基于Frank-Wolfe的带约束自监督组合优化
Self-Supervised Combinatorial Optimization with Constraints via Frank-Wolfe
- NYU（纽约大学）
- MIT（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出一种基于Frank-Wolfe的通用框架，允许神经网络预测可行域外的连续向量，通过几何分解近似为可行解的稀疏凸组合，实现自监督训练和自动舍入，在多个组合优化问题上表现优异。
AI中文摘要：
自监督学习用于组合优化已成为用神经网络解决离散优化问题的一种有前景的范式，但一个核心挑战仍然存在：在连续的基于梯度的训练中处理硬组合约束。将组合目标连续扩展到凸域是一种强大的技术，然而现有方法通常需要投影步骤，将神经网络输出限制在可行多面体内，并依赖于特设且特定于问题的构造。我们提出了一个通用框架，其中允许神经网络预测可能位于可行多面体之外的任意连续向量。然后，这些预测通过基于Frank-Wolfe方法和近似Caratheodory结果的几何分解算法，用可行解的稀疏凸组合来近似。这种分解产生了一个几乎处处可微的自监督损失，定义为离散目标的期望值。同一过程在推理时提供了自动舍入保证。我们在多个组合问题上展示了强大的实证性能，包括二次分配问题、最大覆盖问题和旅行商问题。
英文摘要：
Self-supervised learning for combinatorial optimization has emerged as a promising paradigm for solving discrete optimization problems with neural networks, but a central challenge remains: handling hard combinatorial constraints within continuous, gradient-based training. Continuously extending combinatorial objectives to convex domains is a powerful technique, yet existing approaches often require projection steps that constrain neural network outputs to lie inside the feasible polytope and rely on ad-hoc and problem-specific constructions. We propose a general framework in which the neural network is allowed to predict arbitrary continuous vectors that could potentially lie outside of the feasible polytope. These predictions are then approximated by sparse convex combinations of feasible solutions using a geometric decomposition algorithm based on Frank--Wolfe methods and approximate Caratheodory results. This decomposition induces an a.e.-differentiable, self-supervised loss defined as the expected value of the discrete objective. The same procedure provides an automatic rounding guarantee at inference time. We demonstrate strong empirical performance across multiple combinatorial problems, including the Quadratic Assignment Problem, Maximum Coverage, and the Traveling Salesperson Problem.
