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

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) recently been credited with some theoretical properties for estimating matrices. CONCORD obtains its solution a coordinate descent algorithm (CONCORD-CD) based on convexity of objective function. However, since coordinate-wise update in CONCORD-CD is inherently serial, scale-up nontrivial. In this paper, we propose novel parallelization CONCORD-CD, namely, CONCORD-PCD. CONCORD-PCD partitions off-diagonal elements into several groups and updates each group simultaneously without harming computational convergence CONCORD-CD. We guarantee employing notion edge coloring graph theory. Specifically, establish nontrivial correspondence between scheduling edges complete graph. It turns out that simultanoeusly which associated are colorable same color. As result, number steps required updating reduces $$p(p-1)/2$$ to $$p-1$$ (for even p) or p odd p), where denotes variables. prove such irreducible addition, tailored single-instruction multiple-data (SIMD) parallelism. A numerical study shows SIMD-parallelized PCD implemented graphics processing units boosts multiple times. available package pcdconcord.