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

TL;DR

This paper introduces a parallelized algorithm for large-scale precision matrix estimation that leverages GPU computing, significantly accelerating the existing coordinate descent method without compromising convergence.

Contribution

It proposes CONCORD-PCD, a novel parallelization of the CONCORD coordinate descent algorithm using graph edge coloring, enabling efficient GPU implementation for large-scale problems.

Findings

GPU implementation accelerates convergence significantly

Parallel updates maintain theoretical convergence guarantees

Reduces computational steps from p(p-1)/2 to p-1 or p for even/odd p

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. (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 simultaneously 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 to 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 (GPUs) boosts the CONCORD-CD algorithm multiple times.