Learning the Travelling Salesperson Problem Requires Rethinking Generalization (2006.07054v6)

Abstract: End-to-end training of neural network solvers for graph combinatorial optimization problems such as the Travelling Salesperson Problem (TSP) have seen a surge of interest recently, but remain intractable and inefficient beyond graphs with few hundreds of nodes. While state-of-the-art learning-driven approaches for TSP perform closely to classical solvers when trained on trivially small sizes, they are unable to generalize the learnt policy to larger instances at practical scales. This work presents an end-to-end neural combinatorial optimization pipeline that unifies several papers in order to identify the inductive biases, model architectures and learning algorithms that promote generalization to instances larger than those seen in training. Our controlled experiments provide the first principled investigation into such zero-shot generalization, revealing that extrapolating beyond training data requires rethinking the neural combinatorial optimization pipeline, from network layers and learning paradigms to evaluation protocols. Additionally, we analyze recent advances in deep learning for routing problems through the lens of our pipeline and provide new directions to stimulate future research.

Insights into Learning the Travelling Salesperson Problem: Rethinking Generalization

The paper "Learning the Travelling Salesperson Problem Requires Rethinking Generalization" presents a nuanced examination of neural network strategies applied to the Travelling Salesperson Problem (TSP), an NP-hard combinatorial optimization challenge. Despite the promising potential of neural network-based approaches for TSP, their applicability remains constrained by an inability to generalize effectively from training on small-scale instances to solving larger, more complex graphs—a key limitation addressed by this research.

Key Contributions

The authors undertake a comprehensive analysis of the existing neural combinatorial optimization landscape, developing a unified experimental framework to scrutinize variables influencing zero-shot generalization capability. This framework incorporates diverse architectural elements, data processing techniques, and learning paradigms. Through controlled experiments, they demonstrate that the prevalent methods, which often evaluate models solely on fixed-size training graphs, perform poorly when applied to out-of-distribution larger graph instances.

The paper is meticulous in its investigation of component-specific impacts within the optimization pipeline:

1. Graph Sparsification: Maintaining uniform graph diameter across training datasets, regardless of graph size, facilitates better generalization than traditional fully-connected graph structures.
2. GNN Aggregation Functions and Normalization: The authors identify that while GNNs with Max or Mean aggregation functions offer superior generalization over Sum aggregation, they highlight the necessity for embedding normalization strategies that can adjust to varying graph statistics—a pivotal aspect for preserving node and global graph embedding stability across sizes.
3. Decoding Approaches: Within the context of TSP, autoregressive (AR) decoders exhibit a strong sequential inductive bias that enhances generalization capabilities significantly more than non-autoregressive (NAR) ones, despite the AR models being computationally more intensive during inference.
4. Learning Paradigms: The paper presents a compelling argument for reinforcement learning (RL) with carefully constructed baselines as a more scalable option versus supervised learning (SL). RL models maintain performance improvement as they process additional samples, whereas SL models' reliance on labeled data constrains their scalability.

Numerical Analysis and Implications

The experimental results reveal that training models on trivially small instances, such as TSP20-50, shows potential for zero-shot generalization towards larger instances if the model and data processing are handled astutely. The paper highlights that a direct training methodology on extremely large graphs, such as TSP200, is impractical due to computational and data inefficiency constraints, embodied by their evaluation demonstrating substantive performance gaps when models are scaled in isolation without accounting for foundational architectural limitations.

The implications of this research are profound for both theoretical and practical advancements in neural combinatorial optimization. It challenges the current paradigm, suggesting that a rethink is needed in designing models that must extrapolate learned knowledge beyond seen data distributions effectively. Furthermore, the insights into graph-specific inductive biases offer pathways for refining neural architectures to improve performance predictively on large-scale real-world graphs from limited small-scale training regimes.

Future Research Directions

The authors propose potential areas of exploration that include enhancing GNN architectures for global graph understanding, exploring alternative graph embedding methods, and developing extensive RL systems to address scalability concerns. Techniques from geometric deep learning, which respect symmetries inherent to TSP formulations, are suggested as a promising trajectory for advancing model generalization capabilities.

In conclusion, this paper foregrounds the importance of carefully reconsidering how neural networks are trained and evaluated in the context of graph combinatorial optimization problems. By identifying and systematically testing the key factors impacting generalization, it provides a foundation on which more effective and scalable neural learning systems for TSP and related problems can be built.