T. & Blanford, Geoffrey J., 2024. "On representation of energy storage in electricity planning models," Energy Economics, Elsevier, vol. 136(C).

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