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Register a new userExact Combinatorial Optimization with Graph Convolutional Neural Networks
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Added by
Maxime Gasse
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Combinatorial optimization problems are typically tackled by the branch-and-bound paradigm. We propose a new graph convolutional neural network model for learning branch-and-bound variable selection policies, which leverages the natural variable-constraint bipartite graph representation of mixed-integer linear programs. We train our model via imitation learning from the strong branching expert rule, and demonstrate on a series of hard problems that our approach produces policies that improve upon state-of-the-art machine-learning methods for branching and generalize to instances significantly larger than seen during training. Moreover, we improve for the first time over expert-designed branching rules implemented in a state-of-the-art solver on large problems. Code for reproducing all the experiments can be found at https://github.com/ds4dm/learn2branch.
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We demonstrate how graph neural networks can be used to solve combinatorial optimization problems. Our approach is broadly applicable to canonical NP-hard problems in the form of quadratic unconstrained binary optimization problems, such as maximum cut, minimum vertex cover, maximum independent set, as well as Ising spin glasses and higher-order generalizations thereof in the form of polynomial unconstrained binary optimization problems. We apply a relaxation strategy to the problem Hamiltonian to generate a differentiable loss function with which we train the graph neural network and apply a simple projection to integer variables once the unsupervised training process has completed. We showcase our approach with numerical results for the canonical maximum cut and maximum independent set problems. We find that the graph neural network optimizer performs on par or outperforms existing solvers, with the ability to scale beyond the state of the art to problems with millions of variables.
Inspired by convolutional neural networks on 1D and 2D data, graph convolutional neural networks (GCNNs) have been developed for various learning tasks on graph data, and have shown superior performance on real-world datasets. Despite their success, there is a dearth of theoretical explorations of GCNN models such as their generalization properties. In this paper, we take a first step towards developing a deeper theoretical understanding of GCNN models by analyzing the stability of single-layer GCNN models and deriving their generalization guarantees in a semi-supervised graph learning setting. In particular, we show that the algorithmic stability of a GCNN model depends upon the largest absolute eigenvalue of its graph convolution filter. Moreover, to ensure the uniform stability needed to provide strong generalization guarantees, the largest absolute eigenvalue must be independent of the graph size. Our results shed new insights on the design of new & improved graph convolution filters with guaranteed algorithmic stability. We evaluate the generalization gap and stability on various real-world graph datasets and show that the empirical results indeed support our theoretical findings. To the best of our knowledge, we are the first to study stability bounds on graph learning in a semi-supervised setting and derive generalization bounds for GCNN models.
Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unclear how to transform the features into hyperbolic embeddings with the right amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural Network (HGCN), the first inductive hyperbolic GCN that leverages both the expressiveness of GCNs and hyperbolic geometry to learn inductive node representations for hierarchical and scale-free graphs. We derive GCN operations in the hyperboloid model of hyperbolic space and map Euclidean input features to embeddings in hyperbolic spaces with different trainable curvature at each layer. Experiments demonstrate that HGCN learns embeddings that preserve hierarchical structure, and leads to improved performance when compared to Euclidean analogs, even with very low dimensional embeddings: compared to state-of-the-art GCNs, HGCN achieves an error reduction of up to 63.1% in ROC AUC for link prediction and of up to 47.5% in F1 score for node classification, also improving state-of-the art on the Pubmed dataset.
Graph convolution networks have recently garnered a lot of attention for representation learning on non-Euclidean feature spaces. Recent research has focused on stacking multiple layers like in convolutional neural networks for the increased expressive power of graph convolution networks. However, simply stacking multiple graph convolution layers lead to issues like vanishing gradient, over-fitting and over-smoothing. Such problems are much less when using shallower networks, even though the shallow networks have lower expressive power. In this work, we propose a novel Multipath Graph convolutional neural network that aggregates the output of multiple different shallow networks. We train and test our model on various benchmarks datasets for the task of node property prediction. Results show that the proposed method not only attains increased test accuracy but also requires fewer training epochs to converge. The full implementation is available at https://github.com/rangan2510/MultiPathGCN
Machine Learning (ML) can help solve combinatorial optimization (CO) problems better. A popular approach is to use a neural net to compute on the parameters of a given CO problem and extract useful information that guides the search for good solutions. Many CO problems of practical importance can be specified in a matrix form of parameters quantifying the relationship between two groups of items. There is currently no neural net model, however, that takes in such matrix-style relationship data as an input. Consequently, these types of CO problems have been out of reach for ML engineers. In this paper, we introduce Matrix Encoding Network (MatNet) and show how conveniently it takes in and processes parameters of such complex CO problems. Using an end-to-end model based on MatNet, we solve asymmetric traveling salesman (ATSP) and flexible flow shop (FFSP) problems as the earliest neural approach. In particular, for a class of FFSP we have tested MatNet on, we demonstrate a far superior empirical performance to any methods (neural or not) known to date.
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