Unsupervised Learning for Combinatorial Optimization Needs Meta-Learning

A general framework of unsupervised learning for combinatorial optimization (CO) is to train a neural network (NN) whose output gives a problem solution by directly optimizing the CO objective. Albeit with some advantages over traditional solvers, the current framework optimizes an averaged performance over the distribution of historical problem instances, which misaligns with the actual goal of CO that looks for a good solution to every future encountered instance. With this observation, we propose a new objective of unsupervised learning for CO where the goal of learning is to search for good initialization for future problem instances rather than give direct solutions. We propose a meta-learning-based training pipeline for this new objective. Our method achieves good empirical performance. We observe that even the initial solution given by our model before fine-tuning can significantly outperform the baselines under various evaluation settings including evaluation across multiple datasets, and the case with big shifts in the problem scale. The reason we conjecture is that meta-learning-based training lets the model be loosely tied to each local optimum for a training instance while being more adaptive to the changes of optimization landscapes across instances. Combinatorial optimization (CO), aiming to find out the optimal solution from discrete search space, has a pivotal position in scientific and engineering fields (Papadimitriou & Steiglitz, 1998; Crama, 1997). Most CO problems are NP-complete or NP-hard. Conventional heuristics or approximation requires insightful comprehension of the particular problem. Starting from the seminal work from Hopfield & Tank (1985), researchers apply neural networks (NNs) (Smith, 1999; Vinyals et al., 2015) to solve CO problems. The motivation is that NNs may learn heuristics through solving historical problems, which could be useful to solve similar problems in the future. Many NN-based methods (Selsam et al., 2018; Joshi et al., 2019; Hudson et al., 2021; Gasse et al., 2019; Khalil et al., 2016) require optimal solutions to the CO problem as supervision in training.