Machine learning pipelines that include a combinatorial optimization layer can give surprisingly efficient heuristics for difficult combinatorial optimization problems. Three questions remain open: which architecture should be used, how should the parameters of the machine learning model be learned, and what performance guarantees can we expect from the resulting algorithms? Following the intuitions of geometric deep learning, we explain why equivariant layers should be used when designing such pipelines, and illustrate how to build such layers on routing, scheduling, and network design applications. We introduce a learning approach that enables to learn such pipelines when the training set contains only instances of the difficult optimization problem and not their optimal solutions, and show its numerical performance on our three applications. Finally, using tools from statistical learning theory, we prove a theorem showing the convergence speed of the estimator. As a corollary, we obtain that, if an approximation algorithm can be encoded by the pipeline for some parametrization, then the learned pipeline will retain the approximation ratio guarantee. On our network design problem, our machine learning pipeline has the approximation ratio guarantee of the best approximation algorithm known and the numerical efficiency of the best heuristic.

翻译：包含组合优化层的机器学习流水线能够为困难的组合优化问题提供出人意料高效的启发式算法。三个问题仍然悬而未决：应使用何种架构，机器学习模型的参数应如何学习，以及由此产生的算法能提供怎样的性能保证？遵循几何深度学习的直觉，我们解释了在设计此类流水线时为何应使用等变层，并阐述了如何在路由、调度和网络设计应用中构建此类层。我们引入了一种学习方法，使得当训练集仅包含困难优化问题的实例（而非其最优解）时能学习此类流水线，并在三个应用上展示了其数值性能。最后，利用统计学习理论中的工具，我们证明了一个定理，展示了估计量的收敛速度。作为推论，我们得到：若存在某种参数化方式使得近似算法可由该流水线编码，则学习得到的流水线将保留近似比保证。在我们的网络设计问题上，机器学习流水线既具有已知最佳近似算法的近似比保证，又具有最佳启发式算法的数值效率。