Graph neural networks improved by physics methods for optimization problems
Graph neural networks and the energetic cavity method for combinatorial optimization
Machine Learning
Summary
Finding the best solution to certain tricky problems can be like trying to find the lowest energy arrangement in a complex system called an Ising model. The authors studied how special computer programs called graph neural networks (GNNs) can help find these low-energy states, but at first, GNNs didn’t do as well as older methods based on physics ideas. By tweaking the GNNs to include some of these physics-based tricks, the authors made them work much better. However, a traditional method called simulated annealing still works about as well as these newer deep learning approaches.
Graph neural networksIsing modelCombinatorial optimizationMean-field approximationsMin-sum algorithmEnergetic cavity methodSimulated annealingHeuristic methods
Authors
Joe Bacchus George, George T. Cantwell
Abstract
We study the use of graph neural networks (GNNs) for finding approximate ground states of Ising models. Efficiently finding these ground states is of broad significance because many combinatorial optimization problems can be formulated as an Ising model with the appropriate choice of couplings and fields. Exactly solving these problems is hard but there are many good heuristic methods. A lineage of these heuristics build from mean-field approximations: one approach uses the leading eigenvector of an appropriately defined matrix, another is the min-sum algorithm, also known as the energetic cavity method. Without modification, GNNs perform worse than both of these methods. We consider small modifications to the GNN to incorporate these heuristics and find that this considerably improves performance. While the modified approach is competitive against other deep-learning approaches, we still find that simulated annealing is reliably at least as good as deep learning methods for the same computational cost.