论文标题
适用于经济学的最佳运输算法
Algorithms for Weak Optimal Transport with an Application to Economics
论文作者
论文摘要
[Gozlan等人，2017年]引入的弱最佳运输理论（WOT）通过允许在一个点之间的运输成本和与非线性相匹配的点来概括经典的Monge-Kantorovich框架。在所谓的Barycentric版本WOT中，运输点$ x $的成本仅取决于$ x $以及与之匹配的点的Barycenter。 WOT的这种汇总特性在机器学习，经济学和金融方面具有吸引力。然而，要计算WOT的算法仅针对二次Barycentric Wot的特殊情况开发，或者依赖于神经网络，但不能保证计算的值和匹配。主要困难在于运输限制，这些限制要付出了高昂的代价。在本文中，我们建议使用镜像下降算法来解决WOT问题的原始版本和双重版本。我们还将算法应用于[Choné等，2022]引入的WOT的变体中，其中质量通过非标准化核（WOTUK）从一个空间分布到另一个空间。我们从经验上将WOT和WOTUK的解决方案与经典OT进行比较。我们将我们的数值方法说明了[Choné和Kramarz，2021]的经济框架，即劳动力市场上的工人与公司之间的匹配。
The theory of weak optimal transport (WOT), introduced by [Gozlan et al., 2017], generalizes the classic Monge-Kantorovich framework by allowing the transport cost between one point and the points it is matched with to be nonlinear. In the so-called barycentric version of WOT, the cost for transporting a point $x$ only depends on $x$ and on the barycenter of the points it is matched with. This aggregation property of WOT is appealing in machine learning, economics and finance. Yet algorithms to compute WOT have only been developed for the special case of quadratic barycentric WOT, or depend on neural networks with no guarantee on the computed value and matching. The main difficulty lies in the transportation constraints which are costly to project onto. In this paper, we propose to use mirror descent algorithms to solve the primal and dual versions of the WOT problem. We also apply our algorithms to the variant of WOT introduced by [Choné et al., 2022] where mass is distributed from one space to another through unnormalized kernels (WOTUK). We empirically compare the solutions of WOT and WOTUK with classical OT. We illustrate our numerical methods to the economic framework of [Choné and Kramarz, 2021], namely the matching between workers and firms on labor markets.
