Erdos Goes Neural: an Unsupervised Learning Framework for Combinatorial Optimization on Graphs (2006.10643v4)

Abstract: Combinatorial optimization problems are notoriously challenging for neural networks, especially in the absence of labeled instances. This work proposes an unsupervised learning framework for CO problems on graphs that can provide integral solutions of certified quality. Inspired by Erdos' probabilistic method, we use a neural network to parametrize a probability distribution over sets. Crucially, we show that when the network is optimized w.r.t. a suitably chosen loss, the learned distribution contains, with controlled probability, a low-cost integral solution that obeys the constraints of the combinatorial problem. The probabilistic proof of existence is then derandomized to decode the desired solutions. We demonstrate the efficacy of this approach to obtain valid solutions to the maximum clique problem and to perform local graph clustering. Our method achieves competitive results on both real datasets and synthetic hard instances.

• The paper introduces an unsupervised neural framework that leverages probabilistic methods to model feasible solution distributions for combinatorial graph problems.
• The methodology employs a graph neural network with a custom probabilistic loss function and conditional expectation for derandomizing solutions.
• The paper demonstrates competitive performance on NP-hard tasks like maximum clique and local graph clustering, outperforming traditional solvers in efficiency and scalability.

Unsupervised Learning for Combinatorial Optimization on Graphs

The paper "Erdős Goes Neural: an Unsupervised Learning Framework for Combinatorial Optimization on Graphs" embarks on an exploration of solving combinatorial optimization (CO) problems using a novel unsupervised learning methodology. Combinatorial optimization, particularly on graphs, introduces challenges due to the absence of explicitly labeled data. This paper's proposed approach innovatively leverages neural networks to discover optimized solutions by interpreting the probabilistic method originally advocated by Paul Erdős.

Summary of the Methodology

This research introduces a framework that integrates the probabilistic inspirations of Erdős into a neural network architecture aiming to solve CO problems without supervised data. The framework consists of three primary stages:

1. Graph Neural Networks-Based Distribution Modeling: At the heart of the method is a Graph Neural Network (GNN) that models a probability distribution over potential solution sets for a given CO problem. The network is designed to parameterize a distribution, which acts as the medium through which probable low-cost and valid solutions are discovered.
2. Probabilistic Loss Function: The neural network is trained by minimizing a purposefully constructed loss function that embodies a probabilistic penalty. This function quantifies the likelihood that a sample within the learned distribution adheres to the problem constraints while minimizing cost. The probabilistic foundation of this loss guarantees the existence of feasible solutions within the distribution, allowing for an innovative training regime that does not rely on conventional labels.
3. Derandomization through Conditional Expectation: Once the network acquires a suitable distribution, a deterministic solution is retrieved using the method of conditional expectation. This step involves systematically decoding a solution that satisfies the probabilistically defined criteria set forth during training.

Key Contributions

This paper provides two exemplary applications to demonstrate the efficacy of the proposed framework: the maximum clique problem and constrained minimum cut problem on graphs.

• Maximum Clique: By harnessing the probabilistic penalty method, the authors effectively model the existence of cliques, providing a direct route to bypass the challenges associated with combinatorial explosions in graph search spaces. The maximum clique problem, a prototypical NP-hard problem, serves as a fitting demonstration of the framework's potential to integrate constraints within graph-based solutions effectively.
• Local Graph Clustering: The paper further explores local graph partitioning, emphasizing graph conductance minimization as a balanced graph partitioning problem. Through rigorous comparisons, the conjectured approach demonstrated the capability to achieve lower conductance than traditional local-label-free strategies.

Comparative Analysis and Results

The paper posits that Erdős' GNN shows competitive performance against existing solvers and comparable methods like RUN-CSP. The commercial solver Gurobi is able to consistently deliver optimal solutions but at the expense of computational inefficiency at scale, reinforcing the scalability and practicality of the proposed unsupervised neural approach for larger or more complex graph structures. Both qualitative and quantitative evaluations display that the proposed GNN framework outperforms other neural networks trained with classical relaxation methods, affording it marked improvements in both solution feasibility and efficiency across the datasets considered.

Implications and Future Perspectives

The implications of this work extend to several domains requiring rapid solution evaluations of recursive combinatorial problems, such as logistics, telecommunications, and bioinformatics. The theoretical underpinning founded in Erdős' probabilistic methods envisions a significant shift in how unsupervised learning can be leveraged in graph-structured data optimization, particularly when traditional supervised learning is infeasible.

Future explorations could refine the architecture's efficiency in decoding additional classes of constraints, potentially broadening the applicability to other combinatorial paradigms like paths or trees within graph data. Enhancing the scalability of the approach to multi-core systems and distributed architectures also stands as a vital direction, affording greater efficiency in real-world applications.

In conclusion, "Erdős Goes Neural" reshapes how neural networks can learn to solve without leveraging supervised signals, pointing towards a promising future for unsupervised deep learning methodologies in operational research and beyond.