Learning Fill-in Reduction Ordering via Graph Policy Optimization for Sparse Matrices

TL;DR

This paper introduces a graph policy optimization approach using graph neural networks to improve matrix reordering for sparse solvers, significantly reducing fill-in and memory usage.

Contribution

It presents a novel reinforcement learning method that models global and local fill-in feedback for better matrix reordering in sparse matrices.

Findings

Achieves 29.3% reduction in fill-ins on SuiteSparse collection.

Reduces peak memory usage by 31.3% compared to baselines.

Uses multi-hop graph neural networks for global and local fill-in modeling.

Abstract

Matrix reordering in large sparse solvers seeks a permutation that minimizes factorization fill-in to reduce memory and computation. Because the minimum fill-in ordering problem is NP-complete and fill-in is implicit in the sparsity pattern, graph-theoretic heuristics are used. Existing reinforcement learning methods either ignore sparsity patterns--missing the global fill-in--or lack local exact fill-in feedback. We propose a graph policy optimization method, modeling fill-ins from global and local views: both the policy and value networks use a multi-hop graph neural backbone to embed global fill-in; the policy further interacts with symbolic factorization over graphs to extract local, step-level fill-ins, and the resulting feedback is aligned with the value network via an adaptive saturation function to improve convergence. On the SuiteSparse Matrix Collection, our method achieves mean reductions of 29.3 in fill-ins and 31.3 in peak memory usage over state-of-the-art baselines.