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

Young-Geun Choi, Seunghwan Lee and Donghyeon Yu ()
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Young-Geun Choi: Sookmyung Women’s University
Seunghwan Lee: Inha University
Donghyeon Yu: Inha University

Computational Statistics, 2022, vol. 37, issue 1, No 18, 419-443

Abstract: 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.

Keywords: CONCORD; Edge coloring; Parallel coordinate descent; Graphical model; GPU-parallel computation (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s00180-021-01127-x

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