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An efficient parallel block coordinate descent algorithm for large-scale precision matrix estimation using graphics processing units

Author

Listed:
  • Young-Geun Choi

    (Sookmyung Women’s University)

  • Seunghwan Lee

    (Inha University)

  • Donghyeon Yu

    (Inha University)

Abstract

Large-scale sparse precision matrix estimation has attracted wide interest from the statistics community. The convex partial correlation selection method (CONCORD) developed by Khare et al. (J R Stat Soc Ser B (Stat Methodol) 77(4):803–825, 2015) has recently been credited with some theoretical properties for estimating sparse precision matrices. The CONCORD obtains its solution by a coordinate descent algorithm (CONCORD-CD) based on the convexity of the objective function. However, since a coordinate-wise update in CONCORD-CD is inherently serial, a scale-up is nontrivial. In this paper, we propose a novel parallelization of CONCORD-CD, namely, CONCORD-PCD. CONCORD-PCD partitions the off-diagonal elements into several groups and updates each group simultaneously without harming the computational convergence of CONCORD-CD. We guarantee this by employing the notion of edge coloring in graph theory. Specifically, we establish a nontrivial correspondence between scheduling the updates of the off-diagonal elements in CONCORD-CD and coloring the edges of a complete graph. It turns out that CONCORD-PCD simultanoeusly updates off-diagonal elements in which the associated edges are colorable with the same color. As a result, the number of steps required for updating off-diagonal elements reduces from $$p(p-1)/2$$ p ( p - 1 ) / 2 to $$p-1$$ p - 1 (for even p) or p (for odd p), where p denotes the number of variables. We prove that the number of such steps is irreducible In addition, CONCORD-PCD is tailored to single-instruction multiple-data (SIMD) parallelism. A numerical study shows that the SIMD-parallelized PCD algorithm implemented in graphics processing units boosts the CONCORD-CD algorithm multiple times. The method is available in the R package pcdconcord.

Suggested Citation

  • Young-Geun Choi & Seunghwan Lee & Donghyeon Yu, 2022. "An efficient parallel block coordinate descent algorithm for large-scale precision matrix estimation using graphics processing units," Computational Statistics, Springer, vol. 37(1), pages 419-443, March.
    • Handle: RePEc:spr:compst:v:37:y:2022:i:1:d:10.1007_s00180-021-01127-x
    DOI: 10.1007/s00180-021-01127-x
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    References listed on IDEAS

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    1. 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.
    2. Kshitij Khare & Sang-Yun Oh & Bala Rajaratnam, 2015. "A convex pseudolikelihood framework for high dimensional partial correlation estimation with convergence guarantees," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(4), pages 803-825, September.
    3. Peng, Jie & Wang, Pei & Zhou, Nengfeng & Zhu, Ji, 2009. "Partial Correlation Estimation by Joint Sparse Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 735-746.
    4. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, vol. 94(1), pages 19-35.
    5. Patrick Danaher & Pei Wang & Daniela M. Witten, 2014. "The joint graphical lasso for inverse covariance estimation across multiple classes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(2), pages 373-397, March.
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